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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Thomason cohomology and Quillen’s Theorem A</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="Cohomology of small categories, Baues-Wirsching cohomology, Thomason cohomology, Grothendieck construction, Quillen’s Theorem A" lang="en" name="keywords"/> <base href="/html/2503.14659v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction and statement of results</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S2" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Modules and cohomology for small categories</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S2.SS1" title="In 2. Modules and cohomology for small categories ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Cohomology of small categories</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S2.SS2" title="In 2. Modules and cohomology for small categories ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Baues-Wirsching Cohomology</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Cohomology of simplicial sets and Thomason Cohomology</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.SS1" title="In 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Coefficient systems for simplicial sets</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.SS2" title="In 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Cohomology of simplicial sets with general coefficients</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.SS3" title="In 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Induced Maps on Cohomology</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.SS4" title="In 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>The Thomason cohomology of a category</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S4" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Bisimplicial objects and the Dold-Puppe theorem</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S4.SS1" title="In 4. Bisimplicial objects and the Dold-Puppe theorem ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Homotopy colimits</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S4.SS2" title="In 4. Bisimplicial objects and the Dold-Puppe theorem ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>The Dold-Puppe Theorem</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S5" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Proof of Theorem <span class="ltx_text ltx_ref_tag">1.2</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S6" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Cohomology of bisimplicial sets with nontrivial coefficients</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S7" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Thomason cohomology of the Grothendieck construction</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Thomason cohomology and Quillen’s Theorem A</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Mehmet Kırtışoğlu </span></span> <span class="ltx_author_before"> and </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ergün Yalçın </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_address">Bilkent University, Department of Mathematics, 06800, Bilkent, Ankara, Türkiye </span> <span class="ltx_contact ltx_role_email"><a href="mailto:yalcine@fen.bilkent.edu.tr,%20m.kirtisoglu@bilkent.edu.tr">yalcine@fen.bilkent.edu.tr, m.kirtisoglu@bilkent.edu.tr</a> </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract.</h6> <p class="ltx_p" id="id9.9">Given a functor <math alttext="\varphi:\mathcal{C}\to\mathcal{D}" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mrow id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml"><mi id="id1.1.m1.1.1.2" xref="id1.1.m1.1.1.2.cmml">φ</mi><mo id="id1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="id1.1.m1.1.1.1.cmml">:</mo><mrow id="id1.1.m1.1.1.3" xref="id1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id1.1.m1.1.1.3.2" xref="id1.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="id1.1.m1.1.1.3.1" stretchy="false" xref="id1.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="id1.1.m1.1.1.3.3" xref="id1.1.m1.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><apply id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1"><ci id="id1.1.m1.1.1.1.cmml" xref="id1.1.m1.1.1.1">:</ci><ci id="id1.1.m1.1.1.2.cmml" xref="id1.1.m1.1.1.2">𝜑</ci><apply id="id1.1.m1.1.1.3.cmml" xref="id1.1.m1.1.1.3"><ci id="id1.1.m1.1.1.3.1.cmml" xref="id1.1.m1.1.1.3.1">→</ci><ci id="id1.1.m1.1.1.3.2.cmml" xref="id1.1.m1.1.1.3.2">𝒞</ci><ci id="id1.1.m1.1.1.3.3.cmml" xref="id1.1.m1.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">\varphi:\mathcal{C}\to\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_φ : caligraphic_C → caligraphic_D</annotation></semantics></math> between two small categories, there is a homotopy equivalence <math alttext="\kappa:\operatorname*{hocolim}_{\mathcal{D}}N(\varphi/-)\to N\mathcal{C}" class="ltx_math_unparsed" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mrow id="id2.2.m2.1b"><mi id="id2.2.m2.1.1">κ</mi><mo id="id2.2.m2.1.2" lspace="0.278em">:</mo><msub id="id2.2.m2.1.3"><mo id="id2.2.m2.1.3.2" lspace="0.111em" rspace="0.167em">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="id2.2.m2.1.3.3">𝒟</mi></msub><mi id="id2.2.m2.1.4">N</mi><mrow id="id2.2.m2.1.5"><mo id="id2.2.m2.1.5.1" stretchy="false">(</mo><mi id="id2.2.m2.1.5.2">φ</mi><mo id="id2.2.m2.1.5.3" rspace="0em">/</mo><mo id="id2.2.m2.1.5.4" lspace="0em" rspace="0em">−</mo><mo id="id2.2.m2.1.5.5" stretchy="false">)</mo></mrow><mo id="id2.2.m2.1.6" stretchy="false">→</mo><mi id="id2.2.m2.1.7">N</mi><mi class="ltx_font_mathcaligraphic" id="id2.2.m2.1.8">𝒞</mi></mrow><annotation encoding="application/x-tex" id="id2.2.m2.1c">\kappa:\operatorname*{hocolim}_{\mathcal{D}}N(\varphi/-)\to N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_κ : roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_N ( italic_φ / - ) → italic_N caligraphic_C</annotation></semantics></math> where <math alttext="N(\varphi/-)" class="ltx_math_unparsed" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mrow id="id3.3.m3.1b"><mi id="id3.3.m3.1.1">N</mi><mrow id="id3.3.m3.1.2"><mo id="id3.3.m3.1.2.1" stretchy="false">(</mo><mi id="id3.3.m3.1.2.2">φ</mi><mo id="id3.3.m3.1.2.3" rspace="0em">/</mo><mo id="id3.3.m3.1.2.4" lspace="0em" rspace="0em">−</mo><mo id="id3.3.m3.1.2.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="id3.3.m3.1c">N(\varphi/-)</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">italic_N ( italic_φ / - )</annotation></semantics></math> is the functor which sends every object <math alttext="d" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mi id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><ci id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.1c">d</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.1d">italic_d</annotation></semantics></math> in <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="id5.5.m5.1"><semantics id="id5.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="id5.5.m5.1b"><ci id="id5.5.m5.1.1.cmml" xref="id5.5.m5.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m5.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="id5.5.m5.1d">caligraphic_D</annotation></semantics></math> to the nerve of the comma category <math alttext="\varphi/d" class="ltx_Math" display="inline" id="id6.6.m6.1"><semantics id="id6.6.m6.1a"><mrow id="id6.6.m6.1.1" xref="id6.6.m6.1.1.cmml"><mi id="id6.6.m6.1.1.2" xref="id6.6.m6.1.1.2.cmml">φ</mi><mo id="id6.6.m6.1.1.1" xref="id6.6.m6.1.1.1.cmml">/</mo><mi id="id6.6.m6.1.1.3" xref="id6.6.m6.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="id6.6.m6.1b"><apply id="id6.6.m6.1.1.cmml" xref="id6.6.m6.1.1"><divide id="id6.6.m6.1.1.1.cmml" xref="id6.6.m6.1.1.1"></divide><ci id="id6.6.m6.1.1.2.cmml" xref="id6.6.m6.1.1.2">𝜑</ci><ci id="id6.6.m6.1.1.3.cmml" xref="id6.6.m6.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id6.6.m6.1c">\varphi/d</annotation><annotation encoding="application/x-llamapun" id="id6.6.m6.1d">italic_φ / italic_d</annotation></semantics></math>. We prove that the homotopy equivalence <math alttext="\kappa" class="ltx_Math" display="inline" id="id7.7.m7.1"><semantics id="id7.7.m7.1a"><mi id="id7.7.m7.1.1" xref="id7.7.m7.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="id7.7.m7.1b"><ci id="id7.7.m7.1.1.cmml" xref="id7.7.m7.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="id7.7.m7.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="id7.7.m7.1d">italic_κ</annotation></semantics></math> induces an isomorphism on cohomology with coefficients in any coefficient system. As a consequence, we obtain a version of Quillen’s Theorem A for the Thomason cohomology of categories. We also construct a spectral sequence for the Thomason cohomology of the Grothendieck construction <math alttext="\int_{\mathcal{D}}F" class="ltx_Math" display="inline" id="id8.8.m8.1"><semantics id="id8.8.m8.1a"><mrow id="id8.8.m8.1.1" xref="id8.8.m8.1.1.cmml"><msub id="id8.8.m8.1.1.1" xref="id8.8.m8.1.1.1.cmml"><mo id="id8.8.m8.1.1.1.2" xref="id8.8.m8.1.1.1.2.cmml">∫</mo><mi class="ltx_font_mathcaligraphic" id="id8.8.m8.1.1.1.3" xref="id8.8.m8.1.1.1.3.cmml">𝒟</mi></msub><mi id="id8.8.m8.1.1.2" xref="id8.8.m8.1.1.2.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="id8.8.m8.1b"><apply id="id8.8.m8.1.1.cmml" xref="id8.8.m8.1.1"><apply id="id8.8.m8.1.1.1.cmml" xref="id8.8.m8.1.1.1"><csymbol cd="ambiguous" id="id8.8.m8.1.1.1.1.cmml" xref="id8.8.m8.1.1.1">subscript</csymbol><int id="id8.8.m8.1.1.1.2.cmml" xref="id8.8.m8.1.1.1.2"></int><ci id="id8.8.m8.1.1.1.3.cmml" xref="id8.8.m8.1.1.1.3">𝒟</ci></apply><ci id="id8.8.m8.1.1.2.cmml" xref="id8.8.m8.1.1.2">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.8.m8.1c">\int_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="id8.8.m8.1d">∫ start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> of a functor <math alttext="F:\mathcal{D}\to Cat" class="ltx_Math" display="inline" id="id9.9.m9.1"><semantics id="id9.9.m9.1a"><mrow id="id9.9.m9.1.1" xref="id9.9.m9.1.1.cmml"><mi id="id9.9.m9.1.1.2" xref="id9.9.m9.1.1.2.cmml">F</mi><mo id="id9.9.m9.1.1.1" lspace="0.278em" rspace="0.278em" xref="id9.9.m9.1.1.1.cmml">:</mo><mrow id="id9.9.m9.1.1.3" xref="id9.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id9.9.m9.1.1.3.2" xref="id9.9.m9.1.1.3.2.cmml">𝒟</mi><mo id="id9.9.m9.1.1.3.1" stretchy="false" xref="id9.9.m9.1.1.3.1.cmml">→</mo><mrow id="id9.9.m9.1.1.3.3" xref="id9.9.m9.1.1.3.3.cmml"><mi id="id9.9.m9.1.1.3.3.2" xref="id9.9.m9.1.1.3.3.2.cmml">C</mi><mo id="id9.9.m9.1.1.3.3.1" xref="id9.9.m9.1.1.3.3.1.cmml">⁢</mo><mi id="id9.9.m9.1.1.3.3.3" xref="id9.9.m9.1.1.3.3.3.cmml">a</mi><mo id="id9.9.m9.1.1.3.3.1a" xref="id9.9.m9.1.1.3.3.1.cmml">⁢</mo><mi id="id9.9.m9.1.1.3.3.4" xref="id9.9.m9.1.1.3.3.4.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="id9.9.m9.1b"><apply id="id9.9.m9.1.1.cmml" xref="id9.9.m9.1.1"><ci id="id9.9.m9.1.1.1.cmml" xref="id9.9.m9.1.1.1">:</ci><ci id="id9.9.m9.1.1.2.cmml" xref="id9.9.m9.1.1.2">𝐹</ci><apply id="id9.9.m9.1.1.3.cmml" xref="id9.9.m9.1.1.3"><ci id="id9.9.m9.1.1.3.1.cmml" xref="id9.9.m9.1.1.3.1">→</ci><ci id="id9.9.m9.1.1.3.2.cmml" xref="id9.9.m9.1.1.3.2">𝒟</ci><apply id="id9.9.m9.1.1.3.3.cmml" xref="id9.9.m9.1.1.3.3"><times id="id9.9.m9.1.1.3.3.1.cmml" xref="id9.9.m9.1.1.3.3.1"></times><ci id="id9.9.m9.1.1.3.3.2.cmml" xref="id9.9.m9.1.1.3.3.2">𝐶</ci><ci id="id9.9.m9.1.1.3.3.3.cmml" xref="id9.9.m9.1.1.3.3.3">𝑎</ci><ci id="id9.9.m9.1.1.3.3.4.cmml" xref="id9.9.m9.1.1.3.3.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id9.9.m9.1c">F:\mathcal{D}\to Cat</annotation><annotation encoding="application/x-llamapun" id="id9.9.m9.1d">italic_F : caligraphic_D → italic_C italic_a italic_t</annotation></semantics></math> using the isomorphism in the main theorem.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Key words and phrases: </h6>Cohomology of small categories, Baues-Wirsching cohomology, Thomason cohomology, Grothendieck construction, Quillen’s Theorem A </div> <div class="ltx_acknowledgements">2020 <span class="ltx_text ltx_font_italic" id="id10.id1">Mathematics Subject Classification.</span> Primary: 18G90; Secondary: 55U10, 18G30, 18G35, 18G40 </div> <nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction and statement of results</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S2" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Modules and cohomology for small categories</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Cohomology of simplicial sets and Thomason Cohomology</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S4" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Bisimplicial objects and the Dold-Puppe theorem</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S5" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Proof of Theorem <span class="ltx_text ltx_ref_tag">1.2</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S6" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Cohomology of bisimplicial sets with nontrivial coefficients</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S7" title="In Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Thomason cohomology of the Grothendieck construction</span></a></li> </ol></nav> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1. </span>Introduction and statement of results</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.13">Let <math alttext="\varphi:\mathcal{C}\to\mathcal{D}" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mrow id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml"><mi id="S1.p1.1.m1.1.1.2" xref="S1.p1.1.m1.1.1.2.cmml">φ</mi><mo id="S1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.p1.1.m1.1.1.3" xref="S1.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.1.m1.1.1.3.2" xref="S1.p1.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="S1.p1.1.m1.1.1.3.1" stretchy="false" xref="S1.p1.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.1.m1.1.1.3.3" xref="S1.p1.1.m1.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><apply id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1"><ci id="S1.p1.1.m1.1.1.1.cmml" xref="S1.p1.1.m1.1.1.1">:</ci><ci id="S1.p1.1.m1.1.1.2.cmml" xref="S1.p1.1.m1.1.1.2">𝜑</ci><apply id="S1.p1.1.m1.1.1.3.cmml" xref="S1.p1.1.m1.1.1.3"><ci id="S1.p1.1.m1.1.1.3.1.cmml" xref="S1.p1.1.m1.1.1.3.1">→</ci><ci id="S1.p1.1.m1.1.1.3.2.cmml" xref="S1.p1.1.m1.1.1.3.2">𝒞</ci><ci id="S1.p1.1.m1.1.1.3.3.cmml" xref="S1.p1.1.m1.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\varphi:\mathcal{C}\to\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_φ : caligraphic_C → caligraphic_D</annotation></semantics></math> be a functor between two small categories. For each <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mrow id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml"><mi id="S1.p1.2.m2.1.1.2" xref="S1.p1.2.m2.1.1.2.cmml">d</mi><mo id="S1.p1.2.m2.1.1.1" xref="S1.p1.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.2.m2.1.1.3" xref="S1.p1.2.m2.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><apply id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1"><in id="S1.p1.2.m2.1.1.1.cmml" xref="S1.p1.2.m2.1.1.1"></in><ci id="S1.p1.2.m2.1.1.2.cmml" xref="S1.p1.2.m2.1.1.2">𝑑</ci><ci id="S1.p1.2.m2.1.1.3.cmml" xref="S1.p1.2.m2.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_d ∈ caligraphic_D</annotation></semantics></math>, let <math alttext="\varphi/d" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mrow id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml"><mi id="S1.p1.3.m3.1.1.2" xref="S1.p1.3.m3.1.1.2.cmml">φ</mi><mo id="S1.p1.3.m3.1.1.1" xref="S1.p1.3.m3.1.1.1.cmml">/</mo><mi id="S1.p1.3.m3.1.1.3" xref="S1.p1.3.m3.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><apply id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1"><divide id="S1.p1.3.m3.1.1.1.cmml" xref="S1.p1.3.m3.1.1.1"></divide><ci id="S1.p1.3.m3.1.1.2.cmml" xref="S1.p1.3.m3.1.1.2">𝜑</ci><ci id="S1.p1.3.m3.1.1.3.cmml" xref="S1.p1.3.m3.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">\varphi/d</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_φ / italic_d</annotation></semantics></math> denote the comma category with objects given by the pairs <math alttext="(c,\mu)" class="ltx_Math" display="inline" id="S1.p1.4.m4.2"><semantics id="S1.p1.4.m4.2a"><mrow id="S1.p1.4.m4.2.3.2" xref="S1.p1.4.m4.2.3.1.cmml"><mo id="S1.p1.4.m4.2.3.2.1" stretchy="false" xref="S1.p1.4.m4.2.3.1.cmml">(</mo><mi id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml">c</mi><mo id="S1.p1.4.m4.2.3.2.2" xref="S1.p1.4.m4.2.3.1.cmml">,</mo><mi id="S1.p1.4.m4.2.2" xref="S1.p1.4.m4.2.2.cmml">μ</mi><mo id="S1.p1.4.m4.2.3.2.3" stretchy="false" xref="S1.p1.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.2b"><interval closure="open" id="S1.p1.4.m4.2.3.1.cmml" xref="S1.p1.4.m4.2.3.2"><ci id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1">𝑐</ci><ci id="S1.p1.4.m4.2.2.cmml" xref="S1.p1.4.m4.2.2">𝜇</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.2c">(c,\mu)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.2d">( italic_c , italic_μ )</annotation></semantics></math> where <math alttext="c\in\mathcal{C}" class="ltx_Math" display="inline" id="S1.p1.5.m5.1"><semantics id="S1.p1.5.m5.1a"><mrow id="S1.p1.5.m5.1.1" xref="S1.p1.5.m5.1.1.cmml"><mi id="S1.p1.5.m5.1.1.2" xref="S1.p1.5.m5.1.1.2.cmml">c</mi><mo id="S1.p1.5.m5.1.1.1" xref="S1.p1.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.5.m5.1.1.3" xref="S1.p1.5.m5.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.5.m5.1b"><apply id="S1.p1.5.m5.1.1.cmml" xref="S1.p1.5.m5.1.1"><in id="S1.p1.5.m5.1.1.1.cmml" xref="S1.p1.5.m5.1.1.1"></in><ci id="S1.p1.5.m5.1.1.2.cmml" xref="S1.p1.5.m5.1.1.2">𝑐</ci><ci id="S1.p1.5.m5.1.1.3.cmml" xref="S1.p1.5.m5.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m5.1c">c\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.1d">italic_c ∈ caligraphic_C</annotation></semantics></math> and <math alttext="\mu:\varphi(c)\to d" class="ltx_Math" display="inline" id="S1.p1.6.m6.1"><semantics id="S1.p1.6.m6.1a"><mrow id="S1.p1.6.m6.1.2" xref="S1.p1.6.m6.1.2.cmml"><mi id="S1.p1.6.m6.1.2.2" xref="S1.p1.6.m6.1.2.2.cmml">μ</mi><mo id="S1.p1.6.m6.1.2.1" lspace="0.278em" rspace="0.278em" xref="S1.p1.6.m6.1.2.1.cmml">:</mo><mrow id="S1.p1.6.m6.1.2.3" xref="S1.p1.6.m6.1.2.3.cmml"><mrow id="S1.p1.6.m6.1.2.3.2" xref="S1.p1.6.m6.1.2.3.2.cmml"><mi id="S1.p1.6.m6.1.2.3.2.2" xref="S1.p1.6.m6.1.2.3.2.2.cmml">φ</mi><mo id="S1.p1.6.m6.1.2.3.2.1" xref="S1.p1.6.m6.1.2.3.2.1.cmml">⁢</mo><mrow id="S1.p1.6.m6.1.2.3.2.3.2" xref="S1.p1.6.m6.1.2.3.2.cmml"><mo id="S1.p1.6.m6.1.2.3.2.3.2.1" stretchy="false" xref="S1.p1.6.m6.1.2.3.2.cmml">(</mo><mi id="S1.p1.6.m6.1.1" xref="S1.p1.6.m6.1.1.cmml">c</mi><mo id="S1.p1.6.m6.1.2.3.2.3.2.2" stretchy="false" xref="S1.p1.6.m6.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S1.p1.6.m6.1.2.3.1" stretchy="false" xref="S1.p1.6.m6.1.2.3.1.cmml">→</mo><mi id="S1.p1.6.m6.1.2.3.3" xref="S1.p1.6.m6.1.2.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.6.m6.1b"><apply id="S1.p1.6.m6.1.2.cmml" xref="S1.p1.6.m6.1.2"><ci id="S1.p1.6.m6.1.2.1.cmml" xref="S1.p1.6.m6.1.2.1">:</ci><ci id="S1.p1.6.m6.1.2.2.cmml" xref="S1.p1.6.m6.1.2.2">𝜇</ci><apply id="S1.p1.6.m6.1.2.3.cmml" xref="S1.p1.6.m6.1.2.3"><ci id="S1.p1.6.m6.1.2.3.1.cmml" xref="S1.p1.6.m6.1.2.3.1">→</ci><apply id="S1.p1.6.m6.1.2.3.2.cmml" xref="S1.p1.6.m6.1.2.3.2"><times id="S1.p1.6.m6.1.2.3.2.1.cmml" xref="S1.p1.6.m6.1.2.3.2.1"></times><ci id="S1.p1.6.m6.1.2.3.2.2.cmml" xref="S1.p1.6.m6.1.2.3.2.2">𝜑</ci><ci id="S1.p1.6.m6.1.1.cmml" xref="S1.p1.6.m6.1.1">𝑐</ci></apply><ci id="S1.p1.6.m6.1.2.3.3.cmml" xref="S1.p1.6.m6.1.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.6.m6.1c">\mu:\varphi(c)\to d</annotation><annotation encoding="application/x-llamapun" id="S1.p1.6.m6.1d">italic_μ : italic_φ ( italic_c ) → italic_d</annotation></semantics></math> is a morphism in <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S1.p1.7.m7.1"><semantics id="S1.p1.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p1.7.m7.1.1" xref="S1.p1.7.m7.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S1.p1.7.m7.1b"><ci id="S1.p1.7.m7.1.1.cmml" xref="S1.p1.7.m7.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.7.m7.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.7.m7.1d">caligraphic_D</annotation></semantics></math>. The assignment <math alttext="d\to\varphi/d" class="ltx_Math" display="inline" id="S1.p1.8.m8.1"><semantics id="S1.p1.8.m8.1a"><mrow id="S1.p1.8.m8.1.1" xref="S1.p1.8.m8.1.1.cmml"><mi id="S1.p1.8.m8.1.1.2" xref="S1.p1.8.m8.1.1.2.cmml">d</mi><mo id="S1.p1.8.m8.1.1.1" stretchy="false" xref="S1.p1.8.m8.1.1.1.cmml">→</mo><mrow id="S1.p1.8.m8.1.1.3" xref="S1.p1.8.m8.1.1.3.cmml"><mi id="S1.p1.8.m8.1.1.3.2" xref="S1.p1.8.m8.1.1.3.2.cmml">φ</mi><mo id="S1.p1.8.m8.1.1.3.1" xref="S1.p1.8.m8.1.1.3.1.cmml">/</mo><mi id="S1.p1.8.m8.1.1.3.3" xref="S1.p1.8.m8.1.1.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.8.m8.1b"><apply id="S1.p1.8.m8.1.1.cmml" xref="S1.p1.8.m8.1.1"><ci id="S1.p1.8.m8.1.1.1.cmml" xref="S1.p1.8.m8.1.1.1">→</ci><ci id="S1.p1.8.m8.1.1.2.cmml" xref="S1.p1.8.m8.1.1.2">𝑑</ci><apply id="S1.p1.8.m8.1.1.3.cmml" xref="S1.p1.8.m8.1.1.3"><divide id="S1.p1.8.m8.1.1.3.1.cmml" xref="S1.p1.8.m8.1.1.3.1"></divide><ci id="S1.p1.8.m8.1.1.3.2.cmml" xref="S1.p1.8.m8.1.1.3.2">𝜑</ci><ci id="S1.p1.8.m8.1.1.3.3.cmml" xref="S1.p1.8.m8.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.8.m8.1c">d\to\varphi/d</annotation><annotation encoding="application/x-llamapun" id="S1.p1.8.m8.1d">italic_d → italic_φ / italic_d</annotation></semantics></math> for every <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.p1.9.m9.1"><semantics id="S1.p1.9.m9.1a"><mrow id="S1.p1.9.m9.1.1" xref="S1.p1.9.m9.1.1.cmml"><mi id="S1.p1.9.m9.1.1.2" xref="S1.p1.9.m9.1.1.2.cmml">d</mi><mo id="S1.p1.9.m9.1.1.1" xref="S1.p1.9.m9.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.9.m9.1.1.3" xref="S1.p1.9.m9.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.9.m9.1b"><apply id="S1.p1.9.m9.1.1.cmml" xref="S1.p1.9.m9.1.1"><in id="S1.p1.9.m9.1.1.1.cmml" xref="S1.p1.9.m9.1.1.1"></in><ci id="S1.p1.9.m9.1.1.2.cmml" xref="S1.p1.9.m9.1.1.2">𝑑</ci><ci id="S1.p1.9.m9.1.1.3.cmml" xref="S1.p1.9.m9.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.9.m9.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.9.m9.1d">italic_d ∈ caligraphic_D</annotation></semantics></math> defines a functor <math alttext="\varphi/-:\mathcal{D}\to Cat" class="ltx_math_unparsed" display="inline" id="S1.p1.10.m10.1"><semantics id="S1.p1.10.m10.1a"><mrow id="S1.p1.10.m10.1b"><mi id="S1.p1.10.m10.1.1">φ</mi><mo id="S1.p1.10.m10.1.2" rspace="0em">/</mo><mo id="S1.p1.10.m10.1.3" lspace="0em" rspace="0em">−</mo><mo id="S1.p1.10.m10.1.4" rspace="0.278em">:</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.10.m10.1.5">𝒟</mi><mo id="S1.p1.10.m10.1.6" stretchy="false">→</mo><mi id="S1.p1.10.m10.1.7">C</mi><mi id="S1.p1.10.m10.1.8">a</mi><mi id="S1.p1.10.m10.1.9">t</mi></mrow><annotation encoding="application/x-tex" id="S1.p1.10.m10.1c">\varphi/-:\mathcal{D}\to Cat</annotation><annotation encoding="application/x-llamapun" id="S1.p1.10.m10.1d">italic_φ / - : caligraphic_D → italic_C italic_a italic_t</annotation></semantics></math>. Composing this functor with the nerve functor we obtain a functor <math alttext="N(\varphi/-):\mathcal{D}\to sSet" class="ltx_math_unparsed" display="inline" id="S1.p1.11.m11.1"><semantics id="S1.p1.11.m11.1a"><mrow id="S1.p1.11.m11.1b"><mi id="S1.p1.11.m11.1.1">N</mi><mrow id="S1.p1.11.m11.1.2"><mo id="S1.p1.11.m11.1.2.1" stretchy="false">(</mo><mi id="S1.p1.11.m11.1.2.2">φ</mi><mo id="S1.p1.11.m11.1.2.3" rspace="0em">/</mo><mo id="S1.p1.11.m11.1.2.4" lspace="0em" rspace="0em">−</mo><mo id="S1.p1.11.m11.1.2.5" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S1.p1.11.m11.1.3" rspace="0.278em">:</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.11.m11.1.4">𝒟</mi><mo id="S1.p1.11.m11.1.5" stretchy="false">→</mo><mi id="S1.p1.11.m11.1.6">s</mi><mi id="S1.p1.11.m11.1.7">S</mi><mi id="S1.p1.11.m11.1.8">e</mi><mi id="S1.p1.11.m11.1.9">t</mi></mrow><annotation encoding="application/x-tex" id="S1.p1.11.m11.1c">N(\varphi/-):\mathcal{D}\to sSet</annotation><annotation encoding="application/x-llamapun" id="S1.p1.11.m11.1d">italic_N ( italic_φ / - ) : caligraphic_D → italic_s italic_S italic_e italic_t</annotation></semantics></math>. We define the homotopy colimit <math alttext="\operatorname*{hocolim}_{\mathcal{D}}N(\varphi/-)" class="ltx_math_unparsed" display="inline" id="S1.p1.12.m12.1"><semantics id="S1.p1.12.m12.1a"><mrow id="S1.p1.12.m12.1b"><msub id="S1.p1.12.m12.1.1"><mo id="S1.p1.12.m12.1.1.2">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.12.m12.1.1.3">𝒟</mi></msub><mi id="S1.p1.12.m12.1.2">N</mi><mrow id="S1.p1.12.m12.1.3"><mo id="S1.p1.12.m12.1.3.1" stretchy="false">(</mo><mi id="S1.p1.12.m12.1.3.2">φ</mi><mo id="S1.p1.12.m12.1.3.3" rspace="0em">/</mo><mo id="S1.p1.12.m12.1.3.4" lspace="0em" rspace="0em">−</mo><mo id="S1.p1.12.m12.1.3.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p1.12.m12.1c">\operatorname*{hocolim}_{\mathcal{D}}N(\varphi/-)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.12.m12.1d">roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_N ( italic_φ / - )</annotation></semantics></math> as the diagonal of the simplicial replacement of the functor <math alttext="N(\varphi/-)" class="ltx_math_unparsed" display="inline" id="S1.p1.13.m13.1"><semantics id="S1.p1.13.m13.1a"><mrow id="S1.p1.13.m13.1b"><mi id="S1.p1.13.m13.1.1">N</mi><mrow id="S1.p1.13.m13.1.2"><mo id="S1.p1.13.m13.1.2.1" stretchy="false">(</mo><mi id="S1.p1.13.m13.1.2.2">φ</mi><mo id="S1.p1.13.m13.1.2.3" rspace="0em">/</mo><mo id="S1.p1.13.m13.1.2.4" lspace="0em" rspace="0em">−</mo><mo id="S1.p1.13.m13.1.2.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p1.13.m13.1c">N(\varphi/-)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.13.m13.1d">italic_N ( italic_φ / - )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S1.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.1.1.1">Definition 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.8">Let <math alttext="\varphi:\mathcal{C}\to\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.1.m1.1"><semantics id="S1.Thmtheorem1.p1.1.m1.1a"><mrow id="S1.Thmtheorem1.p1.1.m1.1.1" xref="S1.Thmtheorem1.p1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem1.p1.1.m1.1.1.2" xref="S1.Thmtheorem1.p1.1.m1.1.1.2.cmml">φ</mi><mo id="S1.Thmtheorem1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem1.p1.1.m1.1.1.3" xref="S1.Thmtheorem1.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.1.m1.1.1.3.2" xref="S1.Thmtheorem1.p1.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="S1.Thmtheorem1.p1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmtheorem1.p1.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.1.m1.1.1.3.3" xref="S1.Thmtheorem1.p1.1.m1.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.1.m1.1b"><apply id="S1.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1"><ci id="S1.Thmtheorem1.p1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.1">:</ci><ci id="S1.Thmtheorem1.p1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.2">𝜑</ci><apply id="S1.Thmtheorem1.p1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.3"><ci id="S1.Thmtheorem1.p1.1.m1.1.1.3.1.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.3.1">→</ci><ci id="S1.Thmtheorem1.p1.1.m1.1.1.3.2.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.3.2">𝒞</ci><ci id="S1.Thmtheorem1.p1.1.m1.1.1.3.3.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.1.m1.1c">\varphi:\mathcal{C}\to\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.1.m1.1d">italic_φ : caligraphic_C → caligraphic_D</annotation></semantics></math> be a functor. For each <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.2.m2.1"><semantics id="S1.Thmtheorem1.p1.2.m2.1a"><mrow id="S1.Thmtheorem1.p1.2.m2.1.1" xref="S1.Thmtheorem1.p1.2.m2.1.1.cmml"><mi id="S1.Thmtheorem1.p1.2.m2.1.1.2" xref="S1.Thmtheorem1.p1.2.m2.1.1.2.cmml">d</mi><mo id="S1.Thmtheorem1.p1.2.m2.1.1.1" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.2.m2.1.1.3" xref="S1.Thmtheorem1.p1.2.m2.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.2.m2.1b"><apply id="S1.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1"><in id="S1.Thmtheorem1.p1.2.m2.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1"></in><ci id="S1.Thmtheorem1.p1.2.m2.1.1.2.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.2">𝑑</ci><ci id="S1.Thmtheorem1.p1.2.m2.1.1.3.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.2.m2.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.2.m2.1d">italic_d ∈ caligraphic_D</annotation></semantics></math>, there is a simplicial map <math alttext="j_{d}:N(\varphi/d)\to N\mathcal{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.3.m3.1"><semantics id="S1.Thmtheorem1.p1.3.m3.1a"><mrow id="S1.Thmtheorem1.p1.3.m3.1.1" xref="S1.Thmtheorem1.p1.3.m3.1.1.cmml"><msub id="S1.Thmtheorem1.p1.3.m3.1.1.3" xref="S1.Thmtheorem1.p1.3.m3.1.1.3.cmml"><mi id="S1.Thmtheorem1.p1.3.m3.1.1.3.2" xref="S1.Thmtheorem1.p1.3.m3.1.1.3.2.cmml">j</mi><mi id="S1.Thmtheorem1.p1.3.m3.1.1.3.3" xref="S1.Thmtheorem1.p1.3.m3.1.1.3.3.cmml">d</mi></msub><mo id="S1.Thmtheorem1.p1.3.m3.1.1.2" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem1.p1.3.m3.1.1.2.cmml">:</mo><mrow id="S1.Thmtheorem1.p1.3.m3.1.1.1" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.cmml"><mrow id="S1.Thmtheorem1.p1.3.m3.1.1.1.1" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.cmml"><mi id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.3" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.3.cmml">N</mi><mo id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.2" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.cmml"><mi id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.2" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.2.cmml">φ</mi><mo id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.1" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.3" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem1.p1.3.m3.1.1.1.2" stretchy="false" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.2.cmml">→</mo><mrow id="S1.Thmtheorem1.p1.3.m3.1.1.1.3" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.3.cmml"><mi id="S1.Thmtheorem1.p1.3.m3.1.1.1.3.2" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.3.2.cmml">N</mi><mo id="S1.Thmtheorem1.p1.3.m3.1.1.1.3.1" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.3.m3.1.1.1.3.3" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.3.3.cmml">𝒞</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.3.m3.1b"><apply id="S1.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1"><ci id="S1.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.2">:</ci><apply id="S1.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.3.m3.1.1.3.1.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S1.Thmtheorem1.p1.3.m3.1.1.3.2.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.3.2">𝑗</ci><ci id="S1.Thmtheorem1.p1.3.m3.1.1.3.3.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.3.3">𝑑</ci></apply><apply id="S1.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1"><ci id="S1.Thmtheorem1.p1.3.m3.1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.2">→</ci><apply id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1"><times id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.2"></times><ci id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.3">𝑁</ci><apply id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1"><divide id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.1"></divide><ci id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.2">𝜑</ci><ci id="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S1.Thmtheorem1.p1.3.m3.1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.3"><times id="S1.Thmtheorem1.p1.3.m3.1.1.1.3.1.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.3.1"></times><ci id="S1.Thmtheorem1.p1.3.m3.1.1.1.3.2.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.3.2">𝑁</ci><ci id="S1.Thmtheorem1.p1.3.m3.1.1.1.3.3.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1.1.3.3">𝒞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.3.m3.1c">j_{d}:N(\varphi/d)\to N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.3.m3.1d">italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT : italic_N ( italic_φ / italic_d ) → italic_N caligraphic_C</annotation></semantics></math> induced by the functor <math alttext="\varphi/d\to\mathcal{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.4.m4.1"><semantics id="S1.Thmtheorem1.p1.4.m4.1a"><mrow id="S1.Thmtheorem1.p1.4.m4.1.1" xref="S1.Thmtheorem1.p1.4.m4.1.1.cmml"><mrow id="S1.Thmtheorem1.p1.4.m4.1.1.2" xref="S1.Thmtheorem1.p1.4.m4.1.1.2.cmml"><mi id="S1.Thmtheorem1.p1.4.m4.1.1.2.2" xref="S1.Thmtheorem1.p1.4.m4.1.1.2.2.cmml">φ</mi><mo id="S1.Thmtheorem1.p1.4.m4.1.1.2.1" xref="S1.Thmtheorem1.p1.4.m4.1.1.2.1.cmml">/</mo><mi id="S1.Thmtheorem1.p1.4.m4.1.1.2.3" xref="S1.Thmtheorem1.p1.4.m4.1.1.2.3.cmml">d</mi></mrow><mo id="S1.Thmtheorem1.p1.4.m4.1.1.1" stretchy="false" xref="S1.Thmtheorem1.p1.4.m4.1.1.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.4.m4.1.1.3" xref="S1.Thmtheorem1.p1.4.m4.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.4.m4.1b"><apply id="S1.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S1.Thmtheorem1.p1.4.m4.1.1"><ci id="S1.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="S1.Thmtheorem1.p1.4.m4.1.1.1">→</ci><apply id="S1.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="S1.Thmtheorem1.p1.4.m4.1.1.2"><divide id="S1.Thmtheorem1.p1.4.m4.1.1.2.1.cmml" xref="S1.Thmtheorem1.p1.4.m4.1.1.2.1"></divide><ci id="S1.Thmtheorem1.p1.4.m4.1.1.2.2.cmml" xref="S1.Thmtheorem1.p1.4.m4.1.1.2.2">𝜑</ci><ci id="S1.Thmtheorem1.p1.4.m4.1.1.2.3.cmml" xref="S1.Thmtheorem1.p1.4.m4.1.1.2.3">𝑑</ci></apply><ci id="S1.Thmtheorem1.p1.4.m4.1.1.3.cmml" xref="S1.Thmtheorem1.p1.4.m4.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.4.m4.1c">\varphi/d\to\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.4.m4.1d">italic_φ / italic_d → caligraphic_C</annotation></semantics></math> that sends the pair <math alttext="(c,\mu)" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.5.m5.2"><semantics id="S1.Thmtheorem1.p1.5.m5.2a"><mrow id="S1.Thmtheorem1.p1.5.m5.2.3.2" xref="S1.Thmtheorem1.p1.5.m5.2.3.1.cmml"><mo id="S1.Thmtheorem1.p1.5.m5.2.3.2.1" stretchy="false" xref="S1.Thmtheorem1.p1.5.m5.2.3.1.cmml">(</mo><mi id="S1.Thmtheorem1.p1.5.m5.1.1" xref="S1.Thmtheorem1.p1.5.m5.1.1.cmml">c</mi><mo id="S1.Thmtheorem1.p1.5.m5.2.3.2.2" xref="S1.Thmtheorem1.p1.5.m5.2.3.1.cmml">,</mo><mi id="S1.Thmtheorem1.p1.5.m5.2.2" xref="S1.Thmtheorem1.p1.5.m5.2.2.cmml">μ</mi><mo id="S1.Thmtheorem1.p1.5.m5.2.3.2.3" stretchy="false" xref="S1.Thmtheorem1.p1.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.5.m5.2b"><interval closure="open" id="S1.Thmtheorem1.p1.5.m5.2.3.1.cmml" xref="S1.Thmtheorem1.p1.5.m5.2.3.2"><ci id="S1.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1">𝑐</ci><ci id="S1.Thmtheorem1.p1.5.m5.2.2.cmml" xref="S1.Thmtheorem1.p1.5.m5.2.2">𝜇</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.5.m5.2c">(c,\mu)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.5.m5.2d">( italic_c , italic_μ )</annotation></semantics></math> in <math alttext="\varphi/d" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.6.m6.1"><semantics id="S1.Thmtheorem1.p1.6.m6.1a"><mrow id="S1.Thmtheorem1.p1.6.m6.1.1" xref="S1.Thmtheorem1.p1.6.m6.1.1.cmml"><mi id="S1.Thmtheorem1.p1.6.m6.1.1.2" xref="S1.Thmtheorem1.p1.6.m6.1.1.2.cmml">φ</mi><mo id="S1.Thmtheorem1.p1.6.m6.1.1.1" xref="S1.Thmtheorem1.p1.6.m6.1.1.1.cmml">/</mo><mi id="S1.Thmtheorem1.p1.6.m6.1.1.3" xref="S1.Thmtheorem1.p1.6.m6.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.6.m6.1b"><apply id="S1.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S1.Thmtheorem1.p1.6.m6.1.1"><divide id="S1.Thmtheorem1.p1.6.m6.1.1.1.cmml" xref="S1.Thmtheorem1.p1.6.m6.1.1.1"></divide><ci id="S1.Thmtheorem1.p1.6.m6.1.1.2.cmml" xref="S1.Thmtheorem1.p1.6.m6.1.1.2">𝜑</ci><ci id="S1.Thmtheorem1.p1.6.m6.1.1.3.cmml" xref="S1.Thmtheorem1.p1.6.m6.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.6.m6.1c">\varphi/d</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.6.m6.1d">italic_φ / italic_d</annotation></semantics></math> to <math alttext="c\in\mathcal{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.7.m7.1"><semantics id="S1.Thmtheorem1.p1.7.m7.1a"><mrow id="S1.Thmtheorem1.p1.7.m7.1.1" xref="S1.Thmtheorem1.p1.7.m7.1.1.cmml"><mi id="S1.Thmtheorem1.p1.7.m7.1.1.2" xref="S1.Thmtheorem1.p1.7.m7.1.1.2.cmml">c</mi><mo id="S1.Thmtheorem1.p1.7.m7.1.1.1" xref="S1.Thmtheorem1.p1.7.m7.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.7.m7.1.1.3" xref="S1.Thmtheorem1.p1.7.m7.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.7.m7.1b"><apply id="S1.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S1.Thmtheorem1.p1.7.m7.1.1"><in id="S1.Thmtheorem1.p1.7.m7.1.1.1.cmml" xref="S1.Thmtheorem1.p1.7.m7.1.1.1"></in><ci id="S1.Thmtheorem1.p1.7.m7.1.1.2.cmml" xref="S1.Thmtheorem1.p1.7.m7.1.1.2">𝑐</ci><ci id="S1.Thmtheorem1.p1.7.m7.1.1.3.cmml" xref="S1.Thmtheorem1.p1.7.m7.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.7.m7.1c">c\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.7.m7.1d">italic_c ∈ caligraphic_C</annotation></semantics></math>. The simplicial maps <math alttext="j_{d}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.8.m8.1"><semantics id="S1.Thmtheorem1.p1.8.m8.1a"><msub id="S1.Thmtheorem1.p1.8.m8.1.1" xref="S1.Thmtheorem1.p1.8.m8.1.1.cmml"><mi id="S1.Thmtheorem1.p1.8.m8.1.1.2" xref="S1.Thmtheorem1.p1.8.m8.1.1.2.cmml">j</mi><mi id="S1.Thmtheorem1.p1.8.m8.1.1.3" xref="S1.Thmtheorem1.p1.8.m8.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.8.m8.1b"><apply id="S1.Thmtheorem1.p1.8.m8.1.1.cmml" xref="S1.Thmtheorem1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.8.m8.1.1.1.cmml" xref="S1.Thmtheorem1.p1.8.m8.1.1">subscript</csymbol><ci id="S1.Thmtheorem1.p1.8.m8.1.1.2.cmml" xref="S1.Thmtheorem1.p1.8.m8.1.1.2">𝑗</ci><ci id="S1.Thmtheorem1.p1.8.m8.1.1.3.cmml" xref="S1.Thmtheorem1.p1.8.m8.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.8.m8.1c">j_{d}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.8.m8.1d">italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> induce a simplicial map</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\kappa:\operatorname*{hocolim}_{\mathcal{D}}N(\varphi/-)\to N\mathcal{C}." class="ltx_math_unparsed" display="block" id="S1.Ex1.m1.1"><semantics id="S1.Ex1.m1.1a"><mrow id="S1.Ex1.m1.1b"><mi id="S1.Ex1.m1.1.1">κ</mi><mo id="S1.Ex1.m1.1.2" lspace="0.278em">:</mo><munder id="S1.Ex1.m1.1.3"><mo id="S1.Ex1.m1.1.3.2" lspace="0.111em" rspace="0.167em">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex1.m1.1.3.3">𝒟</mi></munder><mi id="S1.Ex1.m1.1.4">N</mi><mrow id="S1.Ex1.m1.1.5"><mo id="S1.Ex1.m1.1.5.1" stretchy="false">(</mo><mi id="S1.Ex1.m1.1.5.2">φ</mi><mo id="S1.Ex1.m1.1.5.3" rspace="0em">/</mo><mo id="S1.Ex1.m1.1.5.4" lspace="0em" rspace="0em">−</mo><mo id="S1.Ex1.m1.1.5.5" stretchy="false">)</mo></mrow><mo id="S1.Ex1.m1.1.6" stretchy="false">→</mo><mi id="S1.Ex1.m1.1.7">N</mi><mi class="ltx_font_mathcaligraphic" id="S1.Ex1.m1.1.8">𝒞</mi><mo id="S1.Ex1.m1.1.9" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S1.Ex1.m1.1c">\kappa:\operatorname*{hocolim}_{\mathcal{D}}N(\varphi/-)\to N\mathcal{C}.</annotation><annotation encoding="application/x-llamapun" id="S1.Ex1.m1.1d">italic_κ : roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_N ( italic_φ / - ) → italic_N caligraphic_C .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.2">More details on the above definitions can be found in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S5" title="5. Proof of Theorem 1.2 ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">5</span></a>. It is shown in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib16" title="">16</a>, Theorem A]</cite> that <math alttext="\kappa" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_κ</annotation></semantics></math> is a homotopy equivalence. This homotopy equivalence is the first step in the proof of Quillen’s Theorem A. In this paper we prove that the homotopy equivalence <math alttext="\kappa" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_κ</annotation></semantics></math> induces an isomorphism for the cohomology of the associated simplicial sets with coefficients in most general coefficient systems.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.16">For a small category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">caligraphic_C</annotation></semantics></math>, there are different notions of the cohomology of <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">caligraphic_C</annotation></semantics></math> depending on the type of the coefficient system that is chosen. Let <math alttext="R" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mi id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><ci id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">R</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_R</annotation></semantics></math> be a commutative ring, then a functor <math alttext="M:\mathcal{C}\to R" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mrow id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml"><mi id="S1.p3.4.m4.1.1.2" xref="S1.p3.4.m4.1.1.2.cmml">M</mi><mo id="S1.p3.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p3.4.m4.1.1.1.cmml">:</mo><mrow id="S1.p3.4.m4.1.1.3" xref="S1.p3.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.4.m4.1.1.3.2" xref="S1.p3.4.m4.1.1.3.2.cmml">𝒞</mi><mo id="S1.p3.4.m4.1.1.3.1" stretchy="false" xref="S1.p3.4.m4.1.1.3.1.cmml">→</mo><mi id="S1.p3.4.m4.1.1.3.3" xref="S1.p3.4.m4.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><apply id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1"><ci id="S1.p3.4.m4.1.1.1.cmml" xref="S1.p3.4.m4.1.1.1">:</ci><ci id="S1.p3.4.m4.1.1.2.cmml" xref="S1.p3.4.m4.1.1.2">𝑀</ci><apply id="S1.p3.4.m4.1.1.3.cmml" xref="S1.p3.4.m4.1.1.3"><ci id="S1.p3.4.m4.1.1.3.1.cmml" xref="S1.p3.4.m4.1.1.3.1">→</ci><ci id="S1.p3.4.m4.1.1.3.2.cmml" xref="S1.p3.4.m4.1.1.3.2">𝒞</ci><ci id="S1.p3.4.m4.1.1.3.3.cmml" xref="S1.p3.4.m4.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">M:\mathcal{C}\to R</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">italic_M : caligraphic_C → italic_R</annotation></semantics></math>-Mod is called an <em class="ltx_emph ltx_font_italic" id="S1.p3.5.1"><math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S1.p3.5.1.m1.1"><semantics id="S1.p3.5.1.m1.1a"><mrow id="S1.p3.5.1.m1.1.1" xref="S1.p3.5.1.m1.1.1.cmml"><mi id="S1.p3.5.1.m1.1.1.2" xref="S1.p3.5.1.m1.1.1.2.cmml">R</mi><mo id="S1.p3.5.1.m1.1.1.1" xref="S1.p3.5.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.5.1.m1.1.1.3" xref="S1.p3.5.1.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.5.1.m1.1b"><apply id="S1.p3.5.1.m1.1.1.cmml" xref="S1.p3.5.1.m1.1.1"><times id="S1.p3.5.1.m1.1.1.1.cmml" xref="S1.p3.5.1.m1.1.1.1"></times><ci id="S1.p3.5.1.m1.1.1.2.cmml" xref="S1.p3.5.1.m1.1.1.2">𝑅</ci><ci id="S1.p3.5.1.m1.1.1.3.cmml" xref="S1.p3.5.1.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.1.m1.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.1.m1.1d">italic_R caligraphic_C</annotation></semantics></math>-module</em>. The constant functor with value <math alttext="R" class="ltx_Math" display="inline" id="S1.p3.6.m5.1"><semantics id="S1.p3.6.m5.1a"><mi id="S1.p3.6.m5.1.1" xref="S1.p3.6.m5.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S1.p3.6.m5.1b"><ci id="S1.p3.6.m5.1.1.cmml" xref="S1.p3.6.m5.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m5.1c">R</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m5.1d">italic_R</annotation></semantics></math> is denoted by <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S1.p3.7.m6.1"><semantics id="S1.p3.7.m6.1a"><munder accentunder="true" id="S1.p3.7.m6.1.1" xref="S1.p3.7.m6.1.1.cmml"><mi id="S1.p3.7.m6.1.1.2" xref="S1.p3.7.m6.1.1.2.cmml">R</mi><mo id="S1.p3.7.m6.1.1.1" xref="S1.p3.7.m6.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S1.p3.7.m6.1b"><apply id="S1.p3.7.m6.1.1.cmml" xref="S1.p3.7.m6.1.1"><ci id="S1.p3.7.m6.1.1.1.cmml" xref="S1.p3.7.m6.1.1.1">¯</ci><ci id="S1.p3.7.m6.1.1.2.cmml" xref="S1.p3.7.m6.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.7.m6.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.7.m6.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math>. The cohomology <math alttext="H^{*}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S1.p3.8.m7.2"><semantics id="S1.p3.8.m7.2a"><mrow id="S1.p3.8.m7.2.3" xref="S1.p3.8.m7.2.3.cmml"><msup id="S1.p3.8.m7.2.3.2" xref="S1.p3.8.m7.2.3.2.cmml"><mi id="S1.p3.8.m7.2.3.2.2" xref="S1.p3.8.m7.2.3.2.2.cmml">H</mi><mo id="S1.p3.8.m7.2.3.2.3" xref="S1.p3.8.m7.2.3.2.3.cmml">∗</mo></msup><mo id="S1.p3.8.m7.2.3.1" xref="S1.p3.8.m7.2.3.1.cmml">⁢</mo><mrow id="S1.p3.8.m7.2.3.3.2" xref="S1.p3.8.m7.2.3.3.1.cmml"><mo id="S1.p3.8.m7.2.3.3.2.1" stretchy="false" xref="S1.p3.8.m7.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.8.m7.1.1" xref="S1.p3.8.m7.1.1.cmml">𝒞</mi><mo id="S1.p3.8.m7.2.3.3.2.2" xref="S1.p3.8.m7.2.3.3.1.cmml">;</mo><mi id="S1.p3.8.m7.2.2" xref="S1.p3.8.m7.2.2.cmml">M</mi><mo id="S1.p3.8.m7.2.3.3.2.3" stretchy="false" xref="S1.p3.8.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.8.m7.2b"><apply id="S1.p3.8.m7.2.3.cmml" xref="S1.p3.8.m7.2.3"><times id="S1.p3.8.m7.2.3.1.cmml" xref="S1.p3.8.m7.2.3.1"></times><apply id="S1.p3.8.m7.2.3.2.cmml" xref="S1.p3.8.m7.2.3.2"><csymbol cd="ambiguous" id="S1.p3.8.m7.2.3.2.1.cmml" xref="S1.p3.8.m7.2.3.2">superscript</csymbol><ci id="S1.p3.8.m7.2.3.2.2.cmml" xref="S1.p3.8.m7.2.3.2.2">𝐻</ci><times id="S1.p3.8.m7.2.3.2.3.cmml" xref="S1.p3.8.m7.2.3.2.3"></times></apply><list id="S1.p3.8.m7.2.3.3.1.cmml" xref="S1.p3.8.m7.2.3.3.2"><ci id="S1.p3.8.m7.1.1.cmml" xref="S1.p3.8.m7.1.1">𝒞</ci><ci id="S1.p3.8.m7.2.2.cmml" xref="S1.p3.8.m7.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.8.m7.2c">H^{*}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S1.p3.8.m7.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> of a small category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p3.9.m8.1"><semantics id="S1.p3.9.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p3.9.m8.1.1" xref="S1.p3.9.m8.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p3.9.m8.1b"><ci id="S1.p3.9.m8.1.1.cmml" xref="S1.p3.9.m8.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.9.m8.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.9.m8.1d">caligraphic_C</annotation></semantics></math> with coefficients in an <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S1.p3.10.m9.1"><semantics id="S1.p3.10.m9.1a"><mrow id="S1.p3.10.m9.1.1" xref="S1.p3.10.m9.1.1.cmml"><mi id="S1.p3.10.m9.1.1.2" xref="S1.p3.10.m9.1.1.2.cmml">R</mi><mo id="S1.p3.10.m9.1.1.1" xref="S1.p3.10.m9.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.10.m9.1.1.3" xref="S1.p3.10.m9.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.10.m9.1b"><apply id="S1.p3.10.m9.1.1.cmml" xref="S1.p3.10.m9.1.1"><times id="S1.p3.10.m9.1.1.1.cmml" xref="S1.p3.10.m9.1.1.1"></times><ci id="S1.p3.10.m9.1.1.2.cmml" xref="S1.p3.10.m9.1.1.2">𝑅</ci><ci id="S1.p3.10.m9.1.1.3.cmml" xref="S1.p3.10.m9.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.10.m9.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.10.m9.1d">italic_R caligraphic_C</annotation></semantics></math>-module <math alttext="M" class="ltx_Math" display="inline" id="S1.p3.11.m10.1"><semantics id="S1.p3.11.m10.1a"><mi id="S1.p3.11.m10.1.1" xref="S1.p3.11.m10.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S1.p3.11.m10.1b"><ci id="S1.p3.11.m10.1.1.cmml" xref="S1.p3.11.m10.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.11.m10.1c">M</annotation><annotation encoding="application/x-llamapun" id="S1.p3.11.m10.1d">italic_M</annotation></semantics></math> is defined as the ext-group <math alttext="\mathrm{Ext}_{R\mathcal{C}}(\underline{R};M)" class="ltx_Math" display="inline" id="S1.p3.12.m11.2"><semantics id="S1.p3.12.m11.2a"><mrow id="S1.p3.12.m11.2.3" xref="S1.p3.12.m11.2.3.cmml"><msub id="S1.p3.12.m11.2.3.2" xref="S1.p3.12.m11.2.3.2.cmml"><mi id="S1.p3.12.m11.2.3.2.2" xref="S1.p3.12.m11.2.3.2.2.cmml">Ext</mi><mrow id="S1.p3.12.m11.2.3.2.3" xref="S1.p3.12.m11.2.3.2.3.cmml"><mi id="S1.p3.12.m11.2.3.2.3.2" xref="S1.p3.12.m11.2.3.2.3.2.cmml">R</mi><mo id="S1.p3.12.m11.2.3.2.3.1" xref="S1.p3.12.m11.2.3.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.12.m11.2.3.2.3.3" xref="S1.p3.12.m11.2.3.2.3.3.cmml">𝒞</mi></mrow></msub><mo id="S1.p3.12.m11.2.3.1" xref="S1.p3.12.m11.2.3.1.cmml">⁢</mo><mrow id="S1.p3.12.m11.2.3.3.2" xref="S1.p3.12.m11.2.3.3.1.cmml"><mo id="S1.p3.12.m11.2.3.3.2.1" stretchy="false" xref="S1.p3.12.m11.2.3.3.1.cmml">(</mo><munder accentunder="true" id="S1.p3.12.m11.1.1" xref="S1.p3.12.m11.1.1.cmml"><mi id="S1.p3.12.m11.1.1.2" xref="S1.p3.12.m11.1.1.2.cmml">R</mi><mo id="S1.p3.12.m11.1.1.1" xref="S1.p3.12.m11.1.1.1.cmml">¯</mo></munder><mo id="S1.p3.12.m11.2.3.3.2.2" xref="S1.p3.12.m11.2.3.3.1.cmml">;</mo><mi id="S1.p3.12.m11.2.2" xref="S1.p3.12.m11.2.2.cmml">M</mi><mo id="S1.p3.12.m11.2.3.3.2.3" stretchy="false" xref="S1.p3.12.m11.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.12.m11.2b"><apply id="S1.p3.12.m11.2.3.cmml" xref="S1.p3.12.m11.2.3"><times id="S1.p3.12.m11.2.3.1.cmml" xref="S1.p3.12.m11.2.3.1"></times><apply id="S1.p3.12.m11.2.3.2.cmml" xref="S1.p3.12.m11.2.3.2"><csymbol cd="ambiguous" id="S1.p3.12.m11.2.3.2.1.cmml" xref="S1.p3.12.m11.2.3.2">subscript</csymbol><ci id="S1.p3.12.m11.2.3.2.2.cmml" xref="S1.p3.12.m11.2.3.2.2">Ext</ci><apply id="S1.p3.12.m11.2.3.2.3.cmml" xref="S1.p3.12.m11.2.3.2.3"><times id="S1.p3.12.m11.2.3.2.3.1.cmml" xref="S1.p3.12.m11.2.3.2.3.1"></times><ci id="S1.p3.12.m11.2.3.2.3.2.cmml" xref="S1.p3.12.m11.2.3.2.3.2">𝑅</ci><ci id="S1.p3.12.m11.2.3.2.3.3.cmml" xref="S1.p3.12.m11.2.3.2.3.3">𝒞</ci></apply></apply><list id="S1.p3.12.m11.2.3.3.1.cmml" xref="S1.p3.12.m11.2.3.3.2"><apply id="S1.p3.12.m11.1.1.cmml" xref="S1.p3.12.m11.1.1"><ci id="S1.p3.12.m11.1.1.1.cmml" xref="S1.p3.12.m11.1.1.1">¯</ci><ci id="S1.p3.12.m11.1.1.2.cmml" xref="S1.p3.12.m11.1.1.2">𝑅</ci></apply><ci id="S1.p3.12.m11.2.2.cmml" xref="S1.p3.12.m11.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.12.m11.2c">\mathrm{Ext}_{R\mathcal{C}}(\underline{R};M)</annotation><annotation encoding="application/x-llamapun" id="S1.p3.12.m11.2d">roman_Ext start_POSTSUBSCRIPT italic_R caligraphic_C end_POSTSUBSCRIPT ( under¯ start_ARG italic_R end_ARG ; italic_M )</annotation></semantics></math> over the module category of <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S1.p3.13.m12.1"><semantics id="S1.p3.13.m12.1a"><mrow id="S1.p3.13.m12.1.1" xref="S1.p3.13.m12.1.1.cmml"><mi id="S1.p3.13.m12.1.1.2" xref="S1.p3.13.m12.1.1.2.cmml">R</mi><mo id="S1.p3.13.m12.1.1.1" xref="S1.p3.13.m12.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.13.m12.1.1.3" xref="S1.p3.13.m12.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.13.m12.1b"><apply id="S1.p3.13.m12.1.1.cmml" xref="S1.p3.13.m12.1.1"><times id="S1.p3.13.m12.1.1.1.cmml" xref="S1.p3.13.m12.1.1.1"></times><ci id="S1.p3.13.m12.1.1.2.cmml" xref="S1.p3.13.m12.1.1.2">𝑅</ci><ci id="S1.p3.13.m12.1.1.3.cmml" xref="S1.p3.13.m12.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.13.m12.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.13.m12.1d">italic_R caligraphic_C</annotation></semantics></math>-modules (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S2.SS1" title="2.1. Cohomology of small categories ‣ 2. Modules and cohomology for small categories ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">2.1</span></a>). There is a standard resolution for <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S1.p3.14.m13.1"><semantics id="S1.p3.14.m13.1a"><munder accentunder="true" id="S1.p3.14.m13.1.1" xref="S1.p3.14.m13.1.1.cmml"><mi id="S1.p3.14.m13.1.1.2" xref="S1.p3.14.m13.1.1.2.cmml">R</mi><mo id="S1.p3.14.m13.1.1.1" xref="S1.p3.14.m13.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S1.p3.14.m13.1b"><apply id="S1.p3.14.m13.1.1.cmml" xref="S1.p3.14.m13.1.1"><ci id="S1.p3.14.m13.1.1.1.cmml" xref="S1.p3.14.m13.1.1.1">¯</ci><ci id="S1.p3.14.m13.1.1.2.cmml" xref="S1.p3.14.m13.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.14.m13.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.14.m13.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S1.p3.15.m14.1"><semantics id="S1.p3.15.m14.1a"><mrow id="S1.p3.15.m14.1.1" xref="S1.p3.15.m14.1.1.cmml"><mi id="S1.p3.15.m14.1.1.2" xref="S1.p3.15.m14.1.1.2.cmml">R</mi><mo id="S1.p3.15.m14.1.1.1" xref="S1.p3.15.m14.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.15.m14.1.1.3" xref="S1.p3.15.m14.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.15.m14.1b"><apply id="S1.p3.15.m14.1.1.cmml" xref="S1.p3.15.m14.1.1"><times id="S1.p3.15.m14.1.1.1.cmml" xref="S1.p3.15.m14.1.1.1"></times><ci id="S1.p3.15.m14.1.1.2.cmml" xref="S1.p3.15.m14.1.1.2">𝑅</ci><ci id="S1.p3.15.m14.1.1.3.cmml" xref="S1.p3.15.m14.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.15.m14.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.15.m14.1d">italic_R caligraphic_C</annotation></semantics></math>-module that allows us to express the cohomology groups <math alttext="H^{*}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S1.p3.16.m15.2"><semantics id="S1.p3.16.m15.2a"><mrow id="S1.p3.16.m15.2.3" xref="S1.p3.16.m15.2.3.cmml"><msup id="S1.p3.16.m15.2.3.2" xref="S1.p3.16.m15.2.3.2.cmml"><mi id="S1.p3.16.m15.2.3.2.2" xref="S1.p3.16.m15.2.3.2.2.cmml">H</mi><mo id="S1.p3.16.m15.2.3.2.3" xref="S1.p3.16.m15.2.3.2.3.cmml">∗</mo></msup><mo id="S1.p3.16.m15.2.3.1" xref="S1.p3.16.m15.2.3.1.cmml">⁢</mo><mrow id="S1.p3.16.m15.2.3.3.2" xref="S1.p3.16.m15.2.3.3.1.cmml"><mo id="S1.p3.16.m15.2.3.3.2.1" stretchy="false" xref="S1.p3.16.m15.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.16.m15.1.1" xref="S1.p3.16.m15.1.1.cmml">𝒞</mi><mo id="S1.p3.16.m15.2.3.3.2.2" xref="S1.p3.16.m15.2.3.3.1.cmml">;</mo><mi id="S1.p3.16.m15.2.2" xref="S1.p3.16.m15.2.2.cmml">M</mi><mo id="S1.p3.16.m15.2.3.3.2.3" stretchy="false" xref="S1.p3.16.m15.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.16.m15.2b"><apply id="S1.p3.16.m15.2.3.cmml" xref="S1.p3.16.m15.2.3"><times id="S1.p3.16.m15.2.3.1.cmml" xref="S1.p3.16.m15.2.3.1"></times><apply id="S1.p3.16.m15.2.3.2.cmml" xref="S1.p3.16.m15.2.3.2"><csymbol cd="ambiguous" id="S1.p3.16.m15.2.3.2.1.cmml" xref="S1.p3.16.m15.2.3.2">superscript</csymbol><ci id="S1.p3.16.m15.2.3.2.2.cmml" xref="S1.p3.16.m15.2.3.2.2">𝐻</ci><times id="S1.p3.16.m15.2.3.2.3.cmml" xref="S1.p3.16.m15.2.3.2.3"></times></apply><list id="S1.p3.16.m15.2.3.3.1.cmml" xref="S1.p3.16.m15.2.3.3.2"><ci id="S1.p3.16.m15.1.1.cmml" xref="S1.p3.16.m15.1.1">𝒞</ci><ci id="S1.p3.16.m15.2.2.cmml" xref="S1.p3.16.m15.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.16.m15.2c">H^{*}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S1.p3.16.m15.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> as cohomology groups of an explicit cochain complex. We refer to this cohomology theory as the <em class="ltx_emph ltx_font_italic" id="S1.p3.16.2">Quillen cohomology</em> of a small category.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.14">Given a small category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">caligraphic_C</annotation></semantics></math>, the <em class="ltx_emph ltx_font_italic" id="S1.p4.14.1">category of factorizations</em> <math alttext="\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mrow id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml"><mi id="S1.p4.2.m2.1.1.2" xref="S1.p4.2.m2.1.1.2.cmml">𝔉</mi><mo id="S1.p4.2.m2.1.1.1" xref="S1.p4.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p4.2.m2.1.1.3" xref="S1.p4.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><apply id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1"><times id="S1.p4.2.m2.1.1.1.cmml" xref="S1.p4.2.m2.1.1.1"></times><ci id="S1.p4.2.m2.1.1.2.cmml" xref="S1.p4.2.m2.1.1.2">𝔉</ci><ci id="S1.p4.2.m2.1.1.3.cmml" xref="S1.p4.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">fraktur_F caligraphic_C</annotation></semantics></math> is the category whose objects are the morphisms in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">caligraphic_C</annotation></semantics></math> and whose morphisms <math alttext="f\to g" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><mrow id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml"><mi id="S1.p4.4.m4.1.1.2" xref="S1.p4.4.m4.1.1.2.cmml">f</mi><mo id="S1.p4.4.m4.1.1.1" stretchy="false" xref="S1.p4.4.m4.1.1.1.cmml">→</mo><mi id="S1.p4.4.m4.1.1.3" xref="S1.p4.4.m4.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><apply id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1"><ci id="S1.p4.4.m4.1.1.1.cmml" xref="S1.p4.4.m4.1.1.1">→</ci><ci id="S1.p4.4.m4.1.1.2.cmml" xref="S1.p4.4.m4.1.1.2">𝑓</ci><ci id="S1.p4.4.m4.1.1.3.cmml" xref="S1.p4.4.m4.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">f\to g</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">italic_f → italic_g</annotation></semantics></math> are given by a pair of morphisms <math alttext="(u,v)" class="ltx_Math" display="inline" id="S1.p4.5.m5.2"><semantics id="S1.p4.5.m5.2a"><mrow id="S1.p4.5.m5.2.3.2" xref="S1.p4.5.m5.2.3.1.cmml"><mo id="S1.p4.5.m5.2.3.2.1" stretchy="false" xref="S1.p4.5.m5.2.3.1.cmml">(</mo><mi id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml">u</mi><mo id="S1.p4.5.m5.2.3.2.2" xref="S1.p4.5.m5.2.3.1.cmml">,</mo><mi id="S1.p4.5.m5.2.2" xref="S1.p4.5.m5.2.2.cmml">v</mi><mo id="S1.p4.5.m5.2.3.2.3" stretchy="false" xref="S1.p4.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.2b"><interval closure="open" id="S1.p4.5.m5.2.3.1.cmml" xref="S1.p4.5.m5.2.3.2"><ci id="S1.p4.5.m5.1.1.cmml" xref="S1.p4.5.m5.1.1">𝑢</ci><ci id="S1.p4.5.m5.2.2.cmml" xref="S1.p4.5.m5.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.2d">( italic_u , italic_v )</annotation></semantics></math> in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p4.6.m6.1"><semantics id="S1.p4.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p4.6.m6.1b"><ci id="S1.p4.6.m6.1.1.cmml" xref="S1.p4.6.m6.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m6.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m6.1d">caligraphic_C</annotation></semantics></math> such that <math alttext="ufv=g" class="ltx_Math" display="inline" id="S1.p4.7.m7.1"><semantics id="S1.p4.7.m7.1a"><mrow id="S1.p4.7.m7.1.1" xref="S1.p4.7.m7.1.1.cmml"><mrow id="S1.p4.7.m7.1.1.2" xref="S1.p4.7.m7.1.1.2.cmml"><mi id="S1.p4.7.m7.1.1.2.2" xref="S1.p4.7.m7.1.1.2.2.cmml">u</mi><mo id="S1.p4.7.m7.1.1.2.1" xref="S1.p4.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S1.p4.7.m7.1.1.2.3" xref="S1.p4.7.m7.1.1.2.3.cmml">f</mi><mo id="S1.p4.7.m7.1.1.2.1a" xref="S1.p4.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S1.p4.7.m7.1.1.2.4" xref="S1.p4.7.m7.1.1.2.4.cmml">v</mi></mrow><mo id="S1.p4.7.m7.1.1.1" xref="S1.p4.7.m7.1.1.1.cmml">=</mo><mi id="S1.p4.7.m7.1.1.3" xref="S1.p4.7.m7.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.7.m7.1b"><apply id="S1.p4.7.m7.1.1.cmml" xref="S1.p4.7.m7.1.1"><eq id="S1.p4.7.m7.1.1.1.cmml" xref="S1.p4.7.m7.1.1.1"></eq><apply id="S1.p4.7.m7.1.1.2.cmml" xref="S1.p4.7.m7.1.1.2"><times id="S1.p4.7.m7.1.1.2.1.cmml" xref="S1.p4.7.m7.1.1.2.1"></times><ci id="S1.p4.7.m7.1.1.2.2.cmml" xref="S1.p4.7.m7.1.1.2.2">𝑢</ci><ci id="S1.p4.7.m7.1.1.2.3.cmml" xref="S1.p4.7.m7.1.1.2.3">𝑓</ci><ci id="S1.p4.7.m7.1.1.2.4.cmml" xref="S1.p4.7.m7.1.1.2.4">𝑣</ci></apply><ci id="S1.p4.7.m7.1.1.3.cmml" xref="S1.p4.7.m7.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.7.m7.1c">ufv=g</annotation><annotation encoding="application/x-llamapun" id="S1.p4.7.m7.1d">italic_u italic_f italic_v = italic_g</annotation></semantics></math>. A functor <math alttext="M:\mathfrak{F}\mathcal{C}\to R" class="ltx_Math" display="inline" id="S1.p4.8.m8.1"><semantics id="S1.p4.8.m8.1a"><mrow id="S1.p4.8.m8.1.1" xref="S1.p4.8.m8.1.1.cmml"><mi id="S1.p4.8.m8.1.1.2" xref="S1.p4.8.m8.1.1.2.cmml">M</mi><mo id="S1.p4.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p4.8.m8.1.1.1.cmml">:</mo><mrow id="S1.p4.8.m8.1.1.3" xref="S1.p4.8.m8.1.1.3.cmml"><mrow id="S1.p4.8.m8.1.1.3.2" xref="S1.p4.8.m8.1.1.3.2.cmml"><mi id="S1.p4.8.m8.1.1.3.2.2" xref="S1.p4.8.m8.1.1.3.2.2.cmml">𝔉</mi><mo id="S1.p4.8.m8.1.1.3.2.1" xref="S1.p4.8.m8.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p4.8.m8.1.1.3.2.3" xref="S1.p4.8.m8.1.1.3.2.3.cmml">𝒞</mi></mrow><mo id="S1.p4.8.m8.1.1.3.1" stretchy="false" xref="S1.p4.8.m8.1.1.3.1.cmml">→</mo><mi id="S1.p4.8.m8.1.1.3.3" xref="S1.p4.8.m8.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.8.m8.1b"><apply id="S1.p4.8.m8.1.1.cmml" xref="S1.p4.8.m8.1.1"><ci id="S1.p4.8.m8.1.1.1.cmml" xref="S1.p4.8.m8.1.1.1">:</ci><ci id="S1.p4.8.m8.1.1.2.cmml" xref="S1.p4.8.m8.1.1.2">𝑀</ci><apply id="S1.p4.8.m8.1.1.3.cmml" xref="S1.p4.8.m8.1.1.3"><ci id="S1.p4.8.m8.1.1.3.1.cmml" xref="S1.p4.8.m8.1.1.3.1">→</ci><apply id="S1.p4.8.m8.1.1.3.2.cmml" xref="S1.p4.8.m8.1.1.3.2"><times id="S1.p4.8.m8.1.1.3.2.1.cmml" xref="S1.p4.8.m8.1.1.3.2.1"></times><ci id="S1.p4.8.m8.1.1.3.2.2.cmml" xref="S1.p4.8.m8.1.1.3.2.2">𝔉</ci><ci id="S1.p4.8.m8.1.1.3.2.3.cmml" xref="S1.p4.8.m8.1.1.3.2.3">𝒞</ci></apply><ci id="S1.p4.8.m8.1.1.3.3.cmml" xref="S1.p4.8.m8.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.8.m8.1c">M:\mathfrak{F}\mathcal{C}\to R</annotation><annotation encoding="application/x-llamapun" id="S1.p4.8.m8.1d">italic_M : fraktur_F caligraphic_C → italic_R</annotation></semantics></math>-Mod is called a <em class="ltx_emph ltx_font_italic" id="S1.p4.14.2">natural system</em> for <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p4.9.m9.1"><semantics id="S1.p4.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.9.m9.1.1" xref="S1.p4.9.m9.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p4.9.m9.1b"><ci id="S1.p4.9.m9.1.1.cmml" xref="S1.p4.9.m9.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.9.m9.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.9.m9.1d">caligraphic_C</annotation></semantics></math> over <math alttext="R" class="ltx_Math" display="inline" id="S1.p4.10.m10.1"><semantics id="S1.p4.10.m10.1a"><mi id="S1.p4.10.m10.1.1" xref="S1.p4.10.m10.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S1.p4.10.m10.1b"><ci id="S1.p4.10.m10.1.1.cmml" xref="S1.p4.10.m10.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.10.m10.1c">R</annotation><annotation encoding="application/x-llamapun" id="S1.p4.10.m10.1d">italic_R</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S1.p4.14.3">Baues-Wirsching cohomology</em> <math alttext="H^{*}_{BW}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S1.p4.11.m11.2"><semantics id="S1.p4.11.m11.2a"><mrow id="S1.p4.11.m11.2.3" xref="S1.p4.11.m11.2.3.cmml"><msubsup id="S1.p4.11.m11.2.3.2" xref="S1.p4.11.m11.2.3.2.cmml"><mi id="S1.p4.11.m11.2.3.2.2.2" xref="S1.p4.11.m11.2.3.2.2.2.cmml">H</mi><mrow id="S1.p4.11.m11.2.3.2.3" xref="S1.p4.11.m11.2.3.2.3.cmml"><mi id="S1.p4.11.m11.2.3.2.3.2" xref="S1.p4.11.m11.2.3.2.3.2.cmml">B</mi><mo id="S1.p4.11.m11.2.3.2.3.1" xref="S1.p4.11.m11.2.3.2.3.1.cmml">⁢</mo><mi id="S1.p4.11.m11.2.3.2.3.3" xref="S1.p4.11.m11.2.3.2.3.3.cmml">W</mi></mrow><mo id="S1.p4.11.m11.2.3.2.2.3" xref="S1.p4.11.m11.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S1.p4.11.m11.2.3.1" xref="S1.p4.11.m11.2.3.1.cmml">⁢</mo><mrow id="S1.p4.11.m11.2.3.3.2" xref="S1.p4.11.m11.2.3.3.1.cmml"><mo id="S1.p4.11.m11.2.3.3.2.1" stretchy="false" xref="S1.p4.11.m11.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p4.11.m11.1.1" xref="S1.p4.11.m11.1.1.cmml">𝒞</mi><mo id="S1.p4.11.m11.2.3.3.2.2" xref="S1.p4.11.m11.2.3.3.1.cmml">;</mo><mi id="S1.p4.11.m11.2.2" xref="S1.p4.11.m11.2.2.cmml">M</mi><mo id="S1.p4.11.m11.2.3.3.2.3" stretchy="false" xref="S1.p4.11.m11.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.11.m11.2b"><apply id="S1.p4.11.m11.2.3.cmml" xref="S1.p4.11.m11.2.3"><times id="S1.p4.11.m11.2.3.1.cmml" xref="S1.p4.11.m11.2.3.1"></times><apply id="S1.p4.11.m11.2.3.2.cmml" xref="S1.p4.11.m11.2.3.2"><csymbol cd="ambiguous" id="S1.p4.11.m11.2.3.2.1.cmml" xref="S1.p4.11.m11.2.3.2">subscript</csymbol><apply id="S1.p4.11.m11.2.3.2.2.cmml" xref="S1.p4.11.m11.2.3.2"><csymbol cd="ambiguous" id="S1.p4.11.m11.2.3.2.2.1.cmml" xref="S1.p4.11.m11.2.3.2">superscript</csymbol><ci id="S1.p4.11.m11.2.3.2.2.2.cmml" xref="S1.p4.11.m11.2.3.2.2.2">𝐻</ci><times id="S1.p4.11.m11.2.3.2.2.3.cmml" xref="S1.p4.11.m11.2.3.2.2.3"></times></apply><apply id="S1.p4.11.m11.2.3.2.3.cmml" xref="S1.p4.11.m11.2.3.2.3"><times id="S1.p4.11.m11.2.3.2.3.1.cmml" xref="S1.p4.11.m11.2.3.2.3.1"></times><ci id="S1.p4.11.m11.2.3.2.3.2.cmml" xref="S1.p4.11.m11.2.3.2.3.2">𝐵</ci><ci id="S1.p4.11.m11.2.3.2.3.3.cmml" xref="S1.p4.11.m11.2.3.2.3.3">𝑊</ci></apply></apply><list id="S1.p4.11.m11.2.3.3.1.cmml" xref="S1.p4.11.m11.2.3.3.2"><ci id="S1.p4.11.m11.1.1.cmml" xref="S1.p4.11.m11.1.1">𝒞</ci><ci id="S1.p4.11.m11.2.2.cmml" xref="S1.p4.11.m11.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.11.m11.2c">H^{*}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.11.m11.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> of the category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p4.12.m12.1"><semantics id="S1.p4.12.m12.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.12.m12.1.1" xref="S1.p4.12.m12.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p4.12.m12.1b"><ci id="S1.p4.12.m12.1.1.cmml" xref="S1.p4.12.m12.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.12.m12.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.12.m12.1d">caligraphic_C</annotation></semantics></math> with coefficients in a natural system <math alttext="M" class="ltx_Math" display="inline" id="S1.p4.13.m13.1"><semantics id="S1.p4.13.m13.1a"><mi id="S1.p4.13.m13.1.1" xref="S1.p4.13.m13.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S1.p4.13.m13.1b"><ci id="S1.p4.13.m13.1.1.cmml" xref="S1.p4.13.m13.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.13.m13.1c">M</annotation><annotation encoding="application/x-llamapun" id="S1.p4.13.m13.1d">italic_M</annotation></semantics></math> is either defined using an explicit cochain complex or defined as the ext-group <math alttext="\mathrm{Ext}_{R\mathfrak{F}\mathcal{C}}(\underline{R},M)" class="ltx_Math" display="inline" id="S1.p4.14.m14.2"><semantics id="S1.p4.14.m14.2a"><mrow id="S1.p4.14.m14.2.3" xref="S1.p4.14.m14.2.3.cmml"><msub id="S1.p4.14.m14.2.3.2" xref="S1.p4.14.m14.2.3.2.cmml"><mi id="S1.p4.14.m14.2.3.2.2" xref="S1.p4.14.m14.2.3.2.2.cmml">Ext</mi><mrow id="S1.p4.14.m14.2.3.2.3" xref="S1.p4.14.m14.2.3.2.3.cmml"><mi id="S1.p4.14.m14.2.3.2.3.2" xref="S1.p4.14.m14.2.3.2.3.2.cmml">R</mi><mo id="S1.p4.14.m14.2.3.2.3.1" xref="S1.p4.14.m14.2.3.2.3.1.cmml">⁢</mo><mi id="S1.p4.14.m14.2.3.2.3.3" xref="S1.p4.14.m14.2.3.2.3.3.cmml">𝔉</mi><mo id="S1.p4.14.m14.2.3.2.3.1a" xref="S1.p4.14.m14.2.3.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p4.14.m14.2.3.2.3.4" xref="S1.p4.14.m14.2.3.2.3.4.cmml">𝒞</mi></mrow></msub><mo id="S1.p4.14.m14.2.3.1" xref="S1.p4.14.m14.2.3.1.cmml">⁢</mo><mrow id="S1.p4.14.m14.2.3.3.2" xref="S1.p4.14.m14.2.3.3.1.cmml"><mo id="S1.p4.14.m14.2.3.3.2.1" stretchy="false" xref="S1.p4.14.m14.2.3.3.1.cmml">(</mo><munder accentunder="true" id="S1.p4.14.m14.1.1" xref="S1.p4.14.m14.1.1.cmml"><mi id="S1.p4.14.m14.1.1.2" xref="S1.p4.14.m14.1.1.2.cmml">R</mi><mo id="S1.p4.14.m14.1.1.1" xref="S1.p4.14.m14.1.1.1.cmml">¯</mo></munder><mo id="S1.p4.14.m14.2.3.3.2.2" xref="S1.p4.14.m14.2.3.3.1.cmml">,</mo><mi id="S1.p4.14.m14.2.2" xref="S1.p4.14.m14.2.2.cmml">M</mi><mo id="S1.p4.14.m14.2.3.3.2.3" stretchy="false" xref="S1.p4.14.m14.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.14.m14.2b"><apply id="S1.p4.14.m14.2.3.cmml" xref="S1.p4.14.m14.2.3"><times id="S1.p4.14.m14.2.3.1.cmml" xref="S1.p4.14.m14.2.3.1"></times><apply id="S1.p4.14.m14.2.3.2.cmml" xref="S1.p4.14.m14.2.3.2"><csymbol cd="ambiguous" id="S1.p4.14.m14.2.3.2.1.cmml" xref="S1.p4.14.m14.2.3.2">subscript</csymbol><ci id="S1.p4.14.m14.2.3.2.2.cmml" xref="S1.p4.14.m14.2.3.2.2">Ext</ci><apply id="S1.p4.14.m14.2.3.2.3.cmml" xref="S1.p4.14.m14.2.3.2.3"><times id="S1.p4.14.m14.2.3.2.3.1.cmml" xref="S1.p4.14.m14.2.3.2.3.1"></times><ci id="S1.p4.14.m14.2.3.2.3.2.cmml" xref="S1.p4.14.m14.2.3.2.3.2">𝑅</ci><ci id="S1.p4.14.m14.2.3.2.3.3.cmml" xref="S1.p4.14.m14.2.3.2.3.3">𝔉</ci><ci id="S1.p4.14.m14.2.3.2.3.4.cmml" xref="S1.p4.14.m14.2.3.2.3.4">𝒞</ci></apply></apply><interval closure="open" id="S1.p4.14.m14.2.3.3.1.cmml" xref="S1.p4.14.m14.2.3.3.2"><apply id="S1.p4.14.m14.1.1.cmml" xref="S1.p4.14.m14.1.1"><ci id="S1.p4.14.m14.1.1.1.cmml" xref="S1.p4.14.m14.1.1.1">¯</ci><ci id="S1.p4.14.m14.1.1.2.cmml" xref="S1.p4.14.m14.1.1.2">𝑅</ci></apply><ci id="S1.p4.14.m14.2.2.cmml" xref="S1.p4.14.m14.2.2">𝑀</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.14.m14.2c">\mathrm{Ext}_{R\mathfrak{F}\mathcal{C}}(\underline{R},M)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.14.m14.2d">roman_Ext start_POSTSUBSCRIPT italic_R fraktur_F caligraphic_C end_POSTSUBSCRIPT ( under¯ start_ARG italic_R end_ARG , italic_M )</annotation></semantics></math> (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S2.SS2" title="2.2. Baues-Wirsching Cohomology ‣ 2. Modules and cohomology for small categories ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">2.2</span></a>).</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.20">For a small category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><ci id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">caligraphic_C</annotation></semantics></math>, let <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mrow id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><mi id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2.cmml">N</mi><mo id="S1.p5.2.m2.1.1.1" xref="S1.p5.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><apply id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1"><times id="S1.p5.2.m2.1.1.1.cmml" xref="S1.p5.2.m2.1.1.1"></times><ci id="S1.p5.2.m2.1.1.2.cmml" xref="S1.p5.2.m2.1.1.2">𝑁</ci><ci id="S1.p5.2.m2.1.1.3.cmml" xref="S1.p5.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">italic_N caligraphic_C</annotation></semantics></math> denote the nerve of the category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><ci id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">caligraphic_C</annotation></semantics></math>. We can consider the simplex category <math alttext="\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mrow id="S1.p5.4.m4.1.1" xref="S1.p5.4.m4.1.1.cmml"><mi id="S1.p5.4.m4.1.1.3" mathvariant="normal" xref="S1.p5.4.m4.1.1.3.cmml">Δ</mi><mo id="S1.p5.4.m4.1.1.2" xref="S1.p5.4.m4.1.1.2.cmml">⁢</mo><mrow id="S1.p5.4.m4.1.1.1.1" xref="S1.p5.4.m4.1.1.1.1.1.cmml"><mo id="S1.p5.4.m4.1.1.1.1.2" stretchy="false" xref="S1.p5.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.4.m4.1.1.1.1.1" xref="S1.p5.4.m4.1.1.1.1.1.cmml"><mi id="S1.p5.4.m4.1.1.1.1.1.2" xref="S1.p5.4.m4.1.1.1.1.1.2.cmml">N</mi><mo id="S1.p5.4.m4.1.1.1.1.1.1" xref="S1.p5.4.m4.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.4.m4.1.1.1.1.1.3" xref="S1.p5.4.m4.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S1.p5.4.m4.1.1.1.1.3" stretchy="false" xref="S1.p5.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.1b"><apply id="S1.p5.4.m4.1.1.cmml" xref="S1.p5.4.m4.1.1"><times id="S1.p5.4.m4.1.1.2.cmml" xref="S1.p5.4.m4.1.1.2"></times><ci id="S1.p5.4.m4.1.1.3.cmml" xref="S1.p5.4.m4.1.1.3">Δ</ci><apply id="S1.p5.4.m4.1.1.1.1.1.cmml" xref="S1.p5.4.m4.1.1.1.1"><times id="S1.p5.4.m4.1.1.1.1.1.1.cmml" xref="S1.p5.4.m4.1.1.1.1.1.1"></times><ci id="S1.p5.4.m4.1.1.1.1.1.2.cmml" xref="S1.p5.4.m4.1.1.1.1.1.2">𝑁</ci><ci id="S1.p5.4.m4.1.1.1.1.1.3.cmml" xref="S1.p5.4.m4.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math> of the simplicial set <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.5.m5.1"><semantics id="S1.p5.5.m5.1a"><mrow id="S1.p5.5.m5.1.1" xref="S1.p5.5.m5.1.1.cmml"><mi id="S1.p5.5.m5.1.1.2" xref="S1.p5.5.m5.1.1.2.cmml">N</mi><mo id="S1.p5.5.m5.1.1.1" xref="S1.p5.5.m5.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.5.m5.1.1.3" xref="S1.p5.5.m5.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.5.m5.1b"><apply id="S1.p5.5.m5.1.1.cmml" xref="S1.p5.5.m5.1.1"><times id="S1.p5.5.m5.1.1.1.cmml" xref="S1.p5.5.m5.1.1.1"></times><ci id="S1.p5.5.m5.1.1.2.cmml" xref="S1.p5.5.m5.1.1.2">𝑁</ci><ci id="S1.p5.5.m5.1.1.3.cmml" xref="S1.p5.5.m5.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.5.m5.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.5.m5.1d">italic_N caligraphic_C</annotation></semantics></math>. A functor <math alttext="\Delta(N\mathcal{C})\to R" class="ltx_Math" display="inline" id="S1.p5.6.m6.1"><semantics id="S1.p5.6.m6.1a"><mrow id="S1.p5.6.m6.1.1" xref="S1.p5.6.m6.1.1.cmml"><mrow id="S1.p5.6.m6.1.1.1" xref="S1.p5.6.m6.1.1.1.cmml"><mi id="S1.p5.6.m6.1.1.1.3" mathvariant="normal" xref="S1.p5.6.m6.1.1.1.3.cmml">Δ</mi><mo id="S1.p5.6.m6.1.1.1.2" xref="S1.p5.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S1.p5.6.m6.1.1.1.1.1" xref="S1.p5.6.m6.1.1.1.1.1.1.cmml"><mo id="S1.p5.6.m6.1.1.1.1.1.2" stretchy="false" xref="S1.p5.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.6.m6.1.1.1.1.1.1" xref="S1.p5.6.m6.1.1.1.1.1.1.cmml"><mi id="S1.p5.6.m6.1.1.1.1.1.1.2" xref="S1.p5.6.m6.1.1.1.1.1.1.2.cmml">N</mi><mo id="S1.p5.6.m6.1.1.1.1.1.1.1" xref="S1.p5.6.m6.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.6.m6.1.1.1.1.1.1.3" xref="S1.p5.6.m6.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S1.p5.6.m6.1.1.1.1.1.3" stretchy="false" xref="S1.p5.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p5.6.m6.1.1.2" stretchy="false" xref="S1.p5.6.m6.1.1.2.cmml">→</mo><mi id="S1.p5.6.m6.1.1.3" xref="S1.p5.6.m6.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.6.m6.1b"><apply id="S1.p5.6.m6.1.1.cmml" xref="S1.p5.6.m6.1.1"><ci id="S1.p5.6.m6.1.1.2.cmml" xref="S1.p5.6.m6.1.1.2">→</ci><apply id="S1.p5.6.m6.1.1.1.cmml" xref="S1.p5.6.m6.1.1.1"><times id="S1.p5.6.m6.1.1.1.2.cmml" xref="S1.p5.6.m6.1.1.1.2"></times><ci id="S1.p5.6.m6.1.1.1.3.cmml" xref="S1.p5.6.m6.1.1.1.3">Δ</ci><apply id="S1.p5.6.m6.1.1.1.1.1.1.cmml" xref="S1.p5.6.m6.1.1.1.1.1"><times id="S1.p5.6.m6.1.1.1.1.1.1.1.cmml" xref="S1.p5.6.m6.1.1.1.1.1.1.1"></times><ci id="S1.p5.6.m6.1.1.1.1.1.1.2.cmml" xref="S1.p5.6.m6.1.1.1.1.1.1.2">𝑁</ci><ci id="S1.p5.6.m6.1.1.1.1.1.1.3.cmml" xref="S1.p5.6.m6.1.1.1.1.1.1.3">𝒞</ci></apply></apply><ci id="S1.p5.6.m6.1.1.3.cmml" xref="S1.p5.6.m6.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.6.m6.1c">\Delta(N\mathcal{C})\to R</annotation><annotation encoding="application/x-llamapun" id="S1.p5.6.m6.1d">roman_Δ ( italic_N caligraphic_C ) → italic_R</annotation></semantics></math>-Mod is called a <em class="ltx_emph ltx_font_italic" id="S1.p5.7.1">coefficient system on <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.7.1.m1.1"><semantics id="S1.p5.7.1.m1.1a"><mrow id="S1.p5.7.1.m1.1.1" xref="S1.p5.7.1.m1.1.1.cmml"><mi id="S1.p5.7.1.m1.1.1.2" xref="S1.p5.7.1.m1.1.1.2.cmml">N</mi><mo id="S1.p5.7.1.m1.1.1.1" xref="S1.p5.7.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.7.1.m1.1.1.3" xref="S1.p5.7.1.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.7.1.m1.1b"><apply id="S1.p5.7.1.m1.1.1.cmml" xref="S1.p5.7.1.m1.1.1"><times id="S1.p5.7.1.m1.1.1.1.cmml" xref="S1.p5.7.1.m1.1.1.1"></times><ci id="S1.p5.7.1.m1.1.1.2.cmml" xref="S1.p5.7.1.m1.1.1.2">𝑁</ci><ci id="S1.p5.7.1.m1.1.1.3.cmml" xref="S1.p5.7.1.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.7.1.m1.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.7.1.m1.1d">italic_N caligraphic_C</annotation></semantics></math></em> (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.SS2" title="3.2. Cohomology of simplicial sets with general coefficients ‣ 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">3.2</span></a> for details). The <em class="ltx_emph ltx_font_italic" id="S1.p5.20.2">Thomason cohomology</em> <math alttext="H^{*}_{Th}(\mathcal{C};\mathcal{M})" class="ltx_Math" display="inline" id="S1.p5.8.m7.2"><semantics id="S1.p5.8.m7.2a"><mrow id="S1.p5.8.m7.2.3" xref="S1.p5.8.m7.2.3.cmml"><msubsup id="S1.p5.8.m7.2.3.2" xref="S1.p5.8.m7.2.3.2.cmml"><mi id="S1.p5.8.m7.2.3.2.2.2" xref="S1.p5.8.m7.2.3.2.2.2.cmml">H</mi><mrow id="S1.p5.8.m7.2.3.2.3" xref="S1.p5.8.m7.2.3.2.3.cmml"><mi id="S1.p5.8.m7.2.3.2.3.2" xref="S1.p5.8.m7.2.3.2.3.2.cmml">T</mi><mo id="S1.p5.8.m7.2.3.2.3.1" xref="S1.p5.8.m7.2.3.2.3.1.cmml">⁢</mo><mi id="S1.p5.8.m7.2.3.2.3.3" xref="S1.p5.8.m7.2.3.2.3.3.cmml">h</mi></mrow><mo id="S1.p5.8.m7.2.3.2.2.3" xref="S1.p5.8.m7.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S1.p5.8.m7.2.3.1" xref="S1.p5.8.m7.2.3.1.cmml">⁢</mo><mrow id="S1.p5.8.m7.2.3.3.2" xref="S1.p5.8.m7.2.3.3.1.cmml"><mo id="S1.p5.8.m7.2.3.3.2.1" stretchy="false" xref="S1.p5.8.m7.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.8.m7.1.1" xref="S1.p5.8.m7.1.1.cmml">𝒞</mi><mo id="S1.p5.8.m7.2.3.3.2.2" xref="S1.p5.8.m7.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.8.m7.2.2" xref="S1.p5.8.m7.2.2.cmml">ℳ</mi><mo id="S1.p5.8.m7.2.3.3.2.3" stretchy="false" xref="S1.p5.8.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.8.m7.2b"><apply id="S1.p5.8.m7.2.3.cmml" xref="S1.p5.8.m7.2.3"><times id="S1.p5.8.m7.2.3.1.cmml" xref="S1.p5.8.m7.2.3.1"></times><apply id="S1.p5.8.m7.2.3.2.cmml" xref="S1.p5.8.m7.2.3.2"><csymbol cd="ambiguous" id="S1.p5.8.m7.2.3.2.1.cmml" xref="S1.p5.8.m7.2.3.2">subscript</csymbol><apply id="S1.p5.8.m7.2.3.2.2.cmml" xref="S1.p5.8.m7.2.3.2"><csymbol cd="ambiguous" id="S1.p5.8.m7.2.3.2.2.1.cmml" xref="S1.p5.8.m7.2.3.2">superscript</csymbol><ci id="S1.p5.8.m7.2.3.2.2.2.cmml" xref="S1.p5.8.m7.2.3.2.2.2">𝐻</ci><times id="S1.p5.8.m7.2.3.2.2.3.cmml" xref="S1.p5.8.m7.2.3.2.2.3"></times></apply><apply id="S1.p5.8.m7.2.3.2.3.cmml" xref="S1.p5.8.m7.2.3.2.3"><times id="S1.p5.8.m7.2.3.2.3.1.cmml" xref="S1.p5.8.m7.2.3.2.3.1"></times><ci id="S1.p5.8.m7.2.3.2.3.2.cmml" xref="S1.p5.8.m7.2.3.2.3.2">𝑇</ci><ci id="S1.p5.8.m7.2.3.2.3.3.cmml" xref="S1.p5.8.m7.2.3.2.3.3">ℎ</ci></apply></apply><list id="S1.p5.8.m7.2.3.3.1.cmml" xref="S1.p5.8.m7.2.3.3.2"><ci id="S1.p5.8.m7.1.1.cmml" xref="S1.p5.8.m7.1.1">𝒞</ci><ci id="S1.p5.8.m7.2.2.cmml" xref="S1.p5.8.m7.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.8.m7.2c">H^{*}_{Th}(\mathcal{C};\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.8.m7.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; caligraphic_M )</annotation></semantics></math> of a category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.9.m8.1"><semantics id="S1.p5.9.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.9.m8.1.1" xref="S1.p5.9.m8.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p5.9.m8.1b"><ci id="S1.p5.9.m8.1.1.cmml" xref="S1.p5.9.m8.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.9.m8.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.9.m8.1d">caligraphic_C</annotation></semantics></math> with coefficients in <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S1.p5.10.m9.1"><semantics id="S1.p5.10.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.10.m9.1.1" xref="S1.p5.10.m9.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S1.p5.10.m9.1b"><ci id="S1.p5.10.m9.1.1.cmml" xref="S1.p5.10.m9.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.10.m9.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.10.m9.1d">caligraphic_M</annotation></semantics></math> is defined as the cohomology of the simplicial set <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.11.m10.1"><semantics id="S1.p5.11.m10.1a"><mrow id="S1.p5.11.m10.1.1" xref="S1.p5.11.m10.1.1.cmml"><mi id="S1.p5.11.m10.1.1.2" xref="S1.p5.11.m10.1.1.2.cmml">N</mi><mo id="S1.p5.11.m10.1.1.1" xref="S1.p5.11.m10.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.11.m10.1.1.3" xref="S1.p5.11.m10.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.11.m10.1b"><apply id="S1.p5.11.m10.1.1.cmml" xref="S1.p5.11.m10.1.1"><times id="S1.p5.11.m10.1.1.1.cmml" xref="S1.p5.11.m10.1.1.1"></times><ci id="S1.p5.11.m10.1.1.2.cmml" xref="S1.p5.11.m10.1.1.2">𝑁</ci><ci id="S1.p5.11.m10.1.1.3.cmml" xref="S1.p5.11.m10.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.11.m10.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.11.m10.1d">italic_N caligraphic_C</annotation></semantics></math> with coefficients in <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S1.p5.12.m11.1"><semantics id="S1.p5.12.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.12.m11.1.1" xref="S1.p5.12.m11.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S1.p5.12.m11.1b"><ci id="S1.p5.12.m11.1.1.cmml" xref="S1.p5.12.m11.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.12.m11.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.12.m11.1d">caligraphic_M</annotation></semantics></math> which is isomorphic to the cohomology of the category of simplices <math alttext="\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S1.p5.13.m12.1"><semantics id="S1.p5.13.m12.1a"><mrow id="S1.p5.13.m12.1.1" xref="S1.p5.13.m12.1.1.cmml"><mi id="S1.p5.13.m12.1.1.3" mathvariant="normal" xref="S1.p5.13.m12.1.1.3.cmml">Δ</mi><mo id="S1.p5.13.m12.1.1.2" xref="S1.p5.13.m12.1.1.2.cmml">⁢</mo><mrow id="S1.p5.13.m12.1.1.1.1" xref="S1.p5.13.m12.1.1.1.1.1.cmml"><mo id="S1.p5.13.m12.1.1.1.1.2" stretchy="false" xref="S1.p5.13.m12.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.13.m12.1.1.1.1.1" xref="S1.p5.13.m12.1.1.1.1.1.cmml"><mi id="S1.p5.13.m12.1.1.1.1.1.2" xref="S1.p5.13.m12.1.1.1.1.1.2.cmml">N</mi><mo id="S1.p5.13.m12.1.1.1.1.1.1" xref="S1.p5.13.m12.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.13.m12.1.1.1.1.1.3" xref="S1.p5.13.m12.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S1.p5.13.m12.1.1.1.1.3" stretchy="false" xref="S1.p5.13.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.13.m12.1b"><apply id="S1.p5.13.m12.1.1.cmml" xref="S1.p5.13.m12.1.1"><times id="S1.p5.13.m12.1.1.2.cmml" xref="S1.p5.13.m12.1.1.2"></times><ci id="S1.p5.13.m12.1.1.3.cmml" xref="S1.p5.13.m12.1.1.3">Δ</ci><apply id="S1.p5.13.m12.1.1.1.1.1.cmml" xref="S1.p5.13.m12.1.1.1.1"><times id="S1.p5.13.m12.1.1.1.1.1.1.cmml" xref="S1.p5.13.m12.1.1.1.1.1.1"></times><ci id="S1.p5.13.m12.1.1.1.1.1.2.cmml" xref="S1.p5.13.m12.1.1.1.1.1.2">𝑁</ci><ci id="S1.p5.13.m12.1.1.1.1.1.3.cmml" xref="S1.p5.13.m12.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.13.m12.1c">\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.13.m12.1d">roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math> of the nerve <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.14.m13.1"><semantics id="S1.p5.14.m13.1a"><mrow id="S1.p5.14.m13.1.1" xref="S1.p5.14.m13.1.1.cmml"><mi id="S1.p5.14.m13.1.1.2" xref="S1.p5.14.m13.1.1.2.cmml">N</mi><mo id="S1.p5.14.m13.1.1.1" xref="S1.p5.14.m13.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.14.m13.1.1.3" xref="S1.p5.14.m13.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.14.m13.1b"><apply id="S1.p5.14.m13.1.1.cmml" xref="S1.p5.14.m13.1.1"><times id="S1.p5.14.m13.1.1.1.cmml" xref="S1.p5.14.m13.1.1.1"></times><ci id="S1.p5.14.m13.1.1.2.cmml" xref="S1.p5.14.m13.1.1.2">𝑁</ci><ci id="S1.p5.14.m13.1.1.3.cmml" xref="S1.p5.14.m13.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.14.m13.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.14.m13.1d">italic_N caligraphic_C</annotation></semantics></math> with coefficients in <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S1.p5.15.m14.1"><semantics id="S1.p5.15.m14.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.15.m14.1.1" xref="S1.p5.15.m14.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S1.p5.15.m14.1b"><ci id="S1.p5.15.m14.1.1.cmml" xref="S1.p5.15.m14.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.15.m14.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.15.m14.1d">caligraphic_M</annotation></semantics></math> (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.SS4" title="3.4. The Thomason cohomology of a category ‣ 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">3.4</span></a>). The cohomology groups <math alttext="H^{*}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S1.p5.16.m15.2"><semantics id="S1.p5.16.m15.2a"><mrow id="S1.p5.16.m15.2.3" xref="S1.p5.16.m15.2.3.cmml"><msup id="S1.p5.16.m15.2.3.2" xref="S1.p5.16.m15.2.3.2.cmml"><mi id="S1.p5.16.m15.2.3.2.2" xref="S1.p5.16.m15.2.3.2.2.cmml">H</mi><mo id="S1.p5.16.m15.2.3.2.3" xref="S1.p5.16.m15.2.3.2.3.cmml">∗</mo></msup><mo id="S1.p5.16.m15.2.3.1" xref="S1.p5.16.m15.2.3.1.cmml">⁢</mo><mrow id="S1.p5.16.m15.2.3.3.2" xref="S1.p5.16.m15.2.3.3.1.cmml"><mo id="S1.p5.16.m15.2.3.3.2.1" stretchy="false" xref="S1.p5.16.m15.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.16.m15.1.1" xref="S1.p5.16.m15.1.1.cmml">𝒞</mi><mo id="S1.p5.16.m15.2.3.3.2.2" xref="S1.p5.16.m15.2.3.3.1.cmml">;</mo><mi id="S1.p5.16.m15.2.2" xref="S1.p5.16.m15.2.2.cmml">M</mi><mo id="S1.p5.16.m15.2.3.3.2.3" stretchy="false" xref="S1.p5.16.m15.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.16.m15.2b"><apply id="S1.p5.16.m15.2.3.cmml" xref="S1.p5.16.m15.2.3"><times id="S1.p5.16.m15.2.3.1.cmml" xref="S1.p5.16.m15.2.3.1"></times><apply id="S1.p5.16.m15.2.3.2.cmml" xref="S1.p5.16.m15.2.3.2"><csymbol cd="ambiguous" id="S1.p5.16.m15.2.3.2.1.cmml" xref="S1.p5.16.m15.2.3.2">superscript</csymbol><ci id="S1.p5.16.m15.2.3.2.2.cmml" xref="S1.p5.16.m15.2.3.2.2">𝐻</ci><times id="S1.p5.16.m15.2.3.2.3.cmml" xref="S1.p5.16.m15.2.3.2.3"></times></apply><list id="S1.p5.16.m15.2.3.3.1.cmml" xref="S1.p5.16.m15.2.3.3.2"><ci id="S1.p5.16.m15.1.1.cmml" xref="S1.p5.16.m15.1.1">𝒞</ci><ci id="S1.p5.16.m15.2.2.cmml" xref="S1.p5.16.m15.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.16.m15.2c">H^{*}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.16.m15.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> and <math alttext="H^{*}_{BW}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S1.p5.17.m16.2"><semantics id="S1.p5.17.m16.2a"><mrow id="S1.p5.17.m16.2.3" xref="S1.p5.17.m16.2.3.cmml"><msubsup id="S1.p5.17.m16.2.3.2" xref="S1.p5.17.m16.2.3.2.cmml"><mi id="S1.p5.17.m16.2.3.2.2.2" xref="S1.p5.17.m16.2.3.2.2.2.cmml">H</mi><mrow id="S1.p5.17.m16.2.3.2.3" xref="S1.p5.17.m16.2.3.2.3.cmml"><mi id="S1.p5.17.m16.2.3.2.3.2" xref="S1.p5.17.m16.2.3.2.3.2.cmml">B</mi><mo id="S1.p5.17.m16.2.3.2.3.1" xref="S1.p5.17.m16.2.3.2.3.1.cmml">⁢</mo><mi id="S1.p5.17.m16.2.3.2.3.3" xref="S1.p5.17.m16.2.3.2.3.3.cmml">W</mi></mrow><mo id="S1.p5.17.m16.2.3.2.2.3" xref="S1.p5.17.m16.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S1.p5.17.m16.2.3.1" xref="S1.p5.17.m16.2.3.1.cmml">⁢</mo><mrow id="S1.p5.17.m16.2.3.3.2" xref="S1.p5.17.m16.2.3.3.1.cmml"><mo id="S1.p5.17.m16.2.3.3.2.1" stretchy="false" xref="S1.p5.17.m16.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.17.m16.1.1" xref="S1.p5.17.m16.1.1.cmml">𝒞</mi><mo id="S1.p5.17.m16.2.3.3.2.2" xref="S1.p5.17.m16.2.3.3.1.cmml">;</mo><mi id="S1.p5.17.m16.2.2" xref="S1.p5.17.m16.2.2.cmml">M</mi><mo id="S1.p5.17.m16.2.3.3.2.3" stretchy="false" xref="S1.p5.17.m16.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.17.m16.2b"><apply id="S1.p5.17.m16.2.3.cmml" xref="S1.p5.17.m16.2.3"><times id="S1.p5.17.m16.2.3.1.cmml" xref="S1.p5.17.m16.2.3.1"></times><apply id="S1.p5.17.m16.2.3.2.cmml" xref="S1.p5.17.m16.2.3.2"><csymbol cd="ambiguous" id="S1.p5.17.m16.2.3.2.1.cmml" xref="S1.p5.17.m16.2.3.2">subscript</csymbol><apply id="S1.p5.17.m16.2.3.2.2.cmml" xref="S1.p5.17.m16.2.3.2"><csymbol cd="ambiguous" id="S1.p5.17.m16.2.3.2.2.1.cmml" xref="S1.p5.17.m16.2.3.2">superscript</csymbol><ci id="S1.p5.17.m16.2.3.2.2.2.cmml" xref="S1.p5.17.m16.2.3.2.2.2">𝐻</ci><times id="S1.p5.17.m16.2.3.2.2.3.cmml" xref="S1.p5.17.m16.2.3.2.2.3"></times></apply><apply id="S1.p5.17.m16.2.3.2.3.cmml" xref="S1.p5.17.m16.2.3.2.3"><times id="S1.p5.17.m16.2.3.2.3.1.cmml" xref="S1.p5.17.m16.2.3.2.3.1"></times><ci id="S1.p5.17.m16.2.3.2.3.2.cmml" xref="S1.p5.17.m16.2.3.2.3.2">𝐵</ci><ci id="S1.p5.17.m16.2.3.2.3.3.cmml" xref="S1.p5.17.m16.2.3.2.3.3">𝑊</ci></apply></apply><list id="S1.p5.17.m16.2.3.3.1.cmml" xref="S1.p5.17.m16.2.3.3.2"><ci id="S1.p5.17.m16.1.1.cmml" xref="S1.p5.17.m16.1.1">𝒞</ci><ci id="S1.p5.17.m16.2.2.cmml" xref="S1.p5.17.m16.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.17.m16.2c">H^{*}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.17.m16.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> can be realized as the Thomason cohomology <math alttext="H^{*}_{Th}(\mathcal{C};\mathcal{M})" class="ltx_Math" display="inline" id="S1.p5.18.m17.2"><semantics id="S1.p5.18.m17.2a"><mrow id="S1.p5.18.m17.2.3" xref="S1.p5.18.m17.2.3.cmml"><msubsup id="S1.p5.18.m17.2.3.2" xref="S1.p5.18.m17.2.3.2.cmml"><mi id="S1.p5.18.m17.2.3.2.2.2" xref="S1.p5.18.m17.2.3.2.2.2.cmml">H</mi><mrow id="S1.p5.18.m17.2.3.2.3" xref="S1.p5.18.m17.2.3.2.3.cmml"><mi id="S1.p5.18.m17.2.3.2.3.2" xref="S1.p5.18.m17.2.3.2.3.2.cmml">T</mi><mo id="S1.p5.18.m17.2.3.2.3.1" xref="S1.p5.18.m17.2.3.2.3.1.cmml">⁢</mo><mi id="S1.p5.18.m17.2.3.2.3.3" xref="S1.p5.18.m17.2.3.2.3.3.cmml">h</mi></mrow><mo id="S1.p5.18.m17.2.3.2.2.3" xref="S1.p5.18.m17.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S1.p5.18.m17.2.3.1" xref="S1.p5.18.m17.2.3.1.cmml">⁢</mo><mrow id="S1.p5.18.m17.2.3.3.2" xref="S1.p5.18.m17.2.3.3.1.cmml"><mo id="S1.p5.18.m17.2.3.3.2.1" stretchy="false" xref="S1.p5.18.m17.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.18.m17.1.1" xref="S1.p5.18.m17.1.1.cmml">𝒞</mi><mo id="S1.p5.18.m17.2.3.3.2.2" xref="S1.p5.18.m17.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S1.p5.18.m17.2.2" xref="S1.p5.18.m17.2.2.cmml">ℳ</mi><mo id="S1.p5.18.m17.2.3.3.2.3" stretchy="false" xref="S1.p5.18.m17.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.18.m17.2b"><apply id="S1.p5.18.m17.2.3.cmml" xref="S1.p5.18.m17.2.3"><times id="S1.p5.18.m17.2.3.1.cmml" xref="S1.p5.18.m17.2.3.1"></times><apply id="S1.p5.18.m17.2.3.2.cmml" xref="S1.p5.18.m17.2.3.2"><csymbol cd="ambiguous" id="S1.p5.18.m17.2.3.2.1.cmml" xref="S1.p5.18.m17.2.3.2">subscript</csymbol><apply id="S1.p5.18.m17.2.3.2.2.cmml" xref="S1.p5.18.m17.2.3.2"><csymbol cd="ambiguous" id="S1.p5.18.m17.2.3.2.2.1.cmml" xref="S1.p5.18.m17.2.3.2">superscript</csymbol><ci id="S1.p5.18.m17.2.3.2.2.2.cmml" xref="S1.p5.18.m17.2.3.2.2.2">𝐻</ci><times id="S1.p5.18.m17.2.3.2.2.3.cmml" xref="S1.p5.18.m17.2.3.2.2.3"></times></apply><apply id="S1.p5.18.m17.2.3.2.3.cmml" xref="S1.p5.18.m17.2.3.2.3"><times id="S1.p5.18.m17.2.3.2.3.1.cmml" xref="S1.p5.18.m17.2.3.2.3.1"></times><ci id="S1.p5.18.m17.2.3.2.3.2.cmml" xref="S1.p5.18.m17.2.3.2.3.2">𝑇</ci><ci id="S1.p5.18.m17.2.3.2.3.3.cmml" xref="S1.p5.18.m17.2.3.2.3.3">ℎ</ci></apply></apply><list id="S1.p5.18.m17.2.3.3.1.cmml" xref="S1.p5.18.m17.2.3.3.2"><ci id="S1.p5.18.m17.1.1.cmml" xref="S1.p5.18.m17.1.1">𝒞</ci><ci id="S1.p5.18.m17.2.2.cmml" xref="S1.p5.18.m17.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.18.m17.2c">H^{*}_{Th}(\mathcal{C};\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.18.m17.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; caligraphic_M )</annotation></semantics></math> of <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p5.19.m18.1"><semantics id="S1.p5.19.m18.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.19.m18.1.1" xref="S1.p5.19.m18.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p5.19.m18.1b"><ci id="S1.p5.19.m18.1.1.cmml" xref="S1.p5.19.m18.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.19.m18.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.19.m18.1d">caligraphic_C</annotation></semantics></math> with suitable choices of coefficient systems <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S1.p5.20.m19.1"><semantics id="S1.p5.20.m19.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.20.m19.1.1" xref="S1.p5.20.m19.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S1.p5.20.m19.1b"><ci id="S1.p5.20.m19.1.1.cmml" xref="S1.p5.20.m19.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.20.m19.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.20.m19.1d">caligraphic_M</annotation></semantics></math> (see Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.Thmtheorem7" title="Lemma 3.7 ([7, Thm 2.1]). ‣ 3.4. The Thomason cohomology of a category ‣ 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">3.7</span></a>).</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">The main theorem of the paper is the following:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.1.1.1">Theorem 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.4"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.4.4">Let <math alttext="\varphi:\mathcal{C}\rightarrow\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.1.1.m1.1"><semantics id="S1.Thmtheorem2.p1.1.1.m1.1a"><mrow id="S1.Thmtheorem2.p1.1.1.m1.1.1" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem2.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.2.cmml">φ</mi><mo id="S1.Thmtheorem2.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.1.1.m1.1b"><apply id="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1"><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.1">:</ci><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.2">𝜑</ci><apply id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3"><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.1">→</ci><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2">𝒞</ci><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.1.1.m1.1c">\varphi:\mathcal{C}\rightarrow\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.1.1.m1.1d">italic_φ : caligraphic_C → caligraphic_D</annotation></semantics></math> be a functor between two small categories, and let <math alttext="X=\operatorname*{hocolim}_{\mathcal{D}}N(\varphi/-)" class="ltx_math_unparsed" display="inline" id="S1.Thmtheorem2.p1.2.2.m2.1"><semantics id="S1.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem2.p1.2.2.m2.1b"><mi id="S1.Thmtheorem2.p1.2.2.m2.1.1">X</mi><mo id="S1.Thmtheorem2.p1.2.2.m2.1.2" rspace="0.1389em">=</mo><msub id="S1.Thmtheorem2.p1.2.2.m2.1.3"><mo id="S1.Thmtheorem2.p1.2.2.m2.1.3.2" lspace="0.1389em" rspace="0.167em">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.2.2.m2.1.3.3">𝒟</mi></msub><mi id="S1.Thmtheorem2.p1.2.2.m2.1.4">N</mi><mrow id="S1.Thmtheorem2.p1.2.2.m2.1.5"><mo id="S1.Thmtheorem2.p1.2.2.m2.1.5.1" stretchy="false">(</mo><mi id="S1.Thmtheorem2.p1.2.2.m2.1.5.2">φ</mi><mo id="S1.Thmtheorem2.p1.2.2.m2.1.5.3" rspace="0em">/</mo><mo id="S1.Thmtheorem2.p1.2.2.m2.1.5.4" lspace="0em" rspace="0em">−</mo><mo id="S1.Thmtheorem2.p1.2.2.m2.1.5.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.2.2.m2.1c">X=\operatorname*{hocolim}_{\mathcal{D}}N(\varphi/-)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.2.2.m2.1d">italic_X = roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_N ( italic_φ / - )</annotation></semantics></math> and <math alttext="\kappa:X\to N\mathcal{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.3.3.m3.1"><semantics id="S1.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">κ</mi><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.cmml"><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml">X</mi><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.1" stretchy="false" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.1.cmml">→</mo><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml"><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.cmml">N</mi><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.3.cmml">𝒞</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.3.3.m3.1b"><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1"><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1">:</ci><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.2">𝜅</ci><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3"><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.1">→</ci><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.2">𝑋</ci><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3"><times id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.1"></times><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.2">𝑁</ci><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.3.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.3">𝒞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.3.3.m3.1c">\kappa:X\to N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.3.3.m3.1d">italic_κ : italic_X → italic_N caligraphic_C</annotation></semantics></math> be the homotopy equivalence defined in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem1" title="Definition 1.1. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.1</span></a>. Then for every coefficient system <math alttext="\mathcal{M}:\Delta(N\mathcal{C})\rightarrow R" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.4.4.m4.1"><semantics id="S1.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S1.Thmtheorem2.p1.4.4.m4.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">ℳ</mi><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.2" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">:</mo><mrow id="S1.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.cmml"><mrow id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.cmml"><mi id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.3" mathvariant="normal" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.3.cmml">Δ</mi><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.2" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.2" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.3" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.2" stretchy="false" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.2.cmml">→</mo><mi id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.3" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.4.4.m4.1b"><apply id="S1.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1"><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.2">:</ci><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3">ℳ</ci><apply id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1"><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.2">→</ci><apply id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1"><times id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.2"></times><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.3">Δ</ci><apply id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1"><times id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.1"></times><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.2">𝑁</ci><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.3">𝒞</ci></apply></apply><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.4.4.m4.1c">\mathcal{M}:\Delta(N\mathcal{C})\rightarrow R</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.4.4.m4.1d">caligraphic_M : roman_Δ ( italic_N caligraphic_C ) → italic_R</annotation></semantics></math>-Mod, the homomorphism</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\kappa^{*}:H^{*}_{Th}(\mathcal{C};\mathcal{M})\smash{\,\mathop{\longrightarrow% }\limits^{\cong}\,}H^{*}(X;\kappa^{*}\mathcal{M})" class="ltx_Math" display="block" id="S1.Ex2.m1.4"><semantics id="S1.Ex2.m1.4a"><mrow id="S1.Ex2.m1.4.4" xref="S1.Ex2.m1.4.4.cmml"><msup id="S1.Ex2.m1.4.4.3" xref="S1.Ex2.m1.4.4.3.cmml"><mi id="S1.Ex2.m1.4.4.3.2" xref="S1.Ex2.m1.4.4.3.2.cmml">κ</mi><mo id="S1.Ex2.m1.4.4.3.3" xref="S1.Ex2.m1.4.4.3.3.cmml">∗</mo></msup><mo id="S1.Ex2.m1.4.4.2" lspace="0.278em" rspace="0.278em" xref="S1.Ex2.m1.4.4.2.cmml">:</mo><mrow id="S1.Ex2.m1.4.4.1" xref="S1.Ex2.m1.4.4.1.cmml"><msubsup id="S1.Ex2.m1.4.4.1.3" xref="S1.Ex2.m1.4.4.1.3.cmml"><mi id="S1.Ex2.m1.4.4.1.3.2.2" xref="S1.Ex2.m1.4.4.1.3.2.2.cmml">H</mi><mrow id="S1.Ex2.m1.4.4.1.3.3" xref="S1.Ex2.m1.4.4.1.3.3.cmml"><mi id="S1.Ex2.m1.4.4.1.3.3.2" xref="S1.Ex2.m1.4.4.1.3.3.2.cmml">T</mi><mo id="S1.Ex2.m1.4.4.1.3.3.1" xref="S1.Ex2.m1.4.4.1.3.3.1.cmml">⁢</mo><mi id="S1.Ex2.m1.4.4.1.3.3.3" xref="S1.Ex2.m1.4.4.1.3.3.3.cmml">h</mi></mrow><mo id="S1.Ex2.m1.4.4.1.3.2.3" xref="S1.Ex2.m1.4.4.1.3.2.3.cmml">∗</mo></msubsup><mo id="S1.Ex2.m1.4.4.1.2" xref="S1.Ex2.m1.4.4.1.2.cmml">⁢</mo><mrow id="S1.Ex2.m1.4.4.1.4.2" xref="S1.Ex2.m1.4.4.1.4.1.cmml"><mo id="S1.Ex2.m1.4.4.1.4.2.1" stretchy="false" xref="S1.Ex2.m1.4.4.1.4.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.1.1" xref="S1.Ex2.m1.1.1.cmml">𝒞</mi><mo id="S1.Ex2.m1.4.4.1.4.2.2" xref="S1.Ex2.m1.4.4.1.4.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.2.2" xref="S1.Ex2.m1.2.2.cmml">ℳ</mi><mo id="S1.Ex2.m1.4.4.1.4.2.3" stretchy="false" xref="S1.Ex2.m1.4.4.1.4.1.cmml">)</mo></mrow><mo id="S1.Ex2.m1.4.4.1.2a" lspace="0.337em" xref="S1.Ex2.m1.4.4.1.2.cmml">⁢</mo><mrow id="S1.Ex2.m1.4.4.1.1" xref="S1.Ex2.m1.4.4.1.1.cmml"><mover id="S1.Ex2.m1.4.4.1.1.2" xref="S1.Ex2.m1.4.4.1.1.2.cmml"><mo id="S1.Ex2.m1.4.4.1.1.2.2" movablelimits="false" rspace="0.167em" xref="S1.Ex2.m1.4.4.1.1.2.2.cmml">⟶</mo><mo id="S1.Ex2.m1.4.4.1.1.2.3" xref="S1.Ex2.m1.4.4.1.1.2.3.cmml">≅</mo></mover><mrow id="S1.Ex2.m1.4.4.1.1.1" xref="S1.Ex2.m1.4.4.1.1.1.cmml"><msup id="S1.Ex2.m1.4.4.1.1.1.3" xref="S1.Ex2.m1.4.4.1.1.1.3.cmml"><mi id="S1.Ex2.m1.4.4.1.1.1.3.2" xref="S1.Ex2.m1.4.4.1.1.1.3.2.cmml">H</mi><mo id="S1.Ex2.m1.4.4.1.1.1.3.3" xref="S1.Ex2.m1.4.4.1.1.1.3.3.cmml">∗</mo></msup><mo id="S1.Ex2.m1.4.4.1.1.1.2" xref="S1.Ex2.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex2.m1.4.4.1.1.1.1.1" xref="S1.Ex2.m1.4.4.1.1.1.1.2.cmml"><mo id="S1.Ex2.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S1.Ex2.m1.4.4.1.1.1.1.2.cmml">(</mo><mi id="S1.Ex2.m1.3.3" xref="S1.Ex2.m1.3.3.cmml">X</mi><mo id="S1.Ex2.m1.4.4.1.1.1.1.1.3" xref="S1.Ex2.m1.4.4.1.1.1.1.2.cmml">;</mo><mrow id="S1.Ex2.m1.4.4.1.1.1.1.1.1" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.cmml"><msup id="S1.Ex2.m1.4.4.1.1.1.1.1.1.2" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.cmml"><mi id="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.2" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.2.cmml">κ</mi><mo id="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.3" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S1.Ex2.m1.4.4.1.1.1.1.1.1.1" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.4.4.1.1.1.1.1.1.3" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.3.cmml">ℳ</mi></mrow><mo id="S1.Ex2.m1.4.4.1.1.1.1.1.4" stretchy="false" xref="S1.Ex2.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex2.m1.4b"><apply id="S1.Ex2.m1.4.4.cmml" xref="S1.Ex2.m1.4.4"><ci id="S1.Ex2.m1.4.4.2.cmml" xref="S1.Ex2.m1.4.4.2">:</ci><apply id="S1.Ex2.m1.4.4.3.cmml" xref="S1.Ex2.m1.4.4.3"><csymbol cd="ambiguous" id="S1.Ex2.m1.4.4.3.1.cmml" xref="S1.Ex2.m1.4.4.3">superscript</csymbol><ci id="S1.Ex2.m1.4.4.3.2.cmml" xref="S1.Ex2.m1.4.4.3.2">𝜅</ci><times id="S1.Ex2.m1.4.4.3.3.cmml" xref="S1.Ex2.m1.4.4.3.3"></times></apply><apply id="S1.Ex2.m1.4.4.1.cmml" 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id="S1.Ex2.m1.3.3.cmml" xref="S1.Ex2.m1.3.3">𝑋</ci><apply id="S1.Ex2.m1.4.4.1.1.1.1.1.1.cmml" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1"><times id="S1.Ex2.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.1"></times><apply id="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.1.cmml" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.2">superscript</csymbol><ci id="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.2.cmml" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.2">𝜅</ci><times id="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.3.cmml" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.2.3"></times></apply><ci id="S1.Ex2.m1.4.4.1.1.1.1.1.1.3.cmml" xref="S1.Ex2.m1.4.4.1.1.1.1.1.1.3">ℳ</ci></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex2.m1.4c">\kappa^{*}:H^{*}_{Th}(\mathcal{C};\mathcal{M})\smash{\,\mathop{\longrightarrow% }\limits^{\cong}\,}H^{*}(X;\kappa^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S1.Ex2.m1.4d">italic_κ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; caligraphic_M ) ⟶ start_POSTSUPERSCRIPT ≅ end_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; italic_κ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.Thmtheorem2.p1.5"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.5.1">induced by <math alttext="\kappa" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.5.1.m1.1"><semantics id="S1.Thmtheorem2.p1.5.1.m1.1a"><mi id="S1.Thmtheorem2.p1.5.1.m1.1.1" xref="S1.Thmtheorem2.p1.5.1.m1.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.5.1.m1.1b"><ci id="S1.Thmtheorem2.p1.5.1.m1.1.1.cmml" xref="S1.Thmtheorem2.p1.5.1.m1.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.5.1.m1.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.5.1.m1.1d">italic_κ</annotation></semantics></math> is an isomorphism.</span></p> </div> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.5">Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a> is proved by constructing a double complex of projective <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S1.p7.1.m1.1"><semantics id="S1.p7.1.m1.1a"><mrow id="S1.p7.1.m1.1.1" xref="S1.p7.1.m1.1.1.cmml"><mi id="S1.p7.1.m1.1.1.3" xref="S1.p7.1.m1.1.1.3.cmml">R</mi><mo id="S1.p7.1.m1.1.1.2" xref="S1.p7.1.m1.1.1.2.cmml">⁢</mo><mi id="S1.p7.1.m1.1.1.4" mathvariant="normal" xref="S1.p7.1.m1.1.1.4.cmml">Δ</mi><mo id="S1.p7.1.m1.1.1.2a" xref="S1.p7.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.p7.1.m1.1.1.1.1" xref="S1.p7.1.m1.1.1.1.1.1.cmml"><mo id="S1.p7.1.m1.1.1.1.1.2" stretchy="false" xref="S1.p7.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p7.1.m1.1.1.1.1.1" xref="S1.p7.1.m1.1.1.1.1.1.cmml"><mi id="S1.p7.1.m1.1.1.1.1.1.2" xref="S1.p7.1.m1.1.1.1.1.1.2.cmml">N</mi><mo id="S1.p7.1.m1.1.1.1.1.1.1" xref="S1.p7.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p7.1.m1.1.1.1.1.1.3" xref="S1.p7.1.m1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S1.p7.1.m1.1.1.1.1.3" stretchy="false" xref="S1.p7.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.1.m1.1b"><apply id="S1.p7.1.m1.1.1.cmml" xref="S1.p7.1.m1.1.1"><times id="S1.p7.1.m1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.2"></times><ci id="S1.p7.1.m1.1.1.3.cmml" xref="S1.p7.1.m1.1.1.3">𝑅</ci><ci id="S1.p7.1.m1.1.1.4.cmml" xref="S1.p7.1.m1.1.1.4">Δ</ci><apply id="S1.p7.1.m1.1.1.1.1.1.cmml" xref="S1.p7.1.m1.1.1.1.1"><times id="S1.p7.1.m1.1.1.1.1.1.1.cmml" xref="S1.p7.1.m1.1.1.1.1.1.1"></times><ci id="S1.p7.1.m1.1.1.1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.1.1.1.2">𝑁</ci><ci id="S1.p7.1.m1.1.1.1.1.1.3.cmml" xref="S1.p7.1.m1.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.1.m1.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S1.p7.1.m1.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-modules whose total complex gives a projective resolution for the constant functor <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S1.p7.2.m2.1"><semantics id="S1.p7.2.m2.1a"><munder accentunder="true" id="S1.p7.2.m2.1.1" xref="S1.p7.2.m2.1.1.cmml"><mi id="S1.p7.2.m2.1.1.2" xref="S1.p7.2.m2.1.1.2.cmml">R</mi><mo id="S1.p7.2.m2.1.1.1" xref="S1.p7.2.m2.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S1.p7.2.m2.1b"><apply id="S1.p7.2.m2.1.1.cmml" xref="S1.p7.2.m2.1.1"><ci id="S1.p7.2.m2.1.1.1.cmml" xref="S1.p7.2.m2.1.1.1">¯</ci><ci id="S1.p7.2.m2.1.1.2.cmml" xref="S1.p7.2.m2.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.2.m2.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p7.2.m2.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S1.p7.3.m3.1"><semantics id="S1.p7.3.m3.1a"><mrow id="S1.p7.3.m3.1.1" xref="S1.p7.3.m3.1.1.cmml"><mi id="S1.p7.3.m3.1.1.3" xref="S1.p7.3.m3.1.1.3.cmml">R</mi><mo id="S1.p7.3.m3.1.1.2" xref="S1.p7.3.m3.1.1.2.cmml">⁢</mo><mi id="S1.p7.3.m3.1.1.4" mathvariant="normal" xref="S1.p7.3.m3.1.1.4.cmml">Δ</mi><mo id="S1.p7.3.m3.1.1.2a" xref="S1.p7.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.p7.3.m3.1.1.1.1" xref="S1.p7.3.m3.1.1.1.1.1.cmml"><mo id="S1.p7.3.m3.1.1.1.1.2" stretchy="false" xref="S1.p7.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.p7.3.m3.1.1.1.1.1" xref="S1.p7.3.m3.1.1.1.1.1.cmml"><mi id="S1.p7.3.m3.1.1.1.1.1.2" xref="S1.p7.3.m3.1.1.1.1.1.2.cmml">N</mi><mo id="S1.p7.3.m3.1.1.1.1.1.1" xref="S1.p7.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p7.3.m3.1.1.1.1.1.3" xref="S1.p7.3.m3.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S1.p7.3.m3.1.1.1.1.3" stretchy="false" xref="S1.p7.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.3.m3.1b"><apply id="S1.p7.3.m3.1.1.cmml" xref="S1.p7.3.m3.1.1"><times id="S1.p7.3.m3.1.1.2.cmml" xref="S1.p7.3.m3.1.1.2"></times><ci id="S1.p7.3.m3.1.1.3.cmml" xref="S1.p7.3.m3.1.1.3">𝑅</ci><ci id="S1.p7.3.m3.1.1.4.cmml" xref="S1.p7.3.m3.1.1.4">Δ</ci><apply id="S1.p7.3.m3.1.1.1.1.1.cmml" xref="S1.p7.3.m3.1.1.1.1"><times id="S1.p7.3.m3.1.1.1.1.1.1.cmml" xref="S1.p7.3.m3.1.1.1.1.1.1"></times><ci id="S1.p7.3.m3.1.1.1.1.1.2.cmml" xref="S1.p7.3.m3.1.1.1.1.1.2">𝑁</ci><ci id="S1.p7.3.m3.1.1.1.1.1.3.cmml" xref="S1.p7.3.m3.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.3.m3.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S1.p7.3.m3.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-module. This idea of constructing a projective resolution as a double complex was also used by Cegarra in the proof of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib3" title="">3</a>, Theorem 1]</cite>. As a special case of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a>, we obtain that the simplicial map <math alttext="\kappa" class="ltx_Math" display="inline" id="S1.p7.4.m4.1"><semantics id="S1.p7.4.m4.1a"><mi id="S1.p7.4.m4.1.1" xref="S1.p7.4.m4.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S1.p7.4.m4.1b"><ci id="S1.p7.4.m4.1.1.cmml" xref="S1.p7.4.m4.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.4.m4.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S1.p7.4.m4.1d">italic_κ</annotation></semantics></math> gives isomorphisms for the Baues-Wirsching and Quillen cohomologies of <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p7.5.m5.1"><semantics id="S1.p7.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p7.5.m5.1.1" xref="S1.p7.5.m5.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p7.5.m5.1b"><ci id="S1.p7.5.m5.1.1.cmml" xref="S1.p7.5.m5.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.5.m5.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p7.5.m5.1d">caligraphic_C</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.6">As a consequence of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a>, we also prove a version of Quillen’s Theorem A for Thomason cohomology. Let</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="U:\varphi/-\to\mathrm{id}/-" class="ltx_math_unparsed" display="block" id="S1.Ex3.m1.1"><semantics id="S1.Ex3.m1.1a"><mrow id="S1.Ex3.m1.1b"><mi id="S1.Ex3.m1.1.1">U</mi><mo id="S1.Ex3.m1.1.2" lspace="0.278em" rspace="0.278em">:</mo><mi id="S1.Ex3.m1.1.3">φ</mi><mo id="S1.Ex3.m1.1.4" rspace="0em">/</mo><mo id="S1.Ex3.m1.1.5" lspace="0em" rspace="0em">−</mo><mo id="S1.Ex3.m1.1.6" lspace="0em" stretchy="false">→</mo><mi id="S1.Ex3.m1.1.7">id</mi><mo id="S1.Ex3.m1.1.8" rspace="0em">/</mo><mo id="S1.Ex3.m1.1.9" lspace="0em">−</mo></mrow><annotation encoding="application/x-tex" id="S1.Ex3.m1.1c">U:\varphi/-\to\mathrm{id}/-</annotation><annotation encoding="application/x-llamapun" id="S1.Ex3.m1.1d">italic_U : italic_φ / - → roman_id / -</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p8.3">be the natural transformation defined by sending <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.p8.1.m1.1"><semantics id="S1.p8.1.m1.1a"><mrow id="S1.p8.1.m1.1.1" xref="S1.p8.1.m1.1.1.cmml"><mi id="S1.p8.1.m1.1.1.2" xref="S1.p8.1.m1.1.1.2.cmml">d</mi><mo id="S1.p8.1.m1.1.1.1" xref="S1.p8.1.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p8.1.m1.1.1.3" xref="S1.p8.1.m1.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.1.m1.1b"><apply id="S1.p8.1.m1.1.1.cmml" xref="S1.p8.1.m1.1.1"><in id="S1.p8.1.m1.1.1.1.cmml" xref="S1.p8.1.m1.1.1.1"></in><ci id="S1.p8.1.m1.1.1.2.cmml" xref="S1.p8.1.m1.1.1.2">𝑑</ci><ci id="S1.p8.1.m1.1.1.3.cmml" xref="S1.p8.1.m1.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.1.m1.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.1.m1.1d">italic_d ∈ caligraphic_D</annotation></semantics></math> to the functor <math alttext="U(d):\varphi/d\to\mathrm{id}/d:(c,\mu)\to(\varphi(c),\mu)" class="ltx_Math" display="inline" id="S1.p8.2.m2.6"><semantics id="S1.p8.2.m2.6a"><mrow id="S1.p8.2.m2.6.6" xref="S1.p8.2.m2.6.6.cmml"><mrow id="S1.p8.2.m2.6.6.3" xref="S1.p8.2.m2.6.6.3.cmml"><mi id="S1.p8.2.m2.6.6.3.2" xref="S1.p8.2.m2.6.6.3.2.cmml">U</mi><mo id="S1.p8.2.m2.6.6.3.1" xref="S1.p8.2.m2.6.6.3.1.cmml">⁢</mo><mrow id="S1.p8.2.m2.6.6.3.3.2" xref="S1.p8.2.m2.6.6.3.cmml"><mo id="S1.p8.2.m2.6.6.3.3.2.1" stretchy="false" xref="S1.p8.2.m2.6.6.3.cmml">(</mo><mi id="S1.p8.2.m2.1.1" xref="S1.p8.2.m2.1.1.cmml">d</mi><mo id="S1.p8.2.m2.6.6.3.3.2.2" rspace="0.278em" stretchy="false" xref="S1.p8.2.m2.6.6.3.cmml">)</mo></mrow></mrow><mo id="S1.p8.2.m2.6.6.4" rspace="0.278em" xref="S1.p8.2.m2.6.6.4.cmml">:</mo><mrow id="S1.p8.2.m2.6.6.5" xref="S1.p8.2.m2.6.6.5.cmml"><mrow id="S1.p8.2.m2.6.6.5.2" xref="S1.p8.2.m2.6.6.5.2.cmml"><mi id="S1.p8.2.m2.6.6.5.2.2" xref="S1.p8.2.m2.6.6.5.2.2.cmml">φ</mi><mo id="S1.p8.2.m2.6.6.5.2.1" xref="S1.p8.2.m2.6.6.5.2.1.cmml">/</mo><mi id="S1.p8.2.m2.6.6.5.2.3" xref="S1.p8.2.m2.6.6.5.2.3.cmml">d</mi></mrow><mo id="S1.p8.2.m2.6.6.5.1" stretchy="false" xref="S1.p8.2.m2.6.6.5.1.cmml">→</mo><mrow id="S1.p8.2.m2.6.6.5.3" xref="S1.p8.2.m2.6.6.5.3.cmml"><mi id="S1.p8.2.m2.6.6.5.3.2" xref="S1.p8.2.m2.6.6.5.3.2.cmml">id</mi><mo id="S1.p8.2.m2.6.6.5.3.1" xref="S1.p8.2.m2.6.6.5.3.1.cmml">/</mo><mi id="S1.p8.2.m2.6.6.5.3.3" xref="S1.p8.2.m2.6.6.5.3.3.cmml">d</mi></mrow></mrow><mo id="S1.p8.2.m2.6.6.6" lspace="0.278em" rspace="0.278em" xref="S1.p8.2.m2.6.6.6.cmml">:</mo><mrow id="S1.p8.2.m2.6.6.1" xref="S1.p8.2.m2.6.6.1.cmml"><mrow id="S1.p8.2.m2.6.6.1.3.2" xref="S1.p8.2.m2.6.6.1.3.1.cmml"><mo id="S1.p8.2.m2.6.6.1.3.2.1" stretchy="false" xref="S1.p8.2.m2.6.6.1.3.1.cmml">(</mo><mi id="S1.p8.2.m2.2.2" xref="S1.p8.2.m2.2.2.cmml">c</mi><mo id="S1.p8.2.m2.6.6.1.3.2.2" xref="S1.p8.2.m2.6.6.1.3.1.cmml">,</mo><mi id="S1.p8.2.m2.3.3" xref="S1.p8.2.m2.3.3.cmml">μ</mi><mo id="S1.p8.2.m2.6.6.1.3.2.3" stretchy="false" xref="S1.p8.2.m2.6.6.1.3.1.cmml">)</mo></mrow><mo id="S1.p8.2.m2.6.6.1.2" stretchy="false" xref="S1.p8.2.m2.6.6.1.2.cmml">→</mo><mrow id="S1.p8.2.m2.6.6.1.1.1" xref="S1.p8.2.m2.6.6.1.1.2.cmml"><mo id="S1.p8.2.m2.6.6.1.1.1.2" stretchy="false" xref="S1.p8.2.m2.6.6.1.1.2.cmml">(</mo><mrow id="S1.p8.2.m2.6.6.1.1.1.1" xref="S1.p8.2.m2.6.6.1.1.1.1.cmml"><mi id="S1.p8.2.m2.6.6.1.1.1.1.2" xref="S1.p8.2.m2.6.6.1.1.1.1.2.cmml">φ</mi><mo id="S1.p8.2.m2.6.6.1.1.1.1.1" xref="S1.p8.2.m2.6.6.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.p8.2.m2.6.6.1.1.1.1.3.2" xref="S1.p8.2.m2.6.6.1.1.1.1.cmml"><mo id="S1.p8.2.m2.6.6.1.1.1.1.3.2.1" stretchy="false" xref="S1.p8.2.m2.6.6.1.1.1.1.cmml">(</mo><mi id="S1.p8.2.m2.4.4" xref="S1.p8.2.m2.4.4.cmml">c</mi><mo id="S1.p8.2.m2.6.6.1.1.1.1.3.2.2" stretchy="false" xref="S1.p8.2.m2.6.6.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p8.2.m2.6.6.1.1.1.3" xref="S1.p8.2.m2.6.6.1.1.2.cmml">,</mo><mi id="S1.p8.2.m2.5.5" xref="S1.p8.2.m2.5.5.cmml">μ</mi><mo id="S1.p8.2.m2.6.6.1.1.1.4" stretchy="false" xref="S1.p8.2.m2.6.6.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.2.m2.6b"><apply id="S1.p8.2.m2.6.6.cmml" xref="S1.p8.2.m2.6.6"><and id="S1.p8.2.m2.6.6a.cmml" xref="S1.p8.2.m2.6.6"></and><apply id="S1.p8.2.m2.6.6b.cmml" xref="S1.p8.2.m2.6.6"><ci id="S1.p8.2.m2.6.6.4.cmml" xref="S1.p8.2.m2.6.6.4">:</ci><apply id="S1.p8.2.m2.6.6.3.cmml" xref="S1.p8.2.m2.6.6.3"><times id="S1.p8.2.m2.6.6.3.1.cmml" xref="S1.p8.2.m2.6.6.3.1"></times><ci id="S1.p8.2.m2.6.6.3.2.cmml" xref="S1.p8.2.m2.6.6.3.2">𝑈</ci><ci id="S1.p8.2.m2.1.1.cmml" xref="S1.p8.2.m2.1.1">𝑑</ci></apply><apply id="S1.p8.2.m2.6.6.5.cmml" xref="S1.p8.2.m2.6.6.5"><ci id="S1.p8.2.m2.6.6.5.1.cmml" xref="S1.p8.2.m2.6.6.5.1">→</ci><apply 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xref="S1.p8.2.m2.6.6.1.3.2"><ci id="S1.p8.2.m2.2.2.cmml" xref="S1.p8.2.m2.2.2">𝑐</ci><ci id="S1.p8.2.m2.3.3.cmml" xref="S1.p8.2.m2.3.3">𝜇</ci></interval><interval closure="open" id="S1.p8.2.m2.6.6.1.1.2.cmml" xref="S1.p8.2.m2.6.6.1.1.1"><apply id="S1.p8.2.m2.6.6.1.1.1.1.cmml" xref="S1.p8.2.m2.6.6.1.1.1.1"><times id="S1.p8.2.m2.6.6.1.1.1.1.1.cmml" xref="S1.p8.2.m2.6.6.1.1.1.1.1"></times><ci id="S1.p8.2.m2.6.6.1.1.1.1.2.cmml" xref="S1.p8.2.m2.6.6.1.1.1.1.2">𝜑</ci><ci id="S1.p8.2.m2.4.4.cmml" xref="S1.p8.2.m2.4.4">𝑐</ci></apply><ci id="S1.p8.2.m2.5.5.cmml" xref="S1.p8.2.m2.5.5">𝜇</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.2.m2.6c">U(d):\varphi/d\to\mathrm{id}/d:(c,\mu)\to(\varphi(c),\mu)</annotation><annotation encoding="application/x-llamapun" id="S1.p8.2.m2.6d">italic_U ( italic_d ) : italic_φ / italic_d → roman_id / italic_d : ( italic_c , italic_μ ) → ( italic_φ ( italic_c ) , italic_μ )</annotation></semantics></math>. For every <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.p8.3.m3.1"><semantics id="S1.p8.3.m3.1a"><mrow id="S1.p8.3.m3.1.1" xref="S1.p8.3.m3.1.1.cmml"><mi id="S1.p8.3.m3.1.1.2" xref="S1.p8.3.m3.1.1.2.cmml">d</mi><mo id="S1.p8.3.m3.1.1.1" xref="S1.p8.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p8.3.m3.1.1.3" xref="S1.p8.3.m3.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.3.m3.1b"><apply id="S1.p8.3.m3.1.1.cmml" xref="S1.p8.3.m3.1.1"><in id="S1.p8.3.m3.1.1.1.cmml" xref="S1.p8.3.m3.1.1.1"></in><ci id="S1.p8.3.m3.1.1.2.cmml" xref="S1.p8.3.m3.1.1.2">𝑑</ci><ci id="S1.p8.3.m3.1.1.3.cmml" xref="S1.p8.3.m3.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.3.m3.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.3.m3.1d">italic_d ∈ caligraphic_D</annotation></semantics></math>, there is a commuting diagram of simplicial maps</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><svg class="ltx_picture ltx_markedasmath" height="79.57" id="S1.Ex4.m1.1.1.pic1" overflow="visible" version="1.1" width="140.33"><g transform="matrix(1 0 0 -1 37.38 20.07) translate(37.38,0)"><g transform="translate(-30.51,0) translate(4.15,0)"><foreignobject height="13.84" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="52.71"><math alttext="\textstyle{N(\varphi/d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces}" class="ltx_Math" display="inline" id="S1.Ex4.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S1.Ex4.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mrow id="S1.Ex4.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" 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id="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.cmml" xref="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.2.cmml" xref="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.2">𝑗</ci><ci id="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.3.cmml" xref="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.3">𝑑</ci></apply><ci id="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\scriptstyle{j_{d}^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S1.Ex4.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math></foreignobject></g><g transform="translate(63.71,0) translate(0,-55.35)"><path d="M 0 0 A 13.84 13.84 45 0 0 -6.92 2.77" fill="none" stroke="#000000"></path><path d="M 0 0 A 13.84 13.84 45 0 1 -6.92 -2.77" fill="none" stroke="#000000"></path></g><path class="droprule" d="M 30.21 -55.62 L 63.71 -55.07" fill="none" stroke="#000000"></path><path class="droprule" d="M 63.71 -55.62 L 63.71 -55.07" fill="none" stroke="#000000"></path><g transform="translate(63.71,0) translate(0,-55.35) translate(4.15,0)"><foreignobject height="9.46" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="23.2"><math alttext="\textstyle{N\mathcal{D}}" class="ltx_Math" display="inline" id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1a"><mrow id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">N</mi><mo id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1" xref="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1"><times id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1"></times><ci id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑁</ci><ci id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1c">\textstyle{N\mathcal{D}}</annotation><annotation encoding="application/x-llamapun" id="S1.Ex4.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1d">italic_N caligraphic_D</annotation></semantics></math></foreignobject></g></g></svg></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p8.5">where <math alttext="j_{d}" class="ltx_Math" display="inline" id="S1.p8.4.m1.1"><semantics id="S1.p8.4.m1.1a"><msub id="S1.p8.4.m1.1.1" xref="S1.p8.4.m1.1.1.cmml"><mi id="S1.p8.4.m1.1.1.2" xref="S1.p8.4.m1.1.1.2.cmml">j</mi><mi id="S1.p8.4.m1.1.1.3" xref="S1.p8.4.m1.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p8.4.m1.1b"><apply id="S1.p8.4.m1.1.1.cmml" xref="S1.p8.4.m1.1.1"><csymbol cd="ambiguous" id="S1.p8.4.m1.1.1.1.cmml" xref="S1.p8.4.m1.1.1">subscript</csymbol><ci id="S1.p8.4.m1.1.1.2.cmml" xref="S1.p8.4.m1.1.1.2">𝑗</ci><ci id="S1.p8.4.m1.1.1.3.cmml" xref="S1.p8.4.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.4.m1.1c">j_{d}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.4.m1.1d">italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="j_{d}^{\prime}" class="ltx_Math" display="inline" id="S1.p8.5.m2.1"><semantics id="S1.p8.5.m2.1a"><msubsup id="S1.p8.5.m2.1.1" xref="S1.p8.5.m2.1.1.cmml"><mi id="S1.p8.5.m2.1.1.2.2" xref="S1.p8.5.m2.1.1.2.2.cmml">j</mi><mi id="S1.p8.5.m2.1.1.2.3" xref="S1.p8.5.m2.1.1.2.3.cmml">d</mi><mo id="S1.p8.5.m2.1.1.3" xref="S1.p8.5.m2.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S1.p8.5.m2.1b"><apply id="S1.p8.5.m2.1.1.cmml" xref="S1.p8.5.m2.1.1"><csymbol cd="ambiguous" id="S1.p8.5.m2.1.1.1.cmml" xref="S1.p8.5.m2.1.1">superscript</csymbol><apply id="S1.p8.5.m2.1.1.2.cmml" xref="S1.p8.5.m2.1.1"><csymbol cd="ambiguous" id="S1.p8.5.m2.1.1.2.1.cmml" xref="S1.p8.5.m2.1.1">subscript</csymbol><ci id="S1.p8.5.m2.1.1.2.2.cmml" xref="S1.p8.5.m2.1.1.2.2">𝑗</ci><ci id="S1.p8.5.m2.1.1.2.3.cmml" xref="S1.p8.5.m2.1.1.2.3">𝑑</ci></apply><ci id="S1.p8.5.m2.1.1.3.cmml" xref="S1.p8.5.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.5.m2.1c">j_{d}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.5.m2.1d">italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are the simplicial maps defined in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem1" title="Definition 1.1. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.1</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.1.1.1">Theorem 1.3</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem3.p1"> <p class="ltx_p" id="S1.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem3.p1.5.5">Let <math alttext="\varphi:\mathcal{C}\to\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.1.1.m1.1"><semantics id="S1.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S1.Thmtheorem3.p1.1.1.m1.1.1" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">φ</mi><mo id="S1.Thmtheorem3.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.1.1.m1.1.1.3.2" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="S1.Thmtheorem3.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.1.1.m1.1.1.3.3" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.1.1.m1.1b"><apply id="S1.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.1.1"><ci id="S1.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.1">:</ci><ci id="S1.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.2">𝜑</ci><apply id="S1.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.3"><ci id="S1.Thmtheorem3.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.3.1">→</ci><ci id="S1.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.3.2">𝒞</ci><ci id="S1.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.1.1.m1.1c">\varphi:\mathcal{C}\to\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.1.1.m1.1d">italic_φ : caligraphic_C → caligraphic_D</annotation></semantics></math> be a functor and <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.2.2.m2.1"><semantics id="S1.Thmtheorem3.p1.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.2.2.m2.1.1" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.2.2.m2.1b"><ci id="S1.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem3.p1.2.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.2.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.2.2.m2.1d">caligraphic_M</annotation></semantics></math> be any coefficient system for <math alttext="N\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.3.3.m3.1"><semantics id="S1.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem3.p1.3.3.m3.1.1" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi id="S1.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">N</mi><mo id="S1.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.3.3.m3.1b"><apply id="S1.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1"><times id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1"></times><ci id="S1.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.2">𝑁</ci><ci id="S1.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.3.3.m3.1c">N\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.3.3.m3.1d">italic_N caligraphic_D</annotation></semantics></math>. Suppose that for each <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.4.4.m4.1"><semantics id="S1.Thmtheorem3.p1.4.4.m4.1a"><mrow id="S1.Thmtheorem3.p1.4.4.m4.1.1" xref="S1.Thmtheorem3.p1.4.4.m4.1.1.cmml"><mi id="S1.Thmtheorem3.p1.4.4.m4.1.1.2" xref="S1.Thmtheorem3.p1.4.4.m4.1.1.2.cmml">d</mi><mo id="S1.Thmtheorem3.p1.4.4.m4.1.1.1" xref="S1.Thmtheorem3.p1.4.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.4.4.m4.1.1.3" xref="S1.Thmtheorem3.p1.4.4.m4.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.4.4.m4.1b"><apply id="S1.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem3.p1.4.4.m4.1.1"><in id="S1.Thmtheorem3.p1.4.4.m4.1.1.1.cmml" xref="S1.Thmtheorem3.p1.4.4.m4.1.1.1"></in><ci id="S1.Thmtheorem3.p1.4.4.m4.1.1.2.cmml" xref="S1.Thmtheorem3.p1.4.4.m4.1.1.2">𝑑</ci><ci id="S1.Thmtheorem3.p1.4.4.m4.1.1.3.cmml" xref="S1.Thmtheorem3.p1.4.4.m4.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.4.4.m4.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.4.4.m4.1d">italic_d ∈ caligraphic_D</annotation></semantics></math>, the simplicial map <math alttext="NU(d):N(\varphi/d)\to N(\mathrm{id}/d)" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.5.5.m5.3"><semantics id="S1.Thmtheorem3.p1.5.5.m5.3a"><mrow id="S1.Thmtheorem3.p1.5.5.m5.3.3" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.cmml"><mrow id="S1.Thmtheorem3.p1.5.5.m5.3.3.4" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.cmml"><mi id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.2" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.2.cmml">N</mi><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.1" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.1.cmml">⁢</mo><mi id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.3" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.3.cmml">U</mi><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.1a" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.1.cmml">⁢</mo><mrow id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.4.2" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.cmml"><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.4.2.1" stretchy="false" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.cmml">(</mo><mi id="S1.Thmtheorem3.p1.5.5.m5.1.1" xref="S1.Thmtheorem3.p1.5.5.m5.1.1.cmml">d</mi><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.4.2.2" rspace="0.278em" stretchy="false" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.3" rspace="0.278em" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.3.cmml">:</mo><mrow id="S1.Thmtheorem3.p1.5.5.m5.3.3.2" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.cmml"><mrow id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.cmml"><mi id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.3" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.3.cmml">N</mi><mo id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.2" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.cmml"><mi id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.2" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.2.cmml">φ</mi><mo id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.1" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.1.cmml">/</mo><mi id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.3" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.3" stretchy="false" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.3.cmml">→</mo><mrow id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.cmml"><mi id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.3" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.3.cmml">N</mi><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.2" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.2.cmml">⁢</mo><mrow id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.cmml"><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.2" stretchy="false" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.cmml"><mi id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.2" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.2.cmml">id</mi><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.1" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.1.cmml">/</mo><mi id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.3" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.3.cmml">d</mi></mrow><mo id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.3" stretchy="false" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.5.5.m5.3b"><apply id="S1.Thmtheorem3.p1.5.5.m5.3.3.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3"><ci id="S1.Thmtheorem3.p1.5.5.m5.3.3.3.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.3">:</ci><apply id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4"><times id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.1.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.1"></times><ci id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.2.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.2">𝑁</ci><ci id="S1.Thmtheorem3.p1.5.5.m5.3.3.4.3.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.4.3">𝑈</ci><ci id="S1.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.1.1">𝑑</ci></apply><apply id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2"><ci id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.3.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.3">→</ci><apply id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1"><times id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.2.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.2"></times><ci id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.3.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.3">𝑁</ci><apply id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1"><divide id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.1"></divide><ci id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.2">𝜑</ci><ci id="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2"><times id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.2.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.2"></times><ci id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.3.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.3">𝑁</ci><apply id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1"><divide id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.1.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.1"></divide><ci id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.2.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.2">id</ci><ci id="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.5.5.m5.3.3.2.2.1.1.1.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.5.5.m5.3c">NU(d):N(\varphi/d)\to N(\mathrm{id}/d)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.5.5.m5.3d">italic_N italic_U ( italic_d ) : italic_N ( italic_φ / italic_d ) → italic_N ( roman_id / italic_d )</annotation></semantics></math> induces an isomorphism</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{*}(N(\mathrm{id}/d);\mathcal{M}_{d})\cong H^{*}(N(\varphi/d);NU(d)^{*}% \mathcal{M}_{d})" class="ltx_Math" display="block" id="S1.Ex5.m1.5"><semantics id="S1.Ex5.m1.5a"><mrow id="S1.Ex5.m1.5.5" xref="S1.Ex5.m1.5.5.cmml"><mrow id="S1.Ex5.m1.3.3.2" xref="S1.Ex5.m1.3.3.2.cmml"><msup id="S1.Ex5.m1.3.3.2.4" xref="S1.Ex5.m1.3.3.2.4.cmml"><mi id="S1.Ex5.m1.3.3.2.4.2" xref="S1.Ex5.m1.3.3.2.4.2.cmml">H</mi><mo id="S1.Ex5.m1.3.3.2.4.3" xref="S1.Ex5.m1.3.3.2.4.3.cmml">∗</mo></msup><mo id="S1.Ex5.m1.3.3.2.3" xref="S1.Ex5.m1.3.3.2.3.cmml">⁢</mo><mrow id="S1.Ex5.m1.3.3.2.2.2" xref="S1.Ex5.m1.3.3.2.2.3.cmml"><mo id="S1.Ex5.m1.3.3.2.2.2.3" stretchy="false" xref="S1.Ex5.m1.3.3.2.2.3.cmml">(</mo><mrow id="S1.Ex5.m1.2.2.1.1.1.1" xref="S1.Ex5.m1.2.2.1.1.1.1.cmml"><mi id="S1.Ex5.m1.2.2.1.1.1.1.3" xref="S1.Ex5.m1.2.2.1.1.1.1.3.cmml">N</mi><mo id="S1.Ex5.m1.2.2.1.1.1.1.2" xref="S1.Ex5.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex5.m1.2.2.1.1.1.1.1.1" xref="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S1.Ex5.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.Ex5.m1.2.2.1.1.1.1.1.1.1" xref="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.2" xref="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.2.cmml">id</mi><mo id="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.1" xref="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.3" xref="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S1.Ex5.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S1.Ex5.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Ex5.m1.3.3.2.2.2.4" xref="S1.Ex5.m1.3.3.2.2.3.cmml">;</mo><msub id="S1.Ex5.m1.3.3.2.2.2.2" xref="S1.Ex5.m1.3.3.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex5.m1.3.3.2.2.2.2.2" xref="S1.Ex5.m1.3.3.2.2.2.2.2.cmml">ℳ</mi><mi id="S1.Ex5.m1.3.3.2.2.2.2.3" 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xref="S1.Ex5.m1.5.5.4.4.3"></times></apply><list id="S1.Ex5.m1.5.5.4.2.3.cmml" xref="S1.Ex5.m1.5.5.4.2.2"><apply id="S1.Ex5.m1.4.4.3.1.1.1.cmml" xref="S1.Ex5.m1.4.4.3.1.1.1"><times id="S1.Ex5.m1.4.4.3.1.1.1.2.cmml" xref="S1.Ex5.m1.4.4.3.1.1.1.2"></times><ci id="S1.Ex5.m1.4.4.3.1.1.1.3.cmml" xref="S1.Ex5.m1.4.4.3.1.1.1.3">𝑁</ci><apply id="S1.Ex5.m1.4.4.3.1.1.1.1.1.1.cmml" xref="S1.Ex5.m1.4.4.3.1.1.1.1.1"><divide id="S1.Ex5.m1.4.4.3.1.1.1.1.1.1.1.cmml" xref="S1.Ex5.m1.4.4.3.1.1.1.1.1.1.1"></divide><ci id="S1.Ex5.m1.4.4.3.1.1.1.1.1.1.2.cmml" xref="S1.Ex5.m1.4.4.3.1.1.1.1.1.1.2">𝜑</ci><ci id="S1.Ex5.m1.4.4.3.1.1.1.1.1.1.3.cmml" xref="S1.Ex5.m1.4.4.3.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S1.Ex5.m1.5.5.4.2.2.2.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2"><times id="S1.Ex5.m1.5.5.4.2.2.2.1.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.1"></times><ci id="S1.Ex5.m1.5.5.4.2.2.2.2.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.2">𝑁</ci><ci id="S1.Ex5.m1.5.5.4.2.2.2.3.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.3">𝑈</ci><apply id="S1.Ex5.m1.5.5.4.2.2.2.4.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.4"><csymbol cd="ambiguous" id="S1.Ex5.m1.5.5.4.2.2.2.4.1.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.4">superscript</csymbol><ci id="S1.Ex5.m1.1.1.cmml" xref="S1.Ex5.m1.1.1">𝑑</ci><times id="S1.Ex5.m1.5.5.4.2.2.2.4.3.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.4.3"></times></apply><apply id="S1.Ex5.m1.5.5.4.2.2.2.5.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.5"><csymbol cd="ambiguous" id="S1.Ex5.m1.5.5.4.2.2.2.5.1.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.5">subscript</csymbol><ci id="S1.Ex5.m1.5.5.4.2.2.2.5.2.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.5.2">ℳ</ci><ci id="S1.Ex5.m1.5.5.4.2.2.2.5.3.cmml" xref="S1.Ex5.m1.5.5.4.2.2.2.5.3">𝑑</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex5.m1.5c">H^{*}(N(\mathrm{id}/d);\mathcal{M}_{d})\cong H^{*}(N(\varphi/d);NU(d)^{*}% \mathcal{M}_{d})</annotation><annotation encoding="application/x-llamapun" id="S1.Ex5.m1.5d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_N ( roman_id / italic_d ) ; caligraphic_M start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ≅ italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_N ( italic_φ / italic_d ) ; italic_N italic_U ( italic_d ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.Thmtheorem3.p1.6"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem3.p1.6.1">where <math alttext="\mathcal{M}_{d}=(j_{d}^{\prime})^{*}\mathcal{M}" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.6.1.m1.1"><semantics id="S1.Thmtheorem3.p1.6.1.m1.1a"><mrow id="S1.Thmtheorem3.p1.6.1.m1.1.1" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.cmml"><msub id="S1.Thmtheorem3.p1.6.1.m1.1.1.3" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.6.1.m1.1.1.3.2" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.3.2.cmml">ℳ</mi><mi id="S1.Thmtheorem3.p1.6.1.m1.1.1.3.3" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.3.3.cmml">d</mi></msub><mo id="S1.Thmtheorem3.p1.6.1.m1.1.1.2" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.2.cmml">=</mo><mrow id="S1.Thmtheorem3.p1.6.1.m1.1.1.1" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.cmml"><msup id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.cmml"><mrow id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.2" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.2.cmml">j</mi><mi id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.3" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.3.cmml">d</mi><mo id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.3" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.3" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.2" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.3" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.3.cmml">ℳ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.6.1.m1.1b"><apply id="S1.Thmtheorem3.p1.6.1.m1.1.1.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1"><eq id="S1.Thmtheorem3.p1.6.1.m1.1.1.2.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.2"></eq><apply id="S1.Thmtheorem3.p1.6.1.m1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem3.p1.6.1.m1.1.1.3.1.cmml" 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xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.2">𝑗</ci><ci id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.3.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.2.3">𝑑</ci></apply><ci id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.1.1.1.3">′</ci></apply><times id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.1.3"></times></apply><ci id="S1.Thmtheorem3.p1.6.1.m1.1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.6.1.m1.1.1.1.3">ℳ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.6.1.m1.1c">\mathcal{M}_{d}=(j_{d}^{\prime})^{*}\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.6.1.m1.1d">caligraphic_M start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = ( italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M</annotation></semantics></math>. Then the induced map</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\varphi^{*}:H^{*}_{Th}(\mathcal{D};\mathcal{M})\to H^{*}_{Th}(\mathcal{C};% \varphi^{*}\mathcal{M})" class="ltx_Math" display="block" id="S1.Ex6.m1.4"><semantics id="S1.Ex6.m1.4a"><mrow id="S1.Ex6.m1.4.4" xref="S1.Ex6.m1.4.4.cmml"><msup id="S1.Ex6.m1.4.4.3" xref="S1.Ex6.m1.4.4.3.cmml"><mi id="S1.Ex6.m1.4.4.3.2" xref="S1.Ex6.m1.4.4.3.2.cmml">φ</mi><mo id="S1.Ex6.m1.4.4.3.3" xref="S1.Ex6.m1.4.4.3.3.cmml">∗</mo></msup><mo id="S1.Ex6.m1.4.4.2" lspace="0.278em" rspace="0.278em" xref="S1.Ex6.m1.4.4.2.cmml">:</mo><mrow id="S1.Ex6.m1.4.4.1" xref="S1.Ex6.m1.4.4.1.cmml"><mrow id="S1.Ex6.m1.4.4.1.3" xref="S1.Ex6.m1.4.4.1.3.cmml"><msubsup id="S1.Ex6.m1.4.4.1.3.2" xref="S1.Ex6.m1.4.4.1.3.2.cmml"><mi id="S1.Ex6.m1.4.4.1.3.2.2.2" 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id="S1.Ex6.m1.4.4.1.3.2.3.cmml" xref="S1.Ex6.m1.4.4.1.3.2.3"><times id="S1.Ex6.m1.4.4.1.3.2.3.1.cmml" xref="S1.Ex6.m1.4.4.1.3.2.3.1"></times><ci id="S1.Ex6.m1.4.4.1.3.2.3.2.cmml" xref="S1.Ex6.m1.4.4.1.3.2.3.2">𝑇</ci><ci id="S1.Ex6.m1.4.4.1.3.2.3.3.cmml" xref="S1.Ex6.m1.4.4.1.3.2.3.3">ℎ</ci></apply></apply><list id="S1.Ex6.m1.4.4.1.3.3.1.cmml" xref="S1.Ex6.m1.4.4.1.3.3.2"><ci id="S1.Ex6.m1.1.1.cmml" xref="S1.Ex6.m1.1.1">𝒟</ci><ci id="S1.Ex6.m1.2.2.cmml" xref="S1.Ex6.m1.2.2">ℳ</ci></list></apply><apply id="S1.Ex6.m1.4.4.1.1.cmml" xref="S1.Ex6.m1.4.4.1.1"><times id="S1.Ex6.m1.4.4.1.1.2.cmml" xref="S1.Ex6.m1.4.4.1.1.2"></times><apply id="S1.Ex6.m1.4.4.1.1.3.cmml" xref="S1.Ex6.m1.4.4.1.1.3"><csymbol cd="ambiguous" id="S1.Ex6.m1.4.4.1.1.3.1.cmml" xref="S1.Ex6.m1.4.4.1.1.3">subscript</csymbol><apply id="S1.Ex6.m1.4.4.1.1.3.2.cmml" xref="S1.Ex6.m1.4.4.1.1.3"><csymbol cd="ambiguous" id="S1.Ex6.m1.4.4.1.1.3.2.1.cmml" xref="S1.Ex6.m1.4.4.1.1.3">superscript</csymbol><ci id="S1.Ex6.m1.4.4.1.1.3.2.2.cmml" xref="S1.Ex6.m1.4.4.1.1.3.2.2">𝐻</ci><times id="S1.Ex6.m1.4.4.1.1.3.2.3.cmml" xref="S1.Ex6.m1.4.4.1.1.3.2.3"></times></apply><apply id="S1.Ex6.m1.4.4.1.1.3.3.cmml" xref="S1.Ex6.m1.4.4.1.1.3.3"><times id="S1.Ex6.m1.4.4.1.1.3.3.1.cmml" xref="S1.Ex6.m1.4.4.1.1.3.3.1"></times><ci id="S1.Ex6.m1.4.4.1.1.3.3.2.cmml" xref="S1.Ex6.m1.4.4.1.1.3.3.2">𝑇</ci><ci id="S1.Ex6.m1.4.4.1.1.3.3.3.cmml" xref="S1.Ex6.m1.4.4.1.1.3.3.3">ℎ</ci></apply></apply><list id="S1.Ex6.m1.4.4.1.1.1.2.cmml" xref="S1.Ex6.m1.4.4.1.1.1.1"><ci id="S1.Ex6.m1.3.3.cmml" xref="S1.Ex6.m1.3.3">𝒞</ci><apply id="S1.Ex6.m1.4.4.1.1.1.1.1.cmml" xref="S1.Ex6.m1.4.4.1.1.1.1.1"><times id="S1.Ex6.m1.4.4.1.1.1.1.1.1.cmml" xref="S1.Ex6.m1.4.4.1.1.1.1.1.1"></times><apply id="S1.Ex6.m1.4.4.1.1.1.1.1.2.cmml" xref="S1.Ex6.m1.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.Ex6.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S1.Ex6.m1.4.4.1.1.1.1.1.2">superscript</csymbol><ci id="S1.Ex6.m1.4.4.1.1.1.1.1.2.2.cmml" xref="S1.Ex6.m1.4.4.1.1.1.1.1.2.2">𝜑</ci><times id="S1.Ex6.m1.4.4.1.1.1.1.1.2.3.cmml" xref="S1.Ex6.m1.4.4.1.1.1.1.1.2.3"></times></apply><ci id="S1.Ex6.m1.4.4.1.1.1.1.1.3.cmml" xref="S1.Ex6.m1.4.4.1.1.1.1.1.3">ℳ</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex6.m1.4c">\varphi^{*}:H^{*}_{Th}(\mathcal{D};\mathcal{M})\to H^{*}_{Th}(\mathcal{C};% \varphi^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S1.Ex6.m1.4d">italic_φ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_D ; caligraphic_M ) → italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; italic_φ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.Thmtheorem3.p1.7"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem3.p1.7.1">is an isomorphism.</span></p> </div> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.3">To prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem3" title="Theorem 1.3. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.3</span></a>, we construct a spectral sequence that converges to the Thomason cohomology of the category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S1.p9.1.m1.1"><semantics id="S1.p9.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p9.1.m1.1.1" xref="S1.p9.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S1.p9.1.m1.1b"><ci id="S1.p9.1.m1.1.1.cmml" xref="S1.p9.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p9.1.m1.1d">caligraphic_C</annotation></semantics></math>. As a special case of this spectral sequence we obtain a spectral sequence for the Thomason cohomology of the Grothendieck construction <math alttext="\int_{\mathcal{D}}F" class="ltx_Math" display="inline" id="S1.p9.2.m2.1"><semantics id="S1.p9.2.m2.1a"><mrow id="S1.p9.2.m2.1.1" xref="S1.p9.2.m2.1.1.cmml"><msub id="S1.p9.2.m2.1.1.1" xref="S1.p9.2.m2.1.1.1.cmml"><mo id="S1.p9.2.m2.1.1.1.2" xref="S1.p9.2.m2.1.1.1.2.cmml">∫</mo><mi class="ltx_font_mathcaligraphic" id="S1.p9.2.m2.1.1.1.3" xref="S1.p9.2.m2.1.1.1.3.cmml">𝒟</mi></msub><mi id="S1.p9.2.m2.1.1.2" xref="S1.p9.2.m2.1.1.2.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.2.m2.1b"><apply id="S1.p9.2.m2.1.1.cmml" xref="S1.p9.2.m2.1.1"><apply id="S1.p9.2.m2.1.1.1.cmml" xref="S1.p9.2.m2.1.1.1"><csymbol cd="ambiguous" id="S1.p9.2.m2.1.1.1.1.cmml" xref="S1.p9.2.m2.1.1.1">subscript</csymbol><int id="S1.p9.2.m2.1.1.1.2.cmml" xref="S1.p9.2.m2.1.1.1.2"></int><ci id="S1.p9.2.m2.1.1.1.3.cmml" xref="S1.p9.2.m2.1.1.1.3">𝒟</ci></apply><ci id="S1.p9.2.m2.1.1.2.cmml" xref="S1.p9.2.m2.1.1.2">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.2.m2.1c">\int_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="S1.p9.2.m2.1d">∫ start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> of a functor <math alttext="F:\mathcal{D}\to Cat" class="ltx_Math" display="inline" id="S1.p9.3.m3.1"><semantics id="S1.p9.3.m3.1a"><mrow id="S1.p9.3.m3.1.1" xref="S1.p9.3.m3.1.1.cmml"><mi id="S1.p9.3.m3.1.1.2" xref="S1.p9.3.m3.1.1.2.cmml">F</mi><mo id="S1.p9.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p9.3.m3.1.1.1.cmml">:</mo><mrow id="S1.p9.3.m3.1.1.3" xref="S1.p9.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p9.3.m3.1.1.3.2" xref="S1.p9.3.m3.1.1.3.2.cmml">𝒟</mi><mo id="S1.p9.3.m3.1.1.3.1" stretchy="false" xref="S1.p9.3.m3.1.1.3.1.cmml">→</mo><mrow id="S1.p9.3.m3.1.1.3.3" xref="S1.p9.3.m3.1.1.3.3.cmml"><mi id="S1.p9.3.m3.1.1.3.3.2" xref="S1.p9.3.m3.1.1.3.3.2.cmml">C</mi><mo id="S1.p9.3.m3.1.1.3.3.1" xref="S1.p9.3.m3.1.1.3.3.1.cmml">⁢</mo><mi id="S1.p9.3.m3.1.1.3.3.3" xref="S1.p9.3.m3.1.1.3.3.3.cmml">a</mi><mo id="S1.p9.3.m3.1.1.3.3.1a" xref="S1.p9.3.m3.1.1.3.3.1.cmml">⁢</mo><mi id="S1.p9.3.m3.1.1.3.3.4" xref="S1.p9.3.m3.1.1.3.3.4.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.3.m3.1b"><apply id="S1.p9.3.m3.1.1.cmml" xref="S1.p9.3.m3.1.1"><ci id="S1.p9.3.m3.1.1.1.cmml" xref="S1.p9.3.m3.1.1.1">:</ci><ci id="S1.p9.3.m3.1.1.2.cmml" xref="S1.p9.3.m3.1.1.2">𝐹</ci><apply id="S1.p9.3.m3.1.1.3.cmml" xref="S1.p9.3.m3.1.1.3"><ci id="S1.p9.3.m3.1.1.3.1.cmml" xref="S1.p9.3.m3.1.1.3.1">→</ci><ci id="S1.p9.3.m3.1.1.3.2.cmml" xref="S1.p9.3.m3.1.1.3.2">𝒟</ci><apply id="S1.p9.3.m3.1.1.3.3.cmml" xref="S1.p9.3.m3.1.1.3.3"><times id="S1.p9.3.m3.1.1.3.3.1.cmml" xref="S1.p9.3.m3.1.1.3.3.1"></times><ci id="S1.p9.3.m3.1.1.3.3.2.cmml" xref="S1.p9.3.m3.1.1.3.3.2">𝐶</ci><ci id="S1.p9.3.m3.1.1.3.3.3.cmml" xref="S1.p9.3.m3.1.1.3.3.3">𝑎</ci><ci id="S1.p9.3.m3.1.1.3.3.4.cmml" xref="S1.p9.3.m3.1.1.3.3.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.3.m3.1c">F:\mathcal{D}\to Cat</annotation><annotation encoding="application/x-llamapun" id="S1.p9.3.m3.1d">italic_F : caligraphic_D → italic_C italic_a italic_t</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.1.1.1">Theorem 1.4</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem4.p1"> <p class="ltx_p" id="S1.Thmtheorem4.p1.11"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem4.p1.11.11">Let <math alttext="F:\mathcal{D}\to Cat" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.1.1.m1.1"><semantics id="S1.Thmtheorem4.p1.1.1.m1.1a"><mrow id="S1.Thmtheorem4.p1.1.1.m1.1.1" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">F</mi><mo id="S1.Thmtheorem4.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem4.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.2" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml">𝒟</mi><mo id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml">→</mo><mrow id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml"><mi id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.2" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.2.cmml">C</mi><mo id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.1" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.3" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.3.cmml">a</mi><mo id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.1a" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.4" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.4.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.1.1.m1.1b"><apply id="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1"><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.1">:</ci><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.2">𝐹</ci><apply id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3"><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.1">→</ci><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.2">𝒟</ci><apply id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3"><times id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.1"></times><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.2.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.2">𝐶</ci><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.3.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.3">𝑎</ci><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.4.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.3.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.1.1.m1.1c">F:\mathcal{D}\to Cat</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.1.1.m1.1d">italic_F : caligraphic_D → italic_C italic_a italic_t</annotation></semantics></math> be a functor and <math alttext="\int_{\mathcal{D}}F" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.2.2.m2.1"><semantics id="S1.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem4.p1.2.2.m2.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.cmml"><msub id="S1.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.cmml"><mo id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2.cmml">∫</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3.cmml">𝒟</mi></msub><mi id="S1.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.2.2.m2.1b"><apply id="S1.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1"><apply id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1">subscript</csymbol><int id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2"></int><ci id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3">𝒟</ci></apply><ci id="S1.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.2">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.2.2.m2.1c">\int_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.2.2.m2.1d">∫ start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> be the Grothendieck construction of <math alttext="F" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.3.3.m3.1"><semantics id="S1.Thmtheorem4.p1.3.3.m3.1a"><mi id="S1.Thmtheorem4.p1.3.3.m3.1.1" xref="S1.Thmtheorem4.p1.3.3.m3.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.3.3.m3.1b"><ci id="S1.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem4.p1.3.3.m3.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.3.3.m3.1c">F</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.3.3.m3.1d">italic_F</annotation></semantics></math>. Let <math alttext="\pi:\int_{\mathcal{D}}F\to\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.4.4.m4.1"><semantics id="S1.Thmtheorem4.p1.4.4.m4.1a"><mrow id="S1.Thmtheorem4.p1.4.4.m4.1.1" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mi id="S1.Thmtheorem4.p1.4.4.m4.1.1.2" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.2.cmml">π</mi><mo id="S1.Thmtheorem4.p1.4.4.m4.1.1.1" lspace="0.278em" rspace="0.111em" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem4.p1.4.4.m4.1.1.3" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.cmml"><mrow id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.cmml"><msub id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.cmml"><mo id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.2" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.2.cmml">∫</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.3" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.3.cmml">𝒟</mi></msub><mi id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.2" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.2.cmml">F</mi></mrow><mo id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.1" stretchy="false" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.3" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.4.4.m4.1b"><apply id="S1.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1"><ci id="S1.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.1">:</ci><ci id="S1.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.2">𝜋</ci><apply id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3"><ci id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.1.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.1">→</ci><apply id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2"><apply id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.1.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1">subscript</csymbol><int id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.2.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.2"></int><ci id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.3.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.3">𝒟</ci></apply><ci id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.2.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.2.2">𝐹</ci></apply><ci id="S1.Thmtheorem4.p1.4.4.m4.1.1.3.3.cmml" xref="S1.Thmtheorem4.p1.4.4.m4.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.4.4.m4.1c">\pi:\int_{\mathcal{D}}F\to\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.4.4.m4.1d">italic_π : ∫ start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F → caligraphic_D</annotation></semantics></math> be the canonical functor that sends a pair <math alttext="(d,x)\in\int_{\mathcal{D}}F" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.5.5.m5.2"><semantics id="S1.Thmtheorem4.p1.5.5.m5.2a"><mrow id="S1.Thmtheorem4.p1.5.5.m5.2.3" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.cmml"><mrow id="S1.Thmtheorem4.p1.5.5.m5.2.3.2.2" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.2.1.cmml"><mo id="S1.Thmtheorem4.p1.5.5.m5.2.3.2.2.1" stretchy="false" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.2.1.cmml">(</mo><mi id="S1.Thmtheorem4.p1.5.5.m5.1.1" xref="S1.Thmtheorem4.p1.5.5.m5.1.1.cmml">d</mi><mo id="S1.Thmtheorem4.p1.5.5.m5.2.3.2.2.2" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.2.1.cmml">,</mo><mi id="S1.Thmtheorem4.p1.5.5.m5.2.2" xref="S1.Thmtheorem4.p1.5.5.m5.2.2.cmml">x</mi><mo id="S1.Thmtheorem4.p1.5.5.m5.2.3.2.2.3" stretchy="false" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.2.1.cmml">)</mo></mrow><mo id="S1.Thmtheorem4.p1.5.5.m5.2.3.1" rspace="0.111em" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.1.cmml">∈</mo><mrow id="S1.Thmtheorem4.p1.5.5.m5.2.3.3" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.cmml"><msub id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.cmml"><mo id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.2" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.2.cmml">∫</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.3" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.3.cmml">𝒟</mi></msub><mi id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.2" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.2.cmml">F</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.5.5.m5.2b"><apply id="S1.Thmtheorem4.p1.5.5.m5.2.3.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3"><in id="S1.Thmtheorem4.p1.5.5.m5.2.3.1.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.1"></in><interval closure="open" id="S1.Thmtheorem4.p1.5.5.m5.2.3.2.1.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.2.2"><ci id="S1.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.1.1">𝑑</ci><ci id="S1.Thmtheorem4.p1.5.5.m5.2.2.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.2">𝑥</ci></interval><apply id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3"><apply id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.1.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1">subscript</csymbol><int id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.2.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.2"></int><ci id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.3.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.1.3">𝒟</ci></apply><ci id="S1.Thmtheorem4.p1.5.5.m5.2.3.3.2.cmml" xref="S1.Thmtheorem4.p1.5.5.m5.2.3.3.2">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.5.5.m5.2c">(d,x)\in\int_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.5.5.m5.2d">( italic_d , italic_x ) ∈ ∫ start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> to <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.6.6.m6.1"><semantics id="S1.Thmtheorem4.p1.6.6.m6.1a"><mrow id="S1.Thmtheorem4.p1.6.6.m6.1.1" xref="S1.Thmtheorem4.p1.6.6.m6.1.1.cmml"><mi id="S1.Thmtheorem4.p1.6.6.m6.1.1.2" xref="S1.Thmtheorem4.p1.6.6.m6.1.1.2.cmml">d</mi><mo id="S1.Thmtheorem4.p1.6.6.m6.1.1.1" xref="S1.Thmtheorem4.p1.6.6.m6.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.6.6.m6.1.1.3" xref="S1.Thmtheorem4.p1.6.6.m6.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.6.6.m6.1b"><apply id="S1.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="S1.Thmtheorem4.p1.6.6.m6.1.1"><in id="S1.Thmtheorem4.p1.6.6.m6.1.1.1.cmml" xref="S1.Thmtheorem4.p1.6.6.m6.1.1.1"></in><ci id="S1.Thmtheorem4.p1.6.6.m6.1.1.2.cmml" xref="S1.Thmtheorem4.p1.6.6.m6.1.1.2">𝑑</ci><ci id="S1.Thmtheorem4.p1.6.6.m6.1.1.3.cmml" xref="S1.Thmtheorem4.p1.6.6.m6.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.6.6.m6.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.6.6.m6.1d">italic_d ∈ caligraphic_D</annotation></semantics></math>, and <math alttext="\widetilde{F}:\mathcal{D}\to Cat" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.7.7.m7.1"><semantics id="S1.Thmtheorem4.p1.7.7.m7.1a"><mrow id="S1.Thmtheorem4.p1.7.7.m7.1.1" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.cmml"><mover accent="true" id="S1.Thmtheorem4.p1.7.7.m7.1.1.2" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.2.cmml"><mi id="S1.Thmtheorem4.p1.7.7.m7.1.1.2.2" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.2.2.cmml">F</mi><mo id="S1.Thmtheorem4.p1.7.7.m7.1.1.2.1" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.2.1.cmml">~</mo></mover><mo id="S1.Thmtheorem4.p1.7.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem4.p1.7.7.m7.1.1.3" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.2" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.2.cmml">𝒟</mi><mo id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.1" stretchy="false" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.1.cmml">→</mo><mrow id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.cmml"><mi id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.2" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.2.cmml">C</mi><mo id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.1" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.1.cmml">⁢</mo><mi id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.3" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.3.cmml">a</mi><mo id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.1a" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.1.cmml">⁢</mo><mi id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.4" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.4.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.7.7.m7.1b"><apply id="S1.Thmtheorem4.p1.7.7.m7.1.1.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1"><ci id="S1.Thmtheorem4.p1.7.7.m7.1.1.1.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.1">:</ci><apply id="S1.Thmtheorem4.p1.7.7.m7.1.1.2.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.2"><ci id="S1.Thmtheorem4.p1.7.7.m7.1.1.2.1.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.2.1">~</ci><ci id="S1.Thmtheorem4.p1.7.7.m7.1.1.2.2.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.2.2">𝐹</ci></apply><apply id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3"><ci id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.1.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.1">→</ci><ci id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.2.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.2">𝒟</ci><apply id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3"><times id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.1.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.1"></times><ci id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.2.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.2">𝐶</ci><ci id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.3.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.3">𝑎</ci><ci id="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.4.cmml" xref="S1.Thmtheorem4.p1.7.7.m7.1.1.3.3.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.7.7.m7.1c">\widetilde{F}:\mathcal{D}\to Cat</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.7.7.m7.1d">over~ start_ARG italic_F end_ARG : caligraphic_D → italic_C italic_a italic_t</annotation></semantics></math> be the functor that sends <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.8.8.m8.1"><semantics id="S1.Thmtheorem4.p1.8.8.m8.1a"><mrow id="S1.Thmtheorem4.p1.8.8.m8.1.1" xref="S1.Thmtheorem4.p1.8.8.m8.1.1.cmml"><mi id="S1.Thmtheorem4.p1.8.8.m8.1.1.2" xref="S1.Thmtheorem4.p1.8.8.m8.1.1.2.cmml">d</mi><mo id="S1.Thmtheorem4.p1.8.8.m8.1.1.1" xref="S1.Thmtheorem4.p1.8.8.m8.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.8.8.m8.1.1.3" xref="S1.Thmtheorem4.p1.8.8.m8.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.8.8.m8.1b"><apply id="S1.Thmtheorem4.p1.8.8.m8.1.1.cmml" xref="S1.Thmtheorem4.p1.8.8.m8.1.1"><in id="S1.Thmtheorem4.p1.8.8.m8.1.1.1.cmml" xref="S1.Thmtheorem4.p1.8.8.m8.1.1.1"></in><ci id="S1.Thmtheorem4.p1.8.8.m8.1.1.2.cmml" xref="S1.Thmtheorem4.p1.8.8.m8.1.1.2">𝑑</ci><ci id="S1.Thmtheorem4.p1.8.8.m8.1.1.3.cmml" xref="S1.Thmtheorem4.p1.8.8.m8.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.8.8.m8.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.8.8.m8.1d">italic_d ∈ caligraphic_D</annotation></semantics></math> to the comma category <math alttext="\pi/d" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.9.9.m9.1"><semantics id="S1.Thmtheorem4.p1.9.9.m9.1a"><mrow id="S1.Thmtheorem4.p1.9.9.m9.1.1" xref="S1.Thmtheorem4.p1.9.9.m9.1.1.cmml"><mi id="S1.Thmtheorem4.p1.9.9.m9.1.1.2" xref="S1.Thmtheorem4.p1.9.9.m9.1.1.2.cmml">π</mi><mo id="S1.Thmtheorem4.p1.9.9.m9.1.1.1" xref="S1.Thmtheorem4.p1.9.9.m9.1.1.1.cmml">/</mo><mi id="S1.Thmtheorem4.p1.9.9.m9.1.1.3" xref="S1.Thmtheorem4.p1.9.9.m9.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.9.9.m9.1b"><apply id="S1.Thmtheorem4.p1.9.9.m9.1.1.cmml" xref="S1.Thmtheorem4.p1.9.9.m9.1.1"><divide id="S1.Thmtheorem4.p1.9.9.m9.1.1.1.cmml" xref="S1.Thmtheorem4.p1.9.9.m9.1.1.1"></divide><ci id="S1.Thmtheorem4.p1.9.9.m9.1.1.2.cmml" xref="S1.Thmtheorem4.p1.9.9.m9.1.1.2">𝜋</ci><ci id="S1.Thmtheorem4.p1.9.9.m9.1.1.3.cmml" xref="S1.Thmtheorem4.p1.9.9.m9.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.9.9.m9.1c">\pi/d</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.9.9.m9.1d">italic_π / italic_d</annotation></semantics></math>. Then for every coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.10.10.m10.1"><semantics id="S1.Thmtheorem4.p1.10.10.m10.1a"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.10.10.m10.1.1" xref="S1.Thmtheorem4.p1.10.10.m10.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.10.10.m10.1b"><ci id="S1.Thmtheorem4.p1.10.10.m10.1.1.cmml" xref="S1.Thmtheorem4.p1.10.10.m10.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.10.10.m10.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.10.10.m10.1d">caligraphic_M</annotation></semantics></math> for <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.11.11.m11.1"><semantics id="S1.Thmtheorem4.p1.11.11.m11.1a"><mrow id="S1.Thmtheorem4.p1.11.11.m11.1.1" xref="S1.Thmtheorem4.p1.11.11.m11.1.1.cmml"><mi id="S1.Thmtheorem4.p1.11.11.m11.1.1.2" xref="S1.Thmtheorem4.p1.11.11.m11.1.1.2.cmml">N</mi><mo id="S1.Thmtheorem4.p1.11.11.m11.1.1.1" xref="S1.Thmtheorem4.p1.11.11.m11.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.11.11.m11.1.1.3" xref="S1.Thmtheorem4.p1.11.11.m11.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.11.11.m11.1b"><apply id="S1.Thmtheorem4.p1.11.11.m11.1.1.cmml" xref="S1.Thmtheorem4.p1.11.11.m11.1.1"><times id="S1.Thmtheorem4.p1.11.11.m11.1.1.1.cmml" xref="S1.Thmtheorem4.p1.11.11.m11.1.1.1"></times><ci id="S1.Thmtheorem4.p1.11.11.m11.1.1.2.cmml" xref="S1.Thmtheorem4.p1.11.11.m11.1.1.2">𝑁</ci><ci id="S1.Thmtheorem4.p1.11.11.m11.1.1.3.cmml" xref="S1.Thmtheorem4.p1.11.11.m11.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.11.11.m11.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.11.11.m11.1d">italic_N caligraphic_C</annotation></semantics></math>, there is a spectral sequence</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="E_{2}^{p,q}=H^{p}_{Th}(\mathcal{D};\mathcal{H}^{q}_{\mathcal{M}})\Rightarrow H% ^{p+q}_{Th}(\textstyle\int_{\mathcal{D}}F;\mathcal{M})" class="ltx_Math" display="block" id="S1.Ex7.m1.6"><semantics id="S1.Ex7.m1.6a"><mrow id="S1.Ex7.m1.6.6" xref="S1.Ex7.m1.6.6.cmml"><msubsup id="S1.Ex7.m1.6.6.4" xref="S1.Ex7.m1.6.6.4.cmml"><mi id="S1.Ex7.m1.6.6.4.2.2" xref="S1.Ex7.m1.6.6.4.2.2.cmml">E</mi><mn id="S1.Ex7.m1.6.6.4.2.3" xref="S1.Ex7.m1.6.6.4.2.3.cmml">2</mn><mrow id="S1.Ex7.m1.2.2.2.4" xref="S1.Ex7.m1.2.2.2.3.cmml"><mi id="S1.Ex7.m1.1.1.1.1" xref="S1.Ex7.m1.1.1.1.1.cmml">p</mi><mo id="S1.Ex7.m1.2.2.2.4.1" xref="S1.Ex7.m1.2.2.2.3.cmml">,</mo><mi id="S1.Ex7.m1.2.2.2.2" xref="S1.Ex7.m1.2.2.2.2.cmml">q</mi></mrow></msubsup><mo id="S1.Ex7.m1.6.6.5" xref="S1.Ex7.m1.6.6.5.cmml">=</mo><mrow id="S1.Ex7.m1.5.5.1" xref="S1.Ex7.m1.5.5.1.cmml"><msubsup id="S1.Ex7.m1.5.5.1.3" xref="S1.Ex7.m1.5.5.1.3.cmml"><mi id="S1.Ex7.m1.5.5.1.3.2.2" xref="S1.Ex7.m1.5.5.1.3.2.2.cmml">H</mi><mrow id="S1.Ex7.m1.5.5.1.3.3" xref="S1.Ex7.m1.5.5.1.3.3.cmml"><mi id="S1.Ex7.m1.5.5.1.3.3.2" xref="S1.Ex7.m1.5.5.1.3.3.2.cmml">T</mi><mo id="S1.Ex7.m1.5.5.1.3.3.1" xref="S1.Ex7.m1.5.5.1.3.3.1.cmml">⁢</mo><mi id="S1.Ex7.m1.5.5.1.3.3.3" xref="S1.Ex7.m1.5.5.1.3.3.3.cmml">h</mi></mrow><mi id="S1.Ex7.m1.5.5.1.3.2.3" xref="S1.Ex7.m1.5.5.1.3.2.3.cmml">p</mi></msubsup><mo id="S1.Ex7.m1.5.5.1.2" xref="S1.Ex7.m1.5.5.1.2.cmml">⁢</mo><mrow id="S1.Ex7.m1.5.5.1.1.1" xref="S1.Ex7.m1.5.5.1.1.2.cmml"><mo id="S1.Ex7.m1.5.5.1.1.1.2" stretchy="false" xref="S1.Ex7.m1.5.5.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex7.m1.3.3" xref="S1.Ex7.m1.3.3.cmml">𝒟</mi><mo id="S1.Ex7.m1.5.5.1.1.1.3" xref="S1.Ex7.m1.5.5.1.1.2.cmml">;</mo><msubsup id="S1.Ex7.m1.5.5.1.1.1.1" xref="S1.Ex7.m1.5.5.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex7.m1.5.5.1.1.1.1.2.2" xref="S1.Ex7.m1.5.5.1.1.1.1.2.2.cmml">ℋ</mi><mi class="ltx_font_mathcaligraphic" id="S1.Ex7.m1.5.5.1.1.1.1.3" xref="S1.Ex7.m1.5.5.1.1.1.1.3.cmml">ℳ</mi><mi id="S1.Ex7.m1.5.5.1.1.1.1.2.3" xref="S1.Ex7.m1.5.5.1.1.1.1.2.3.cmml">q</mi></msubsup><mo id="S1.Ex7.m1.5.5.1.1.1.4" stretchy="false" xref="S1.Ex7.m1.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.Ex7.m1.6.6.6" stretchy="false" xref="S1.Ex7.m1.6.6.6.cmml">⇒</mo><mrow id="S1.Ex7.m1.6.6.2" xref="S1.Ex7.m1.6.6.2.cmml"><msubsup id="S1.Ex7.m1.6.6.2.3" xref="S1.Ex7.m1.6.6.2.3.cmml"><mi id="S1.Ex7.m1.6.6.2.3.2.2" xref="S1.Ex7.m1.6.6.2.3.2.2.cmml">H</mi><mrow id="S1.Ex7.m1.6.6.2.3.3" xref="S1.Ex7.m1.6.6.2.3.3.cmml"><mi id="S1.Ex7.m1.6.6.2.3.3.2" xref="S1.Ex7.m1.6.6.2.3.3.2.cmml">T</mi><mo id="S1.Ex7.m1.6.6.2.3.3.1" xref="S1.Ex7.m1.6.6.2.3.3.1.cmml">⁢</mo><mi id="S1.Ex7.m1.6.6.2.3.3.3" xref="S1.Ex7.m1.6.6.2.3.3.3.cmml">h</mi></mrow><mrow id="S1.Ex7.m1.6.6.2.3.2.3" xref="S1.Ex7.m1.6.6.2.3.2.3.cmml"><mi id="S1.Ex7.m1.6.6.2.3.2.3.2" xref="S1.Ex7.m1.6.6.2.3.2.3.2.cmml">p</mi><mo id="S1.Ex7.m1.6.6.2.3.2.3.1" xref="S1.Ex7.m1.6.6.2.3.2.3.1.cmml">+</mo><mi id="S1.Ex7.m1.6.6.2.3.2.3.3" xref="S1.Ex7.m1.6.6.2.3.2.3.3.cmml">q</mi></mrow></msubsup><mo id="S1.Ex7.m1.6.6.2.2" xref="S1.Ex7.m1.6.6.2.2.cmml">⁢</mo><mrow id="S1.Ex7.m1.6.6.2.1.1" xref="S1.Ex7.m1.6.6.2.1.2.cmml"><mo id="S1.Ex7.m1.6.6.2.1.1.2" stretchy="false" xref="S1.Ex7.m1.6.6.2.1.2.cmml">(</mo><mrow id="S1.Ex7.m1.6.6.2.1.1.1" xref="S1.Ex7.m1.6.6.2.1.1.1.cmml"><mstyle displaystyle="false" id="S1.Ex7.m1.6.6.2.1.1.1.1" xref="S1.Ex7.m1.6.6.2.1.1.1.1.cmml"><msub id="S1.Ex7.m1.6.6.2.1.1.1.1a" xref="S1.Ex7.m1.6.6.2.1.1.1.1.cmml"><mo id="S1.Ex7.m1.6.6.2.1.1.1.1.2" xref="S1.Ex7.m1.6.6.2.1.1.1.1.2.cmml">∫</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex7.m1.6.6.2.1.1.1.1.3" xref="S1.Ex7.m1.6.6.2.1.1.1.1.3.cmml">𝒟</mi></msub></mstyle><mi id="S1.Ex7.m1.6.6.2.1.1.1.2" xref="S1.Ex7.m1.6.6.2.1.1.1.2.cmml">F</mi></mrow><mo id="S1.Ex7.m1.6.6.2.1.1.3" xref="S1.Ex7.m1.6.6.2.1.2.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex7.m1.4.4" xref="S1.Ex7.m1.4.4.cmml">ℳ</mi><mo id="S1.Ex7.m1.6.6.2.1.1.4" stretchy="false" xref="S1.Ex7.m1.6.6.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex7.m1.6b"><apply id="S1.Ex7.m1.6.6.cmml" xref="S1.Ex7.m1.6.6"><and id="S1.Ex7.m1.6.6a.cmml" xref="S1.Ex7.m1.6.6"></and><apply id="S1.Ex7.m1.6.6b.cmml" xref="S1.Ex7.m1.6.6"><eq id="S1.Ex7.m1.6.6.5.cmml" xref="S1.Ex7.m1.6.6.5"></eq><apply id="S1.Ex7.m1.6.6.4.cmml" xref="S1.Ex7.m1.6.6.4"><csymbol cd="ambiguous" id="S1.Ex7.m1.6.6.4.1.cmml" xref="S1.Ex7.m1.6.6.4">superscript</csymbol><apply id="S1.Ex7.m1.6.6.4.2.cmml" xref="S1.Ex7.m1.6.6.4"><csymbol cd="ambiguous" id="S1.Ex7.m1.6.6.4.2.1.cmml" xref="S1.Ex7.m1.6.6.4">subscript</csymbol><ci id="S1.Ex7.m1.6.6.4.2.2.cmml" xref="S1.Ex7.m1.6.6.4.2.2">𝐸</ci><cn id="S1.Ex7.m1.6.6.4.2.3.cmml" type="integer" xref="S1.Ex7.m1.6.6.4.2.3">2</cn></apply><list id="S1.Ex7.m1.2.2.2.3.cmml" xref="S1.Ex7.m1.2.2.2.4"><ci id="S1.Ex7.m1.1.1.1.1.cmml" xref="S1.Ex7.m1.1.1.1.1">𝑝</ci><ci id="S1.Ex7.m1.2.2.2.2.cmml" xref="S1.Ex7.m1.2.2.2.2">𝑞</ci></list></apply><apply id="S1.Ex7.m1.5.5.1.cmml" xref="S1.Ex7.m1.5.5.1"><times id="S1.Ex7.m1.5.5.1.2.cmml" xref="S1.Ex7.m1.5.5.1.2"></times><apply id="S1.Ex7.m1.5.5.1.3.cmml" xref="S1.Ex7.m1.5.5.1.3"><csymbol cd="ambiguous" id="S1.Ex7.m1.5.5.1.3.1.cmml" xref="S1.Ex7.m1.5.5.1.3">subscript</csymbol><apply id="S1.Ex7.m1.5.5.1.3.2.cmml" xref="S1.Ex7.m1.5.5.1.3"><csymbol cd="ambiguous" id="S1.Ex7.m1.5.5.1.3.2.1.cmml" xref="S1.Ex7.m1.5.5.1.3">superscript</csymbol><ci id="S1.Ex7.m1.5.5.1.3.2.2.cmml" xref="S1.Ex7.m1.5.5.1.3.2.2">𝐻</ci><ci id="S1.Ex7.m1.5.5.1.3.2.3.cmml" xref="S1.Ex7.m1.5.5.1.3.2.3">𝑝</ci></apply><apply id="S1.Ex7.m1.5.5.1.3.3.cmml" xref="S1.Ex7.m1.5.5.1.3.3"><times id="S1.Ex7.m1.5.5.1.3.3.1.cmml" xref="S1.Ex7.m1.5.5.1.3.3.1"></times><ci id="S1.Ex7.m1.5.5.1.3.3.2.cmml" xref="S1.Ex7.m1.5.5.1.3.3.2">𝑇</ci><ci id="S1.Ex7.m1.5.5.1.3.3.3.cmml" xref="S1.Ex7.m1.5.5.1.3.3.3">ℎ</ci></apply></apply><list id="S1.Ex7.m1.5.5.1.1.2.cmml" xref="S1.Ex7.m1.5.5.1.1.1"><ci id="S1.Ex7.m1.3.3.cmml" xref="S1.Ex7.m1.3.3">𝒟</ci><apply id="S1.Ex7.m1.5.5.1.1.1.1.cmml" xref="S1.Ex7.m1.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S1.Ex7.m1.5.5.1.1.1.1.1.cmml" xref="S1.Ex7.m1.5.5.1.1.1.1">subscript</csymbol><apply id="S1.Ex7.m1.5.5.1.1.1.1.2.cmml" xref="S1.Ex7.m1.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S1.Ex7.m1.5.5.1.1.1.1.2.1.cmml" xref="S1.Ex7.m1.5.5.1.1.1.1">superscript</csymbol><ci id="S1.Ex7.m1.5.5.1.1.1.1.2.2.cmml" xref="S1.Ex7.m1.5.5.1.1.1.1.2.2">ℋ</ci><ci id="S1.Ex7.m1.5.5.1.1.1.1.2.3.cmml" xref="S1.Ex7.m1.5.5.1.1.1.1.2.3">𝑞</ci></apply><ci id="S1.Ex7.m1.5.5.1.1.1.1.3.cmml" xref="S1.Ex7.m1.5.5.1.1.1.1.3">ℳ</ci></apply></list></apply></apply><apply id="S1.Ex7.m1.6.6c.cmml" xref="S1.Ex7.m1.6.6"><ci id="S1.Ex7.m1.6.6.6.cmml" xref="S1.Ex7.m1.6.6.6">⇒</ci><share href="https://arxiv.org/html/2503.14659v1#S1.Ex7.m1.5.5.1.cmml" id="S1.Ex7.m1.6.6d.cmml" xref="S1.Ex7.m1.6.6"></share><apply id="S1.Ex7.m1.6.6.2.cmml" xref="S1.Ex7.m1.6.6.2"><times id="S1.Ex7.m1.6.6.2.2.cmml" xref="S1.Ex7.m1.6.6.2.2"></times><apply id="S1.Ex7.m1.6.6.2.3.cmml" xref="S1.Ex7.m1.6.6.2.3"><csymbol cd="ambiguous" id="S1.Ex7.m1.6.6.2.3.1.cmml" xref="S1.Ex7.m1.6.6.2.3">subscript</csymbol><apply id="S1.Ex7.m1.6.6.2.3.2.cmml" xref="S1.Ex7.m1.6.6.2.3"><csymbol cd="ambiguous" id="S1.Ex7.m1.6.6.2.3.2.1.cmml" xref="S1.Ex7.m1.6.6.2.3">superscript</csymbol><ci id="S1.Ex7.m1.6.6.2.3.2.2.cmml" xref="S1.Ex7.m1.6.6.2.3.2.2">𝐻</ci><apply id="S1.Ex7.m1.6.6.2.3.2.3.cmml" xref="S1.Ex7.m1.6.6.2.3.2.3"><plus id="S1.Ex7.m1.6.6.2.3.2.3.1.cmml" xref="S1.Ex7.m1.6.6.2.3.2.3.1"></plus><ci id="S1.Ex7.m1.6.6.2.3.2.3.2.cmml" xref="S1.Ex7.m1.6.6.2.3.2.3.2">𝑝</ci><ci id="S1.Ex7.m1.6.6.2.3.2.3.3.cmml" xref="S1.Ex7.m1.6.6.2.3.2.3.3">𝑞</ci></apply></apply><apply id="S1.Ex7.m1.6.6.2.3.3.cmml" xref="S1.Ex7.m1.6.6.2.3.3"><times id="S1.Ex7.m1.6.6.2.3.3.1.cmml" xref="S1.Ex7.m1.6.6.2.3.3.1"></times><ci id="S1.Ex7.m1.6.6.2.3.3.2.cmml" xref="S1.Ex7.m1.6.6.2.3.3.2">𝑇</ci><ci id="S1.Ex7.m1.6.6.2.3.3.3.cmml" xref="S1.Ex7.m1.6.6.2.3.3.3">ℎ</ci></apply></apply><list id="S1.Ex7.m1.6.6.2.1.2.cmml" xref="S1.Ex7.m1.6.6.2.1.1"><apply id="S1.Ex7.m1.6.6.2.1.1.1.cmml" xref="S1.Ex7.m1.6.6.2.1.1.1"><apply id="S1.Ex7.m1.6.6.2.1.1.1.1.cmml" xref="S1.Ex7.m1.6.6.2.1.1.1.1"><csymbol cd="ambiguous" id="S1.Ex7.m1.6.6.2.1.1.1.1.1.cmml" xref="S1.Ex7.m1.6.6.2.1.1.1.1">subscript</csymbol><int id="S1.Ex7.m1.6.6.2.1.1.1.1.2.cmml" xref="S1.Ex7.m1.6.6.2.1.1.1.1.2"></int><ci id="S1.Ex7.m1.6.6.2.1.1.1.1.3.cmml" xref="S1.Ex7.m1.6.6.2.1.1.1.1.3">𝒟</ci></apply><ci id="S1.Ex7.m1.6.6.2.1.1.1.2.cmml" xref="S1.Ex7.m1.6.6.2.1.1.1.2">𝐹</ci></apply><ci id="S1.Ex7.m1.4.4.cmml" xref="S1.Ex7.m1.4.4">ℳ</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex7.m1.6c">E_{2}^{p,q}=H^{p}_{Th}(\mathcal{D};\mathcal{H}^{q}_{\mathcal{M}})\Rightarrow H% ^{p+q}_{Th}(\textstyle\int_{\mathcal{D}}F;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S1.Ex7.m1.6d">italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p , italic_q end_POSTSUPERSCRIPT = italic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_D ; caligraphic_H start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT ) ⇒ italic_H start_POSTSUPERSCRIPT italic_p + italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( ∫ start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F ; caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.Thmtheorem4.p1.15"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem4.p1.15.4">where <math alttext="\mathcal{H}^{q}_{\mathcal{M}}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.12.1.m1.1"><semantics id="S1.Thmtheorem4.p1.12.1.m1.1a"><msubsup id="S1.Thmtheorem4.p1.12.1.m1.1.1" xref="S1.Thmtheorem4.p1.12.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.12.1.m1.1.1.2.2" xref="S1.Thmtheorem4.p1.12.1.m1.1.1.2.2.cmml">ℋ</mi><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.12.1.m1.1.1.3" xref="S1.Thmtheorem4.p1.12.1.m1.1.1.3.cmml">ℳ</mi><mi id="S1.Thmtheorem4.p1.12.1.m1.1.1.2.3" xref="S1.Thmtheorem4.p1.12.1.m1.1.1.2.3.cmml">q</mi></msubsup><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.12.1.m1.1b"><apply id="S1.Thmtheorem4.p1.12.1.m1.1.1.cmml" xref="S1.Thmtheorem4.p1.12.1.m1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.12.1.m1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.12.1.m1.1.1">subscript</csymbol><apply id="S1.Thmtheorem4.p1.12.1.m1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.12.1.m1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.12.1.m1.1.1.2.1.cmml" xref="S1.Thmtheorem4.p1.12.1.m1.1.1">superscript</csymbol><ci id="S1.Thmtheorem4.p1.12.1.m1.1.1.2.2.cmml" xref="S1.Thmtheorem4.p1.12.1.m1.1.1.2.2">ℋ</ci><ci id="S1.Thmtheorem4.p1.12.1.m1.1.1.2.3.cmml" xref="S1.Thmtheorem4.p1.12.1.m1.1.1.2.3">𝑞</ci></apply><ci id="S1.Thmtheorem4.p1.12.1.m1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.12.1.m1.1.1.3">ℳ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.12.1.m1.1c">\mathcal{H}^{q}_{\mathcal{M}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.12.1.m1.1d">caligraphic_H start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT</annotation></semantics></math> is the coefficient system on <math alttext="N\mathcal{D}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.13.2.m2.1"><semantics id="S1.Thmtheorem4.p1.13.2.m2.1a"><mrow id="S1.Thmtheorem4.p1.13.2.m2.1.1" xref="S1.Thmtheorem4.p1.13.2.m2.1.1.cmml"><mi id="S1.Thmtheorem4.p1.13.2.m2.1.1.2" xref="S1.Thmtheorem4.p1.13.2.m2.1.1.2.cmml">N</mi><mo id="S1.Thmtheorem4.p1.13.2.m2.1.1.1" xref="S1.Thmtheorem4.p1.13.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.13.2.m2.1.1.3" xref="S1.Thmtheorem4.p1.13.2.m2.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.13.2.m2.1b"><apply id="S1.Thmtheorem4.p1.13.2.m2.1.1.cmml" xref="S1.Thmtheorem4.p1.13.2.m2.1.1"><times id="S1.Thmtheorem4.p1.13.2.m2.1.1.1.cmml" xref="S1.Thmtheorem4.p1.13.2.m2.1.1.1"></times><ci id="S1.Thmtheorem4.p1.13.2.m2.1.1.2.cmml" xref="S1.Thmtheorem4.p1.13.2.m2.1.1.2">𝑁</ci><ci id="S1.Thmtheorem4.p1.13.2.m2.1.1.3.cmml" xref="S1.Thmtheorem4.p1.13.2.m2.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.13.2.m2.1c">N\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.13.2.m2.1d">italic_N caligraphic_D</annotation></semantics></math> that sends a simplex <math alttext="\sigma=(d_{0}\to\cdots\to d_{p})\in N\mathcal{D}_{p}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.14.3.m3.1"><semantics id="S1.Thmtheorem4.p1.14.3.m3.1a"><mrow id="S1.Thmtheorem4.p1.14.3.m3.1.1" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.cmml"><mi id="S1.Thmtheorem4.p1.14.3.m3.1.1.3" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.3.cmml">σ</mi><mo id="S1.Thmtheorem4.p1.14.3.m3.1.1.4" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.4.cmml">=</mo><mrow id="S1.Thmtheorem4.p1.14.3.m3.1.1.1.1" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem4.p1.14.3.m3.1.1.1.1.2" stretchy="false" 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xref="S1.Thmtheorem4.p1.14.3.m3.1.1.6.1"></times><ci id="S1.Thmtheorem4.p1.14.3.m3.1.1.6.2.cmml" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.6.2">𝑁</ci><apply id="S1.Thmtheorem4.p1.14.3.m3.1.1.6.3.cmml" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.6.3"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.14.3.m3.1.1.6.3.1.cmml" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.6.3">subscript</csymbol><ci id="S1.Thmtheorem4.p1.14.3.m3.1.1.6.3.2.cmml" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.6.3.2">𝒟</ci><ci id="S1.Thmtheorem4.p1.14.3.m3.1.1.6.3.3.cmml" xref="S1.Thmtheorem4.p1.14.3.m3.1.1.6.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.14.3.m3.1c">\sigma=(d_{0}\to\cdots\to d_{p})\in N\mathcal{D}_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.14.3.m3.1d">italic_σ = ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ⋯ → italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ∈ italic_N caligraphic_D start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> to the cohomology module <math alttext="H^{q}(N\widetilde{F}(d_{0});j_{d_{0}}^{*}\mathcal{M})" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.15.4.m4.2"><semantics id="S1.Thmtheorem4.p1.15.4.m4.2a"><mrow id="S1.Thmtheorem4.p1.15.4.m4.2.2" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.cmml"><msup id="S1.Thmtheorem4.p1.15.4.m4.2.2.4" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.4.cmml"><mi id="S1.Thmtheorem4.p1.15.4.m4.2.2.4.2" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.4.2.cmml">H</mi><mi id="S1.Thmtheorem4.p1.15.4.m4.2.2.4.3" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.4.3.cmml">q</mi></msup><mo id="S1.Thmtheorem4.p1.15.4.m4.2.2.3" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.3.cmml">⁢</mo><mrow id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.3.cmml"><mo id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.3" stretchy="false" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.3.cmml">(</mo><mrow id="S1.Thmtheorem4.p1.15.4.m4.1.1.1.1.1" 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xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2">superscript</csymbol><apply id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.cmml" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.1.cmml" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.2.cmml" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.2">𝑗</ci><apply id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.3.cmml" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.3.1.cmml" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.3">subscript</csymbol><ci id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.3.2.cmml" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.3.2">𝑑</ci><cn id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.2.3.3">0</cn></apply></apply><times id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.3.cmml" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.2.3"></times></apply><ci id="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.3.cmml" xref="S1.Thmtheorem4.p1.15.4.m4.2.2.2.2.2.3">ℳ</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.15.4.m4.2c">H^{q}(N\widetilde{F}(d_{0});j_{d_{0}}^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.15.4.m4.2d">italic_H start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ( italic_N over~ start_ARG italic_F end_ARG ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ; italic_j start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.p10"> <p class="ltx_p" id="S1.p10.2">A similar spectral sequence that converges to the Baues-Wirsching cohomology of the Grothendieck construction <math alttext="\int_{\mathcal{D}}F" class="ltx_Math" display="inline" id="S1.p10.1.m1.1"><semantics id="S1.p10.1.m1.1a"><mrow id="S1.p10.1.m1.1.1" xref="S1.p10.1.m1.1.1.cmml"><msub id="S1.p10.1.m1.1.1.1" xref="S1.p10.1.m1.1.1.1.cmml"><mo id="S1.p10.1.m1.1.1.1.2" xref="S1.p10.1.m1.1.1.1.2.cmml">∫</mo><mi class="ltx_font_mathcaligraphic" id="S1.p10.1.m1.1.1.1.3" xref="S1.p10.1.m1.1.1.1.3.cmml">𝒟</mi></msub><mi id="S1.p10.1.m1.1.1.2" xref="S1.p10.1.m1.1.1.2.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p10.1.m1.1b"><apply id="S1.p10.1.m1.1.1.cmml" xref="S1.p10.1.m1.1.1"><apply id="S1.p10.1.m1.1.1.1.cmml" xref="S1.p10.1.m1.1.1.1"><csymbol cd="ambiguous" id="S1.p10.1.m1.1.1.1.1.cmml" xref="S1.p10.1.m1.1.1.1">subscript</csymbol><int id="S1.p10.1.m1.1.1.1.2.cmml" xref="S1.p10.1.m1.1.1.1.2"></int><ci id="S1.p10.1.m1.1.1.1.3.cmml" xref="S1.p10.1.m1.1.1.1.3">𝒟</ci></apply><ci id="S1.p10.1.m1.1.1.2.cmml" xref="S1.p10.1.m1.1.1.2">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.1.m1.1c">\int_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="S1.p10.1.m1.1d">∫ start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> was constructed earlier by Pirashvili and Redondo <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib15" title="">15</a>]</cite> and by Cegarra <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib3" title="">3</a>]</cite> using different methods under certain conditions on the coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S1.p10.2.m2.1"><semantics id="S1.p10.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p10.2.m2.1.1" xref="S1.p10.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S1.p10.2.m2.1b"><ci id="S1.p10.2.m2.1.1.cmml" xref="S1.p10.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.p10.2.m2.1d">caligraphic_M</annotation></semantics></math>. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem4" title="Theorem 1.4. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.4</span></a> can be considered as a generalization of these spectral sequences to the Thomason cohomology of small categories.</p> </div> <div class="ltx_para" id="S1.p11"> <p class="ltx_p" id="S1.p11.3"><em class="ltx_emph ltx_font_italic" id="S1.p11.3.1">Notation and Conventions:</em> All the categories are assumed to be small (or equivalent to a small category). We write <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S1.p11.1.m1.1"><semantics id="S1.p11.1.m1.1a"><mrow id="S1.p11.1.m1.1.1" xref="S1.p11.1.m1.1.1.cmml"><mi id="S1.p11.1.m1.1.1.2" xref="S1.p11.1.m1.1.1.2.cmml">d</mi><mo id="S1.p11.1.m1.1.1.1" xref="S1.p11.1.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p11.1.m1.1.1.3" xref="S1.p11.1.m1.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p11.1.m1.1b"><apply id="S1.p11.1.m1.1.1.cmml" xref="S1.p11.1.m1.1.1"><in id="S1.p11.1.m1.1.1.1.cmml" xref="S1.p11.1.m1.1.1.1"></in><ci id="S1.p11.1.m1.1.1.2.cmml" xref="S1.p11.1.m1.1.1.2">𝑑</ci><ci id="S1.p11.1.m1.1.1.3.cmml" xref="S1.p11.1.m1.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p11.1.m1.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p11.1.m1.1d">italic_d ∈ caligraphic_D</annotation></semantics></math> when <math alttext="d" class="ltx_Math" display="inline" id="S1.p11.2.m2.1"><semantics id="S1.p11.2.m2.1a"><mi id="S1.p11.2.m2.1.1" xref="S1.p11.2.m2.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.p11.2.m2.1b"><ci id="S1.p11.2.m2.1.1.cmml" xref="S1.p11.2.m2.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p11.2.m2.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.p11.2.m2.1d">italic_d</annotation></semantics></math> is an object of the category <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S1.p11.3.m3.1"><semantics id="S1.p11.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p11.3.m3.1.1" xref="S1.p11.3.m3.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S1.p11.3.m3.1b"><ci id="S1.p11.3.m3.1.1.cmml" xref="S1.p11.3.m3.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p11.3.m3.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S1.p11.3.m3.1d">caligraphic_D</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p12"> <p class="ltx_p" id="S1.p12.1"><span class="ltx_text ltx_font_bold" id="S1.p12.1.1">Acknowledgements:</span> The first author is supported by TÜBİTAK grant 2211-A National PhD Scholarship Program. The second author is supported by TÜBİTAK grant 2219-International Postdoctoral Research Fellowship Program (2023, 2nd term). We thank TÜBİTAK for their support of this research. We also thank Matthew Gelvin and Caroline Yalçın for reading an earlier version of the paper and for their suggestions.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2. </span>Modules and cohomology for small categories</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.9">Let <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">caligraphic_C</annotation></semantics></math> be a small category and <math alttext="R" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_R</annotation></semantics></math> a commutative ring. A functor <math alttext="M:\mathcal{C}\to R" class="ltx_Math" display="inline" id="S2.p1.3.m3.1"><semantics id="S2.p1.3.m3.1a"><mrow id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml"><mi id="S2.p1.3.m3.1.1.2" xref="S2.p1.3.m3.1.1.2.cmml">M</mi><mo id="S2.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S2.p1.3.m3.1.1.3" xref="S2.p1.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p1.3.m3.1.1.3.2" xref="S2.p1.3.m3.1.1.3.2.cmml">𝒞</mi><mo id="S2.p1.3.m3.1.1.3.1" stretchy="false" xref="S2.p1.3.m3.1.1.3.1.cmml">→</mo><mi id="S2.p1.3.m3.1.1.3.3" xref="S2.p1.3.m3.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.1b"><apply id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1"><ci id="S2.p1.3.m3.1.1.1.cmml" xref="S2.p1.3.m3.1.1.1">:</ci><ci id="S2.p1.3.m3.1.1.2.cmml" xref="S2.p1.3.m3.1.1.2">𝑀</ci><apply id="S2.p1.3.m3.1.1.3.cmml" xref="S2.p1.3.m3.1.1.3"><ci id="S2.p1.3.m3.1.1.3.1.cmml" xref="S2.p1.3.m3.1.1.3.1">→</ci><ci id="S2.p1.3.m3.1.1.3.2.cmml" xref="S2.p1.3.m3.1.1.3.2">𝒞</ci><ci id="S2.p1.3.m3.1.1.3.3.cmml" xref="S2.p1.3.m3.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">M:\mathcal{C}\to R</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">italic_M : caligraphic_C → italic_R</annotation></semantics></math>-Mod is called a (covariant) <em class="ltx_emph ltx_font_italic" id="S2.p1.4.1"><math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p1.4.1.m1.1"><semantics id="S2.p1.4.1.m1.1a"><mrow id="S2.p1.4.1.m1.1.1" xref="S2.p1.4.1.m1.1.1.cmml"><mi id="S2.p1.4.1.m1.1.1.2" xref="S2.p1.4.1.m1.1.1.2.cmml">R</mi><mo id="S2.p1.4.1.m1.1.1.1" xref="S2.p1.4.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p1.4.1.m1.1.1.3" xref="S2.p1.4.1.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.1.m1.1b"><apply id="S2.p1.4.1.m1.1.1.cmml" xref="S2.p1.4.1.m1.1.1"><times id="S2.p1.4.1.m1.1.1.1.cmml" xref="S2.p1.4.1.m1.1.1.1"></times><ci id="S2.p1.4.1.m1.1.1.2.cmml" xref="S2.p1.4.1.m1.1.1.2">𝑅</ci><ci id="S2.p1.4.1.m1.1.1.3.cmml" xref="S2.p1.4.1.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.1.m1.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.1.m1.1d">italic_R caligraphic_C</annotation></semantics></math>-module</em>. The category of <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p1.5.m4.1"><semantics id="S2.p1.5.m4.1a"><mrow id="S2.p1.5.m4.1.1" xref="S2.p1.5.m4.1.1.cmml"><mi id="S2.p1.5.m4.1.1.2" xref="S2.p1.5.m4.1.1.2.cmml">R</mi><mo id="S2.p1.5.m4.1.1.1" xref="S2.p1.5.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p1.5.m4.1.1.3" xref="S2.p1.5.m4.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m4.1b"><apply id="S2.p1.5.m4.1.1.cmml" xref="S2.p1.5.m4.1.1"><times id="S2.p1.5.m4.1.1.1.cmml" xref="S2.p1.5.m4.1.1.1"></times><ci id="S2.p1.5.m4.1.1.2.cmml" xref="S2.p1.5.m4.1.1.2">𝑅</ci><ci id="S2.p1.5.m4.1.1.3.cmml" xref="S2.p1.5.m4.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m4.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m4.1d">italic_R caligraphic_C</annotation></semantics></math>-modules is an abelian category, which has enough projective and injective modules. The exactness of a short exact sequence <math alttext="M_{1}\to M_{2}\to M_{3}" class="ltx_Math" display="inline" id="S2.p1.6.m5.1"><semantics id="S2.p1.6.m5.1a"><mrow id="S2.p1.6.m5.1.1" xref="S2.p1.6.m5.1.1.cmml"><msub id="S2.p1.6.m5.1.1.2" xref="S2.p1.6.m5.1.1.2.cmml"><mi id="S2.p1.6.m5.1.1.2.2" xref="S2.p1.6.m5.1.1.2.2.cmml">M</mi><mn id="S2.p1.6.m5.1.1.2.3" xref="S2.p1.6.m5.1.1.2.3.cmml">1</mn></msub><mo id="S2.p1.6.m5.1.1.3" stretchy="false" xref="S2.p1.6.m5.1.1.3.cmml">→</mo><msub id="S2.p1.6.m5.1.1.4" xref="S2.p1.6.m5.1.1.4.cmml"><mi id="S2.p1.6.m5.1.1.4.2" xref="S2.p1.6.m5.1.1.4.2.cmml">M</mi><mn id="S2.p1.6.m5.1.1.4.3" xref="S2.p1.6.m5.1.1.4.3.cmml">2</mn></msub><mo id="S2.p1.6.m5.1.1.5" stretchy="false" xref="S2.p1.6.m5.1.1.5.cmml">→</mo><msub id="S2.p1.6.m5.1.1.6" xref="S2.p1.6.m5.1.1.6.cmml"><mi id="S2.p1.6.m5.1.1.6.2" xref="S2.p1.6.m5.1.1.6.2.cmml">M</mi><mn id="S2.p1.6.m5.1.1.6.3" xref="S2.p1.6.m5.1.1.6.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.6.m5.1b"><apply id="S2.p1.6.m5.1.1.cmml" xref="S2.p1.6.m5.1.1"><and id="S2.p1.6.m5.1.1a.cmml" xref="S2.p1.6.m5.1.1"></and><apply id="S2.p1.6.m5.1.1b.cmml" xref="S2.p1.6.m5.1.1"><ci id="S2.p1.6.m5.1.1.3.cmml" xref="S2.p1.6.m5.1.1.3">→</ci><apply id="S2.p1.6.m5.1.1.2.cmml" xref="S2.p1.6.m5.1.1.2"><csymbol cd="ambiguous" id="S2.p1.6.m5.1.1.2.1.cmml" xref="S2.p1.6.m5.1.1.2">subscript</csymbol><ci id="S2.p1.6.m5.1.1.2.2.cmml" xref="S2.p1.6.m5.1.1.2.2">𝑀</ci><cn id="S2.p1.6.m5.1.1.2.3.cmml" type="integer" xref="S2.p1.6.m5.1.1.2.3">1</cn></apply><apply id="S2.p1.6.m5.1.1.4.cmml" xref="S2.p1.6.m5.1.1.4"><csymbol cd="ambiguous" id="S2.p1.6.m5.1.1.4.1.cmml" xref="S2.p1.6.m5.1.1.4">subscript</csymbol><ci id="S2.p1.6.m5.1.1.4.2.cmml" xref="S2.p1.6.m5.1.1.4.2">𝑀</ci><cn id="S2.p1.6.m5.1.1.4.3.cmml" type="integer" xref="S2.p1.6.m5.1.1.4.3">2</cn></apply></apply><apply id="S2.p1.6.m5.1.1c.cmml" xref="S2.p1.6.m5.1.1"><ci id="S2.p1.6.m5.1.1.5.cmml" xref="S2.p1.6.m5.1.1.5">→</ci><share href="https://arxiv.org/html/2503.14659v1#S2.p1.6.m5.1.1.4.cmml" id="S2.p1.6.m5.1.1d.cmml" xref="S2.p1.6.m5.1.1"></share><apply id="S2.p1.6.m5.1.1.6.cmml" xref="S2.p1.6.m5.1.1.6"><csymbol cd="ambiguous" id="S2.p1.6.m5.1.1.6.1.cmml" xref="S2.p1.6.m5.1.1.6">subscript</csymbol><ci id="S2.p1.6.m5.1.1.6.2.cmml" xref="S2.p1.6.m5.1.1.6.2">𝑀</ci><cn id="S2.p1.6.m5.1.1.6.3.cmml" type="integer" xref="S2.p1.6.m5.1.1.6.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m5.1c">M_{1}\to M_{2}\to M_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m5.1d">italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p1.7.m6.1"><semantics id="S2.p1.7.m6.1a"><mrow id="S2.p1.7.m6.1.1" xref="S2.p1.7.m6.1.1.cmml"><mi id="S2.p1.7.m6.1.1.2" xref="S2.p1.7.m6.1.1.2.cmml">R</mi><mo id="S2.p1.7.m6.1.1.1" xref="S2.p1.7.m6.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p1.7.m6.1.1.3" xref="S2.p1.7.m6.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.7.m6.1b"><apply id="S2.p1.7.m6.1.1.cmml" xref="S2.p1.7.m6.1.1"><times id="S2.p1.7.m6.1.1.1.cmml" xref="S2.p1.7.m6.1.1.1"></times><ci id="S2.p1.7.m6.1.1.2.cmml" xref="S2.p1.7.m6.1.1.2">𝑅</ci><ci id="S2.p1.7.m6.1.1.3.cmml" xref="S2.p1.7.m6.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m6.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m6.1d">italic_R caligraphic_C</annotation></semantics></math>-Mod is defined by component-wise exactness of the corresponding <math alttext="R" class="ltx_Math" display="inline" id="S2.p1.8.m7.1"><semantics id="S2.p1.8.m7.1a"><mi id="S2.p1.8.m7.1.1" xref="S2.p1.8.m7.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.p1.8.m7.1b"><ci id="S2.p1.8.m7.1.1.cmml" xref="S2.p1.8.m7.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m7.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m7.1d">italic_R</annotation></semantics></math>-modules <math alttext="M_{1}(x)\to M_{2}(x)\to M_{3}(x)" class="ltx_Math" display="inline" id="S2.p1.9.m8.3"><semantics id="S2.p1.9.m8.3a"><mrow id="S2.p1.9.m8.3.4" xref="S2.p1.9.m8.3.4.cmml"><mrow id="S2.p1.9.m8.3.4.2" xref="S2.p1.9.m8.3.4.2.cmml"><msub id="S2.p1.9.m8.3.4.2.2" xref="S2.p1.9.m8.3.4.2.2.cmml"><mi id="S2.p1.9.m8.3.4.2.2.2" xref="S2.p1.9.m8.3.4.2.2.2.cmml">M</mi><mn id="S2.p1.9.m8.3.4.2.2.3" xref="S2.p1.9.m8.3.4.2.2.3.cmml">1</mn></msub><mo id="S2.p1.9.m8.3.4.2.1" xref="S2.p1.9.m8.3.4.2.1.cmml">⁢</mo><mrow id="S2.p1.9.m8.3.4.2.3.2" xref="S2.p1.9.m8.3.4.2.cmml"><mo id="S2.p1.9.m8.3.4.2.3.2.1" stretchy="false" xref="S2.p1.9.m8.3.4.2.cmml">(</mo><mi id="S2.p1.9.m8.1.1" xref="S2.p1.9.m8.1.1.cmml">x</mi><mo id="S2.p1.9.m8.3.4.2.3.2.2" stretchy="false" xref="S2.p1.9.m8.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.9.m8.3.4.3" stretchy="false" xref="S2.p1.9.m8.3.4.3.cmml">→</mo><mrow id="S2.p1.9.m8.3.4.4" xref="S2.p1.9.m8.3.4.4.cmml"><msub id="S2.p1.9.m8.3.4.4.2" xref="S2.p1.9.m8.3.4.4.2.cmml"><mi id="S2.p1.9.m8.3.4.4.2.2" xref="S2.p1.9.m8.3.4.4.2.2.cmml">M</mi><mn id="S2.p1.9.m8.3.4.4.2.3" xref="S2.p1.9.m8.3.4.4.2.3.cmml">2</mn></msub><mo id="S2.p1.9.m8.3.4.4.1" xref="S2.p1.9.m8.3.4.4.1.cmml">⁢</mo><mrow id="S2.p1.9.m8.3.4.4.3.2" xref="S2.p1.9.m8.3.4.4.cmml"><mo id="S2.p1.9.m8.3.4.4.3.2.1" stretchy="false" xref="S2.p1.9.m8.3.4.4.cmml">(</mo><mi id="S2.p1.9.m8.2.2" xref="S2.p1.9.m8.2.2.cmml">x</mi><mo id="S2.p1.9.m8.3.4.4.3.2.2" stretchy="false" xref="S2.p1.9.m8.3.4.4.cmml">)</mo></mrow></mrow><mo id="S2.p1.9.m8.3.4.5" stretchy="false" xref="S2.p1.9.m8.3.4.5.cmml">→</mo><mrow id="S2.p1.9.m8.3.4.6" xref="S2.p1.9.m8.3.4.6.cmml"><msub id="S2.p1.9.m8.3.4.6.2" xref="S2.p1.9.m8.3.4.6.2.cmml"><mi id="S2.p1.9.m8.3.4.6.2.2" xref="S2.p1.9.m8.3.4.6.2.2.cmml">M</mi><mn id="S2.p1.9.m8.3.4.6.2.3" xref="S2.p1.9.m8.3.4.6.2.3.cmml">3</mn></msub><mo id="S2.p1.9.m8.3.4.6.1" xref="S2.p1.9.m8.3.4.6.1.cmml">⁢</mo><mrow id="S2.p1.9.m8.3.4.6.3.2" xref="S2.p1.9.m8.3.4.6.cmml"><mo id="S2.p1.9.m8.3.4.6.3.2.1" stretchy="false" xref="S2.p1.9.m8.3.4.6.cmml">(</mo><mi id="S2.p1.9.m8.3.3" xref="S2.p1.9.m8.3.3.cmml">x</mi><mo id="S2.p1.9.m8.3.4.6.3.2.2" stretchy="false" xref="S2.p1.9.m8.3.4.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m8.3b"><apply id="S2.p1.9.m8.3.4.cmml" xref="S2.p1.9.m8.3.4"><and id="S2.p1.9.m8.3.4a.cmml" xref="S2.p1.9.m8.3.4"></and><apply id="S2.p1.9.m8.3.4b.cmml" xref="S2.p1.9.m8.3.4"><ci id="S2.p1.9.m8.3.4.3.cmml" xref="S2.p1.9.m8.3.4.3">→</ci><apply id="S2.p1.9.m8.3.4.2.cmml" xref="S2.p1.9.m8.3.4.2"><times id="S2.p1.9.m8.3.4.2.1.cmml" xref="S2.p1.9.m8.3.4.2.1"></times><apply id="S2.p1.9.m8.3.4.2.2.cmml" xref="S2.p1.9.m8.3.4.2.2"><csymbol cd="ambiguous" id="S2.p1.9.m8.3.4.2.2.1.cmml" xref="S2.p1.9.m8.3.4.2.2">subscript</csymbol><ci id="S2.p1.9.m8.3.4.2.2.2.cmml" xref="S2.p1.9.m8.3.4.2.2.2">𝑀</ci><cn id="S2.p1.9.m8.3.4.2.2.3.cmml" type="integer" xref="S2.p1.9.m8.3.4.2.2.3">1</cn></apply><ci id="S2.p1.9.m8.1.1.cmml" xref="S2.p1.9.m8.1.1">𝑥</ci></apply><apply id="S2.p1.9.m8.3.4.4.cmml" xref="S2.p1.9.m8.3.4.4"><times id="S2.p1.9.m8.3.4.4.1.cmml" xref="S2.p1.9.m8.3.4.4.1"></times><apply id="S2.p1.9.m8.3.4.4.2.cmml" xref="S2.p1.9.m8.3.4.4.2"><csymbol cd="ambiguous" id="S2.p1.9.m8.3.4.4.2.1.cmml" xref="S2.p1.9.m8.3.4.4.2">subscript</csymbol><ci id="S2.p1.9.m8.3.4.4.2.2.cmml" xref="S2.p1.9.m8.3.4.4.2.2">𝑀</ci><cn id="S2.p1.9.m8.3.4.4.2.3.cmml" type="integer" xref="S2.p1.9.m8.3.4.4.2.3">2</cn></apply><ci id="S2.p1.9.m8.2.2.cmml" xref="S2.p1.9.m8.2.2">𝑥</ci></apply></apply><apply id="S2.p1.9.m8.3.4c.cmml" xref="S2.p1.9.m8.3.4"><ci id="S2.p1.9.m8.3.4.5.cmml" xref="S2.p1.9.m8.3.4.5">→</ci><share href="https://arxiv.org/html/2503.14659v1#S2.p1.9.m8.3.4.4.cmml" id="S2.p1.9.m8.3.4d.cmml" xref="S2.p1.9.m8.3.4"></share><apply id="S2.p1.9.m8.3.4.6.cmml" xref="S2.p1.9.m8.3.4.6"><times id="S2.p1.9.m8.3.4.6.1.cmml" xref="S2.p1.9.m8.3.4.6.1"></times><apply id="S2.p1.9.m8.3.4.6.2.cmml" xref="S2.p1.9.m8.3.4.6.2"><csymbol cd="ambiguous" id="S2.p1.9.m8.3.4.6.2.1.cmml" xref="S2.p1.9.m8.3.4.6.2">subscript</csymbol><ci id="S2.p1.9.m8.3.4.6.2.2.cmml" xref="S2.p1.9.m8.3.4.6.2.2">𝑀</ci><cn id="S2.p1.9.m8.3.4.6.2.3.cmml" type="integer" xref="S2.p1.9.m8.3.4.6.2.3">3</cn></apply><ci id="S2.p1.9.m8.3.3.cmml" xref="S2.p1.9.m8.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m8.3c">M_{1}(x)\to M_{2}(x)\to M_{3}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m8.3d">italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) → italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) → italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.9">There is an explicit construction of projective <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mrow id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml"><mi id="S2.p2.1.m1.1.1.2" xref="S2.p2.1.m1.1.1.2.cmml">R</mi><mo id="S2.p2.1.m1.1.1.1" xref="S2.p2.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p2.1.m1.1.1.3" xref="S2.p2.1.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><apply id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1"><times id="S2.p2.1.m1.1.1.1.cmml" xref="S2.p2.1.m1.1.1.1"></times><ci id="S2.p2.1.m1.1.1.2.cmml" xref="S2.p2.1.m1.1.1.2">𝑅</ci><ci id="S2.p2.1.m1.1.1.3.cmml" xref="S2.p2.1.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">italic_R caligraphic_C</annotation></semantics></math>-modules using the Yoneda Lemma: For every <math alttext="x\in\mathcal{C}" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mrow id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml"><mi id="S2.p2.2.m2.1.1.2" xref="S2.p2.2.m2.1.1.2.cmml">x</mi><mo id="S2.p2.2.m2.1.1.1" xref="S2.p2.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.p2.2.m2.1.1.3" xref="S2.p2.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><apply id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1"><in id="S2.p2.2.m2.1.1.1.cmml" xref="S2.p2.2.m2.1.1.1"></in><ci id="S2.p2.2.m2.1.1.2.cmml" xref="S2.p2.2.m2.1.1.2">𝑥</ci><ci id="S2.p2.2.m2.1.1.3.cmml" xref="S2.p2.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">x\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">italic_x ∈ caligraphic_C</annotation></semantics></math>, let <math alttext="R\mathrm{Mor}_{\mathcal{C}}(x,?)" class="ltx_Math" display="inline" id="S2.p2.3.m3.2"><semantics id="S2.p2.3.m3.2a"><mrow id="S2.p2.3.m3.2.3" xref="S2.p2.3.m3.2.3.cmml"><mi id="S2.p2.3.m3.2.3.2" xref="S2.p2.3.m3.2.3.2.cmml">R</mi><mo id="S2.p2.3.m3.2.3.1" xref="S2.p2.3.m3.2.3.1.cmml">⁢</mo><msub id="S2.p2.3.m3.2.3.3" xref="S2.p2.3.m3.2.3.3.cmml"><mi id="S2.p2.3.m3.2.3.3.2" xref="S2.p2.3.m3.2.3.3.2.cmml">Mor</mi><mi class="ltx_font_mathcaligraphic" id="S2.p2.3.m3.2.3.3.3" xref="S2.p2.3.m3.2.3.3.3.cmml">𝒞</mi></msub><mo id="S2.p2.3.m3.2.3.1a" xref="S2.p2.3.m3.2.3.1.cmml">⁢</mo><mrow id="S2.p2.3.m3.2.3.4.2" xref="S2.p2.3.m3.2.3.4.1.cmml"><mo id="S2.p2.3.m3.2.3.4.2.1" stretchy="false" xref="S2.p2.3.m3.2.3.4.1.cmml">(</mo><mi id="S2.p2.3.m3.1.1" xref="S2.p2.3.m3.1.1.cmml">x</mi><mo id="S2.p2.3.m3.2.3.4.2.2" xref="S2.p2.3.m3.2.3.4.1.cmml">,</mo><mi id="S2.p2.3.m3.2.2" mathvariant="normal" xref="S2.p2.3.m3.2.2.cmml">?</mi><mo id="S2.p2.3.m3.2.3.4.2.3" stretchy="false" xref="S2.p2.3.m3.2.3.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.3.m3.2b"><apply id="S2.p2.3.m3.2.3.cmml" xref="S2.p2.3.m3.2.3"><times id="S2.p2.3.m3.2.3.1.cmml" xref="S2.p2.3.m3.2.3.1"></times><ci id="S2.p2.3.m3.2.3.2.cmml" xref="S2.p2.3.m3.2.3.2">𝑅</ci><apply id="S2.p2.3.m3.2.3.3.cmml" xref="S2.p2.3.m3.2.3.3"><csymbol cd="ambiguous" id="S2.p2.3.m3.2.3.3.1.cmml" xref="S2.p2.3.m3.2.3.3">subscript</csymbol><ci id="S2.p2.3.m3.2.3.3.2.cmml" xref="S2.p2.3.m3.2.3.3.2">Mor</ci><ci id="S2.p2.3.m3.2.3.3.3.cmml" xref="S2.p2.3.m3.2.3.3.3">𝒞</ci></apply><interval closure="open" id="S2.p2.3.m3.2.3.4.1.cmml" xref="S2.p2.3.m3.2.3.4.2"><ci id="S2.p2.3.m3.1.1.cmml" xref="S2.p2.3.m3.1.1">𝑥</ci><ci id="S2.p2.3.m3.2.2.cmml" xref="S2.p2.3.m3.2.2">?</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.2c">R\mathrm{Mor}_{\mathcal{C}}(x,?)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m3.2d">italic_R roman_Mor start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_x , ? )</annotation></semantics></math> denote the functor <math alttext="\mathcal{C}\to R" class="ltx_Math" display="inline" id="S2.p2.4.m4.1"><semantics id="S2.p2.4.m4.1a"><mrow id="S2.p2.4.m4.1.1" xref="S2.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p2.4.m4.1.1.2" xref="S2.p2.4.m4.1.1.2.cmml">𝒞</mi><mo id="S2.p2.4.m4.1.1.1" stretchy="false" xref="S2.p2.4.m4.1.1.1.cmml">→</mo><mi id="S2.p2.4.m4.1.1.3" xref="S2.p2.4.m4.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.4.m4.1b"><apply id="S2.p2.4.m4.1.1.cmml" xref="S2.p2.4.m4.1.1"><ci id="S2.p2.4.m4.1.1.1.cmml" xref="S2.p2.4.m4.1.1.1">→</ci><ci id="S2.p2.4.m4.1.1.2.cmml" xref="S2.p2.4.m4.1.1.2">𝒞</ci><ci id="S2.p2.4.m4.1.1.3.cmml" xref="S2.p2.4.m4.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m4.1c">\mathcal{C}\to R</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.m4.1d">caligraphic_C → italic_R</annotation></semantics></math>-Mod that sends an object <math alttext="c\in\mathcal{C}" class="ltx_Math" display="inline" id="S2.p2.5.m5.1"><semantics id="S2.p2.5.m5.1a"><mrow id="S2.p2.5.m5.1.1" xref="S2.p2.5.m5.1.1.cmml"><mi id="S2.p2.5.m5.1.1.2" xref="S2.p2.5.m5.1.1.2.cmml">c</mi><mo id="S2.p2.5.m5.1.1.1" xref="S2.p2.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.p2.5.m5.1.1.3" xref="S2.p2.5.m5.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.5.m5.1b"><apply id="S2.p2.5.m5.1.1.cmml" xref="S2.p2.5.m5.1.1"><in id="S2.p2.5.m5.1.1.1.cmml" xref="S2.p2.5.m5.1.1.1"></in><ci id="S2.p2.5.m5.1.1.2.cmml" xref="S2.p2.5.m5.1.1.2">𝑐</ci><ci id="S2.p2.5.m5.1.1.3.cmml" xref="S2.p2.5.m5.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.5.m5.1c">c\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.5.m5.1d">italic_c ∈ caligraphic_C</annotation></semantics></math> to the free <math alttext="R" class="ltx_Math" display="inline" id="S2.p2.6.m6.1"><semantics id="S2.p2.6.m6.1a"><mi id="S2.p2.6.m6.1.1" xref="S2.p2.6.m6.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.p2.6.m6.1b"><ci id="S2.p2.6.m6.1.1.cmml" xref="S2.p2.6.m6.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.6.m6.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.p2.6.m6.1d">italic_R</annotation></semantics></math>-module with basis <math alttext="\mathrm{Mor}_{\mathcal{C}}(x,c)" class="ltx_Math" display="inline" id="S2.p2.7.m7.2"><semantics id="S2.p2.7.m7.2a"><mrow id="S2.p2.7.m7.2.3" xref="S2.p2.7.m7.2.3.cmml"><msub id="S2.p2.7.m7.2.3.2" xref="S2.p2.7.m7.2.3.2.cmml"><mi id="S2.p2.7.m7.2.3.2.2" xref="S2.p2.7.m7.2.3.2.2.cmml">Mor</mi><mi class="ltx_font_mathcaligraphic" id="S2.p2.7.m7.2.3.2.3" xref="S2.p2.7.m7.2.3.2.3.cmml">𝒞</mi></msub><mo id="S2.p2.7.m7.2.3.1" xref="S2.p2.7.m7.2.3.1.cmml">⁢</mo><mrow id="S2.p2.7.m7.2.3.3.2" xref="S2.p2.7.m7.2.3.3.1.cmml"><mo id="S2.p2.7.m7.2.3.3.2.1" stretchy="false" xref="S2.p2.7.m7.2.3.3.1.cmml">(</mo><mi id="S2.p2.7.m7.1.1" xref="S2.p2.7.m7.1.1.cmml">x</mi><mo id="S2.p2.7.m7.2.3.3.2.2" xref="S2.p2.7.m7.2.3.3.1.cmml">,</mo><mi id="S2.p2.7.m7.2.2" xref="S2.p2.7.m7.2.2.cmml">c</mi><mo id="S2.p2.7.m7.2.3.3.2.3" stretchy="false" xref="S2.p2.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.7.m7.2b"><apply id="S2.p2.7.m7.2.3.cmml" xref="S2.p2.7.m7.2.3"><times id="S2.p2.7.m7.2.3.1.cmml" xref="S2.p2.7.m7.2.3.1"></times><apply id="S2.p2.7.m7.2.3.2.cmml" xref="S2.p2.7.m7.2.3.2"><csymbol cd="ambiguous" id="S2.p2.7.m7.2.3.2.1.cmml" xref="S2.p2.7.m7.2.3.2">subscript</csymbol><ci id="S2.p2.7.m7.2.3.2.2.cmml" xref="S2.p2.7.m7.2.3.2.2">Mor</ci><ci id="S2.p2.7.m7.2.3.2.3.cmml" xref="S2.p2.7.m7.2.3.2.3">𝒞</ci></apply><interval closure="open" id="S2.p2.7.m7.2.3.3.1.cmml" xref="S2.p2.7.m7.2.3.3.2"><ci id="S2.p2.7.m7.1.1.cmml" xref="S2.p2.7.m7.1.1">𝑥</ci><ci id="S2.p2.7.m7.2.2.cmml" xref="S2.p2.7.m7.2.2">𝑐</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.7.m7.2c">\mathrm{Mor}_{\mathcal{C}}(x,c)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.7.m7.2d">roman_Mor start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_x , italic_c )</annotation></semantics></math>. By the Yoneda Lemma, for every <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p2.8.m8.1"><semantics id="S2.p2.8.m8.1a"><mrow id="S2.p2.8.m8.1.1" xref="S2.p2.8.m8.1.1.cmml"><mi id="S2.p2.8.m8.1.1.2" xref="S2.p2.8.m8.1.1.2.cmml">R</mi><mo id="S2.p2.8.m8.1.1.1" xref="S2.p2.8.m8.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p2.8.m8.1.1.3" xref="S2.p2.8.m8.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.8.m8.1b"><apply id="S2.p2.8.m8.1.1.cmml" xref="S2.p2.8.m8.1.1"><times id="S2.p2.8.m8.1.1.1.cmml" xref="S2.p2.8.m8.1.1.1"></times><ci id="S2.p2.8.m8.1.1.2.cmml" xref="S2.p2.8.m8.1.1.2">𝑅</ci><ci id="S2.p2.8.m8.1.1.3.cmml" xref="S2.p2.8.m8.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.8.m8.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.8.m8.1d">italic_R caligraphic_C</annotation></semantics></math>-module <math alttext="M" class="ltx_Math" display="inline" id="S2.p2.9.m9.1"><semantics id="S2.p2.9.m9.1a"><mi id="S2.p2.9.m9.1.1" xref="S2.p2.9.m9.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.p2.9.m9.1b"><ci id="S2.p2.9.m9.1.1.cmml" xref="S2.p2.9.m9.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.9.m9.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.p2.9.m9.1d">italic_M</annotation></semantics></math></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathrm{Hom}_{R\mathcal{C}}(R\mathrm{Mor}_{\mathcal{C}}(x,?),M)\cong M(x)." class="ltx_Math" display="block" id="S2.Ex8.m1.5"><semantics id="S2.Ex8.m1.5a"><mrow id="S2.Ex8.m1.5.5.1" xref="S2.Ex8.m1.5.5.1.1.cmml"><mrow id="S2.Ex8.m1.5.5.1.1" xref="S2.Ex8.m1.5.5.1.1.cmml"><mrow id="S2.Ex8.m1.5.5.1.1.1" xref="S2.Ex8.m1.5.5.1.1.1.cmml"><msub id="S2.Ex8.m1.5.5.1.1.1.3" xref="S2.Ex8.m1.5.5.1.1.1.3.cmml"><mi id="S2.Ex8.m1.5.5.1.1.1.3.2" xref="S2.Ex8.m1.5.5.1.1.1.3.2.cmml">Hom</mi><mrow id="S2.Ex8.m1.5.5.1.1.1.3.3" xref="S2.Ex8.m1.5.5.1.1.1.3.3.cmml"><mi id="S2.Ex8.m1.5.5.1.1.1.3.3.2" xref="S2.Ex8.m1.5.5.1.1.1.3.3.2.cmml">R</mi><mo id="S2.Ex8.m1.5.5.1.1.1.3.3.1" xref="S2.Ex8.m1.5.5.1.1.1.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex8.m1.5.5.1.1.1.3.3.3" xref="S2.Ex8.m1.5.5.1.1.1.3.3.3.cmml">𝒞</mi></mrow></msub><mo id="S2.Ex8.m1.5.5.1.1.1.2" xref="S2.Ex8.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex8.m1.5.5.1.1.1.1.1" xref="S2.Ex8.m1.5.5.1.1.1.1.2.cmml"><mo id="S2.Ex8.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S2.Ex8.m1.5.5.1.1.1.1.2.cmml">(</mo><mrow id="S2.Ex8.m1.5.5.1.1.1.1.1.1" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.cmml"><mi id="S2.Ex8.m1.5.5.1.1.1.1.1.1.2" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.2.cmml">R</mi><mo id="S2.Ex8.m1.5.5.1.1.1.1.1.1.1" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S2.Ex8.m1.5.5.1.1.1.1.1.1.3" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.cmml"><mi id="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.2" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.2.cmml">Mor</mi><mi class="ltx_font_mathcaligraphic" id="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.3" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.3.cmml">𝒞</mi></msub><mo id="S2.Ex8.m1.5.5.1.1.1.1.1.1.1a" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.2" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.1.cmml"><mo id="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.2.1" stretchy="false" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.1.cmml">(</mo><mi id="S2.Ex8.m1.1.1" xref="S2.Ex8.m1.1.1.cmml">x</mi><mo id="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.2.2" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.1.cmml">,</mo><mi id="S2.Ex8.m1.2.2" mathvariant="normal" xref="S2.Ex8.m1.2.2.cmml">?</mi><mo id="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.2.3" stretchy="false" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex8.m1.5.5.1.1.1.1.1.3" xref="S2.Ex8.m1.5.5.1.1.1.1.2.cmml">,</mo><mi id="S2.Ex8.m1.3.3" xref="S2.Ex8.m1.3.3.cmml">M</mi><mo id="S2.Ex8.m1.5.5.1.1.1.1.1.4" stretchy="false" xref="S2.Ex8.m1.5.5.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex8.m1.5.5.1.1.2" xref="S2.Ex8.m1.5.5.1.1.2.cmml">≅</mo><mrow id="S2.Ex8.m1.5.5.1.1.3" xref="S2.Ex8.m1.5.5.1.1.3.cmml"><mi id="S2.Ex8.m1.5.5.1.1.3.2" xref="S2.Ex8.m1.5.5.1.1.3.2.cmml">M</mi><mo id="S2.Ex8.m1.5.5.1.1.3.1" xref="S2.Ex8.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex8.m1.5.5.1.1.3.3.2" xref="S2.Ex8.m1.5.5.1.1.3.cmml"><mo id="S2.Ex8.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S2.Ex8.m1.5.5.1.1.3.cmml">(</mo><mi id="S2.Ex8.m1.4.4" xref="S2.Ex8.m1.4.4.cmml">x</mi><mo id="S2.Ex8.m1.5.5.1.1.3.3.2.2" stretchy="false" xref="S2.Ex8.m1.5.5.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex8.m1.5.5.1.2" lspace="0em" xref="S2.Ex8.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex8.m1.5b"><apply id="S2.Ex8.m1.5.5.1.1.cmml" xref="S2.Ex8.m1.5.5.1"><approx id="S2.Ex8.m1.5.5.1.1.2.cmml" xref="S2.Ex8.m1.5.5.1.1.2"></approx><apply id="S2.Ex8.m1.5.5.1.1.1.cmml" xref="S2.Ex8.m1.5.5.1.1.1"><times id="S2.Ex8.m1.5.5.1.1.1.2.cmml" xref="S2.Ex8.m1.5.5.1.1.1.2"></times><apply id="S2.Ex8.m1.5.5.1.1.1.3.cmml" xref="S2.Ex8.m1.5.5.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex8.m1.5.5.1.1.1.3.1.cmml" xref="S2.Ex8.m1.5.5.1.1.1.3">subscript</csymbol><ci id="S2.Ex8.m1.5.5.1.1.1.3.2.cmml" xref="S2.Ex8.m1.5.5.1.1.1.3.2">Hom</ci><apply id="S2.Ex8.m1.5.5.1.1.1.3.3.cmml" xref="S2.Ex8.m1.5.5.1.1.1.3.3"><times id="S2.Ex8.m1.5.5.1.1.1.3.3.1.cmml" xref="S2.Ex8.m1.5.5.1.1.1.3.3.1"></times><ci id="S2.Ex8.m1.5.5.1.1.1.3.3.2.cmml" xref="S2.Ex8.m1.5.5.1.1.1.3.3.2">𝑅</ci><ci id="S2.Ex8.m1.5.5.1.1.1.3.3.3.cmml" xref="S2.Ex8.m1.5.5.1.1.1.3.3.3">𝒞</ci></apply></apply><interval closure="open" id="S2.Ex8.m1.5.5.1.1.1.1.2.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1"><apply id="S2.Ex8.m1.5.5.1.1.1.1.1.1.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1"><times id="S2.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.1"></times><ci id="S2.Ex8.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.2">𝑅</ci><apply id="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.1.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.2.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.2">Mor</ci><ci id="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.3.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.3.3">𝒞</ci></apply><interval closure="open" id="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.1.cmml" xref="S2.Ex8.m1.5.5.1.1.1.1.1.1.4.2"><ci id="S2.Ex8.m1.1.1.cmml" xref="S2.Ex8.m1.1.1">𝑥</ci><ci id="S2.Ex8.m1.2.2.cmml" xref="S2.Ex8.m1.2.2">?</ci></interval></apply><ci id="S2.Ex8.m1.3.3.cmml" xref="S2.Ex8.m1.3.3">𝑀</ci></interval></apply><apply id="S2.Ex8.m1.5.5.1.1.3.cmml" xref="S2.Ex8.m1.5.5.1.1.3"><times id="S2.Ex8.m1.5.5.1.1.3.1.cmml" xref="S2.Ex8.m1.5.5.1.1.3.1"></times><ci id="S2.Ex8.m1.5.5.1.1.3.2.cmml" xref="S2.Ex8.m1.5.5.1.1.3.2">𝑀</ci><ci id="S2.Ex8.m1.4.4.cmml" xref="S2.Ex8.m1.4.4">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m1.5c">\mathrm{Hom}_{R\mathcal{C}}(R\mathrm{Mor}_{\mathcal{C}}(x,?),M)\cong M(x).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m1.5d">roman_Hom start_POSTSUBSCRIPT italic_R caligraphic_C end_POSTSUBSCRIPT ( italic_R roman_Mor start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_x , ? ) , italic_M ) ≅ italic_M ( italic_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p2.12">This shows that the <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p2.10.m1.1"><semantics id="S2.p2.10.m1.1a"><mrow id="S2.p2.10.m1.1.1" xref="S2.p2.10.m1.1.1.cmml"><mi id="S2.p2.10.m1.1.1.2" xref="S2.p2.10.m1.1.1.2.cmml">R</mi><mo id="S2.p2.10.m1.1.1.1" xref="S2.p2.10.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p2.10.m1.1.1.3" xref="S2.p2.10.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.10.m1.1b"><apply id="S2.p2.10.m1.1.1.cmml" xref="S2.p2.10.m1.1.1"><times id="S2.p2.10.m1.1.1.1.cmml" xref="S2.p2.10.m1.1.1.1"></times><ci id="S2.p2.10.m1.1.1.2.cmml" xref="S2.p2.10.m1.1.1.2">𝑅</ci><ci id="S2.p2.10.m1.1.1.3.cmml" xref="S2.p2.10.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.10.m1.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.10.m1.1d">italic_R caligraphic_C</annotation></semantics></math>-module <math alttext="R\mathrm{Mor}_{\mathcal{C}}(x,?)" class="ltx_Math" display="inline" id="S2.p2.11.m2.2"><semantics id="S2.p2.11.m2.2a"><mrow id="S2.p2.11.m2.2.3" xref="S2.p2.11.m2.2.3.cmml"><mi id="S2.p2.11.m2.2.3.2" xref="S2.p2.11.m2.2.3.2.cmml">R</mi><mo id="S2.p2.11.m2.2.3.1" xref="S2.p2.11.m2.2.3.1.cmml">⁢</mo><msub id="S2.p2.11.m2.2.3.3" xref="S2.p2.11.m2.2.3.3.cmml"><mi id="S2.p2.11.m2.2.3.3.2" xref="S2.p2.11.m2.2.3.3.2.cmml">Mor</mi><mi class="ltx_font_mathcaligraphic" id="S2.p2.11.m2.2.3.3.3" xref="S2.p2.11.m2.2.3.3.3.cmml">𝒞</mi></msub><mo id="S2.p2.11.m2.2.3.1a" xref="S2.p2.11.m2.2.3.1.cmml">⁢</mo><mrow id="S2.p2.11.m2.2.3.4.2" xref="S2.p2.11.m2.2.3.4.1.cmml"><mo id="S2.p2.11.m2.2.3.4.2.1" stretchy="false" xref="S2.p2.11.m2.2.3.4.1.cmml">(</mo><mi id="S2.p2.11.m2.1.1" xref="S2.p2.11.m2.1.1.cmml">x</mi><mo id="S2.p2.11.m2.2.3.4.2.2" xref="S2.p2.11.m2.2.3.4.1.cmml">,</mo><mi id="S2.p2.11.m2.2.2" mathvariant="normal" xref="S2.p2.11.m2.2.2.cmml">?</mi><mo id="S2.p2.11.m2.2.3.4.2.3" stretchy="false" xref="S2.p2.11.m2.2.3.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.11.m2.2b"><apply id="S2.p2.11.m2.2.3.cmml" xref="S2.p2.11.m2.2.3"><times id="S2.p2.11.m2.2.3.1.cmml" xref="S2.p2.11.m2.2.3.1"></times><ci id="S2.p2.11.m2.2.3.2.cmml" xref="S2.p2.11.m2.2.3.2">𝑅</ci><apply id="S2.p2.11.m2.2.3.3.cmml" xref="S2.p2.11.m2.2.3.3"><csymbol cd="ambiguous" id="S2.p2.11.m2.2.3.3.1.cmml" xref="S2.p2.11.m2.2.3.3">subscript</csymbol><ci id="S2.p2.11.m2.2.3.3.2.cmml" xref="S2.p2.11.m2.2.3.3.2">Mor</ci><ci id="S2.p2.11.m2.2.3.3.3.cmml" xref="S2.p2.11.m2.2.3.3.3">𝒞</ci></apply><interval closure="open" id="S2.p2.11.m2.2.3.4.1.cmml" xref="S2.p2.11.m2.2.3.4.2"><ci id="S2.p2.11.m2.1.1.cmml" xref="S2.p2.11.m2.1.1">𝑥</ci><ci id="S2.p2.11.m2.2.2.cmml" xref="S2.p2.11.m2.2.2">?</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.11.m2.2c">R\mathrm{Mor}_{\mathcal{C}}(x,?)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.11.m2.2d">italic_R roman_Mor start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_x , ? )</annotation></semantics></math> is projective (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib14" title="">14</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib20" title="">20</a>]</cite> for details). Having enough projective modules allows us to construct projective resolutions and define ext-groups over <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p2.12.m3.1"><semantics id="S2.p2.12.m3.1a"><mrow id="S2.p2.12.m3.1.1" xref="S2.p2.12.m3.1.1.cmml"><mi id="S2.p2.12.m3.1.1.2" xref="S2.p2.12.m3.1.1.2.cmml">R</mi><mo id="S2.p2.12.m3.1.1.1" xref="S2.p2.12.m3.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p2.12.m3.1.1.3" xref="S2.p2.12.m3.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.12.m3.1b"><apply id="S2.p2.12.m3.1.1.cmml" xref="S2.p2.12.m3.1.1"><times id="S2.p2.12.m3.1.1.1.cmml" xref="S2.p2.12.m3.1.1.1"></times><ci id="S2.p2.12.m3.1.1.2.cmml" xref="S2.p2.12.m3.1.1.2">𝑅</ci><ci id="S2.p2.12.m3.1.1.3.cmml" xref="S2.p2.12.m3.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.12.m3.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.12.m3.1d">italic_R caligraphic_C</annotation></semantics></math>-modules.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.1.1.1">Definition 2.1</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.9">For every pair of <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.m1.1"><semantics id="S2.Thmtheorem1.p1.1.m1.1a"><mrow id="S2.Thmtheorem1.p1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.1.m1.1.1.2" xref="S2.Thmtheorem1.p1.1.m1.1.1.2.cmml">R</mi><mo id="S2.Thmtheorem1.p1.1.m1.1.1.1" xref="S2.Thmtheorem1.p1.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.1.m1.1.1.3" xref="S2.Thmtheorem1.p1.1.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.m1.1b"><apply id="S2.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1"><times id="S2.Thmtheorem1.p1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1.1"></times><ci id="S2.Thmtheorem1.p1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1.2">𝑅</ci><ci id="S2.Thmtheorem1.p1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.m1.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.m1.1d">italic_R caligraphic_C</annotation></semantics></math>-modules <math alttext="M" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.m2.1"><semantics id="S2.Thmtheorem1.p1.2.m2.1a"><mi id="S2.Thmtheorem1.p1.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.2.m2.1b"><ci id="S2.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.m2.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.m2.1d">italic_M</annotation></semantics></math> and <math alttext="N" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.m3.1"><semantics id="S2.Thmtheorem1.p1.3.m3.1a"><mi id="S2.Thmtheorem1.p1.3.m3.1.1" xref="S2.Thmtheorem1.p1.3.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.m3.1b"><ci id="S2.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.m3.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.m3.1d">italic_N</annotation></semantics></math>, the <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem1.p1.9.1">ext-group</em> <math alttext="\mathrm{Ext}^{n}_{R\mathcal{C}}(N,M)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.m4.2"><semantics id="S2.Thmtheorem1.p1.4.m4.2a"><mrow id="S2.Thmtheorem1.p1.4.m4.2.3" xref="S2.Thmtheorem1.p1.4.m4.2.3.cmml"><msubsup id="S2.Thmtheorem1.p1.4.m4.2.3.2" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.cmml"><mi id="S2.Thmtheorem1.p1.4.m4.2.3.2.2.2" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.2.2.cmml">Ext</mi><mrow id="S2.Thmtheorem1.p1.4.m4.2.3.2.3" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.3.cmml"><mi id="S2.Thmtheorem1.p1.4.m4.2.3.2.3.2" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.3.2.cmml">R</mi><mo id="S2.Thmtheorem1.p1.4.m4.2.3.2.3.1" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.4.m4.2.3.2.3.3" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.3.3.cmml">𝒞</mi></mrow><mi id="S2.Thmtheorem1.p1.4.m4.2.3.2.2.3" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.2.3.cmml">n</mi></msubsup><mo id="S2.Thmtheorem1.p1.4.m4.2.3.1" xref="S2.Thmtheorem1.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.4.m4.2.3.3.2" xref="S2.Thmtheorem1.p1.4.m4.2.3.3.1.cmml"><mo id="S2.Thmtheorem1.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.4.m4.2.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem1.p1.4.m4.1.1" xref="S2.Thmtheorem1.p1.4.m4.1.1.cmml">N</mi><mo id="S2.Thmtheorem1.p1.4.m4.2.3.3.2.2" xref="S2.Thmtheorem1.p1.4.m4.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem1.p1.4.m4.2.2" xref="S2.Thmtheorem1.p1.4.m4.2.2.cmml">M</mi><mo id="S2.Thmtheorem1.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.m4.2b"><apply id="S2.Thmtheorem1.p1.4.m4.2.3.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3"><times id="S2.Thmtheorem1.p1.4.m4.2.3.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.1"></times><apply id="S2.Thmtheorem1.p1.4.m4.2.3.2.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.4.m4.2.3.2.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2">subscript</csymbol><apply id="S2.Thmtheorem1.p1.4.m4.2.3.2.2.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.4.m4.2.3.2.2.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem1.p1.4.m4.2.3.2.2.2.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.2.2">Ext</ci><ci id="S2.Thmtheorem1.p1.4.m4.2.3.2.2.3.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.2.3">𝑛</ci></apply><apply id="S2.Thmtheorem1.p1.4.m4.2.3.2.3.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.3"><times id="S2.Thmtheorem1.p1.4.m4.2.3.2.3.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.3.1"></times><ci id="S2.Thmtheorem1.p1.4.m4.2.3.2.3.2.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.3.2">𝑅</ci><ci id="S2.Thmtheorem1.p1.4.m4.2.3.2.3.3.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.2.3.3">𝒞</ci></apply></apply><interval closure="open" id="S2.Thmtheorem1.p1.4.m4.2.3.3.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.2.3.3.2"><ci 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id="S2.Thmtheorem1.p1.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.m5.1d">italic_n</annotation></semantics></math>-th cohomology of the cochain complex <math alttext="\mathrm{Hom}_{R\mathcal{C}}(P_{*},M)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.6.m6.2"><semantics id="S2.Thmtheorem1.p1.6.m6.2a"><mrow id="S2.Thmtheorem1.p1.6.m6.2.2" xref="S2.Thmtheorem1.p1.6.m6.2.2.cmml"><msub id="S2.Thmtheorem1.p1.6.m6.2.2.3" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.cmml"><mi id="S2.Thmtheorem1.p1.6.m6.2.2.3.2" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.2.cmml">Hom</mi><mrow id="S2.Thmtheorem1.p1.6.m6.2.2.3.3" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.3.cmml"><mi id="S2.Thmtheorem1.p1.6.m6.2.2.3.3.2" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.3.2.cmml">R</mi><mo id="S2.Thmtheorem1.p1.6.m6.2.2.3.3.1" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.6.m6.2.2.3.3.3" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.3.3.cmml">𝒞</mi></mrow></msub><mo id="S2.Thmtheorem1.p1.6.m6.2.2.2" xref="S2.Thmtheorem1.p1.6.m6.2.2.2.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.6.m6.2.2.1.1" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.2.cmml"><mo id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.2" stretchy="false" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.2.cmml">(</mo><msub id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.2" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.2.cmml">P</mi><mo id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.3" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.3.cmml">∗</mo></msub><mo id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.3" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.2.cmml">,</mo><mi id="S2.Thmtheorem1.p1.6.m6.1.1" xref="S2.Thmtheorem1.p1.6.m6.1.1.cmml">M</mi><mo id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.4" stretchy="false" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.6.m6.2b"><apply id="S2.Thmtheorem1.p1.6.m6.2.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2"><times id="S2.Thmtheorem1.p1.6.m6.2.2.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.2"></times><apply id="S2.Thmtheorem1.p1.6.m6.2.2.3.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.6.m6.2.2.3.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.3">subscript</csymbol><ci id="S2.Thmtheorem1.p1.6.m6.2.2.3.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.2">Hom</ci><apply id="S2.Thmtheorem1.p1.6.m6.2.2.3.3.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.3"><times id="S2.Thmtheorem1.p1.6.m6.2.2.3.3.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.3.1"></times><ci id="S2.Thmtheorem1.p1.6.m6.2.2.3.3.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.3.2">𝑅</ci><ci id="S2.Thmtheorem1.p1.6.m6.2.2.3.3.3.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.3.3.3">𝒞</ci></apply></apply><interval closure="open" id="S2.Thmtheorem1.p1.6.m6.2.2.1.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.1"><apply id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.2">𝑃</ci><times id="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.6.m6.2.2.1.1.1.3"></times></apply><ci id="S2.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1">𝑀</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.6.m6.2c">\mathrm{Hom}_{R\mathcal{C}}(P_{*},M)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.6.m6.2d">roman_Hom start_POSTSUBSCRIPT italic_R caligraphic_C end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , italic_M )</annotation></semantics></math> where <math alttext="P_{*}\to N" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.7.m7.1"><semantics id="S2.Thmtheorem1.p1.7.m7.1a"><mrow id="S2.Thmtheorem1.p1.7.m7.1.1" xref="S2.Thmtheorem1.p1.7.m7.1.1.cmml"><msub id="S2.Thmtheorem1.p1.7.m7.1.1.2" xref="S2.Thmtheorem1.p1.7.m7.1.1.2.cmml"><mi id="S2.Thmtheorem1.p1.7.m7.1.1.2.2" xref="S2.Thmtheorem1.p1.7.m7.1.1.2.2.cmml">P</mi><mo id="S2.Thmtheorem1.p1.7.m7.1.1.2.3" xref="S2.Thmtheorem1.p1.7.m7.1.1.2.3.cmml">∗</mo></msub><mo id="S2.Thmtheorem1.p1.7.m7.1.1.1" stretchy="false" xref="S2.Thmtheorem1.p1.7.m7.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem1.p1.7.m7.1.1.3" xref="S2.Thmtheorem1.p1.7.m7.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.7.m7.1b"><apply id="S2.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1"><ci id="S2.Thmtheorem1.p1.7.m7.1.1.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.1">→</ci><apply id="S2.Thmtheorem1.p1.7.m7.1.1.2.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.7.m7.1.1.2.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem1.p1.7.m7.1.1.2.2.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.2.2">𝑃</ci><times id="S2.Thmtheorem1.p1.7.m7.1.1.2.3.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.2.3"></times></apply><ci id="S2.Thmtheorem1.p1.7.m7.1.1.3.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.7.m7.1c">P_{*}\to N</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.7.m7.1d">italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → italic_N</annotation></semantics></math> is any projective resolution of <math alttext="N" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.8.m8.1"><semantics id="S2.Thmtheorem1.p1.8.m8.1a"><mi id="S2.Thmtheorem1.p1.8.m8.1.1" xref="S2.Thmtheorem1.p1.8.m8.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.8.m8.1b"><ci id="S2.Thmtheorem1.p1.8.m8.1.1.cmml" xref="S2.Thmtheorem1.p1.8.m8.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.8.m8.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.8.m8.1d">italic_N</annotation></semantics></math> as an <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.9.m9.1"><semantics id="S2.Thmtheorem1.p1.9.m9.1a"><mrow id="S2.Thmtheorem1.p1.9.m9.1.1" xref="S2.Thmtheorem1.p1.9.m9.1.1.cmml"><mi id="S2.Thmtheorem1.p1.9.m9.1.1.2" xref="S2.Thmtheorem1.p1.9.m9.1.1.2.cmml">R</mi><mo id="S2.Thmtheorem1.p1.9.m9.1.1.1" xref="S2.Thmtheorem1.p1.9.m9.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.9.m9.1.1.3" xref="S2.Thmtheorem1.p1.9.m9.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.9.m9.1b"><apply id="S2.Thmtheorem1.p1.9.m9.1.1.cmml" xref="S2.Thmtheorem1.p1.9.m9.1.1"><times id="S2.Thmtheorem1.p1.9.m9.1.1.1.cmml" xref="S2.Thmtheorem1.p1.9.m9.1.1.1"></times><ci id="S2.Thmtheorem1.p1.9.m9.1.1.2.cmml" xref="S2.Thmtheorem1.p1.9.m9.1.1.2">𝑅</ci><ci id="S2.Thmtheorem1.p1.9.m9.1.1.3.cmml" xref="S2.Thmtheorem1.p1.9.m9.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.9.m9.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.9.m9.1d">italic_R caligraphic_C</annotation></semantics></math>-module.</p> </div> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.13">Instead of covariant functors <math alttext="M:\mathcal{C}\to R" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mrow id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><mi id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml">M</mi><mo id="S2.p3.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.p3.1.m1.1.1.1.cmml">:</mo><mrow id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p3.1.m1.1.1.3.2" xref="S2.p3.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="S2.p3.1.m1.1.1.3.1" stretchy="false" xref="S2.p3.1.m1.1.1.3.1.cmml">→</mo><mi id="S2.p3.1.m1.1.1.3.3" xref="S2.p3.1.m1.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><ci id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1.1">:</ci><ci id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2">𝑀</ci><apply id="S2.p3.1.m1.1.1.3.cmml" xref="S2.p3.1.m1.1.1.3"><ci id="S2.p3.1.m1.1.1.3.1.cmml" xref="S2.p3.1.m1.1.1.3.1">→</ci><ci id="S2.p3.1.m1.1.1.3.2.cmml" xref="S2.p3.1.m1.1.1.3.2">𝒞</ci><ci id="S2.p3.1.m1.1.1.3.3.cmml" xref="S2.p3.1.m1.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">M:\mathcal{C}\to R</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_M : caligraphic_C → italic_R</annotation></semantics></math>-Mod, one can consider the module category of contravariant functors <math alttext="M:\mathcal{C}^{op}\to R" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><mrow id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml"><mi id="S2.p3.2.m2.1.1.2" xref="S2.p3.2.m2.1.1.2.cmml">M</mi><mo id="S2.p3.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.p3.2.m2.1.1.1.cmml">:</mo><mrow id="S2.p3.2.m2.1.1.3" xref="S2.p3.2.m2.1.1.3.cmml"><msup id="S2.p3.2.m2.1.1.3.2" xref="S2.p3.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p3.2.m2.1.1.3.2.2" xref="S2.p3.2.m2.1.1.3.2.2.cmml">𝒞</mi><mrow id="S2.p3.2.m2.1.1.3.2.3" xref="S2.p3.2.m2.1.1.3.2.3.cmml"><mi id="S2.p3.2.m2.1.1.3.2.3.2" xref="S2.p3.2.m2.1.1.3.2.3.2.cmml">o</mi><mo id="S2.p3.2.m2.1.1.3.2.3.1" xref="S2.p3.2.m2.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.p3.2.m2.1.1.3.2.3.3" xref="S2.p3.2.m2.1.1.3.2.3.3.cmml">p</mi></mrow></msup><mo id="S2.p3.2.m2.1.1.3.1" stretchy="false" xref="S2.p3.2.m2.1.1.3.1.cmml">→</mo><mi id="S2.p3.2.m2.1.1.3.3" xref="S2.p3.2.m2.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1"><ci id="S2.p3.2.m2.1.1.1.cmml" xref="S2.p3.2.m2.1.1.1">:</ci><ci id="S2.p3.2.m2.1.1.2.cmml" xref="S2.p3.2.m2.1.1.2">𝑀</ci><apply id="S2.p3.2.m2.1.1.3.cmml" xref="S2.p3.2.m2.1.1.3"><ci id="S2.p3.2.m2.1.1.3.1.cmml" xref="S2.p3.2.m2.1.1.3.1">→</ci><apply id="S2.p3.2.m2.1.1.3.2.cmml" xref="S2.p3.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.1.3.2.1.cmml" xref="S2.p3.2.m2.1.1.3.2">superscript</csymbol><ci id="S2.p3.2.m2.1.1.3.2.2.cmml" xref="S2.p3.2.m2.1.1.3.2.2">𝒞</ci><apply id="S2.p3.2.m2.1.1.3.2.3.cmml" xref="S2.p3.2.m2.1.1.3.2.3"><times id="S2.p3.2.m2.1.1.3.2.3.1.cmml" xref="S2.p3.2.m2.1.1.3.2.3.1"></times><ci id="S2.p3.2.m2.1.1.3.2.3.2.cmml" xref="S2.p3.2.m2.1.1.3.2.3.2">𝑜</ci><ci id="S2.p3.2.m2.1.1.3.2.3.3.cmml" xref="S2.p3.2.m2.1.1.3.2.3.3">𝑝</ci></apply></apply><ci id="S2.p3.2.m2.1.1.3.3.cmml" xref="S2.p3.2.m2.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">M:\mathcal{C}^{op}\to R</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_M : caligraphic_C start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → italic_R</annotation></semantics></math>-Mod. We call such a module a <em class="ltx_emph ltx_font_italic" id="S2.p3.3.1">contravariant <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p3.3.1.m1.1"><semantics id="S2.p3.3.1.m1.1a"><mrow id="S2.p3.3.1.m1.1.1" xref="S2.p3.3.1.m1.1.1.cmml"><mi id="S2.p3.3.1.m1.1.1.2" xref="S2.p3.3.1.m1.1.1.2.cmml">R</mi><mo id="S2.p3.3.1.m1.1.1.1" xref="S2.p3.3.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.3.1.m1.1.1.3" xref="S2.p3.3.1.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.3.1.m1.1b"><apply id="S2.p3.3.1.m1.1.1.cmml" xref="S2.p3.3.1.m1.1.1"><times id="S2.p3.3.1.m1.1.1.1.cmml" xref="S2.p3.3.1.m1.1.1.1"></times><ci id="S2.p3.3.1.m1.1.1.2.cmml" xref="S2.p3.3.1.m1.1.1.2">𝑅</ci><ci id="S2.p3.3.1.m1.1.1.3.cmml" xref="S2.p3.3.1.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.1.m1.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.1.m1.1d">italic_R caligraphic_C</annotation></semantics></math>-module</em> (or a right <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p3.4.m3.1"><semantics id="S2.p3.4.m3.1a"><mrow id="S2.p3.4.m3.1.1" xref="S2.p3.4.m3.1.1.cmml"><mi id="S2.p3.4.m3.1.1.2" xref="S2.p3.4.m3.1.1.2.cmml">R</mi><mo id="S2.p3.4.m3.1.1.1" xref="S2.p3.4.m3.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.4.m3.1.1.3" xref="S2.p3.4.m3.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.4.m3.1b"><apply id="S2.p3.4.m3.1.1.cmml" xref="S2.p3.4.m3.1.1"><times id="S2.p3.4.m3.1.1.1.cmml" xref="S2.p3.4.m3.1.1.1"></times><ci id="S2.p3.4.m3.1.1.2.cmml" xref="S2.p3.4.m3.1.1.2">𝑅</ci><ci id="S2.p3.4.m3.1.1.3.cmml" xref="S2.p3.4.m3.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m3.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m3.1d">italic_R caligraphic_C</annotation></semantics></math>-module). The contravariant <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p3.5.m4.1"><semantics id="S2.p3.5.m4.1a"><mrow id="S2.p3.5.m4.1.1" xref="S2.p3.5.m4.1.1.cmml"><mi id="S2.p3.5.m4.1.1.2" xref="S2.p3.5.m4.1.1.2.cmml">R</mi><mo id="S2.p3.5.m4.1.1.1" xref="S2.p3.5.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.5.m4.1.1.3" xref="S2.p3.5.m4.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.5.m4.1b"><apply id="S2.p3.5.m4.1.1.cmml" xref="S2.p3.5.m4.1.1"><times id="S2.p3.5.m4.1.1.1.cmml" xref="S2.p3.5.m4.1.1.1"></times><ci id="S2.p3.5.m4.1.1.2.cmml" xref="S2.p3.5.m4.1.1.2">𝑅</ci><ci id="S2.p3.5.m4.1.1.3.cmml" xref="S2.p3.5.m4.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m4.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m4.1d">italic_R caligraphic_C</annotation></semantics></math>-modules defined by <math alttext="R\mathrm{Mor}_{\mathcal{C}}(?,x)" class="ltx_Math" display="inline" id="S2.p3.6.m5.2"><semantics id="S2.p3.6.m5.2a"><mrow id="S2.p3.6.m5.2.3" xref="S2.p3.6.m5.2.3.cmml"><mi id="S2.p3.6.m5.2.3.2" xref="S2.p3.6.m5.2.3.2.cmml">R</mi><mo id="S2.p3.6.m5.2.3.1" xref="S2.p3.6.m5.2.3.1.cmml">⁢</mo><msub id="S2.p3.6.m5.2.3.3" xref="S2.p3.6.m5.2.3.3.cmml"><mi id="S2.p3.6.m5.2.3.3.2" xref="S2.p3.6.m5.2.3.3.2.cmml">Mor</mi><mi class="ltx_font_mathcaligraphic" id="S2.p3.6.m5.2.3.3.3" xref="S2.p3.6.m5.2.3.3.3.cmml">𝒞</mi></msub><mo id="S2.p3.6.m5.2.3.1a" xref="S2.p3.6.m5.2.3.1.cmml">⁢</mo><mrow id="S2.p3.6.m5.2.3.4.2" xref="S2.p3.6.m5.2.3.4.1.cmml"><mo id="S2.p3.6.m5.2.3.4.2.1" stretchy="false" xref="S2.p3.6.m5.2.3.4.1.cmml">(</mo><mi id="S2.p3.6.m5.1.1" mathvariant="normal" xref="S2.p3.6.m5.1.1.cmml">?</mi><mo id="S2.p3.6.m5.2.3.4.2.2" xref="S2.p3.6.m5.2.3.4.1.cmml">,</mo><mi id="S2.p3.6.m5.2.2" xref="S2.p3.6.m5.2.2.cmml">x</mi><mo id="S2.p3.6.m5.2.3.4.2.3" stretchy="false" xref="S2.p3.6.m5.2.3.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.6.m5.2b"><apply id="S2.p3.6.m5.2.3.cmml" xref="S2.p3.6.m5.2.3"><times id="S2.p3.6.m5.2.3.1.cmml" xref="S2.p3.6.m5.2.3.1"></times><ci id="S2.p3.6.m5.2.3.2.cmml" xref="S2.p3.6.m5.2.3.2">𝑅</ci><apply id="S2.p3.6.m5.2.3.3.cmml" xref="S2.p3.6.m5.2.3.3"><csymbol cd="ambiguous" id="S2.p3.6.m5.2.3.3.1.cmml" xref="S2.p3.6.m5.2.3.3">subscript</csymbol><ci id="S2.p3.6.m5.2.3.3.2.cmml" xref="S2.p3.6.m5.2.3.3.2">Mor</ci><ci id="S2.p3.6.m5.2.3.3.3.cmml" xref="S2.p3.6.m5.2.3.3.3">𝒞</ci></apply><interval closure="open" id="S2.p3.6.m5.2.3.4.1.cmml" xref="S2.p3.6.m5.2.3.4.2"><ci id="S2.p3.6.m5.1.1.cmml" xref="S2.p3.6.m5.1.1">?</ci><ci id="S2.p3.6.m5.2.2.cmml" xref="S2.p3.6.m5.2.2">𝑥</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.6.m5.2c">R\mathrm{Mor}_{\mathcal{C}}(?,x)</annotation><annotation encoding="application/x-llamapun" id="S2.p3.6.m5.2d">italic_R roman_Mor start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( ? , italic_x )</annotation></semantics></math> for <math alttext="x\in\mathcal{C}" class="ltx_Math" display="inline" id="S2.p3.7.m6.1"><semantics id="S2.p3.7.m6.1a"><mrow id="S2.p3.7.m6.1.1" xref="S2.p3.7.m6.1.1.cmml"><mi id="S2.p3.7.m6.1.1.2" xref="S2.p3.7.m6.1.1.2.cmml">x</mi><mo id="S2.p3.7.m6.1.1.1" xref="S2.p3.7.m6.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.7.m6.1.1.3" xref="S2.p3.7.m6.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.7.m6.1b"><apply id="S2.p3.7.m6.1.1.cmml" xref="S2.p3.7.m6.1.1"><in id="S2.p3.7.m6.1.1.1.cmml" xref="S2.p3.7.m6.1.1.1"></in><ci id="S2.p3.7.m6.1.1.2.cmml" xref="S2.p3.7.m6.1.1.2">𝑥</ci><ci id="S2.p3.7.m6.1.1.3.cmml" xref="S2.p3.7.m6.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.7.m6.1c">x\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.7.m6.1d">italic_x ∈ caligraphic_C</annotation></semantics></math> are projective <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p3.8.m7.1"><semantics id="S2.p3.8.m7.1a"><mrow id="S2.p3.8.m7.1.1" xref="S2.p3.8.m7.1.1.cmml"><mi id="S2.p3.8.m7.1.1.2" xref="S2.p3.8.m7.1.1.2.cmml">R</mi><mo id="S2.p3.8.m7.1.1.1" xref="S2.p3.8.m7.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.8.m7.1.1.3" xref="S2.p3.8.m7.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.8.m7.1b"><apply id="S2.p3.8.m7.1.1.cmml" xref="S2.p3.8.m7.1.1"><times id="S2.p3.8.m7.1.1.1.cmml" xref="S2.p3.8.m7.1.1.1"></times><ci id="S2.p3.8.m7.1.1.2.cmml" xref="S2.p3.8.m7.1.1.2">𝑅</ci><ci id="S2.p3.8.m7.1.1.3.cmml" xref="S2.p3.8.m7.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.8.m7.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.8.m7.1d">italic_R caligraphic_C</annotation></semantics></math>-modules, and the ext-groups <math alttext="\mathrm{Ext}^{*}_{R\mathcal{C}}(M,N)" class="ltx_Math" display="inline" id="S2.p3.9.m8.2"><semantics id="S2.p3.9.m8.2a"><mrow id="S2.p3.9.m8.2.3" xref="S2.p3.9.m8.2.3.cmml"><msubsup id="S2.p3.9.m8.2.3.2" xref="S2.p3.9.m8.2.3.2.cmml"><mi id="S2.p3.9.m8.2.3.2.2.2" xref="S2.p3.9.m8.2.3.2.2.2.cmml">Ext</mi><mrow id="S2.p3.9.m8.2.3.2.3" xref="S2.p3.9.m8.2.3.2.3.cmml"><mi id="S2.p3.9.m8.2.3.2.3.2" xref="S2.p3.9.m8.2.3.2.3.2.cmml">R</mi><mo id="S2.p3.9.m8.2.3.2.3.1" xref="S2.p3.9.m8.2.3.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.9.m8.2.3.2.3.3" xref="S2.p3.9.m8.2.3.2.3.3.cmml">𝒞</mi></mrow><mo id="S2.p3.9.m8.2.3.2.2.3" xref="S2.p3.9.m8.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S2.p3.9.m8.2.3.1" xref="S2.p3.9.m8.2.3.1.cmml">⁢</mo><mrow id="S2.p3.9.m8.2.3.3.2" xref="S2.p3.9.m8.2.3.3.1.cmml"><mo id="S2.p3.9.m8.2.3.3.2.1" stretchy="false" xref="S2.p3.9.m8.2.3.3.1.cmml">(</mo><mi id="S2.p3.9.m8.1.1" xref="S2.p3.9.m8.1.1.cmml">M</mi><mo id="S2.p3.9.m8.2.3.3.2.2" xref="S2.p3.9.m8.2.3.3.1.cmml">,</mo><mi id="S2.p3.9.m8.2.2" xref="S2.p3.9.m8.2.2.cmml">N</mi><mo id="S2.p3.9.m8.2.3.3.2.3" stretchy="false" xref="S2.p3.9.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.9.m8.2b"><apply id="S2.p3.9.m8.2.3.cmml" xref="S2.p3.9.m8.2.3"><times id="S2.p3.9.m8.2.3.1.cmml" xref="S2.p3.9.m8.2.3.1"></times><apply id="S2.p3.9.m8.2.3.2.cmml" xref="S2.p3.9.m8.2.3.2"><csymbol cd="ambiguous" id="S2.p3.9.m8.2.3.2.1.cmml" xref="S2.p3.9.m8.2.3.2">subscript</csymbol><apply id="S2.p3.9.m8.2.3.2.2.cmml" xref="S2.p3.9.m8.2.3.2"><csymbol cd="ambiguous" id="S2.p3.9.m8.2.3.2.2.1.cmml" xref="S2.p3.9.m8.2.3.2">superscript</csymbol><ci id="S2.p3.9.m8.2.3.2.2.2.cmml" xref="S2.p3.9.m8.2.3.2.2.2">Ext</ci><times id="S2.p3.9.m8.2.3.2.2.3.cmml" xref="S2.p3.9.m8.2.3.2.2.3"></times></apply><apply id="S2.p3.9.m8.2.3.2.3.cmml" xref="S2.p3.9.m8.2.3.2.3"><times id="S2.p3.9.m8.2.3.2.3.1.cmml" xref="S2.p3.9.m8.2.3.2.3.1"></times><ci id="S2.p3.9.m8.2.3.2.3.2.cmml" xref="S2.p3.9.m8.2.3.2.3.2">𝑅</ci><ci id="S2.p3.9.m8.2.3.2.3.3.cmml" xref="S2.p3.9.m8.2.3.2.3.3">𝒞</ci></apply></apply><interval closure="open" id="S2.p3.9.m8.2.3.3.1.cmml" xref="S2.p3.9.m8.2.3.3.2"><ci id="S2.p3.9.m8.1.1.cmml" xref="S2.p3.9.m8.1.1">𝑀</ci><ci id="S2.p3.9.m8.2.2.cmml" xref="S2.p3.9.m8.2.2">𝑁</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.9.m8.2c">\mathrm{Ext}^{*}_{R\mathcal{C}}(M,N)</annotation><annotation encoding="application/x-llamapun" id="S2.p3.9.m8.2d">roman_Ext start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_R caligraphic_C end_POSTSUBSCRIPT ( italic_M , italic_N )</annotation></semantics></math> for contravariant <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p3.10.m9.1"><semantics id="S2.p3.10.m9.1a"><mrow id="S2.p3.10.m9.1.1" xref="S2.p3.10.m9.1.1.cmml"><mi id="S2.p3.10.m9.1.1.2" xref="S2.p3.10.m9.1.1.2.cmml">R</mi><mo id="S2.p3.10.m9.1.1.1" xref="S2.p3.10.m9.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.10.m9.1.1.3" xref="S2.p3.10.m9.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.10.m9.1b"><apply id="S2.p3.10.m9.1.1.cmml" xref="S2.p3.10.m9.1.1"><times id="S2.p3.10.m9.1.1.1.cmml" xref="S2.p3.10.m9.1.1.1"></times><ci id="S2.p3.10.m9.1.1.2.cmml" xref="S2.p3.10.m9.1.1.2">𝑅</ci><ci id="S2.p3.10.m9.1.1.3.cmml" xref="S2.p3.10.m9.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.10.m9.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.10.m9.1d">italic_R caligraphic_C</annotation></semantics></math>-modules <math alttext="M" class="ltx_Math" display="inline" id="S2.p3.11.m10.1"><semantics id="S2.p3.11.m10.1a"><mi id="S2.p3.11.m10.1.1" xref="S2.p3.11.m10.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.p3.11.m10.1b"><ci id="S2.p3.11.m10.1.1.cmml" xref="S2.p3.11.m10.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.11.m10.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.p3.11.m10.1d">italic_M</annotation></semantics></math> and <math alttext="N" class="ltx_Math" display="inline" id="S2.p3.12.m11.1"><semantics id="S2.p3.12.m11.1a"><mi id="S2.p3.12.m11.1.1" xref="S2.p3.12.m11.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.p3.12.m11.1b"><ci id="S2.p3.12.m11.1.1.cmml" xref="S2.p3.12.m11.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.12.m11.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.p3.12.m11.1d">italic_N</annotation></semantics></math> are defined in a similar way using contravariant projective <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.p3.13.m12.1"><semantics id="S2.p3.13.m12.1a"><mrow id="S2.p3.13.m12.1.1" xref="S2.p3.13.m12.1.1.cmml"><mi id="S2.p3.13.m12.1.1.2" xref="S2.p3.13.m12.1.1.2.cmml">R</mi><mo id="S2.p3.13.m12.1.1.1" xref="S2.p3.13.m12.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.13.m12.1.1.3" xref="S2.p3.13.m12.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.13.m12.1b"><apply id="S2.p3.13.m12.1.1.cmml" xref="S2.p3.13.m12.1.1"><times id="S2.p3.13.m12.1.1.1.cmml" xref="S2.p3.13.m12.1.1.1"></times><ci id="S2.p3.13.m12.1.1.2.cmml" xref="S2.p3.13.m12.1.1.2">𝑅</ci><ci id="S2.p3.13.m12.1.1.3.cmml" xref="S2.p3.13.m12.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.13.m12.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.13.m12.1d">italic_R caligraphic_C</annotation></semantics></math>-modules.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1. </span>Cohomology of small categories</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.9">For a small category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">caligraphic_C</annotation></semantics></math> and a commutative ring <math alttext="R" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><ci id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_R</annotation></semantics></math>, the constant functor <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><munder accentunder="true" id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml"><mi id="S2.SS1.p1.3.m3.1.1.2" xref="S2.SS1.p1.3.m3.1.1.2.cmml">R</mi><mo id="S2.SS1.p1.3.m3.1.1.1" xref="S2.SS1.p1.3.m3.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><apply id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1"><ci id="S2.SS1.p1.3.m3.1.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1.1">¯</ci><ci id="S2.SS1.p1.3.m3.1.1.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> is the <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mrow id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">R</mi><mo id="S2.SS1.p1.4.m4.1.1.1" xref="S2.SS1.p1.4.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><times id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1.1"></times><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝑅</ci><ci id="S2.SS1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_R caligraphic_C</annotation></semantics></math>-module such that for each <math alttext="c\in\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><mrow id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml"><mi id="S2.SS1.p1.5.m5.1.1.2" xref="S2.SS1.p1.5.m5.1.1.2.cmml">c</mi><mo id="S2.SS1.p1.5.m5.1.1.1" xref="S2.SS1.p1.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.5.m5.1.1.3" xref="S2.SS1.p1.5.m5.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><apply id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1"><in id="S2.SS1.p1.5.m5.1.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1.1"></in><ci id="S2.SS1.p1.5.m5.1.1.2.cmml" xref="S2.SS1.p1.5.m5.1.1.2">𝑐</ci><ci id="S2.SS1.p1.5.m5.1.1.3.cmml" xref="S2.SS1.p1.5.m5.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">c\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">italic_c ∈ caligraphic_C</annotation></semantics></math>, <math alttext="\underline{R}(c)=R" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.1"><semantics id="S2.SS1.p1.6.m6.1a"><mrow id="S2.SS1.p1.6.m6.1.2" xref="S2.SS1.p1.6.m6.1.2.cmml"><mrow id="S2.SS1.p1.6.m6.1.2.2" xref="S2.SS1.p1.6.m6.1.2.2.cmml"><munder accentunder="true" id="S2.SS1.p1.6.m6.1.2.2.2" xref="S2.SS1.p1.6.m6.1.2.2.2.cmml"><mi id="S2.SS1.p1.6.m6.1.2.2.2.2" xref="S2.SS1.p1.6.m6.1.2.2.2.2.cmml">R</mi><mo id="S2.SS1.p1.6.m6.1.2.2.2.1" xref="S2.SS1.p1.6.m6.1.2.2.2.1.cmml">¯</mo></munder><mo id="S2.SS1.p1.6.m6.1.2.2.1" xref="S2.SS1.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.6.m6.1.2.2.3.2" xref="S2.SS1.p1.6.m6.1.2.2.cmml"><mo id="S2.SS1.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.p1.6.m6.1.2.2.cmml">(</mo><mi id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml">c</mi><mo id="S2.SS1.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S2.SS1.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p1.6.m6.1.2.1" xref="S2.SS1.p1.6.m6.1.2.1.cmml">=</mo><mi id="S2.SS1.p1.6.m6.1.2.3" xref="S2.SS1.p1.6.m6.1.2.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.1b"><apply id="S2.SS1.p1.6.m6.1.2.cmml" xref="S2.SS1.p1.6.m6.1.2"><eq id="S2.SS1.p1.6.m6.1.2.1.cmml" xref="S2.SS1.p1.6.m6.1.2.1"></eq><apply id="S2.SS1.p1.6.m6.1.2.2.cmml" xref="S2.SS1.p1.6.m6.1.2.2"><times id="S2.SS1.p1.6.m6.1.2.2.1.cmml" xref="S2.SS1.p1.6.m6.1.2.2.1"></times><apply id="S2.SS1.p1.6.m6.1.2.2.2.cmml" xref="S2.SS1.p1.6.m6.1.2.2.2"><ci id="S2.SS1.p1.6.m6.1.2.2.2.1.cmml" xref="S2.SS1.p1.6.m6.1.2.2.2.1">¯</ci><ci id="S2.SS1.p1.6.m6.1.2.2.2.2.cmml" xref="S2.SS1.p1.6.m6.1.2.2.2.2">𝑅</ci></apply><ci id="S2.SS1.p1.6.m6.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1">𝑐</ci></apply><ci id="S2.SS1.p1.6.m6.1.2.3.cmml" xref="S2.SS1.p1.6.m6.1.2.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.1c">\underline{R}(c)=R</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.1d">under¯ start_ARG italic_R end_ARG ( italic_c ) = italic_R</annotation></semantics></math>, and for every morphism <math alttext="\varphi:c\to c^{\prime}" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><mrow id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml"><mi id="S2.SS1.p1.7.m7.1.1.2" xref="S2.SS1.p1.7.m7.1.1.2.cmml">φ</mi><mo id="S2.SS1.p1.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p1.7.m7.1.1.1.cmml">:</mo><mrow id="S2.SS1.p1.7.m7.1.1.3" xref="S2.SS1.p1.7.m7.1.1.3.cmml"><mi id="S2.SS1.p1.7.m7.1.1.3.2" xref="S2.SS1.p1.7.m7.1.1.3.2.cmml">c</mi><mo id="S2.SS1.p1.7.m7.1.1.3.1" stretchy="false" xref="S2.SS1.p1.7.m7.1.1.3.1.cmml">→</mo><msup id="S2.SS1.p1.7.m7.1.1.3.3" xref="S2.SS1.p1.7.m7.1.1.3.3.cmml"><mi id="S2.SS1.p1.7.m7.1.1.3.3.2" xref="S2.SS1.p1.7.m7.1.1.3.3.2.cmml">c</mi><mo id="S2.SS1.p1.7.m7.1.1.3.3.3" xref="S2.SS1.p1.7.m7.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><apply id="S2.SS1.p1.7.m7.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1"><ci id="S2.SS1.p1.7.m7.1.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1.1">:</ci><ci id="S2.SS1.p1.7.m7.1.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.2">𝜑</ci><apply id="S2.SS1.p1.7.m7.1.1.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3"><ci id="S2.SS1.p1.7.m7.1.1.3.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1">→</ci><ci id="S2.SS1.p1.7.m7.1.1.3.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2">𝑐</ci><apply id="S2.SS1.p1.7.m7.1.1.3.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.3.3.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.3">superscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.3.3.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.3.2">𝑐</ci><ci id="S2.SS1.p1.7.m7.1.1.3.3.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">\varphi:c\to c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">italic_φ : italic_c → italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.1"><semantics id="S2.SS1.p1.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.1b"><ci id="S2.SS1.p1.8.m8.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.1d">caligraphic_C</annotation></semantics></math>, the induced map <math alttext="\underline{R}(\varphi):R\to R" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m9.1"><semantics id="S2.SS1.p1.9.m9.1a"><mrow id="S2.SS1.p1.9.m9.1.2" xref="S2.SS1.p1.9.m9.1.2.cmml"><mrow id="S2.SS1.p1.9.m9.1.2.2" xref="S2.SS1.p1.9.m9.1.2.2.cmml"><munder accentunder="true" id="S2.SS1.p1.9.m9.1.2.2.2" xref="S2.SS1.p1.9.m9.1.2.2.2.cmml"><mi id="S2.SS1.p1.9.m9.1.2.2.2.2" xref="S2.SS1.p1.9.m9.1.2.2.2.2.cmml">R</mi><mo id="S2.SS1.p1.9.m9.1.2.2.2.1" xref="S2.SS1.p1.9.m9.1.2.2.2.1.cmml">¯</mo></munder><mo id="S2.SS1.p1.9.m9.1.2.2.1" xref="S2.SS1.p1.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.9.m9.1.2.2.3.2" xref="S2.SS1.p1.9.m9.1.2.2.cmml"><mo id="S2.SS1.p1.9.m9.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.p1.9.m9.1.2.2.cmml">(</mo><mi id="S2.SS1.p1.9.m9.1.1" xref="S2.SS1.p1.9.m9.1.1.cmml">φ</mi><mo id="S2.SS1.p1.9.m9.1.2.2.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS1.p1.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p1.9.m9.1.2.1" rspace="0.278em" xref="S2.SS1.p1.9.m9.1.2.1.cmml">:</mo><mrow id="S2.SS1.p1.9.m9.1.2.3" xref="S2.SS1.p1.9.m9.1.2.3.cmml"><mi id="S2.SS1.p1.9.m9.1.2.3.2" xref="S2.SS1.p1.9.m9.1.2.3.2.cmml">R</mi><mo id="S2.SS1.p1.9.m9.1.2.3.1" stretchy="false" xref="S2.SS1.p1.9.m9.1.2.3.1.cmml">→</mo><mi id="S2.SS1.p1.9.m9.1.2.3.3" xref="S2.SS1.p1.9.m9.1.2.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m9.1b"><apply id="S2.SS1.p1.9.m9.1.2.cmml" xref="S2.SS1.p1.9.m9.1.2"><ci id="S2.SS1.p1.9.m9.1.2.1.cmml" xref="S2.SS1.p1.9.m9.1.2.1">:</ci><apply id="S2.SS1.p1.9.m9.1.2.2.cmml" xref="S2.SS1.p1.9.m9.1.2.2"><times id="S2.SS1.p1.9.m9.1.2.2.1.cmml" xref="S2.SS1.p1.9.m9.1.2.2.1"></times><apply id="S2.SS1.p1.9.m9.1.2.2.2.cmml" xref="S2.SS1.p1.9.m9.1.2.2.2"><ci id="S2.SS1.p1.9.m9.1.2.2.2.1.cmml" xref="S2.SS1.p1.9.m9.1.2.2.2.1">¯</ci><ci id="S2.SS1.p1.9.m9.1.2.2.2.2.cmml" xref="S2.SS1.p1.9.m9.1.2.2.2.2">𝑅</ci></apply><ci id="S2.SS1.p1.9.m9.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1">𝜑</ci></apply><apply id="S2.SS1.p1.9.m9.1.2.3.cmml" xref="S2.SS1.p1.9.m9.1.2.3"><ci id="S2.SS1.p1.9.m9.1.2.3.1.cmml" xref="S2.SS1.p1.9.m9.1.2.3.1">→</ci><ci id="S2.SS1.p1.9.m9.1.2.3.2.cmml" xref="S2.SS1.p1.9.m9.1.2.3.2">𝑅</ci><ci id="S2.SS1.p1.9.m9.1.2.3.3.cmml" xref="S2.SS1.p1.9.m9.1.2.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m9.1c">\underline{R}(\varphi):R\to R</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m9.1d">under¯ start_ARG italic_R end_ARG ( italic_φ ) : italic_R → italic_R</annotation></semantics></math> is the identity map.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.1.1.1">Definition 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem2.p1"> <p class="ltx_p" id="S2.Thmtheorem2.p1.3">The <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem2.p1.3.1">(Quillen) cohomology of a small category</em> <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.1.m1.1"><semantics id="S2.Thmtheorem2.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem2.p1.1.m1.1.1" xref="S2.Thmtheorem2.p1.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.1.m1.1b"><ci id="S2.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.1.m1.1d">caligraphic_C</annotation></semantics></math> with coefficients in an <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.2.m2.1"><semantics id="S2.Thmtheorem2.p1.2.m2.1a"><mrow id="S2.Thmtheorem2.p1.2.m2.1.1" xref="S2.Thmtheorem2.p1.2.m2.1.1.cmml"><mi id="S2.Thmtheorem2.p1.2.m2.1.1.2" xref="S2.Thmtheorem2.p1.2.m2.1.1.2.cmml">R</mi><mo id="S2.Thmtheorem2.p1.2.m2.1.1.1" xref="S2.Thmtheorem2.p1.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem2.p1.2.m2.1.1.3" xref="S2.Thmtheorem2.p1.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.2.m2.1b"><apply id="S2.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem2.p1.2.m2.1.1"><times id="S2.Thmtheorem2.p1.2.m2.1.1.1.cmml" xref="S2.Thmtheorem2.p1.2.m2.1.1.1"></times><ci id="S2.Thmtheorem2.p1.2.m2.1.1.2.cmml" xref="S2.Thmtheorem2.p1.2.m2.1.1.2">𝑅</ci><ci id="S2.Thmtheorem2.p1.2.m2.1.1.3.cmml" xref="S2.Thmtheorem2.p1.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.2.m2.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.2.m2.1d">italic_R caligraphic_C</annotation></semantics></math>-module <math alttext="M" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.3.m3.1"><semantics id="S2.Thmtheorem2.p1.3.m3.1a"><mi id="S2.Thmtheorem2.p1.3.m3.1.1" xref="S2.Thmtheorem2.p1.3.m3.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.3.m3.1b"><ci id="S2.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem2.p1.3.m3.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.3.m3.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.3.m3.1d">italic_M</annotation></semantics></math> is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{*}(\mathcal{C};M):=\mathrm{Ext}^{*}_{R\mathcal{C}}(\underline{R},M)." class="ltx_Math" display="block" id="S2.Ex9.m1.5"><semantics id="S2.Ex9.m1.5a"><mrow id="S2.Ex9.m1.5.5.1" xref="S2.Ex9.m1.5.5.1.1.cmml"><mrow id="S2.Ex9.m1.5.5.1.1" xref="S2.Ex9.m1.5.5.1.1.cmml"><mrow id="S2.Ex9.m1.5.5.1.1.2" xref="S2.Ex9.m1.5.5.1.1.2.cmml"><msup id="S2.Ex9.m1.5.5.1.1.2.2" xref="S2.Ex9.m1.5.5.1.1.2.2.cmml"><mi id="S2.Ex9.m1.5.5.1.1.2.2.2" xref="S2.Ex9.m1.5.5.1.1.2.2.2.cmml">H</mi><mo id="S2.Ex9.m1.5.5.1.1.2.2.3" xref="S2.Ex9.m1.5.5.1.1.2.2.3.cmml">∗</mo></msup><mo id="S2.Ex9.m1.5.5.1.1.2.1" xref="S2.Ex9.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex9.m1.5.5.1.1.2.3.2" xref="S2.Ex9.m1.5.5.1.1.2.3.1.cmml"><mo id="S2.Ex9.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S2.Ex9.m1.5.5.1.1.2.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex9.m1.1.1" xref="S2.Ex9.m1.1.1.cmml">𝒞</mi><mo id="S2.Ex9.m1.5.5.1.1.2.3.2.2" xref="S2.Ex9.m1.5.5.1.1.2.3.1.cmml">;</mo><mi id="S2.Ex9.m1.2.2" xref="S2.Ex9.m1.2.2.cmml">M</mi><mo id="S2.Ex9.m1.5.5.1.1.2.3.2.3" rspace="0.278em" stretchy="false" xref="S2.Ex9.m1.5.5.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex9.m1.5.5.1.1.1" rspace="0.278em" xref="S2.Ex9.m1.5.5.1.1.1.cmml">:=</mo><mrow id="S2.Ex9.m1.5.5.1.1.3" xref="S2.Ex9.m1.5.5.1.1.3.cmml"><msubsup id="S2.Ex9.m1.5.5.1.1.3.2" xref="S2.Ex9.m1.5.5.1.1.3.2.cmml"><mi id="S2.Ex9.m1.5.5.1.1.3.2.2.2" xref="S2.Ex9.m1.5.5.1.1.3.2.2.2.cmml">Ext</mi><mrow id="S2.Ex9.m1.5.5.1.1.3.2.3" xref="S2.Ex9.m1.5.5.1.1.3.2.3.cmml"><mi id="S2.Ex9.m1.5.5.1.1.3.2.3.2" xref="S2.Ex9.m1.5.5.1.1.3.2.3.2.cmml">R</mi><mo id="S2.Ex9.m1.5.5.1.1.3.2.3.1" xref="S2.Ex9.m1.5.5.1.1.3.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex9.m1.5.5.1.1.3.2.3.3" xref="S2.Ex9.m1.5.5.1.1.3.2.3.3.cmml">𝒞</mi></mrow><mo id="S2.Ex9.m1.5.5.1.1.3.2.2.3" xref="S2.Ex9.m1.5.5.1.1.3.2.2.3.cmml">∗</mo></msubsup><mo id="S2.Ex9.m1.5.5.1.1.3.1" xref="S2.Ex9.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex9.m1.5.5.1.1.3.3.2" xref="S2.Ex9.m1.5.5.1.1.3.3.1.cmml"><mo id="S2.Ex9.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S2.Ex9.m1.5.5.1.1.3.3.1.cmml">(</mo><munder accentunder="true" id="S2.Ex9.m1.3.3" xref="S2.Ex9.m1.3.3.cmml"><mi id="S2.Ex9.m1.3.3.2" xref="S2.Ex9.m1.3.3.2.cmml">R</mi><mo id="S2.Ex9.m1.3.3.1" xref="S2.Ex9.m1.3.3.1.cmml">¯</mo></munder><mo id="S2.Ex9.m1.5.5.1.1.3.3.2.2" xref="S2.Ex9.m1.5.5.1.1.3.3.1.cmml">,</mo><mi id="S2.Ex9.m1.4.4" xref="S2.Ex9.m1.4.4.cmml">M</mi><mo id="S2.Ex9.m1.5.5.1.1.3.3.2.3" stretchy="false" xref="S2.Ex9.m1.5.5.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex9.m1.5.5.1.2" lspace="0em" xref="S2.Ex9.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex9.m1.5b"><apply id="S2.Ex9.m1.5.5.1.1.cmml" xref="S2.Ex9.m1.5.5.1"><csymbol cd="latexml" id="S2.Ex9.m1.5.5.1.1.1.cmml" xref="S2.Ex9.m1.5.5.1.1.1">assign</csymbol><apply id="S2.Ex9.m1.5.5.1.1.2.cmml" xref="S2.Ex9.m1.5.5.1.1.2"><times id="S2.Ex9.m1.5.5.1.1.2.1.cmml" xref="S2.Ex9.m1.5.5.1.1.2.1"></times><apply id="S2.Ex9.m1.5.5.1.1.2.2.cmml" xref="S2.Ex9.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex9.m1.5.5.1.1.2.2.1.cmml" xref="S2.Ex9.m1.5.5.1.1.2.2">superscript</csymbol><ci id="S2.Ex9.m1.5.5.1.1.2.2.2.cmml" xref="S2.Ex9.m1.5.5.1.1.2.2.2">𝐻</ci><times id="S2.Ex9.m1.5.5.1.1.2.2.3.cmml" xref="S2.Ex9.m1.5.5.1.1.2.2.3"></times></apply><list id="S2.Ex9.m1.5.5.1.1.2.3.1.cmml" xref="S2.Ex9.m1.5.5.1.1.2.3.2"><ci id="S2.Ex9.m1.1.1.cmml" xref="S2.Ex9.m1.1.1">𝒞</ci><ci id="S2.Ex9.m1.2.2.cmml" xref="S2.Ex9.m1.2.2">𝑀</ci></list></apply><apply id="S2.Ex9.m1.5.5.1.1.3.cmml" xref="S2.Ex9.m1.5.5.1.1.3"><times id="S2.Ex9.m1.5.5.1.1.3.1.cmml" xref="S2.Ex9.m1.5.5.1.1.3.1"></times><apply id="S2.Ex9.m1.5.5.1.1.3.2.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2"><csymbol cd="ambiguous" id="S2.Ex9.m1.5.5.1.1.3.2.1.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2">subscript</csymbol><apply id="S2.Ex9.m1.5.5.1.1.3.2.2.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2"><csymbol cd="ambiguous" id="S2.Ex9.m1.5.5.1.1.3.2.2.1.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2">superscript</csymbol><ci id="S2.Ex9.m1.5.5.1.1.3.2.2.2.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2.2.2">Ext</ci><times id="S2.Ex9.m1.5.5.1.1.3.2.2.3.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2.2.3"></times></apply><apply id="S2.Ex9.m1.5.5.1.1.3.2.3.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2.3"><times id="S2.Ex9.m1.5.5.1.1.3.2.3.1.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2.3.1"></times><ci id="S2.Ex9.m1.5.5.1.1.3.2.3.2.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2.3.2">𝑅</ci><ci id="S2.Ex9.m1.5.5.1.1.3.2.3.3.cmml" xref="S2.Ex9.m1.5.5.1.1.3.2.3.3">𝒞</ci></apply></apply><interval closure="open" id="S2.Ex9.m1.5.5.1.1.3.3.1.cmml" xref="S2.Ex9.m1.5.5.1.1.3.3.2"><apply id="S2.Ex9.m1.3.3.cmml" xref="S2.Ex9.m1.3.3"><ci id="S2.Ex9.m1.3.3.1.cmml" xref="S2.Ex9.m1.3.3.1">¯</ci><ci id="S2.Ex9.m1.3.3.2.cmml" xref="S2.Ex9.m1.3.3.2">𝑅</ci></apply><ci id="S2.Ex9.m1.4.4.cmml" xref="S2.Ex9.m1.4.4">𝑀</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex9.m1.5c">H^{*}(\mathcal{C};M):=\mathrm{Ext}^{*}_{R\mathcal{C}}(\underline{R},M).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m1.5d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M ) := roman_Ext start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_R caligraphic_C end_POSTSUBSCRIPT ( under¯ start_ARG italic_R end_ARG , italic_M ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S2.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.1.1.1">Remark 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem3.p1"> <p class="ltx_p" id="S2.Thmtheorem3.p1.1">There is no standard terminology for the above definition of the cohomology of a small category. Throughout the paper we refer this cohomology definition the Quillen cohomology of a small category to distinguish it from the Baues-Wirsching and the Thomason cohomology of a small category.</p> </div> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.13">The nerve <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mrow id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml"><mi id="S2.SS1.p2.1.m1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.2.cmml">N</mi><mo id="S2.SS1.p2.1.m1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.1.m1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><apply id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1"><times id="S2.SS1.p2.1.m1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1"></times><ci id="S2.SS1.p2.1.m1.1.1.2.cmml" xref="S2.SS1.p2.1.m1.1.1.2">𝑁</ci><ci id="S2.SS1.p2.1.m1.1.1.3.cmml" xref="S2.SS1.p2.1.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">italic_N caligraphic_C</annotation></semantics></math> of the category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">caligraphic_C</annotation></semantics></math> is a simplicial set whose <math alttext="n" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mi id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><ci id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_n</annotation></semantics></math>-simplices, <math alttext="N\mathcal{C}_{n}" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mrow id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml"><mi id="S2.SS1.p2.4.m4.1.1.2" xref="S2.SS1.p2.4.m4.1.1.2.cmml">N</mi><mo id="S2.SS1.p2.4.m4.1.1.1" xref="S2.SS1.p2.4.m4.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p2.4.m4.1.1.3" xref="S2.SS1.p2.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.4.m4.1.1.3.2" xref="S2.SS1.p2.4.m4.1.1.3.2.cmml">𝒞</mi><mi id="S2.SS1.p2.4.m4.1.1.3.3" xref="S2.SS1.p2.4.m4.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><apply id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1"><times id="S2.SS1.p2.4.m4.1.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1.1"></times><ci id="S2.SS1.p2.4.m4.1.1.2.cmml" xref="S2.SS1.p2.4.m4.1.1.2">𝑁</ci><apply id="S2.SS1.p2.4.m4.1.1.3.cmml" xref="S2.SS1.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.1.1.3.1.cmml" xref="S2.SS1.p2.4.m4.1.1.3">subscript</csymbol><ci id="S2.SS1.p2.4.m4.1.1.3.2.cmml" xref="S2.SS1.p2.4.m4.1.1.3.2">𝒞</ci><ci id="S2.SS1.p2.4.m4.1.1.3.3.cmml" xref="S2.SS1.p2.4.m4.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">N\mathcal{C}_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, are given by the length <math alttext="n" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><mi id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><ci id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">italic_n</annotation></semantics></math> chains of composable morphisms <math alttext="c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n}" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m6.1"><semantics id="S2.SS1.p2.6.m6.1a"><mrow id="S2.SS1.p2.6.m6.1.1" xref="S2.SS1.p2.6.m6.1.1.cmml"><msub id="S2.SS1.p2.6.m6.1.1.2" xref="S2.SS1.p2.6.m6.1.1.2.cmml"><mi id="S2.SS1.p2.6.m6.1.1.2.2" xref="S2.SS1.p2.6.m6.1.1.2.2.cmml">c</mi><mn id="S2.SS1.p2.6.m6.1.1.2.3" 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xref="S2.SS1.p2.6.m6.1.1.3.2.3.1.3.3">𝑛</ci></apply></apply><apply id="S2.SS1.p2.6.m6.1.1.3.2.3.2.cmml" xref="S2.SS1.p2.6.m6.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.1.1.3.2.3.2.1.cmml" xref="S2.SS1.p2.6.m6.1.1.3.2.3.2">subscript</csymbol><ci id="S2.SS1.p2.6.m6.1.1.3.2.3.2.2.cmml" xref="S2.SS1.p2.6.m6.1.1.3.2.3.2.2">𝑐</ci><ci id="S2.SS1.p2.6.m6.1.1.3.2.3.2.3.cmml" xref="S2.SS1.p2.6.m6.1.1.3.2.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.1c">c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.1d">italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.1"><semantics id="S2.SS1.p2.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.7.m7.1.1" xref="S2.SS1.p2.7.m7.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m7.1b"><ci id="S2.SS1.p2.7.m7.1.1.cmml" xref="S2.SS1.p2.7.m7.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m7.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.1d">caligraphic_C</annotation></semantics></math>. For <math alttext="n=0" class="ltx_Math" display="inline" id="S2.SS1.p2.8.m8.1"><semantics id="S2.SS1.p2.8.m8.1a"><mrow id="S2.SS1.p2.8.m8.1.1" xref="S2.SS1.p2.8.m8.1.1.cmml"><mi id="S2.SS1.p2.8.m8.1.1.2" xref="S2.SS1.p2.8.m8.1.1.2.cmml">n</mi><mo id="S2.SS1.p2.8.m8.1.1.1" xref="S2.SS1.p2.8.m8.1.1.1.cmml">=</mo><mn id="S2.SS1.p2.8.m8.1.1.3" xref="S2.SS1.p2.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.8.m8.1b"><apply id="S2.SS1.p2.8.m8.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1"><eq id="S2.SS1.p2.8.m8.1.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1.1"></eq><ci id="S2.SS1.p2.8.m8.1.1.2.cmml" xref="S2.SS1.p2.8.m8.1.1.2">𝑛</ci><cn id="S2.SS1.p2.8.m8.1.1.3.cmml" type="integer" xref="S2.SS1.p2.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.8.m8.1c">n=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m8.1d">italic_n = 0</annotation></semantics></math>, we take <math alttext="NC_{0}" class="ltx_Math" display="inline" id="S2.SS1.p2.9.m9.1"><semantics id="S2.SS1.p2.9.m9.1a"><mrow id="S2.SS1.p2.9.m9.1.1" xref="S2.SS1.p2.9.m9.1.1.cmml"><mi id="S2.SS1.p2.9.m9.1.1.2" xref="S2.SS1.p2.9.m9.1.1.2.cmml">N</mi><mo id="S2.SS1.p2.9.m9.1.1.1" xref="S2.SS1.p2.9.m9.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p2.9.m9.1.1.3" xref="S2.SS1.p2.9.m9.1.1.3.cmml"><mi id="S2.SS1.p2.9.m9.1.1.3.2" xref="S2.SS1.p2.9.m9.1.1.3.2.cmml">C</mi><mn id="S2.SS1.p2.9.m9.1.1.3.3" xref="S2.SS1.p2.9.m9.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.9.m9.1b"><apply id="S2.SS1.p2.9.m9.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1"><times id="S2.SS1.p2.9.m9.1.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1.1"></times><ci id="S2.SS1.p2.9.m9.1.1.2.cmml" xref="S2.SS1.p2.9.m9.1.1.2">𝑁</ci><apply id="S2.SS1.p2.9.m9.1.1.3.cmml" xref="S2.SS1.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p2.9.m9.1.1.3.1.cmml" xref="S2.SS1.p2.9.m9.1.1.3">subscript</csymbol><ci id="S2.SS1.p2.9.m9.1.1.3.2.cmml" xref="S2.SS1.p2.9.m9.1.1.3.2">𝐶</ci><cn id="S2.SS1.p2.9.m9.1.1.3.3.cmml" type="integer" xref="S2.SS1.p2.9.m9.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.9.m9.1c">NC_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.9.m9.1d">italic_N italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> to be the set of objects in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p2.10.m10.1"><semantics id="S2.SS1.p2.10.m10.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.10.m10.1.1" xref="S2.SS1.p2.10.m10.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.10.m10.1b"><ci id="S2.SS1.p2.10.m10.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.10.m10.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.10.m10.1d">caligraphic_C</annotation></semantics></math>. For <math alttext="n\geq 1" class="ltx_Math" display="inline" id="S2.SS1.p2.11.m11.1"><semantics id="S2.SS1.p2.11.m11.1a"><mrow id="S2.SS1.p2.11.m11.1.1" xref="S2.SS1.p2.11.m11.1.1.cmml"><mi id="S2.SS1.p2.11.m11.1.1.2" xref="S2.SS1.p2.11.m11.1.1.2.cmml">n</mi><mo id="S2.SS1.p2.11.m11.1.1.1" xref="S2.SS1.p2.11.m11.1.1.1.cmml">≥</mo><mn id="S2.SS1.p2.11.m11.1.1.3" xref="S2.SS1.p2.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.11.m11.1b"><apply id="S2.SS1.p2.11.m11.1.1.cmml" xref="S2.SS1.p2.11.m11.1.1"><geq id="S2.SS1.p2.11.m11.1.1.1.cmml" xref="S2.SS1.p2.11.m11.1.1.1"></geq><ci id="S2.SS1.p2.11.m11.1.1.2.cmml" xref="S2.SS1.p2.11.m11.1.1.2">𝑛</ci><cn id="S2.SS1.p2.11.m11.1.1.3.cmml" type="integer" xref="S2.SS1.p2.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.11.m11.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.11.m11.1d">italic_n ≥ 1</annotation></semantics></math>, the boundary map <math alttext="d_{i}:N\mathcal{C}_{n}\to N\mathcal{C}_{n-1}" class="ltx_Math" display="inline" id="S2.SS1.p2.12.m12.1"><semantics id="S2.SS1.p2.12.m12.1a"><mrow id="S2.SS1.p2.12.m12.1.1" xref="S2.SS1.p2.12.m12.1.1.cmml"><msub id="S2.SS1.p2.12.m12.1.1.2" xref="S2.SS1.p2.12.m12.1.1.2.cmml"><mi id="S2.SS1.p2.12.m12.1.1.2.2" xref="S2.SS1.p2.12.m12.1.1.2.2.cmml">d</mi><mi id="S2.SS1.p2.12.m12.1.1.2.3" xref="S2.SS1.p2.12.m12.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p2.12.m12.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.12.m12.1.1.1.cmml">:</mo><mrow id="S2.SS1.p2.12.m12.1.1.3" xref="S2.SS1.p2.12.m12.1.1.3.cmml"><mrow id="S2.SS1.p2.12.m12.1.1.3.2" xref="S2.SS1.p2.12.m12.1.1.3.2.cmml"><mi id="S2.SS1.p2.12.m12.1.1.3.2.2" xref="S2.SS1.p2.12.m12.1.1.3.2.2.cmml">N</mi><mo id="S2.SS1.p2.12.m12.1.1.3.2.1" xref="S2.SS1.p2.12.m12.1.1.3.2.1.cmml">⁢</mo><msub id="S2.SS1.p2.12.m12.1.1.3.2.3" xref="S2.SS1.p2.12.m12.1.1.3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.12.m12.1.1.3.2.3.2" xref="S2.SS1.p2.12.m12.1.1.3.2.3.2.cmml">𝒞</mi><mi id="S2.SS1.p2.12.m12.1.1.3.2.3.3" xref="S2.SS1.p2.12.m12.1.1.3.2.3.3.cmml">n</mi></msub></mrow><mo id="S2.SS1.p2.12.m12.1.1.3.1" stretchy="false" xref="S2.SS1.p2.12.m12.1.1.3.1.cmml">→</mo><mrow id="S2.SS1.p2.12.m12.1.1.3.3" xref="S2.SS1.p2.12.m12.1.1.3.3.cmml"><mi id="S2.SS1.p2.12.m12.1.1.3.3.2" xref="S2.SS1.p2.12.m12.1.1.3.3.2.cmml">N</mi><mo id="S2.SS1.p2.12.m12.1.1.3.3.1" xref="S2.SS1.p2.12.m12.1.1.3.3.1.cmml">⁢</mo><msub id="S2.SS1.p2.12.m12.1.1.3.3.3" xref="S2.SS1.p2.12.m12.1.1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.12.m12.1.1.3.3.3.2" xref="S2.SS1.p2.12.m12.1.1.3.3.3.2.cmml">𝒞</mi><mrow id="S2.SS1.p2.12.m12.1.1.3.3.3.3" xref="S2.SS1.p2.12.m12.1.1.3.3.3.3.cmml"><mi id="S2.SS1.p2.12.m12.1.1.3.3.3.3.2" xref="S2.SS1.p2.12.m12.1.1.3.3.3.3.2.cmml">n</mi><mo id="S2.SS1.p2.12.m12.1.1.3.3.3.3.1" xref="S2.SS1.p2.12.m12.1.1.3.3.3.3.1.cmml">−</mo><mn id="S2.SS1.p2.12.m12.1.1.3.3.3.3.3" xref="S2.SS1.p2.12.m12.1.1.3.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.12.m12.1b"><apply id="S2.SS1.p2.12.m12.1.1.cmml" xref="S2.SS1.p2.12.m12.1.1"><ci id="S2.SS1.p2.12.m12.1.1.1.cmml" xref="S2.SS1.p2.12.m12.1.1.1">:</ci><apply id="S2.SS1.p2.12.m12.1.1.2.cmml" xref="S2.SS1.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.12.m12.1.1.2.1.cmml" xref="S2.SS1.p2.12.m12.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.12.m12.1.1.2.2.cmml" xref="S2.SS1.p2.12.m12.1.1.2.2">𝑑</ci><ci id="S2.SS1.p2.12.m12.1.1.2.3.cmml" xref="S2.SS1.p2.12.m12.1.1.2.3">𝑖</ci></apply><apply id="S2.SS1.p2.12.m12.1.1.3.cmml" xref="S2.SS1.p2.12.m12.1.1.3"><ci id="S2.SS1.p2.12.m12.1.1.3.1.cmml" xref="S2.SS1.p2.12.m12.1.1.3.1">→</ci><apply id="S2.SS1.p2.12.m12.1.1.3.2.cmml" xref="S2.SS1.p2.12.m12.1.1.3.2"><times id="S2.SS1.p2.12.m12.1.1.3.2.1.cmml" xref="S2.SS1.p2.12.m12.1.1.3.2.1"></times><ci id="S2.SS1.p2.12.m12.1.1.3.2.2.cmml" xref="S2.SS1.p2.12.m12.1.1.3.2.2">𝑁</ci><apply id="S2.SS1.p2.12.m12.1.1.3.2.3.cmml" xref="S2.SS1.p2.12.m12.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS1.p2.12.m12.1.1.3.2.3.1.cmml" xref="S2.SS1.p2.12.m12.1.1.3.2.3">subscript</csymbol><ci id="S2.SS1.p2.12.m12.1.1.3.2.3.2.cmml" xref="S2.SS1.p2.12.m12.1.1.3.2.3.2">𝒞</ci><ci id="S2.SS1.p2.12.m12.1.1.3.2.3.3.cmml" xref="S2.SS1.p2.12.m12.1.1.3.2.3.3">𝑛</ci></apply></apply><apply id="S2.SS1.p2.12.m12.1.1.3.3.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3"><times id="S2.SS1.p2.12.m12.1.1.3.3.1.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3.1"></times><ci id="S2.SS1.p2.12.m12.1.1.3.3.2.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3.2">𝑁</ci><apply id="S2.SS1.p2.12.m12.1.1.3.3.3.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p2.12.m12.1.1.3.3.3.1.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3.3">subscript</csymbol><ci id="S2.SS1.p2.12.m12.1.1.3.3.3.2.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3.3.2">𝒞</ci><apply id="S2.SS1.p2.12.m12.1.1.3.3.3.3.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3.3.3"><minus id="S2.SS1.p2.12.m12.1.1.3.3.3.3.1.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3.3.3.1"></minus><ci id="S2.SS1.p2.12.m12.1.1.3.3.3.3.2.cmml" xref="S2.SS1.p2.12.m12.1.1.3.3.3.3.2">𝑛</ci><cn id="S2.SS1.p2.12.m12.1.1.3.3.3.3.3.cmml" type="integer" xref="S2.SS1.p2.12.m12.1.1.3.3.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.12.m12.1c">d_{i}:N\mathcal{C}_{n}\to N\mathcal{C}_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.12.m12.1d">italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_N caligraphic_C start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="0\leq i\leq n" class="ltx_Math" display="inline" id="S2.SS1.p2.13.m13.1"><semantics id="S2.SS1.p2.13.m13.1a"><mrow id="S2.SS1.p2.13.m13.1.1" xref="S2.SS1.p2.13.m13.1.1.cmml"><mn id="S2.SS1.p2.13.m13.1.1.2" xref="S2.SS1.p2.13.m13.1.1.2.cmml">0</mn><mo id="S2.SS1.p2.13.m13.1.1.3" xref="S2.SS1.p2.13.m13.1.1.3.cmml">≤</mo><mi id="S2.SS1.p2.13.m13.1.1.4" xref="S2.SS1.p2.13.m13.1.1.4.cmml">i</mi><mo id="S2.SS1.p2.13.m13.1.1.5" xref="S2.SS1.p2.13.m13.1.1.5.cmml">≤</mo><mi id="S2.SS1.p2.13.m13.1.1.6" xref="S2.SS1.p2.13.m13.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.13.m13.1b"><apply id="S2.SS1.p2.13.m13.1.1.cmml" xref="S2.SS1.p2.13.m13.1.1"><and id="S2.SS1.p2.13.m13.1.1a.cmml" xref="S2.SS1.p2.13.m13.1.1"></and><apply id="S2.SS1.p2.13.m13.1.1b.cmml" xref="S2.SS1.p2.13.m13.1.1"><leq id="S2.SS1.p2.13.m13.1.1.3.cmml" xref="S2.SS1.p2.13.m13.1.1.3"></leq><cn id="S2.SS1.p2.13.m13.1.1.2.cmml" type="integer" xref="S2.SS1.p2.13.m13.1.1.2">0</cn><ci id="S2.SS1.p2.13.m13.1.1.4.cmml" xref="S2.SS1.p2.13.m13.1.1.4">𝑖</ci></apply><apply id="S2.SS1.p2.13.m13.1.1c.cmml" xref="S2.SS1.p2.13.m13.1.1"><leq id="S2.SS1.p2.13.m13.1.1.5.cmml" xref="S2.SS1.p2.13.m13.1.1.5"></leq><share href="https://arxiv.org/html/2503.14659v1#S2.SS1.p2.13.m13.1.1.4.cmml" id="S2.SS1.p2.13.m13.1.1d.cmml" xref="S2.SS1.p2.13.m13.1.1"></share><ci id="S2.SS1.p2.13.m13.1.1.6.cmml" xref="S2.SS1.p2.13.m13.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.13.m13.1c">0\leq i\leq n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.13.m13.1d">0 ≤ italic_i ≤ italic_n</annotation></semantics></math>, is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="d_{i}(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})=\begin{cases}c_% {1}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{2}}\,}\cdots\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n}&\text{ if }i=0\\ c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots c_{i-1}% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{i+1}\alpha_{i}}\,}c_{i+1}% \cdots\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n}&\text{ if % }0<i<n\\ c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{n-1}}\,}c_{n-1}&\text{ if }i=n\end{cases}" class="ltx_Math" display="block" id="S2.Ex10.m1.7"><semantics id="S2.Ex10.m1.7a"><mrow id="S2.Ex10.m1.7.7" xref="S2.Ex10.m1.7.7.cmml"><mrow id="S2.Ex10.m1.7.7.1" xref="S2.Ex10.m1.7.7.1.cmml"><msub id="S2.Ex10.m1.7.7.1.3" xref="S2.Ex10.m1.7.7.1.3.cmml"><mi id="S2.Ex10.m1.7.7.1.3.2" xref="S2.Ex10.m1.7.7.1.3.2.cmml">d</mi><mi id="S2.Ex10.m1.7.7.1.3.3" xref="S2.Ex10.m1.7.7.1.3.3.cmml">i</mi></msub><mo id="S2.Ex10.m1.7.7.1.2" xref="S2.Ex10.m1.7.7.1.2.cmml">⁢</mo><mrow id="S2.Ex10.m1.7.7.1.1.1" xref="S2.Ex10.m1.7.7.1.1.1.1.cmml"><mo id="S2.Ex10.m1.7.7.1.1.1.2" stretchy="false" xref="S2.Ex10.m1.7.7.1.1.1.1.cmml">(</mo><mrow id="S2.Ex10.m1.7.7.1.1.1.1" xref="S2.Ex10.m1.7.7.1.1.1.1.cmml"><msub id="S2.Ex10.m1.7.7.1.1.1.1.2" xref="S2.Ex10.m1.7.7.1.1.1.1.2.cmml"><mi id="S2.Ex10.m1.7.7.1.1.1.1.2.2" xref="S2.Ex10.m1.7.7.1.1.1.1.2.2.cmml">c</mi><mn id="S2.Ex10.m1.7.7.1.1.1.1.2.3" xref="S2.Ex10.m1.7.7.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Ex10.m1.7.7.1.1.1.1.1" lspace="0.167em" xref="S2.Ex10.m1.7.7.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.7.7.1.1.1.1.3" xref="S2.Ex10.m1.7.7.1.1.1.1.3.cmml"><mover id="S2.Ex10.m1.7.7.1.1.1.1.3.1" xref="S2.Ex10.m1.7.7.1.1.1.1.3.1.cmml"><mo id="S2.Ex10.m1.7.7.1.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex10.m1.7.7.1.1.1.1.3.1.2.cmml">⟶</mo><msub id="S2.Ex10.m1.7.7.1.1.1.1.3.1.3" xref="S2.Ex10.m1.7.7.1.1.1.1.3.1.3.cmml"><mi id="S2.Ex10.m1.7.7.1.1.1.1.3.1.3.2" xref="S2.Ex10.m1.7.7.1.1.1.1.3.1.3.2.cmml">α</mi><mn id="S2.Ex10.m1.7.7.1.1.1.1.3.1.3.3" xref="S2.Ex10.m1.7.7.1.1.1.1.3.1.3.3.cmml">1</mn></msub></mover><mrow id="S2.Ex10.m1.7.7.1.1.1.1.3.2" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.cmml"><mi id="S2.Ex10.m1.7.7.1.1.1.1.3.2.2" mathvariant="normal" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.2.cmml">⋯</mi><mo id="S2.Ex10.m1.7.7.1.1.1.1.3.2.1" lspace="0.337em" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.cmml"><mover id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.cmml"><mo id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.3" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.3.cmml"><mi id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.3.2" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.3.2.cmml">α</mi><mi id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.3.3" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.1.3.3.cmml">n</mi></msub></mover><msub id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.2" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.2.cmml"><mi id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.2.2" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.2.2.cmml">c</mi><mi id="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.2.3" xref="S2.Ex10.m1.7.7.1.1.1.1.3.2.3.2.3.cmml">n</mi></msub></mrow></mrow></mrow></mrow><mo id="S2.Ex10.m1.7.7.1.1.1.3" stretchy="false" xref="S2.Ex10.m1.7.7.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex10.m1.7.7.2" xref="S2.Ex10.m1.7.7.2.cmml">=</mo><mrow id="S2.Ex10.m1.6.6" xref="S2.Ex10.m1.7.7.3.1.cmml"><mo id="S2.Ex10.m1.6.6.7" xref="S2.Ex10.m1.7.7.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S2.Ex10.m1.6.6.6" rowspacing="0pt" xref="S2.Ex10.m1.7.7.3.1.cmml"><mtr id="S2.Ex10.m1.6.6.6a" xref="S2.Ex10.m1.7.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S2.Ex10.m1.6.6.6b" xref="S2.Ex10.m1.7.7.3.1.cmml"><mrow id="S2.Ex10.m1.1.1.1.1.1.1" xref="S2.Ex10.m1.1.1.1.1.1.1.cmml"><msub id="S2.Ex10.m1.1.1.1.1.1.1.2" xref="S2.Ex10.m1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex10.m1.1.1.1.1.1.1.2.2" xref="S2.Ex10.m1.1.1.1.1.1.1.2.2.cmml">c</mi><mn id="S2.Ex10.m1.1.1.1.1.1.1.2.3" xref="S2.Ex10.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.Ex10.m1.1.1.1.1.1.1.1" lspace="0.337em" xref="S2.Ex10.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.1.1.1.1.1.1.3" xref="S2.Ex10.m1.1.1.1.1.1.1.3.cmml"><mover id="S2.Ex10.m1.1.1.1.1.1.1.3.1" xref="S2.Ex10.m1.1.1.1.1.1.1.3.1.cmml"><mo id="S2.Ex10.m1.1.1.1.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex10.m1.1.1.1.1.1.1.3.1.2.cmml">⟶</mo><msub id="S2.Ex10.m1.1.1.1.1.1.1.3.1.3" xref="S2.Ex10.m1.1.1.1.1.1.1.3.1.3.cmml"><mi id="S2.Ex10.m1.1.1.1.1.1.1.3.1.3.2" xref="S2.Ex10.m1.1.1.1.1.1.1.3.1.3.2.cmml">α</mi><mn id="S2.Ex10.m1.1.1.1.1.1.1.3.1.3.3" xref="S2.Ex10.m1.1.1.1.1.1.1.3.1.3.3.cmml">2</mn></msub></mover><mrow id="S2.Ex10.m1.1.1.1.1.1.1.3.2" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S2.Ex10.m1.1.1.1.1.1.1.3.2.2" mathvariant="normal" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.2.cmml">⋯</mi><mo id="S2.Ex10.m1.1.1.1.1.1.1.3.2.1" lspace="0.337em" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.cmml"><mover id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.cmml"><mo id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.3" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.3.cmml"><mi id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.3.2" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.3.2.cmml">α</mi><mi id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.3.3" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.1.3.3.cmml">n</mi></msub></mover><msub id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.2" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.2.cmml"><mi id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.2.2" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.2.2.cmml">c</mi><mi id="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.2.3" xref="S2.Ex10.m1.1.1.1.1.1.1.3.2.3.2.3.cmml">n</mi></msub></mrow></mrow></mrow></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S2.Ex10.m1.6.6.6c" xref="S2.Ex10.m1.7.7.3.1.cmml"><mrow id="S2.Ex10.m1.2.2.2.2.2.1" xref="S2.Ex10.m1.2.2.2.2.2.1.cmml"><mrow id="S2.Ex10.m1.2.2.2.2.2.1.2" xref="S2.Ex10.m1.2.2.2.2.2.1.2.cmml"><mtext id="S2.Ex10.m1.2.2.2.2.2.1.2.2" xref="S2.Ex10.m1.2.2.2.2.2.1.2.2a.cmml"> if </mtext><mo id="S2.Ex10.m1.2.2.2.2.2.1.2.1" xref="S2.Ex10.m1.2.2.2.2.2.1.2.1.cmml">⁢</mo><mi id="S2.Ex10.m1.2.2.2.2.2.1.2.3" xref="S2.Ex10.m1.2.2.2.2.2.1.2.3.cmml">i</mi></mrow><mo id="S2.Ex10.m1.2.2.2.2.2.1.1" xref="S2.Ex10.m1.2.2.2.2.2.1.1.cmml">=</mo><mn id="S2.Ex10.m1.2.2.2.2.2.1.3" xref="S2.Ex10.m1.2.2.2.2.2.1.3.cmml">0</mn></mrow></mtd></mtr><mtr id="S2.Ex10.m1.6.6.6d" xref="S2.Ex10.m1.7.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S2.Ex10.m1.6.6.6e" xref="S2.Ex10.m1.7.7.3.1.cmml"><mrow id="S2.Ex10.m1.3.3.3.3.1.1" xref="S2.Ex10.m1.3.3.3.3.1.1.cmml"><msub id="S2.Ex10.m1.3.3.3.3.1.1.2" xref="S2.Ex10.m1.3.3.3.3.1.1.2.cmml"><mi id="S2.Ex10.m1.3.3.3.3.1.1.2.2" xref="S2.Ex10.m1.3.3.3.3.1.1.2.2.cmml">c</mi><mn id="S2.Ex10.m1.3.3.3.3.1.1.2.3" xref="S2.Ex10.m1.3.3.3.3.1.1.2.3.cmml">0</mn></msub><mo id="S2.Ex10.m1.3.3.3.3.1.1.1" lspace="0.337em" xref="S2.Ex10.m1.3.3.3.3.1.1.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.3.3.3.3.1.1.3" xref="S2.Ex10.m1.3.3.3.3.1.1.3.cmml"><mover id="S2.Ex10.m1.3.3.3.3.1.1.3.1" xref="S2.Ex10.m1.3.3.3.3.1.1.3.1.cmml"><mo id="S2.Ex10.m1.3.3.3.3.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex10.m1.3.3.3.3.1.1.3.1.2.cmml">⟶</mo><msub id="S2.Ex10.m1.3.3.3.3.1.1.3.1.3" xref="S2.Ex10.m1.3.3.3.3.1.1.3.1.3.cmml"><mi id="S2.Ex10.m1.3.3.3.3.1.1.3.1.3.2" xref="S2.Ex10.m1.3.3.3.3.1.1.3.1.3.2.cmml">α</mi><mn id="S2.Ex10.m1.3.3.3.3.1.1.3.1.3.3" xref="S2.Ex10.m1.3.3.3.3.1.1.3.1.3.3.cmml">1</mn></msub></mover><mrow id="S2.Ex10.m1.3.3.3.3.1.1.3.2" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.cmml"><mi id="S2.Ex10.m1.3.3.3.3.1.1.3.2.2" mathvariant="normal" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.2.cmml">⋯</mi><mo id="S2.Ex10.m1.3.3.3.3.1.1.3.2.1" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.1.cmml">⁢</mo><msub id="S2.Ex10.m1.3.3.3.3.1.1.3.2.3" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.cmml"><mi id="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.2" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.2.cmml">c</mi><mrow id="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.3" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.3.cmml"><mi id="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.3.2" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.3.2.cmml">i</mi><mo id="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.3.1" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.3.1.cmml">−</mo><mn id="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.3.3" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.Ex10.m1.3.3.3.3.1.1.3.2.1a" lspace="0.167em" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.3.3.3.3.1.1.3.2.4" xref="S2.Ex10.m1.3.3.3.3.1.1.3.2.4.cmml"><mover id="S2.Ex10.m1.3.3.3.3.1.1.3.2.4.1" 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xref="S2.Ex10.m1.6.6.6.6.2.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex10.m1.7c">d_{i}(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})=\begin{cases}c_% {1}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{2}}\,}\cdots\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n}&\text{ if }i=0\\ c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots c_{i-1}% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{i+1}\alpha_{i}}\,}c_{i+1}% \cdots\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n}&\text{ if % }0<i<n\\ c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{n-1}}\,}c_{n-1}&\text{ if }i=n\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex10.m1.7d">italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = { start_ROW start_CELL italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_CELL start_CELL if italic_i = 0 end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ italic_c start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_CELL start_CELL if 0 < italic_i < italic_n end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_CELL start_CELL if italic_i = italic_n end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.15">The degeneracy map <math alttext="s_{i}:N\mathcal{C}_{n}\to N\mathcal{C}_{n+1}" class="ltx_Math" display="inline" id="S2.SS1.p2.14.m1.1"><semantics id="S2.SS1.p2.14.m1.1a"><mrow id="S2.SS1.p2.14.m1.1.1" xref="S2.SS1.p2.14.m1.1.1.cmml"><msub id="S2.SS1.p2.14.m1.1.1.2" xref="S2.SS1.p2.14.m1.1.1.2.cmml"><mi id="S2.SS1.p2.14.m1.1.1.2.2" xref="S2.SS1.p2.14.m1.1.1.2.2.cmml">s</mi><mi id="S2.SS1.p2.14.m1.1.1.2.3" xref="S2.SS1.p2.14.m1.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p2.14.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.14.m1.1.1.1.cmml">:</mo><mrow id="S2.SS1.p2.14.m1.1.1.3" xref="S2.SS1.p2.14.m1.1.1.3.cmml"><mrow id="S2.SS1.p2.14.m1.1.1.3.2" xref="S2.SS1.p2.14.m1.1.1.3.2.cmml"><mi id="S2.SS1.p2.14.m1.1.1.3.2.2" xref="S2.SS1.p2.14.m1.1.1.3.2.2.cmml">N</mi><mo id="S2.SS1.p2.14.m1.1.1.3.2.1" xref="S2.SS1.p2.14.m1.1.1.3.2.1.cmml">⁢</mo><msub id="S2.SS1.p2.14.m1.1.1.3.2.3" xref="S2.SS1.p2.14.m1.1.1.3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.14.m1.1.1.3.2.3.2" xref="S2.SS1.p2.14.m1.1.1.3.2.3.2.cmml">𝒞</mi><mi id="S2.SS1.p2.14.m1.1.1.3.2.3.3" xref="S2.SS1.p2.14.m1.1.1.3.2.3.3.cmml">n</mi></msub></mrow><mo id="S2.SS1.p2.14.m1.1.1.3.1" stretchy="false" xref="S2.SS1.p2.14.m1.1.1.3.1.cmml">→</mo><mrow id="S2.SS1.p2.14.m1.1.1.3.3" xref="S2.SS1.p2.14.m1.1.1.3.3.cmml"><mi id="S2.SS1.p2.14.m1.1.1.3.3.2" xref="S2.SS1.p2.14.m1.1.1.3.3.2.cmml">N</mi><mo id="S2.SS1.p2.14.m1.1.1.3.3.1" xref="S2.SS1.p2.14.m1.1.1.3.3.1.cmml">⁢</mo><msub id="S2.SS1.p2.14.m1.1.1.3.3.3" xref="S2.SS1.p2.14.m1.1.1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.14.m1.1.1.3.3.3.2" xref="S2.SS1.p2.14.m1.1.1.3.3.3.2.cmml">𝒞</mi><mrow id="S2.SS1.p2.14.m1.1.1.3.3.3.3" xref="S2.SS1.p2.14.m1.1.1.3.3.3.3.cmml"><mi id="S2.SS1.p2.14.m1.1.1.3.3.3.3.2" xref="S2.SS1.p2.14.m1.1.1.3.3.3.3.2.cmml">n</mi><mo id="S2.SS1.p2.14.m1.1.1.3.3.3.3.1" xref="S2.SS1.p2.14.m1.1.1.3.3.3.3.1.cmml">+</mo><mn id="S2.SS1.p2.14.m1.1.1.3.3.3.3.3" xref="S2.SS1.p2.14.m1.1.1.3.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.14.m1.1b"><apply id="S2.SS1.p2.14.m1.1.1.cmml" xref="S2.SS1.p2.14.m1.1.1"><ci id="S2.SS1.p2.14.m1.1.1.1.cmml" xref="S2.SS1.p2.14.m1.1.1.1">:</ci><apply id="S2.SS1.p2.14.m1.1.1.2.cmml" xref="S2.SS1.p2.14.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.14.m1.1.1.2.1.cmml" xref="S2.SS1.p2.14.m1.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.14.m1.1.1.2.2.cmml" xref="S2.SS1.p2.14.m1.1.1.2.2">𝑠</ci><ci id="S2.SS1.p2.14.m1.1.1.2.3.cmml" xref="S2.SS1.p2.14.m1.1.1.2.3">𝑖</ci></apply><apply id="S2.SS1.p2.14.m1.1.1.3.cmml" xref="S2.SS1.p2.14.m1.1.1.3"><ci id="S2.SS1.p2.14.m1.1.1.3.1.cmml" xref="S2.SS1.p2.14.m1.1.1.3.1">→</ci><apply id="S2.SS1.p2.14.m1.1.1.3.2.cmml" xref="S2.SS1.p2.14.m1.1.1.3.2"><times id="S2.SS1.p2.14.m1.1.1.3.2.1.cmml" xref="S2.SS1.p2.14.m1.1.1.3.2.1"></times><ci id="S2.SS1.p2.14.m1.1.1.3.2.2.cmml" xref="S2.SS1.p2.14.m1.1.1.3.2.2">𝑁</ci><apply id="S2.SS1.p2.14.m1.1.1.3.2.3.cmml" xref="S2.SS1.p2.14.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS1.p2.14.m1.1.1.3.2.3.1.cmml" xref="S2.SS1.p2.14.m1.1.1.3.2.3">subscript</csymbol><ci id="S2.SS1.p2.14.m1.1.1.3.2.3.2.cmml" xref="S2.SS1.p2.14.m1.1.1.3.2.3.2">𝒞</ci><ci id="S2.SS1.p2.14.m1.1.1.3.2.3.3.cmml" xref="S2.SS1.p2.14.m1.1.1.3.2.3.3">𝑛</ci></apply></apply><apply id="S2.SS1.p2.14.m1.1.1.3.3.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3"><times id="S2.SS1.p2.14.m1.1.1.3.3.1.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3.1"></times><ci id="S2.SS1.p2.14.m1.1.1.3.3.2.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3.2">𝑁</ci><apply id="S2.SS1.p2.14.m1.1.1.3.3.3.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p2.14.m1.1.1.3.3.3.1.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3.3">subscript</csymbol><ci id="S2.SS1.p2.14.m1.1.1.3.3.3.2.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3.3.2">𝒞</ci><apply id="S2.SS1.p2.14.m1.1.1.3.3.3.3.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3.3.3"><plus id="S2.SS1.p2.14.m1.1.1.3.3.3.3.1.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3.3.3.1"></plus><ci id="S2.SS1.p2.14.m1.1.1.3.3.3.3.2.cmml" xref="S2.SS1.p2.14.m1.1.1.3.3.3.3.2">𝑛</ci><cn id="S2.SS1.p2.14.m1.1.1.3.3.3.3.3.cmml" type="integer" xref="S2.SS1.p2.14.m1.1.1.3.3.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.14.m1.1c">s_{i}:N\mathcal{C}_{n}\to N\mathcal{C}_{n+1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.14.m1.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_N caligraphic_C start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="0\leq i\leq n" class="ltx_Math" display="inline" id="S2.SS1.p2.15.m2.1"><semantics id="S2.SS1.p2.15.m2.1a"><mrow id="S2.SS1.p2.15.m2.1.1" xref="S2.SS1.p2.15.m2.1.1.cmml"><mn id="S2.SS1.p2.15.m2.1.1.2" xref="S2.SS1.p2.15.m2.1.1.2.cmml">0</mn><mo id="S2.SS1.p2.15.m2.1.1.3" xref="S2.SS1.p2.15.m2.1.1.3.cmml">≤</mo><mi id="S2.SS1.p2.15.m2.1.1.4" xref="S2.SS1.p2.15.m2.1.1.4.cmml">i</mi><mo id="S2.SS1.p2.15.m2.1.1.5" xref="S2.SS1.p2.15.m2.1.1.5.cmml">≤</mo><mi id="S2.SS1.p2.15.m2.1.1.6" xref="S2.SS1.p2.15.m2.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.15.m2.1b"><apply id="S2.SS1.p2.15.m2.1.1.cmml" xref="S2.SS1.p2.15.m2.1.1"><and id="S2.SS1.p2.15.m2.1.1a.cmml" xref="S2.SS1.p2.15.m2.1.1"></and><apply id="S2.SS1.p2.15.m2.1.1b.cmml" xref="S2.SS1.p2.15.m2.1.1"><leq id="S2.SS1.p2.15.m2.1.1.3.cmml" xref="S2.SS1.p2.15.m2.1.1.3"></leq><cn id="S2.SS1.p2.15.m2.1.1.2.cmml" type="integer" xref="S2.SS1.p2.15.m2.1.1.2">0</cn><ci id="S2.SS1.p2.15.m2.1.1.4.cmml" xref="S2.SS1.p2.15.m2.1.1.4">𝑖</ci></apply><apply id="S2.SS1.p2.15.m2.1.1c.cmml" xref="S2.SS1.p2.15.m2.1.1"><leq id="S2.SS1.p2.15.m2.1.1.5.cmml" xref="S2.SS1.p2.15.m2.1.1.5"></leq><share href="https://arxiv.org/html/2503.14659v1#S2.SS1.p2.15.m2.1.1.4.cmml" id="S2.SS1.p2.15.m2.1.1d.cmml" xref="S2.SS1.p2.15.m2.1.1"></share><ci id="S2.SS1.p2.15.m2.1.1.6.cmml" xref="S2.SS1.p2.15.m2.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.15.m2.1c">0\leq i\leq n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.15.m2.1d">0 ≤ italic_i ≤ italic_n</annotation></semantics></math>, is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{i}(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})=c_{0}\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots c_{i}\smash{\,\mathop{% \longrightarrow}\limits^{\mathrm{id}}\,}c_{i}\cdots\smash{\,\mathop{% \longrightarrow}\limits^{\alpha_{n}}\,}c_{n}." class="ltx_Math" display="block" id="S2.Ex11.m1.1"><semantics 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id="S2.Ex11.m1.1.1.1.1.3.3.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2"><times id="S2.Ex11.m1.1.1.1.1.3.3.2.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.1"></times><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.2">⋯</ci><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.3.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S2.Ex11.m1.1.1.1.1.3.3.2.3.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.3">subscript</csymbol><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.3.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.3.2">𝑐</ci><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.3.3.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.3.3">𝑖</ci></apply><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.4.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4"><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.4.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.1"><csymbol cd="ambiguous" id="S2.Ex11.m1.1.1.1.1.3.3.2.4.1.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.1">superscript</csymbol><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.1.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.1.2">⟶</ci><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.1.3.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.1.3">id</ci></apply><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2"><times id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.1"></times><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.2"><csymbol cd="ambiguous" id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.2.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.2">subscript</csymbol><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.2.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.2.2">𝑐</ci><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.2.3.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.2.3">𝑖</ci></apply><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.3.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.3">⋯</ci><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4"><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1"><csymbol cd="ambiguous" id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1">superscript</csymbol><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.2">⟶</ci><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.3.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.3"><csymbol cd="ambiguous" id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.3.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.3">subscript</csymbol><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.3.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.3.2">𝛼</ci><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.3.3.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.1.3.3">𝑛</ci></apply></apply><apply id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.2"><csymbol cd="ambiguous" id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.2.1.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.2">subscript</csymbol><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.2.2.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.2.2">𝑐</ci><ci id="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.2.3.cmml" xref="S2.Ex11.m1.1.1.1.1.3.3.2.4.2.4.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex11.m1.1c">s_{i}(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})=c_{0}\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots c_{i}\smash{\,\mathop{% \longrightarrow}\limits^{\mathrm{id}}\,}c_{i}\cdots\smash{\,\mathop{% \longrightarrow}\limits^{\alpha_{n}}\,}c_{n}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex11.m1.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT roman_id end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.1">For each <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml">n</mi><mo id="S2.SS1.p3.1.m1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.cmml">≥</mo><mn id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><geq id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1.1"></geq><ci id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2">𝑛</ci><cn id="S2.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_n ≥ 0</annotation></semantics></math>, let</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P_{n}=\bigoplus_{\sigma=(c_{0}\to\cdots\to c_{n})\in N\mathcal{C}_{n}}R\mathrm% {Mor}_{\mathcal{C}}(c_{n},?)," class="ltx_Math" display="block" id="S2.Ex12.m1.3"><semantics id="S2.Ex12.m1.3a"><mrow id="S2.Ex12.m1.3.3.1" xref="S2.Ex12.m1.3.3.1.1.cmml"><mrow id="S2.Ex12.m1.3.3.1.1" xref="S2.Ex12.m1.3.3.1.1.cmml"><msub id="S2.Ex12.m1.3.3.1.1.3" xref="S2.Ex12.m1.3.3.1.1.3.cmml"><mi id="S2.Ex12.m1.3.3.1.1.3.2" xref="S2.Ex12.m1.3.3.1.1.3.2.cmml">P</mi><mi id="S2.Ex12.m1.3.3.1.1.3.3" xref="S2.Ex12.m1.3.3.1.1.3.3.cmml">n</mi></msub><mo id="S2.Ex12.m1.3.3.1.1.2" rspace="0.111em" xref="S2.Ex12.m1.3.3.1.1.2.cmml">=</mo><mrow id="S2.Ex12.m1.3.3.1.1.1" xref="S2.Ex12.m1.3.3.1.1.1.cmml"><munder id="S2.Ex12.m1.3.3.1.1.1.2" xref="S2.Ex12.m1.3.3.1.1.1.2.cmml"><mo id="S2.Ex12.m1.3.3.1.1.1.2.2" movablelimits="false" xref="S2.Ex12.m1.3.3.1.1.1.2.2.cmml">⨁</mo><mrow id="S2.Ex12.m1.1.1.1" xref="S2.Ex12.m1.1.1.1.cmml"><mi id="S2.Ex12.m1.1.1.1.3" xref="S2.Ex12.m1.1.1.1.3.cmml">σ</mi><mo id="S2.Ex12.m1.1.1.1.4" xref="S2.Ex12.m1.1.1.1.4.cmml">=</mo><mrow id="S2.Ex12.m1.1.1.1.1.1" xref="S2.Ex12.m1.1.1.1.1.1.1.cmml"><mo id="S2.Ex12.m1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex12.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex12.m1.1.1.1.1.1.1" xref="S2.Ex12.m1.1.1.1.1.1.1.cmml"><msub id="S2.Ex12.m1.1.1.1.1.1.1.2" xref="S2.Ex12.m1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex12.m1.1.1.1.1.1.1.2.2" xref="S2.Ex12.m1.1.1.1.1.1.1.2.2.cmml">c</mi><mn id="S2.Ex12.m1.1.1.1.1.1.1.2.3" xref="S2.Ex12.m1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Ex12.m1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex12.m1.1.1.1.1.1.1.3.cmml">→</mo><mi id="S2.Ex12.m1.1.1.1.1.1.1.4" mathvariant="normal" xref="S2.Ex12.m1.1.1.1.1.1.1.4.cmml">⋯</mi><mo id="S2.Ex12.m1.1.1.1.1.1.1.5" stretchy="false" xref="S2.Ex12.m1.1.1.1.1.1.1.5.cmml">→</mo><msub id="S2.Ex12.m1.1.1.1.1.1.1.6" xref="S2.Ex12.m1.1.1.1.1.1.1.6.cmml"><mi id="S2.Ex12.m1.1.1.1.1.1.1.6.2" xref="S2.Ex12.m1.1.1.1.1.1.1.6.2.cmml">c</mi><mi id="S2.Ex12.m1.1.1.1.1.1.1.6.3" xref="S2.Ex12.m1.1.1.1.1.1.1.6.3.cmml">n</mi></msub></mrow><mo id="S2.Ex12.m1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex12.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex12.m1.1.1.1.5" xref="S2.Ex12.m1.1.1.1.5.cmml">∈</mo><mrow id="S2.Ex12.m1.1.1.1.6" xref="S2.Ex12.m1.1.1.1.6.cmml"><mi id="S2.Ex12.m1.1.1.1.6.2" xref="S2.Ex12.m1.1.1.1.6.2.cmml">N</mi><mo id="S2.Ex12.m1.1.1.1.6.1" xref="S2.Ex12.m1.1.1.1.6.1.cmml">⁢</mo><msub id="S2.Ex12.m1.1.1.1.6.3" xref="S2.Ex12.m1.1.1.1.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex12.m1.1.1.1.6.3.2" xref="S2.Ex12.m1.1.1.1.6.3.2.cmml">𝒞</mi><mi id="S2.Ex12.m1.1.1.1.6.3.3" xref="S2.Ex12.m1.1.1.1.6.3.3.cmml">n</mi></msub></mrow></mrow></munder><mrow id="S2.Ex12.m1.3.3.1.1.1.1" xref="S2.Ex12.m1.3.3.1.1.1.1.cmml"><mi id="S2.Ex12.m1.3.3.1.1.1.1.3" xref="S2.Ex12.m1.3.3.1.1.1.1.3.cmml">R</mi><mo id="S2.Ex12.m1.3.3.1.1.1.1.2" xref="S2.Ex12.m1.3.3.1.1.1.1.2.cmml">⁢</mo><msub id="S2.Ex12.m1.3.3.1.1.1.1.4" xref="S2.Ex12.m1.3.3.1.1.1.1.4.cmml"><mi id="S2.Ex12.m1.3.3.1.1.1.1.4.2" xref="S2.Ex12.m1.3.3.1.1.1.1.4.2.cmml">Mor</mi><mi class="ltx_font_mathcaligraphic" id="S2.Ex12.m1.3.3.1.1.1.1.4.3" xref="S2.Ex12.m1.3.3.1.1.1.1.4.3.cmml">𝒞</mi></msub><mo id="S2.Ex12.m1.3.3.1.1.1.1.2a" xref="S2.Ex12.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex12.m1.3.3.1.1.1.1.1.1" xref="S2.Ex12.m1.3.3.1.1.1.1.1.2.cmml"><mo id="S2.Ex12.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex12.m1.3.3.1.1.1.1.1.2.cmml">(</mo><msub id="S2.Ex12.m1.3.3.1.1.1.1.1.1.1" xref="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.2" xref="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.2.cmml">c</mi><mi id="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.3" xref="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.Ex12.m1.3.3.1.1.1.1.1.1.3" xref="S2.Ex12.m1.3.3.1.1.1.1.1.2.cmml">,</mo><mi id="S2.Ex12.m1.2.2" mathvariant="normal" xref="S2.Ex12.m1.2.2.cmml">?</mi><mo id="S2.Ex12.m1.3.3.1.1.1.1.1.1.4" stretchy="false" xref="S2.Ex12.m1.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.Ex12.m1.3.3.1.2" xref="S2.Ex12.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex12.m1.3b"><apply id="S2.Ex12.m1.3.3.1.1.cmml" xref="S2.Ex12.m1.3.3.1"><eq id="S2.Ex12.m1.3.3.1.1.2.cmml" xref="S2.Ex12.m1.3.3.1.1.2"></eq><apply id="S2.Ex12.m1.3.3.1.1.3.cmml" xref="S2.Ex12.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S2.Ex12.m1.3.3.1.1.3.1.cmml" xref="S2.Ex12.m1.3.3.1.1.3">subscript</csymbol><ci id="S2.Ex12.m1.3.3.1.1.3.2.cmml" xref="S2.Ex12.m1.3.3.1.1.3.2">𝑃</ci><ci id="S2.Ex12.m1.3.3.1.1.3.3.cmml" xref="S2.Ex12.m1.3.3.1.1.3.3">𝑛</ci></apply><apply id="S2.Ex12.m1.3.3.1.1.1.cmml" xref="S2.Ex12.m1.3.3.1.1.1"><apply id="S2.Ex12.m1.3.3.1.1.1.2.cmml" xref="S2.Ex12.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex12.m1.3.3.1.1.1.2.1.cmml" xref="S2.Ex12.m1.3.3.1.1.1.2">subscript</csymbol><csymbol cd="latexml" id="S2.Ex12.m1.3.3.1.1.1.2.2.cmml" xref="S2.Ex12.m1.3.3.1.1.1.2.2">direct-sum</csymbol><apply id="S2.Ex12.m1.1.1.1.cmml" xref="S2.Ex12.m1.1.1.1"><and id="S2.Ex12.m1.1.1.1a.cmml" xref="S2.Ex12.m1.1.1.1"></and><apply id="S2.Ex12.m1.1.1.1b.cmml" xref="S2.Ex12.m1.1.1.1"><eq id="S2.Ex12.m1.1.1.1.4.cmml" xref="S2.Ex12.m1.1.1.1.4"></eq><ci id="S2.Ex12.m1.1.1.1.3.cmml" xref="S2.Ex12.m1.1.1.1.3">𝜎</ci><apply id="S2.Ex12.m1.1.1.1.1.1.1.cmml" xref="S2.Ex12.m1.1.1.1.1.1"><and id="S2.Ex12.m1.1.1.1.1.1.1a.cmml" xref="S2.Ex12.m1.1.1.1.1.1"></and><apply id="S2.Ex12.m1.1.1.1.1.1.1b.cmml" xref="S2.Ex12.m1.1.1.1.1.1"><ci id="S2.Ex12.m1.1.1.1.1.1.1.3.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.3">→</ci><apply id="S2.Ex12.m1.1.1.1.1.1.1.2.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex12.m1.1.1.1.1.1.1.2.1.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.Ex12.m1.1.1.1.1.1.1.2.2.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.2.2">𝑐</ci><cn id="S2.Ex12.m1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.Ex12.m1.1.1.1.1.1.1.2.3">0</cn></apply><ci id="S2.Ex12.m1.1.1.1.1.1.1.4.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.4">⋯</ci></apply><apply id="S2.Ex12.m1.1.1.1.1.1.1c.cmml" xref="S2.Ex12.m1.1.1.1.1.1"><ci id="S2.Ex12.m1.1.1.1.1.1.1.5.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.5">→</ci><share href="https://arxiv.org/html/2503.14659v1#S2.Ex12.m1.1.1.1.1.1.1.4.cmml" id="S2.Ex12.m1.1.1.1.1.1.1d.cmml" xref="S2.Ex12.m1.1.1.1.1.1"></share><apply id="S2.Ex12.m1.1.1.1.1.1.1.6.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.6"><csymbol cd="ambiguous" id="S2.Ex12.m1.1.1.1.1.1.1.6.1.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.6">subscript</csymbol><ci id="S2.Ex12.m1.1.1.1.1.1.1.6.2.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.6.2">𝑐</ci><ci id="S2.Ex12.m1.1.1.1.1.1.1.6.3.cmml" xref="S2.Ex12.m1.1.1.1.1.1.1.6.3">𝑛</ci></apply></apply></apply></apply><apply id="S2.Ex12.m1.1.1.1c.cmml" xref="S2.Ex12.m1.1.1.1"><in id="S2.Ex12.m1.1.1.1.5.cmml" xref="S2.Ex12.m1.1.1.1.5"></in><share href="https://arxiv.org/html/2503.14659v1#S2.Ex12.m1.1.1.1.1.cmml" id="S2.Ex12.m1.1.1.1d.cmml" xref="S2.Ex12.m1.1.1.1"></share><apply id="S2.Ex12.m1.1.1.1.6.cmml" xref="S2.Ex12.m1.1.1.1.6"><times id="S2.Ex12.m1.1.1.1.6.1.cmml" xref="S2.Ex12.m1.1.1.1.6.1"></times><ci id="S2.Ex12.m1.1.1.1.6.2.cmml" xref="S2.Ex12.m1.1.1.1.6.2">𝑁</ci><apply id="S2.Ex12.m1.1.1.1.6.3.cmml" xref="S2.Ex12.m1.1.1.1.6.3"><csymbol cd="ambiguous" id="S2.Ex12.m1.1.1.1.6.3.1.cmml" xref="S2.Ex12.m1.1.1.1.6.3">subscript</csymbol><ci id="S2.Ex12.m1.1.1.1.6.3.2.cmml" xref="S2.Ex12.m1.1.1.1.6.3.2">𝒞</ci><ci id="S2.Ex12.m1.1.1.1.6.3.3.cmml" xref="S2.Ex12.m1.1.1.1.6.3.3">𝑛</ci></apply></apply></apply></apply></apply><apply id="S2.Ex12.m1.3.3.1.1.1.1.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1"><times id="S2.Ex12.m1.3.3.1.1.1.1.2.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.2"></times><ci id="S2.Ex12.m1.3.3.1.1.1.1.3.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.3">𝑅</ci><apply id="S2.Ex12.m1.3.3.1.1.1.1.4.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.Ex12.m1.3.3.1.1.1.1.4.1.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.4">subscript</csymbol><ci id="S2.Ex12.m1.3.3.1.1.1.1.4.2.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.4.2">Mor</ci><ci id="S2.Ex12.m1.3.3.1.1.1.1.4.3.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.4.3">𝒞</ci></apply><interval closure="open" id="S2.Ex12.m1.3.3.1.1.1.1.1.2.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.1.1"><apply id="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.1.1.1.3">𝑛</ci></apply><ci id="S2.Ex12.m1.2.2.cmml" xref="S2.Ex12.m1.2.2">?</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex12.m1.3c">P_{n}=\bigoplus_{\sigma=(c_{0}\to\cdots\to c_{n})\in N\mathcal{C}_{n}}R\mathrm% {Mor}_{\mathcal{C}}(c_{n},?),</annotation><annotation encoding="application/x-llamapun" id="S2.Ex12.m1.3d">italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT italic_σ = ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ⋯ → italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_R roman_Mor start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , ? ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.3">and for every <math alttext="c\in\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m1.1"><semantics id="S2.SS1.p3.2.m1.1a"><mrow id="S2.SS1.p3.2.m1.1.1" xref="S2.SS1.p3.2.m1.1.1.cmml"><mi id="S2.SS1.p3.2.m1.1.1.2" xref="S2.SS1.p3.2.m1.1.1.2.cmml">c</mi><mo id="S2.SS1.p3.2.m1.1.1.1" xref="S2.SS1.p3.2.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.2.m1.1.1.3" xref="S2.SS1.p3.2.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m1.1b"><apply id="S2.SS1.p3.2.m1.1.1.cmml" xref="S2.SS1.p3.2.m1.1.1"><in id="S2.SS1.p3.2.m1.1.1.1.cmml" xref="S2.SS1.p3.2.m1.1.1.1"></in><ci id="S2.SS1.p3.2.m1.1.1.2.cmml" xref="S2.SS1.p3.2.m1.1.1.2">𝑐</ci><ci id="S2.SS1.p3.2.m1.1.1.3.cmml" xref="S2.SS1.p3.2.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m1.1c">c\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m1.1d">italic_c ∈ caligraphic_C</annotation></semantics></math>, let <math alttext="\partial_{n}(c):P_{n}(c)\to P_{n-1}(c)" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m2.3"><semantics id="S2.SS1.p3.3.m2.3a"><mrow id="S2.SS1.p3.3.m2.3.4" xref="S2.SS1.p3.3.m2.3.4.cmml"><mrow id="S2.SS1.p3.3.m2.3.4.2" xref="S2.SS1.p3.3.m2.3.4.2.cmml"><msub id="S2.SS1.p3.3.m2.3.4.2.1" xref="S2.SS1.p3.3.m2.3.4.2.1.cmml"><mo id="S2.SS1.p3.3.m2.3.4.2.1.2" xref="S2.SS1.p3.3.m2.3.4.2.1.2.cmml">∂</mo><mi id="S2.SS1.p3.3.m2.3.4.2.1.3" xref="S2.SS1.p3.3.m2.3.4.2.1.3.cmml">n</mi></msub><mrow id="S2.SS1.p3.3.m2.3.4.2.2.2" xref="S2.SS1.p3.3.m2.3.4.2.cmml"><mo id="S2.SS1.p3.3.m2.3.4.2.2.2.1" lspace="0em" stretchy="false" xref="S2.SS1.p3.3.m2.3.4.2.cmml">(</mo><mi id="S2.SS1.p3.3.m2.1.1" xref="S2.SS1.p3.3.m2.1.1.cmml">c</mi><mo id="S2.SS1.p3.3.m2.3.4.2.2.2.2" rspace="0.278em" stretchy="false" xref="S2.SS1.p3.3.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.3.m2.3.4.1" rspace="0.278em" xref="S2.SS1.p3.3.m2.3.4.1.cmml">:</mo><mrow id="S2.SS1.p3.3.m2.3.4.3" xref="S2.SS1.p3.3.m2.3.4.3.cmml"><mrow id="S2.SS1.p3.3.m2.3.4.3.2" xref="S2.SS1.p3.3.m2.3.4.3.2.cmml"><msub id="S2.SS1.p3.3.m2.3.4.3.2.2" xref="S2.SS1.p3.3.m2.3.4.3.2.2.cmml"><mi id="S2.SS1.p3.3.m2.3.4.3.2.2.2" xref="S2.SS1.p3.3.m2.3.4.3.2.2.2.cmml">P</mi><mi id="S2.SS1.p3.3.m2.3.4.3.2.2.3" xref="S2.SS1.p3.3.m2.3.4.3.2.2.3.cmml">n</mi></msub><mo id="S2.SS1.p3.3.m2.3.4.3.2.1" xref="S2.SS1.p3.3.m2.3.4.3.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.3.m2.3.4.3.2.3.2" xref="S2.SS1.p3.3.m2.3.4.3.2.cmml"><mo id="S2.SS1.p3.3.m2.3.4.3.2.3.2.1" stretchy="false" xref="S2.SS1.p3.3.m2.3.4.3.2.cmml">(</mo><mi id="S2.SS1.p3.3.m2.2.2" xref="S2.SS1.p3.3.m2.2.2.cmml">c</mi><mo id="S2.SS1.p3.3.m2.3.4.3.2.3.2.2" stretchy="false" xref="S2.SS1.p3.3.m2.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.3.m2.3.4.3.1" stretchy="false" xref="S2.SS1.p3.3.m2.3.4.3.1.cmml">→</mo><mrow id="S2.SS1.p3.3.m2.3.4.3.3" xref="S2.SS1.p3.3.m2.3.4.3.3.cmml"><msub id="S2.SS1.p3.3.m2.3.4.3.3.2" xref="S2.SS1.p3.3.m2.3.4.3.3.2.cmml"><mi id="S2.SS1.p3.3.m2.3.4.3.3.2.2" xref="S2.SS1.p3.3.m2.3.4.3.3.2.2.cmml">P</mi><mrow id="S2.SS1.p3.3.m2.3.4.3.3.2.3" xref="S2.SS1.p3.3.m2.3.4.3.3.2.3.cmml"><mi id="S2.SS1.p3.3.m2.3.4.3.3.2.3.2" xref="S2.SS1.p3.3.m2.3.4.3.3.2.3.2.cmml">n</mi><mo id="S2.SS1.p3.3.m2.3.4.3.3.2.3.1" xref="S2.SS1.p3.3.m2.3.4.3.3.2.3.1.cmml">−</mo><mn id="S2.SS1.p3.3.m2.3.4.3.3.2.3.3" xref="S2.SS1.p3.3.m2.3.4.3.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p3.3.m2.3.4.3.3.1" xref="S2.SS1.p3.3.m2.3.4.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.3.m2.3.4.3.3.3.2" xref="S2.SS1.p3.3.m2.3.4.3.3.cmml"><mo id="S2.SS1.p3.3.m2.3.4.3.3.3.2.1" stretchy="false" xref="S2.SS1.p3.3.m2.3.4.3.3.cmml">(</mo><mi id="S2.SS1.p3.3.m2.3.3" xref="S2.SS1.p3.3.m2.3.3.cmml">c</mi><mo id="S2.SS1.p3.3.m2.3.4.3.3.3.2.2" stretchy="false" xref="S2.SS1.p3.3.m2.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m2.3b"><apply id="S2.SS1.p3.3.m2.3.4.cmml" xref="S2.SS1.p3.3.m2.3.4"><ci id="S2.SS1.p3.3.m2.3.4.1.cmml" xref="S2.SS1.p3.3.m2.3.4.1">:</ci><apply id="S2.SS1.p3.3.m2.3.4.2.cmml" xref="S2.SS1.p3.3.m2.3.4.2"><apply id="S2.SS1.p3.3.m2.3.4.2.1.cmml" xref="S2.SS1.p3.3.m2.3.4.2.1"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m2.3.4.2.1.1.cmml" xref="S2.SS1.p3.3.m2.3.4.2.1">subscript</csymbol><partialdiff id="S2.SS1.p3.3.m2.3.4.2.1.2.cmml" xref="S2.SS1.p3.3.m2.3.4.2.1.2"></partialdiff><ci id="S2.SS1.p3.3.m2.3.4.2.1.3.cmml" xref="S2.SS1.p3.3.m2.3.4.2.1.3">𝑛</ci></apply><ci id="S2.SS1.p3.3.m2.1.1.cmml" xref="S2.SS1.p3.3.m2.1.1">𝑐</ci></apply><apply id="S2.SS1.p3.3.m2.3.4.3.cmml" xref="S2.SS1.p3.3.m2.3.4.3"><ci id="S2.SS1.p3.3.m2.3.4.3.1.cmml" xref="S2.SS1.p3.3.m2.3.4.3.1">→</ci><apply id="S2.SS1.p3.3.m2.3.4.3.2.cmml" xref="S2.SS1.p3.3.m2.3.4.3.2"><times id="S2.SS1.p3.3.m2.3.4.3.2.1.cmml" xref="S2.SS1.p3.3.m2.3.4.3.2.1"></times><apply id="S2.SS1.p3.3.m2.3.4.3.2.2.cmml" xref="S2.SS1.p3.3.m2.3.4.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m2.3.4.3.2.2.1.cmml" xref="S2.SS1.p3.3.m2.3.4.3.2.2">subscript</csymbol><ci id="S2.SS1.p3.3.m2.3.4.3.2.2.2.cmml" xref="S2.SS1.p3.3.m2.3.4.3.2.2.2">𝑃</ci><ci id="S2.SS1.p3.3.m2.3.4.3.2.2.3.cmml" xref="S2.SS1.p3.3.m2.3.4.3.2.2.3">𝑛</ci></apply><ci id="S2.SS1.p3.3.m2.2.2.cmml" xref="S2.SS1.p3.3.m2.2.2">𝑐</ci></apply><apply id="S2.SS1.p3.3.m2.3.4.3.3.cmml" xref="S2.SS1.p3.3.m2.3.4.3.3"><times id="S2.SS1.p3.3.m2.3.4.3.3.1.cmml" xref="S2.SS1.p3.3.m2.3.4.3.3.1"></times><apply id="S2.SS1.p3.3.m2.3.4.3.3.2.cmml" xref="S2.SS1.p3.3.m2.3.4.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m2.3.4.3.3.2.1.cmml" xref="S2.SS1.p3.3.m2.3.4.3.3.2">subscript</csymbol><ci id="S2.SS1.p3.3.m2.3.4.3.3.2.2.cmml" xref="S2.SS1.p3.3.m2.3.4.3.3.2.2">𝑃</ci><apply id="S2.SS1.p3.3.m2.3.4.3.3.2.3.cmml" xref="S2.SS1.p3.3.m2.3.4.3.3.2.3"><minus id="S2.SS1.p3.3.m2.3.4.3.3.2.3.1.cmml" xref="S2.SS1.p3.3.m2.3.4.3.3.2.3.1"></minus><ci id="S2.SS1.p3.3.m2.3.4.3.3.2.3.2.cmml" xref="S2.SS1.p3.3.m2.3.4.3.3.2.3.2">𝑛</ci><cn id="S2.SS1.p3.3.m2.3.4.3.3.2.3.3.cmml" type="integer" xref="S2.SS1.p3.3.m2.3.4.3.3.2.3.3">1</cn></apply></apply><ci id="S2.SS1.p3.3.m2.3.3.cmml" xref="S2.SS1.p3.3.m2.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m2.3c">\partial_{n}(c):P_{n}(c)\to P_{n-1}(c)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m2.3d">∂ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_c ) : italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_c ) → italic_P start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math> be the map defined by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S7.EGx1"> <tbody id="S2.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\partial_{n}(\sigma,\alpha)=\sum_{i=0}^{n-1}(-1)^{i}(d_{i}\sigma,% \alpha)+(-1)^{n}(d_{n}\sigma,\alpha\circ\alpha_{n})" class="ltx_Math" display="inline" id="S2.Ex13.m1.8"><semantics id="S2.Ex13.m1.8a"><mrow id="S2.Ex13.m1.8.8" xref="S2.Ex13.m1.8.8.cmml"><mrow id="S2.Ex13.m1.8.8.7" xref="S2.Ex13.m1.8.8.7.cmml"><msub id="S2.Ex13.m1.8.8.7.1" xref="S2.Ex13.m1.8.8.7.1.cmml"><mo id="S2.Ex13.m1.8.8.7.1.2" xref="S2.Ex13.m1.8.8.7.1.2.cmml">∂</mo><mi id="S2.Ex13.m1.8.8.7.1.3" xref="S2.Ex13.m1.8.8.7.1.3.cmml">n</mi></msub><mrow id="S2.Ex13.m1.8.8.7.2.2" xref="S2.Ex13.m1.8.8.7.2.1.cmml"><mo id="S2.Ex13.m1.8.8.7.2.2.1" lspace="0em" stretchy="false" xref="S2.Ex13.m1.8.8.7.2.1.cmml">(</mo><mi id="S2.Ex13.m1.1.1" xref="S2.Ex13.m1.1.1.cmml">σ</mi><mo id="S2.Ex13.m1.8.8.7.2.2.2" xref="S2.Ex13.m1.8.8.7.2.1.cmml">,</mo><mi id="S2.Ex13.m1.2.2" xref="S2.Ex13.m1.2.2.cmml">α</mi><mo id="S2.Ex13.m1.8.8.7.2.2.3" stretchy="false" xref="S2.Ex13.m1.8.8.7.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex13.m1.8.8.6" xref="S2.Ex13.m1.8.8.6.cmml">=</mo><mrow id="S2.Ex13.m1.8.8.5" xref="S2.Ex13.m1.8.8.5.cmml"><mrow id="S2.Ex13.m1.5.5.2.2" xref="S2.Ex13.m1.5.5.2.2.cmml"><mstyle displaystyle="true" id="S2.Ex13.m1.5.5.2.2.3" xref="S2.Ex13.m1.5.5.2.2.3.cmml"><munderover id="S2.Ex13.m1.5.5.2.2.3a" xref="S2.Ex13.m1.5.5.2.2.3.cmml"><mo id="S2.Ex13.m1.5.5.2.2.3.2.2" movablelimits="false" xref="S2.Ex13.m1.5.5.2.2.3.2.2.cmml">∑</mo><mrow id="S2.Ex13.m1.5.5.2.2.3.2.3" 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xref="S2.Ex13.m1.5.5.2.2.2.2.1.1.2.3.cmml">i</mi></msub><mo id="S2.Ex13.m1.5.5.2.2.2.2.1.1.1" xref="S2.Ex13.m1.5.5.2.2.2.2.1.1.1.cmml">⁢</mo><mi id="S2.Ex13.m1.5.5.2.2.2.2.1.1.3" xref="S2.Ex13.m1.5.5.2.2.2.2.1.1.3.cmml">σ</mi></mrow><mo id="S2.Ex13.m1.5.5.2.2.2.2.1.3" xref="S2.Ex13.m1.5.5.2.2.2.2.2.cmml">,</mo><mi id="S2.Ex13.m1.3.3" xref="S2.Ex13.m1.3.3.cmml">α</mi><mo id="S2.Ex13.m1.5.5.2.2.2.2.1.4" stretchy="false" xref="S2.Ex13.m1.5.5.2.2.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex13.m1.8.8.5.6" xref="S2.Ex13.m1.8.8.5.6.cmml">+</mo><mrow id="S2.Ex13.m1.8.8.5.5" xref="S2.Ex13.m1.8.8.5.5.cmml"><msup id="S2.Ex13.m1.6.6.3.3.1" xref="S2.Ex13.m1.6.6.3.3.1.cmml"><mrow id="S2.Ex13.m1.6.6.3.3.1.1.1" xref="S2.Ex13.m1.6.6.3.3.1.1.1.1.cmml"><mo id="S2.Ex13.m1.6.6.3.3.1.1.1.2" stretchy="false" xref="S2.Ex13.m1.6.6.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.Ex13.m1.6.6.3.3.1.1.1.1" xref="S2.Ex13.m1.6.6.3.3.1.1.1.1.cmml"><mo id="S2.Ex13.m1.6.6.3.3.1.1.1.1a" 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end_POSTSUPERSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ , italic_α ) + ( - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_σ , italic_α ∘ italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.7">where <math alttext="\sigma=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m1.1"><semantics id="S2.SS1.p3.4.m1.1a"><mrow id="S2.SS1.p3.4.m1.1.1" xref="S2.SS1.p3.4.m1.1.1.cmml"><mi id="S2.SS1.p3.4.m1.1.1.3" xref="S2.SS1.p3.4.m1.1.1.3.cmml">σ</mi><mo id="S2.SS1.p3.4.m1.1.1.2" xref="S2.SS1.p3.4.m1.1.1.2.cmml">=</mo><mrow id="S2.SS1.p3.4.m1.1.1.1.1" 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id="S2.SS1.p3.4.m1.1.1.1.1.1.3.1.3.2.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.1.3.2">𝛼</ci><cn id="S2.SS1.p3.4.m1.1.1.1.1.1.3.1.3.3.cmml" type="integer" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.1.3.3">1</cn></apply></apply><apply id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2"><times id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.1.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.1"></times><ci id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.2.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.2">⋯</ci><apply id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3"><apply id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.1.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1">superscript</csymbol><ci id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.2.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.2">⟶</ci><apply id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.3.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.3.1.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.3">subscript</csymbol><ci id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.3.2.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.3.2">𝛼</ci><ci id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.3.3.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.1.3.3">𝑛</ci></apply></apply><apply id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.2.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.2.1.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.2">subscript</csymbol><ci id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.2.2.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.2.2">𝑐</ci><ci id="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.2.3.cmml" xref="S2.SS1.p3.4.m1.1.1.1.1.1.3.2.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m1.1c">\sigma=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m1.1d">italic_σ = ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> is a simplex in <math alttext="N\mathcal{C}_{n}" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m2.1"><semantics id="S2.SS1.p3.5.m2.1a"><mrow id="S2.SS1.p3.5.m2.1.1" xref="S2.SS1.p3.5.m2.1.1.cmml"><mi id="S2.SS1.p3.5.m2.1.1.2" xref="S2.SS1.p3.5.m2.1.1.2.cmml">N</mi><mo id="S2.SS1.p3.5.m2.1.1.1" xref="S2.SS1.p3.5.m2.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p3.5.m2.1.1.3" xref="S2.SS1.p3.5.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.5.m2.1.1.3.2" xref="S2.SS1.p3.5.m2.1.1.3.2.cmml">𝒞</mi><mi id="S2.SS1.p3.5.m2.1.1.3.3" xref="S2.SS1.p3.5.m2.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m2.1b"><apply id="S2.SS1.p3.5.m2.1.1.cmml" xref="S2.SS1.p3.5.m2.1.1"><times id="S2.SS1.p3.5.m2.1.1.1.cmml" xref="S2.SS1.p3.5.m2.1.1.1"></times><ci id="S2.SS1.p3.5.m2.1.1.2.cmml" xref="S2.SS1.p3.5.m2.1.1.2">𝑁</ci><apply id="S2.SS1.p3.5.m2.1.1.3.cmml" xref="S2.SS1.p3.5.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.5.m2.1.1.3.1.cmml" xref="S2.SS1.p3.5.m2.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.5.m2.1.1.3.2.cmml" xref="S2.SS1.p3.5.m2.1.1.3.2">𝒞</ci><ci id="S2.SS1.p3.5.m2.1.1.3.3.cmml" xref="S2.SS1.p3.5.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m2.1c">N\mathcal{C}_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m2.1d">italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\alpha:c_{n}\to c" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m3.1"><semantics id="S2.SS1.p3.6.m3.1a"><mrow id="S2.SS1.p3.6.m3.1.1" xref="S2.SS1.p3.6.m3.1.1.cmml"><mi id="S2.SS1.p3.6.m3.1.1.2" xref="S2.SS1.p3.6.m3.1.1.2.cmml">α</mi><mo id="S2.SS1.p3.6.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p3.6.m3.1.1.1.cmml">:</mo><mrow id="S2.SS1.p3.6.m3.1.1.3" xref="S2.SS1.p3.6.m3.1.1.3.cmml"><msub id="S2.SS1.p3.6.m3.1.1.3.2" xref="S2.SS1.p3.6.m3.1.1.3.2.cmml"><mi id="S2.SS1.p3.6.m3.1.1.3.2.2" xref="S2.SS1.p3.6.m3.1.1.3.2.2.cmml">c</mi><mi id="S2.SS1.p3.6.m3.1.1.3.2.3" xref="S2.SS1.p3.6.m3.1.1.3.2.3.cmml">n</mi></msub><mo id="S2.SS1.p3.6.m3.1.1.3.1" stretchy="false" xref="S2.SS1.p3.6.m3.1.1.3.1.cmml">→</mo><mi id="S2.SS1.p3.6.m3.1.1.3.3" xref="S2.SS1.p3.6.m3.1.1.3.3.cmml">c</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m3.1b"><apply id="S2.SS1.p3.6.m3.1.1.cmml" xref="S2.SS1.p3.6.m3.1.1"><ci id="S2.SS1.p3.6.m3.1.1.1.cmml" xref="S2.SS1.p3.6.m3.1.1.1">:</ci><ci id="S2.SS1.p3.6.m3.1.1.2.cmml" xref="S2.SS1.p3.6.m3.1.1.2">𝛼</ci><apply id="S2.SS1.p3.6.m3.1.1.3.cmml" xref="S2.SS1.p3.6.m3.1.1.3"><ci id="S2.SS1.p3.6.m3.1.1.3.1.cmml" xref="S2.SS1.p3.6.m3.1.1.3.1">→</ci><apply id="S2.SS1.p3.6.m3.1.1.3.2.cmml" xref="S2.SS1.p3.6.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.6.m3.1.1.3.2.1.cmml" xref="S2.SS1.p3.6.m3.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p3.6.m3.1.1.3.2.2.cmml" xref="S2.SS1.p3.6.m3.1.1.3.2.2">𝑐</ci><ci id="S2.SS1.p3.6.m3.1.1.3.2.3.cmml" xref="S2.SS1.p3.6.m3.1.1.3.2.3">𝑛</ci></apply><ci id="S2.SS1.p3.6.m3.1.1.3.3.cmml" xref="S2.SS1.p3.6.m3.1.1.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m3.1c">\alpha:c_{n}\to c</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m3.1d">italic_α : italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_c</annotation></semantics></math> is a morphism in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m4.1"><semantics id="S2.SS1.p3.7.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.7.m4.1.1" xref="S2.SS1.p3.7.m4.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m4.1b"><ci id="S2.SS1.p3.7.m4.1.1.cmml" xref="S2.SS1.p3.7.m4.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m4.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m4.1d">caligraphic_C</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.1.1.1">Lemma 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem4.p1"> <p class="ltx_p" id="S2.Thmtheorem4.p1.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem4.p1.4.1">The chain complex</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{P}_{*}:\cdots\to P_{n}\smash{\,\mathop{\longrightarrow}\limits^{% \partial_{n}}\,}P_{n-1}\to\cdots\to P_{0}\smash{\,\mathop{\longrightarrow}% \limits^{\epsilon}\,}\underline{R}\to 0" class="ltx_Math" display="block" id="S2.Ex14.m1.1"><semantics id="S2.Ex14.m1.1a"><mrow id="S2.Ex14.m1.1.1" xref="S2.Ex14.m1.1.1.cmml"><msub id="S2.Ex14.m1.1.1.2" xref="S2.Ex14.m1.1.1.2.cmml"><mover accent="true" id="S2.Ex14.m1.1.1.2.2" xref="S2.Ex14.m1.1.1.2.2.cmml"><mi id="S2.Ex14.m1.1.1.2.2.2" xref="S2.Ex14.m1.1.1.2.2.2.cmml">P</mi><mo id="S2.Ex14.m1.1.1.2.2.1" xref="S2.Ex14.m1.1.1.2.2.1.cmml">~</mo></mover><mo id="S2.Ex14.m1.1.1.2.3" xref="S2.Ex14.m1.1.1.2.3.cmml">∗</mo></msub><mo id="S2.Ex14.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex14.m1.1.1.1.cmml">:</mo><mrow id="S2.Ex14.m1.1.1.3" xref="S2.Ex14.m1.1.1.3.cmml"><mi id="S2.Ex14.m1.1.1.3.2" mathvariant="normal" xref="S2.Ex14.m1.1.1.3.2.cmml">⋯</mi><mo id="S2.Ex14.m1.1.1.3.3" stretchy="false" xref="S2.Ex14.m1.1.1.3.3.cmml">→</mo><mrow id="S2.Ex14.m1.1.1.3.4" xref="S2.Ex14.m1.1.1.3.4.cmml"><msub id="S2.Ex14.m1.1.1.3.4.2" xref="S2.Ex14.m1.1.1.3.4.2.cmml"><mi id="S2.Ex14.m1.1.1.3.4.2.2" xref="S2.Ex14.m1.1.1.3.4.2.2.cmml">P</mi><mi id="S2.Ex14.m1.1.1.3.4.2.3" xref="S2.Ex14.m1.1.1.3.4.2.3.cmml">n</mi></msub><mo id="S2.Ex14.m1.1.1.3.4.1" lspace="0.167em" xref="S2.Ex14.m1.1.1.3.4.1.cmml">⁢</mo><mrow id="S2.Ex14.m1.1.1.3.4.3" xref="S2.Ex14.m1.1.1.3.4.3.cmml"><mover id="S2.Ex14.m1.1.1.3.4.3.1" xref="S2.Ex14.m1.1.1.3.4.3.1.cmml"><mo id="S2.Ex14.m1.1.1.3.4.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex14.m1.1.1.3.4.3.1.2.cmml">⟶</mo><msub id="S2.Ex14.m1.1.1.3.4.3.1.3" xref="S2.Ex14.m1.1.1.3.4.3.1.3.cmml"><mo id="S2.Ex14.m1.1.1.3.4.3.1.3.2" xref="S2.Ex14.m1.1.1.3.4.3.1.3.2.cmml">∂</mo><mi id="S2.Ex14.m1.1.1.3.4.3.1.3.3" xref="S2.Ex14.m1.1.1.3.4.3.1.3.3.cmml">n</mi></msub></mover><msub id="S2.Ex14.m1.1.1.3.4.3.2" xref="S2.Ex14.m1.1.1.3.4.3.2.cmml"><mi id="S2.Ex14.m1.1.1.3.4.3.2.2" xref="S2.Ex14.m1.1.1.3.4.3.2.2.cmml">P</mi><mrow id="S2.Ex14.m1.1.1.3.4.3.2.3" xref="S2.Ex14.m1.1.1.3.4.3.2.3.cmml"><mi id="S2.Ex14.m1.1.1.3.4.3.2.3.2" xref="S2.Ex14.m1.1.1.3.4.3.2.3.2.cmml">n</mi><mo id="S2.Ex14.m1.1.1.3.4.3.2.3.1" xref="S2.Ex14.m1.1.1.3.4.3.2.3.1.cmml">−</mo><mn id="S2.Ex14.m1.1.1.3.4.3.2.3.3" xref="S2.Ex14.m1.1.1.3.4.3.2.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S2.Ex14.m1.1.1.3.5" stretchy="false" xref="S2.Ex14.m1.1.1.3.5.cmml">→</mo><mi id="S2.Ex14.m1.1.1.3.6" mathvariant="normal" xref="S2.Ex14.m1.1.1.3.6.cmml">⋯</mi><mo id="S2.Ex14.m1.1.1.3.7" stretchy="false" xref="S2.Ex14.m1.1.1.3.7.cmml">→</mo><mrow id="S2.Ex14.m1.1.1.3.8" xref="S2.Ex14.m1.1.1.3.8.cmml"><msub id="S2.Ex14.m1.1.1.3.8.2" xref="S2.Ex14.m1.1.1.3.8.2.cmml"><mi id="S2.Ex14.m1.1.1.3.8.2.2" xref="S2.Ex14.m1.1.1.3.8.2.2.cmml">P</mi><mn id="S2.Ex14.m1.1.1.3.8.2.3" xref="S2.Ex14.m1.1.1.3.8.2.3.cmml">0</mn></msub><mo id="S2.Ex14.m1.1.1.3.8.1" lspace="0.167em" xref="S2.Ex14.m1.1.1.3.8.1.cmml">⁢</mo><mrow id="S2.Ex14.m1.1.1.3.8.3" xref="S2.Ex14.m1.1.1.3.8.3.cmml"><mover id="S2.Ex14.m1.1.1.3.8.3.1" xref="S2.Ex14.m1.1.1.3.8.3.1.cmml"><mo id="S2.Ex14.m1.1.1.3.8.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex14.m1.1.1.3.8.3.1.2.cmml">⟶</mo><mi id="S2.Ex14.m1.1.1.3.8.3.1.3" xref="S2.Ex14.m1.1.1.3.8.3.1.3.cmml">ϵ</mi></mover><munder accentunder="true" id="S2.Ex14.m1.1.1.3.8.3.2" xref="S2.Ex14.m1.1.1.3.8.3.2.cmml"><mi id="S2.Ex14.m1.1.1.3.8.3.2.2" xref="S2.Ex14.m1.1.1.3.8.3.2.2.cmml">R</mi><mo id="S2.Ex14.m1.1.1.3.8.3.2.1" xref="S2.Ex14.m1.1.1.3.8.3.2.1.cmml">¯</mo></munder></mrow></mrow><mo id="S2.Ex14.m1.1.1.3.9" stretchy="false" xref="S2.Ex14.m1.1.1.3.9.cmml">→</mo><mn id="S2.Ex14.m1.1.1.3.10" xref="S2.Ex14.m1.1.1.3.10.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex14.m1.1b"><apply id="S2.Ex14.m1.1.1.cmml" xref="S2.Ex14.m1.1.1"><ci id="S2.Ex14.m1.1.1.1.cmml" xref="S2.Ex14.m1.1.1.1">:</ci><apply id="S2.Ex14.m1.1.1.2.cmml" xref="S2.Ex14.m1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex14.m1.1.1.2.1.cmml" xref="S2.Ex14.m1.1.1.2">subscript</csymbol><apply id="S2.Ex14.m1.1.1.2.2.cmml" xref="S2.Ex14.m1.1.1.2.2"><ci id="S2.Ex14.m1.1.1.2.2.1.cmml" xref="S2.Ex14.m1.1.1.2.2.1">~</ci><ci id="S2.Ex14.m1.1.1.2.2.2.cmml" xref="S2.Ex14.m1.1.1.2.2.2">𝑃</ci></apply><times id="S2.Ex14.m1.1.1.2.3.cmml" xref="S2.Ex14.m1.1.1.2.3"></times></apply><apply id="S2.Ex14.m1.1.1.3.cmml" xref="S2.Ex14.m1.1.1.3"><and id="S2.Ex14.m1.1.1.3a.cmml" xref="S2.Ex14.m1.1.1.3"></and><apply id="S2.Ex14.m1.1.1.3b.cmml" xref="S2.Ex14.m1.1.1.3"><ci id="S2.Ex14.m1.1.1.3.3.cmml" xref="S2.Ex14.m1.1.1.3.3">→</ci><ci id="S2.Ex14.m1.1.1.3.2.cmml" xref="S2.Ex14.m1.1.1.3.2">⋯</ci><apply id="S2.Ex14.m1.1.1.3.4.cmml" xref="S2.Ex14.m1.1.1.3.4"><times id="S2.Ex14.m1.1.1.3.4.1.cmml" xref="S2.Ex14.m1.1.1.3.4.1"></times><apply id="S2.Ex14.m1.1.1.3.4.2.cmml" xref="S2.Ex14.m1.1.1.3.4.2"><csymbol cd="ambiguous" id="S2.Ex14.m1.1.1.3.4.2.1.cmml" xref="S2.Ex14.m1.1.1.3.4.2">subscript</csymbol><ci id="S2.Ex14.m1.1.1.3.4.2.2.cmml" xref="S2.Ex14.m1.1.1.3.4.2.2">𝑃</ci><ci id="S2.Ex14.m1.1.1.3.4.2.3.cmml" xref="S2.Ex14.m1.1.1.3.4.2.3">𝑛</ci></apply><apply id="S2.Ex14.m1.1.1.3.4.3.cmml" xref="S2.Ex14.m1.1.1.3.4.3"><apply id="S2.Ex14.m1.1.1.3.4.3.1.cmml" xref="S2.Ex14.m1.1.1.3.4.3.1"><csymbol cd="ambiguous" id="S2.Ex14.m1.1.1.3.4.3.1.1.cmml" xref="S2.Ex14.m1.1.1.3.4.3.1">superscript</csymbol><ci id="S2.Ex14.m1.1.1.3.4.3.1.2.cmml" xref="S2.Ex14.m1.1.1.3.4.3.1.2">⟶</ci><apply id="S2.Ex14.m1.1.1.3.4.3.1.3.cmml" xref="S2.Ex14.m1.1.1.3.4.3.1.3"><csymbol cd="ambiguous" id="S2.Ex14.m1.1.1.3.4.3.1.3.1.cmml" xref="S2.Ex14.m1.1.1.3.4.3.1.3">subscript</csymbol><partialdiff id="S2.Ex14.m1.1.1.3.4.3.1.3.2.cmml" xref="S2.Ex14.m1.1.1.3.4.3.1.3.2"></partialdiff><ci id="S2.Ex14.m1.1.1.3.4.3.1.3.3.cmml" xref="S2.Ex14.m1.1.1.3.4.3.1.3.3">𝑛</ci></apply></apply><apply id="S2.Ex14.m1.1.1.3.4.3.2.cmml" xref="S2.Ex14.m1.1.1.3.4.3.2"><csymbol cd="ambiguous" id="S2.Ex14.m1.1.1.3.4.3.2.1.cmml" xref="S2.Ex14.m1.1.1.3.4.3.2">subscript</csymbol><ci id="S2.Ex14.m1.1.1.3.4.3.2.2.cmml" xref="S2.Ex14.m1.1.1.3.4.3.2.2">𝑃</ci><apply id="S2.Ex14.m1.1.1.3.4.3.2.3.cmml" xref="S2.Ex14.m1.1.1.3.4.3.2.3"><minus id="S2.Ex14.m1.1.1.3.4.3.2.3.1.cmml" xref="S2.Ex14.m1.1.1.3.4.3.2.3.1"></minus><ci id="S2.Ex14.m1.1.1.3.4.3.2.3.2.cmml" xref="S2.Ex14.m1.1.1.3.4.3.2.3.2">𝑛</ci><cn id="S2.Ex14.m1.1.1.3.4.3.2.3.3.cmml" type="integer" xref="S2.Ex14.m1.1.1.3.4.3.2.3.3">1</cn></apply></apply></apply></apply></apply><apply id="S2.Ex14.m1.1.1.3c.cmml" xref="S2.Ex14.m1.1.1.3"><ci id="S2.Ex14.m1.1.1.3.5.cmml" xref="S2.Ex14.m1.1.1.3.5">→</ci><share href="https://arxiv.org/html/2503.14659v1#S2.Ex14.m1.1.1.3.4.cmml" id="S2.Ex14.m1.1.1.3d.cmml" xref="S2.Ex14.m1.1.1.3"></share><ci id="S2.Ex14.m1.1.1.3.6.cmml" xref="S2.Ex14.m1.1.1.3.6">⋯</ci></apply><apply id="S2.Ex14.m1.1.1.3e.cmml" xref="S2.Ex14.m1.1.1.3"><ci id="S2.Ex14.m1.1.1.3.7.cmml" xref="S2.Ex14.m1.1.1.3.7">→</ci><share href="https://arxiv.org/html/2503.14659v1#S2.Ex14.m1.1.1.3.6.cmml" id="S2.Ex14.m1.1.1.3f.cmml" xref="S2.Ex14.m1.1.1.3"></share><apply id="S2.Ex14.m1.1.1.3.8.cmml" xref="S2.Ex14.m1.1.1.3.8"><times id="S2.Ex14.m1.1.1.3.8.1.cmml" xref="S2.Ex14.m1.1.1.3.8.1"></times><apply id="S2.Ex14.m1.1.1.3.8.2.cmml" xref="S2.Ex14.m1.1.1.3.8.2"><csymbol cd="ambiguous" id="S2.Ex14.m1.1.1.3.8.2.1.cmml" xref="S2.Ex14.m1.1.1.3.8.2">subscript</csymbol><ci id="S2.Ex14.m1.1.1.3.8.2.2.cmml" xref="S2.Ex14.m1.1.1.3.8.2.2">𝑃</ci><cn id="S2.Ex14.m1.1.1.3.8.2.3.cmml" type="integer" xref="S2.Ex14.m1.1.1.3.8.2.3">0</cn></apply><apply id="S2.Ex14.m1.1.1.3.8.3.cmml" xref="S2.Ex14.m1.1.1.3.8.3"><apply id="S2.Ex14.m1.1.1.3.8.3.1.cmml" xref="S2.Ex14.m1.1.1.3.8.3.1"><csymbol cd="ambiguous" id="S2.Ex14.m1.1.1.3.8.3.1.1.cmml" xref="S2.Ex14.m1.1.1.3.8.3.1">superscript</csymbol><ci id="S2.Ex14.m1.1.1.3.8.3.1.2.cmml" xref="S2.Ex14.m1.1.1.3.8.3.1.2">⟶</ci><ci id="S2.Ex14.m1.1.1.3.8.3.1.3.cmml" xref="S2.Ex14.m1.1.1.3.8.3.1.3">italic-ϵ</ci></apply><apply id="S2.Ex14.m1.1.1.3.8.3.2.cmml" xref="S2.Ex14.m1.1.1.3.8.3.2"><ci id="S2.Ex14.m1.1.1.3.8.3.2.1.cmml" xref="S2.Ex14.m1.1.1.3.8.3.2.1">¯</ci><ci id="S2.Ex14.m1.1.1.3.8.3.2.2.cmml" xref="S2.Ex14.m1.1.1.3.8.3.2.2">𝑅</ci></apply></apply></apply></apply><apply id="S2.Ex14.m1.1.1.3g.cmml" xref="S2.Ex14.m1.1.1.3"><ci id="S2.Ex14.m1.1.1.3.9.cmml" xref="S2.Ex14.m1.1.1.3.9">→</ci><share href="https://arxiv.org/html/2503.14659v1#S2.Ex14.m1.1.1.3.8.cmml" id="S2.Ex14.m1.1.1.3h.cmml" xref="S2.Ex14.m1.1.1.3"></share><cn id="S2.Ex14.m1.1.1.3.10.cmml" type="integer" xref="S2.Ex14.m1.1.1.3.10">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex14.m1.1c">\widetilde{P}_{*}:\cdots\to P_{n}\smash{\,\mathop{\longrightarrow}\limits^{% \partial_{n}}\,}P_{n-1}\to\cdots\to P_{0}\smash{\,\mathop{\longrightarrow}% \limits^{\epsilon}\,}\underline{R}\to 0</annotation><annotation encoding="application/x-llamapun" id="S2.Ex14.m1.1d">over~ start_ARG italic_P end_ARG start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : ⋯ → italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT → ⋯ → italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT under¯ start_ARG italic_R end_ARG → 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem4.p1.3"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem4.p1.3.3">is a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.1.1.m1.1"><semantics id="S2.Thmtheorem4.p1.1.1.m1.1a"><munder accentunder="true" id="S2.Thmtheorem4.p1.1.1.m1.1.1" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">R</mi><mo id="S2.Thmtheorem4.p1.1.1.m1.1.1.1" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.1.1.m1.1b"><apply id="S2.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1"><ci id="S2.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.1">¯</ci><ci id="S2.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.1.1.m1.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.1.1.m1.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.2.2.m2.1"><semantics id="S2.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem4.p1.2.2.m2.1.1" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">R</mi><mo id="S2.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.2.2.m2.1b"><apply id="S2.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1"><times id="S2.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.1"></times><ci id="S2.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.2">𝑅</ci><ci id="S2.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.2.2.m2.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.2.2.m2.1d">italic_R caligraphic_C</annotation></semantics></math>-module. This resolution is called the <em class="ltx_emph ltx_font_upright" id="S2.Thmtheorem4.p1.3.3.1">standard resolution</em> for <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.3.3.m3.1"><semantics id="S2.Thmtheorem4.p1.3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem4.p1.3.3.m3.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.3.3.m3.1b"><ci id="S2.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.3.3.m3.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.3.3.m3.1d">caligraphic_C</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S2.SS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS1.1.p1"> <p class="ltx_p" id="S2.SS1.1.p1.3">For each <math alttext="c\in\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.1.p1.1.m1.1"><semantics id="S2.SS1.1.p1.1.m1.1a"><mrow id="S2.SS1.1.p1.1.m1.1.1" xref="S2.SS1.1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.1.p1.1.m1.1.1.2" xref="S2.SS1.1.p1.1.m1.1.1.2.cmml">c</mi><mo id="S2.SS1.1.p1.1.m1.1.1.1" xref="S2.SS1.1.p1.1.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.1.p1.1.m1.1.1.3" xref="S2.SS1.1.p1.1.m1.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.1.p1.1.m1.1b"><apply id="S2.SS1.1.p1.1.m1.1.1.cmml" xref="S2.SS1.1.p1.1.m1.1.1"><in id="S2.SS1.1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.1.p1.1.m1.1.1.1"></in><ci id="S2.SS1.1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.1.p1.1.m1.1.1.2">𝑐</ci><ci id="S2.SS1.1.p1.1.m1.1.1.3.cmml" xref="S2.SS1.1.p1.1.m1.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.1.p1.1.m1.1c">c\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.1.p1.1.m1.1d">italic_c ∈ caligraphic_C</annotation></semantics></math>, the chain complex <math alttext="\widetilde{P}_{*}(c)" class="ltx_Math" display="inline" id="S2.SS1.1.p1.2.m2.1"><semantics id="S2.SS1.1.p1.2.m2.1a"><mrow id="S2.SS1.1.p1.2.m2.1.2" xref="S2.SS1.1.p1.2.m2.1.2.cmml"><msub id="S2.SS1.1.p1.2.m2.1.2.2" xref="S2.SS1.1.p1.2.m2.1.2.2.cmml"><mover accent="true" id="S2.SS1.1.p1.2.m2.1.2.2.2" xref="S2.SS1.1.p1.2.m2.1.2.2.2.cmml"><mi id="S2.SS1.1.p1.2.m2.1.2.2.2.2" xref="S2.SS1.1.p1.2.m2.1.2.2.2.2.cmml">P</mi><mo id="S2.SS1.1.p1.2.m2.1.2.2.2.1" xref="S2.SS1.1.p1.2.m2.1.2.2.2.1.cmml">~</mo></mover><mo id="S2.SS1.1.p1.2.m2.1.2.2.3" xref="S2.SS1.1.p1.2.m2.1.2.2.3.cmml">∗</mo></msub><mo id="S2.SS1.1.p1.2.m2.1.2.1" xref="S2.SS1.1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.1.p1.2.m2.1.2.3.2" xref="S2.SS1.1.p1.2.m2.1.2.cmml"><mo id="S2.SS1.1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.1.p1.2.m2.1.2.cmml">(</mo><mi id="S2.SS1.1.p1.2.m2.1.1" xref="S2.SS1.1.p1.2.m2.1.1.cmml">c</mi><mo id="S2.SS1.1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.1.p1.2.m2.1b"><apply id="S2.SS1.1.p1.2.m2.1.2.cmml" xref="S2.SS1.1.p1.2.m2.1.2"><times id="S2.SS1.1.p1.2.m2.1.2.1.cmml" xref="S2.SS1.1.p1.2.m2.1.2.1"></times><apply id="S2.SS1.1.p1.2.m2.1.2.2.cmml" xref="S2.SS1.1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.1.p1.2.m2.1.2.2.1.cmml" xref="S2.SS1.1.p1.2.m2.1.2.2">subscript</csymbol><apply id="S2.SS1.1.p1.2.m2.1.2.2.2.cmml" xref="S2.SS1.1.p1.2.m2.1.2.2.2"><ci id="S2.SS1.1.p1.2.m2.1.2.2.2.1.cmml" xref="S2.SS1.1.p1.2.m2.1.2.2.2.1">~</ci><ci id="S2.SS1.1.p1.2.m2.1.2.2.2.2.cmml" xref="S2.SS1.1.p1.2.m2.1.2.2.2.2">𝑃</ci></apply><times id="S2.SS1.1.p1.2.m2.1.2.2.3.cmml" xref="S2.SS1.1.p1.2.m2.1.2.2.3"></times></apply><ci id="S2.SS1.1.p1.2.m2.1.1.cmml" xref="S2.SS1.1.p1.2.m2.1.1">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.1.p1.2.m2.1c">\widetilde{P}_{*}(c)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.1.p1.2.m2.1d">over~ start_ARG italic_P end_ARG start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math> is exact because there is a splitting <math alttext="s_{*}:P_{*}(c)\to P_{*+1}(c)" class="ltx_Math" display="inline" id="S2.SS1.1.p1.3.m3.2"><semantics id="S2.SS1.1.p1.3.m3.2a"><mrow id="S2.SS1.1.p1.3.m3.2.3" xref="S2.SS1.1.p1.3.m3.2.3.cmml"><msub id="S2.SS1.1.p1.3.m3.2.3.2" xref="S2.SS1.1.p1.3.m3.2.3.2.cmml"><mi id="S2.SS1.1.p1.3.m3.2.3.2.2" xref="S2.SS1.1.p1.3.m3.2.3.2.2.cmml">s</mi><mo id="S2.SS1.1.p1.3.m3.2.3.2.3" xref="S2.SS1.1.p1.3.m3.2.3.2.3.cmml">∗</mo></msub><mo id="S2.SS1.1.p1.3.m3.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.1.p1.3.m3.2.3.1.cmml">:</mo><mrow id="S2.SS1.1.p1.3.m3.2.3.3" xref="S2.SS1.1.p1.3.m3.2.3.3.cmml"><mrow id="S2.SS1.1.p1.3.m3.2.3.3.2" xref="S2.SS1.1.p1.3.m3.2.3.3.2.cmml"><msub id="S2.SS1.1.p1.3.m3.2.3.3.2.2" xref="S2.SS1.1.p1.3.m3.2.3.3.2.2.cmml"><mi id="S2.SS1.1.p1.3.m3.2.3.3.2.2.2" xref="S2.SS1.1.p1.3.m3.2.3.3.2.2.2.cmml">P</mi><mo id="S2.SS1.1.p1.3.m3.2.3.3.2.2.3" xref="S2.SS1.1.p1.3.m3.2.3.3.2.2.3.cmml">∗</mo></msub><mo id="S2.SS1.1.p1.3.m3.2.3.3.2.1" xref="S2.SS1.1.p1.3.m3.2.3.3.2.1.cmml">⁢</mo><mrow id="S2.SS1.1.p1.3.m3.2.3.3.2.3.2" xref="S2.SS1.1.p1.3.m3.2.3.3.2.cmml"><mo id="S2.SS1.1.p1.3.m3.2.3.3.2.3.2.1" stretchy="false" xref="S2.SS1.1.p1.3.m3.2.3.3.2.cmml">(</mo><mi id="S2.SS1.1.p1.3.m3.1.1" xref="S2.SS1.1.p1.3.m3.1.1.cmml">c</mi><mo id="S2.SS1.1.p1.3.m3.2.3.3.2.3.2.2" stretchy="false" xref="S2.SS1.1.p1.3.m3.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.1.p1.3.m3.2.3.3.1" stretchy="false" xref="S2.SS1.1.p1.3.m3.2.3.3.1.cmml">→</mo><mrow id="S2.SS1.1.p1.3.m3.2.3.3.3" xref="S2.SS1.1.p1.3.m3.2.3.3.3.cmml"><msub id="S2.SS1.1.p1.3.m3.2.3.3.3.2" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.cmml"><mi id="S2.SS1.1.p1.3.m3.2.3.3.3.2.2" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.2.cmml">P</mi><mrow id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.cmml"><mi id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.2" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.2.cmml"></mi><mo id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.1.cmml">∗</mo><mrow id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3.cmml"><mo id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3a" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3.cmml">+</mo><mn id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3.2" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3.2.cmml">1</mn></mrow></mrow></msub><mo id="S2.SS1.1.p1.3.m3.2.3.3.3.1" xref="S2.SS1.1.p1.3.m3.2.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.1.p1.3.m3.2.3.3.3.3.2" xref="S2.SS1.1.p1.3.m3.2.3.3.3.cmml"><mo id="S2.SS1.1.p1.3.m3.2.3.3.3.3.2.1" stretchy="false" xref="S2.SS1.1.p1.3.m3.2.3.3.3.cmml">(</mo><mi id="S2.SS1.1.p1.3.m3.2.2" xref="S2.SS1.1.p1.3.m3.2.2.cmml">c</mi><mo id="S2.SS1.1.p1.3.m3.2.3.3.3.3.2.2" stretchy="false" xref="S2.SS1.1.p1.3.m3.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.1.p1.3.m3.2b"><apply id="S2.SS1.1.p1.3.m3.2.3.cmml" xref="S2.SS1.1.p1.3.m3.2.3"><ci id="S2.SS1.1.p1.3.m3.2.3.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.1">:</ci><apply id="S2.SS1.1.p1.3.m3.2.3.2.cmml" xref="S2.SS1.1.p1.3.m3.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.1.p1.3.m3.2.3.2.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.2">subscript</csymbol><ci id="S2.SS1.1.p1.3.m3.2.3.2.2.cmml" xref="S2.SS1.1.p1.3.m3.2.3.2.2">𝑠</ci><times id="S2.SS1.1.p1.3.m3.2.3.2.3.cmml" xref="S2.SS1.1.p1.3.m3.2.3.2.3"></times></apply><apply id="S2.SS1.1.p1.3.m3.2.3.3.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3"><ci id="S2.SS1.1.p1.3.m3.2.3.3.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.1">→</ci><apply id="S2.SS1.1.p1.3.m3.2.3.3.2.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.2"><times id="S2.SS1.1.p1.3.m3.2.3.3.2.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.2.1"></times><apply id="S2.SS1.1.p1.3.m3.2.3.3.2.2.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.1.p1.3.m3.2.3.3.2.2.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.2.2">subscript</csymbol><ci id="S2.SS1.1.p1.3.m3.2.3.3.2.2.2.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.2.2.2">𝑃</ci><times id="S2.SS1.1.p1.3.m3.2.3.3.2.2.3.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.2.2.3"></times></apply><ci id="S2.SS1.1.p1.3.m3.1.1.cmml" xref="S2.SS1.1.p1.3.m3.1.1">𝑐</ci></apply><apply id="S2.SS1.1.p1.3.m3.2.3.3.3.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3"><times id="S2.SS1.1.p1.3.m3.2.3.3.3.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.1"></times><apply id="S2.SS1.1.p1.3.m3.2.3.3.3.2.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.1.p1.3.m3.2.3.3.3.2.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2">subscript</csymbol><ci id="S2.SS1.1.p1.3.m3.2.3.3.3.2.2.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.2">𝑃</ci><apply id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3"><times id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.1"></times><csymbol cd="latexml" id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.2.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.2">absent</csymbol><apply id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3"><plus id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3.1.cmml" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3"></plus><cn id="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3.2.cmml" type="integer" xref="S2.SS1.1.p1.3.m3.2.3.3.3.2.3.3.2">1</cn></apply></apply></apply><ci id="S2.SS1.1.p1.3.m3.2.2.cmml" xref="S2.SS1.1.p1.3.m3.2.2">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.1.p1.3.m3.2c">s_{*}:P_{*}(c)\to P_{*+1}(c)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.1.p1.3.m3.2d">italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_c ) → italic_P start_POSTSUBSCRIPT ∗ + 1 end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math> defined by</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{n}((c_{0}\to\cdots\to c_{n}),\alpha:c_{n}\to c)=(-1)^{n+1}((c_{0}\to\cdots% \to c_{n}\smash{\,\mathop{\longrightarrow}\limits^{\alpha}\,}c),\mathrm{id}:c% \to c)" class="ltx_math_unparsed" display="block" id="S2.Ex15.m1.1"><semantics id="S2.Ex15.m1.1a"><mrow id="S2.Ex15.m1.1b"><msub id="S2.Ex15.m1.1.2"><mi id="S2.Ex15.m1.1.2.2">s</mi><mi id="S2.Ex15.m1.1.2.3">n</mi></msub><mrow id="S2.Ex15.m1.1.3"><mo id="S2.Ex15.m1.1.3.1" stretchy="false">(</mo><mrow id="S2.Ex15.m1.1.3.2"><mo id="S2.Ex15.m1.1.3.2.1" stretchy="false">(</mo><msub id="S2.Ex15.m1.1.3.2.2"><mi id="S2.Ex15.m1.1.3.2.2.2">c</mi><mn id="S2.Ex15.m1.1.3.2.2.3">0</mn></msub><mo id="S2.Ex15.m1.1.3.2.3" stretchy="false">→</mo><mi id="S2.Ex15.m1.1.3.2.4" mathvariant="normal">⋯</mi><mo id="S2.Ex15.m1.1.3.2.5" stretchy="false">→</mo><msub id="S2.Ex15.m1.1.3.2.6"><mi id="S2.Ex15.m1.1.3.2.6.2">c</mi><mi id="S2.Ex15.m1.1.3.2.6.3">n</mi></msub><mo id="S2.Ex15.m1.1.3.2.7" stretchy="false">)</mo></mrow><mo id="S2.Ex15.m1.1.3.3">,</mo><mi id="S2.Ex15.m1.1.1">α</mi><mo id="S2.Ex15.m1.1.3.4" lspace="0.278em" rspace="0.278em">:</mo><msub id="S2.Ex15.m1.1.3.5"><mi id="S2.Ex15.m1.1.3.5.2">c</mi><mi id="S2.Ex15.m1.1.3.5.3">n</mi></msub><mo id="S2.Ex15.m1.1.3.6" stretchy="false">→</mo><mi id="S2.Ex15.m1.1.3.7">c</mi><mo id="S2.Ex15.m1.1.3.8" stretchy="false">)</mo></mrow><mo id="S2.Ex15.m1.1.4">=</mo><msup id="S2.Ex15.m1.1.5"><mrow id="S2.Ex15.m1.1.5.2"><mo id="S2.Ex15.m1.1.5.2.1" stretchy="false">(</mo><mo id="S2.Ex15.m1.1.5.2.2" lspace="0em">−</mo><mn id="S2.Ex15.m1.1.5.2.3">1</mn><mo id="S2.Ex15.m1.1.5.2.4" stretchy="false">)</mo></mrow><mrow id="S2.Ex15.m1.1.5.3"><mi id="S2.Ex15.m1.1.5.3.2">n</mi><mo id="S2.Ex15.m1.1.5.3.1">+</mo><mn id="S2.Ex15.m1.1.5.3.3">1</mn></mrow></msup><mrow id="S2.Ex15.m1.1.6"><mo id="S2.Ex15.m1.1.6.1" stretchy="false">(</mo><mrow id="S2.Ex15.m1.1.6.2"><mo id="S2.Ex15.m1.1.6.2.1" stretchy="false">(</mo><msub id="S2.Ex15.m1.1.6.2.2"><mi id="S2.Ex15.m1.1.6.2.2.2">c</mi><mn id="S2.Ex15.m1.1.6.2.2.3">0</mn></msub><mo id="S2.Ex15.m1.1.6.2.3" stretchy="false">→</mo><mi id="S2.Ex15.m1.1.6.2.4" mathvariant="normal">⋯</mi><mo id="S2.Ex15.m1.1.6.2.5" stretchy="false">→</mo><msub id="S2.Ex15.m1.1.6.2.6"><mi id="S2.Ex15.m1.1.6.2.6.2">c</mi><mi id="S2.Ex15.m1.1.6.2.6.3">n</mi></msub><mover id="S2.Ex15.m1.1.6.2.7"><mo id="S2.Ex15.m1.1.6.2.7.2" lspace="0.167em" movablelimits="false" rspace="0.167em">⟶</mo><mi id="S2.Ex15.m1.1.6.2.7.3">α</mi></mover><mi id="S2.Ex15.m1.1.6.2.8">c</mi><mo id="S2.Ex15.m1.1.6.2.9" stretchy="false">)</mo></mrow><mo id="S2.Ex15.m1.1.6.3">,</mo><mi id="S2.Ex15.m1.1.6.4">id</mi><mo id="S2.Ex15.m1.1.6.5" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.Ex15.m1.1.6.6">c</mi><mo id="S2.Ex15.m1.1.6.7" stretchy="false">→</mo><mi id="S2.Ex15.m1.1.6.8">c</mi><mo id="S2.Ex15.m1.1.6.9" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.Ex15.m1.1c">s_{n}((c_{0}\to\cdots\to c_{n}),\alpha:c_{n}\to c)=(-1)^{n+1}((c_{0}\to\cdots% \to c_{n}\smash{\,\mathop{\longrightarrow}\limits^{\alpha}\,}c),\mathrm{id}:c% \to c)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex15.m1.1d">italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ⋯ → italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) , italic_α : italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_c ) = ( - 1 ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT ( ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ⋯ → italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_c ) , roman_id : italic_c → italic_c )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.1.p1.7">for <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S2.SS1.1.p1.4.m1.1"><semantics id="S2.SS1.1.p1.4.m1.1a"><mrow id="S2.SS1.1.p1.4.m1.1.1" xref="S2.SS1.1.p1.4.m1.1.1.cmml"><mi id="S2.SS1.1.p1.4.m1.1.1.2" xref="S2.SS1.1.p1.4.m1.1.1.2.cmml">n</mi><mo id="S2.SS1.1.p1.4.m1.1.1.1" xref="S2.SS1.1.p1.4.m1.1.1.1.cmml">≥</mo><mn id="S2.SS1.1.p1.4.m1.1.1.3" xref="S2.SS1.1.p1.4.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.1.p1.4.m1.1b"><apply id="S2.SS1.1.p1.4.m1.1.1.cmml" xref="S2.SS1.1.p1.4.m1.1.1"><geq id="S2.SS1.1.p1.4.m1.1.1.1.cmml" xref="S2.SS1.1.p1.4.m1.1.1.1"></geq><ci id="S2.SS1.1.p1.4.m1.1.1.2.cmml" xref="S2.SS1.1.p1.4.m1.1.1.2">𝑛</ci><cn id="S2.SS1.1.p1.4.m1.1.1.3.cmml" type="integer" xref="S2.SS1.1.p1.4.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.1.p1.4.m1.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.1.p1.4.m1.1d">italic_n ≥ 0</annotation></semantics></math>, and <math alttext="s_{-1}(1)=(c,\mathrm{id}:c\to c)" class="ltx_math_unparsed" display="inline" id="S2.SS1.1.p1.5.m2.3"><semantics id="S2.SS1.1.p1.5.m2.3a"><mrow id="S2.SS1.1.p1.5.m2.3b"><msub id="S2.SS1.1.p1.5.m2.3.4"><mi id="S2.SS1.1.p1.5.m2.3.4.2">s</mi><mrow id="S2.SS1.1.p1.5.m2.3.4.3"><mo id="S2.SS1.1.p1.5.m2.3.4.3a">−</mo><mn id="S2.SS1.1.p1.5.m2.3.4.3.2">1</mn></mrow></msub><mrow id="S2.SS1.1.p1.5.m2.3.5"><mo id="S2.SS1.1.p1.5.m2.3.5.1" stretchy="false">(</mo><mn id="S2.SS1.1.p1.5.m2.1.1">1</mn><mo id="S2.SS1.1.p1.5.m2.3.5.2" stretchy="false">)</mo></mrow><mo id="S2.SS1.1.p1.5.m2.3.6">=</mo><mrow id="S2.SS1.1.p1.5.m2.3.7"><mo id="S2.SS1.1.p1.5.m2.3.7.1" stretchy="false">(</mo><mi id="S2.SS1.1.p1.5.m2.2.2">c</mi><mo id="S2.SS1.1.p1.5.m2.3.7.2">,</mo><mi id="S2.SS1.1.p1.5.m2.3.3">id</mi><mo id="S2.SS1.1.p1.5.m2.3.7.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.SS1.1.p1.5.m2.3.7.4">c</mi><mo id="S2.SS1.1.p1.5.m2.3.7.5" stretchy="false">→</mo><mi id="S2.SS1.1.p1.5.m2.3.7.6">c</mi><mo id="S2.SS1.1.p1.5.m2.3.7.7" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS1.1.p1.5.m2.3c">s_{-1}(1)=(c,\mathrm{id}:c\to c)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.1.p1.5.m2.3d">italic_s start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ( 1 ) = ( italic_c , roman_id : italic_c → italic_c )</annotation></semantics></math>. Hence <math alttext="P_{*}" class="ltx_Math" display="inline" id="S2.SS1.1.p1.6.m3.1"><semantics id="S2.SS1.1.p1.6.m3.1a"><msub id="S2.SS1.1.p1.6.m3.1.1" xref="S2.SS1.1.p1.6.m3.1.1.cmml"><mi id="S2.SS1.1.p1.6.m3.1.1.2" xref="S2.SS1.1.p1.6.m3.1.1.2.cmml">P</mi><mo id="S2.SS1.1.p1.6.m3.1.1.3" xref="S2.SS1.1.p1.6.m3.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.1.p1.6.m3.1b"><apply id="S2.SS1.1.p1.6.m3.1.1.cmml" xref="S2.SS1.1.p1.6.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.1.p1.6.m3.1.1.1.cmml" xref="S2.SS1.1.p1.6.m3.1.1">subscript</csymbol><ci id="S2.SS1.1.p1.6.m3.1.1.2.cmml" xref="S2.SS1.1.p1.6.m3.1.1.2">𝑃</ci><times id="S2.SS1.1.p1.6.m3.1.1.3.cmml" xref="S2.SS1.1.p1.6.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.1.p1.6.m3.1c">P_{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.1.p1.6.m3.1d">italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> is a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S2.SS1.1.p1.7.m4.1"><semantics id="S2.SS1.1.p1.7.m4.1a"><munder accentunder="true" id="S2.SS1.1.p1.7.m4.1.1" xref="S2.SS1.1.p1.7.m4.1.1.cmml"><mi id="S2.SS1.1.p1.7.m4.1.1.2" xref="S2.SS1.1.p1.7.m4.1.1.2.cmml">R</mi><mo id="S2.SS1.1.p1.7.m4.1.1.1" xref="S2.SS1.1.p1.7.m4.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S2.SS1.1.p1.7.m4.1b"><apply id="S2.SS1.1.p1.7.m4.1.1.cmml" xref="S2.SS1.1.p1.7.m4.1.1"><ci id="S2.SS1.1.p1.7.m4.1.1.1.cmml" xref="S2.SS1.1.p1.7.m4.1.1.1">¯</ci><ci id="S2.SS1.1.p1.7.m4.1.1.2.cmml" xref="S2.SS1.1.p1.7.m4.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.1.p1.7.m4.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.1.p1.7.m4.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.5">Using the standard resolution, one can describe the cohomology of a category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.1b"><ci id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">caligraphic_C</annotation></semantics></math> with coefficients in an <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p4.2.m2.1"><semantics id="S2.SS1.p4.2.m2.1a"><mrow id="S2.SS1.p4.2.m2.1.1" xref="S2.SS1.p4.2.m2.1.1.cmml"><mi id="S2.SS1.p4.2.m2.1.1.2" xref="S2.SS1.p4.2.m2.1.1.2.cmml">R</mi><mo id="S2.SS1.p4.2.m2.1.1.1" xref="S2.SS1.p4.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.2.m2.1.1.3" xref="S2.SS1.p4.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.2.m2.1b"><apply id="S2.SS1.p4.2.m2.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1"><times id="S2.SS1.p4.2.m2.1.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1.1"></times><ci id="S2.SS1.p4.2.m2.1.1.2.cmml" xref="S2.SS1.p4.2.m2.1.1.2">𝑅</ci><ci id="S2.SS1.p4.2.m2.1.1.3.cmml" xref="S2.SS1.p4.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">italic_R caligraphic_C</annotation></semantics></math>-module <math alttext="M" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><mi id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.1b"><ci id="S2.SS1.p4.3.m3.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.1d">italic_M</annotation></semantics></math> as the cohomology of the cochain complex <math alttext="C^{*}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.SS1.p4.4.m4.2"><semantics id="S2.SS1.p4.4.m4.2a"><mrow id="S2.SS1.p4.4.m4.2.3" xref="S2.SS1.p4.4.m4.2.3.cmml"><msup id="S2.SS1.p4.4.m4.2.3.2" xref="S2.SS1.p4.4.m4.2.3.2.cmml"><mi id="S2.SS1.p4.4.m4.2.3.2.2" xref="S2.SS1.p4.4.m4.2.3.2.2.cmml">C</mi><mo id="S2.SS1.p4.4.m4.2.3.2.3" xref="S2.SS1.p4.4.m4.2.3.2.3.cmml">∗</mo></msup><mo id="S2.SS1.p4.4.m4.2.3.1" xref="S2.SS1.p4.4.m4.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p4.4.m4.2.3.3.2" xref="S2.SS1.p4.4.m4.2.3.3.1.cmml"><mo id="S2.SS1.p4.4.m4.2.3.3.2.1" stretchy="false" xref="S2.SS1.p4.4.m4.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.4.m4.1.1" xref="S2.SS1.p4.4.m4.1.1.cmml">𝒞</mi><mo id="S2.SS1.p4.4.m4.2.3.3.2.2" xref="S2.SS1.p4.4.m4.2.3.3.1.cmml">;</mo><mi id="S2.SS1.p4.4.m4.2.2" xref="S2.SS1.p4.4.m4.2.2.cmml">M</mi><mo id="S2.SS1.p4.4.m4.2.3.3.2.3" stretchy="false" xref="S2.SS1.p4.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.4.m4.2b"><apply id="S2.SS1.p4.4.m4.2.3.cmml" xref="S2.SS1.p4.4.m4.2.3"><times id="S2.SS1.p4.4.m4.2.3.1.cmml" xref="S2.SS1.p4.4.m4.2.3.1"></times><apply id="S2.SS1.p4.4.m4.2.3.2.cmml" xref="S2.SS1.p4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.4.m4.2.3.2.1.cmml" xref="S2.SS1.p4.4.m4.2.3.2">superscript</csymbol><ci id="S2.SS1.p4.4.m4.2.3.2.2.cmml" xref="S2.SS1.p4.4.m4.2.3.2.2">𝐶</ci><times id="S2.SS1.p4.4.m4.2.3.2.3.cmml" xref="S2.SS1.p4.4.m4.2.3.2.3"></times></apply><list id="S2.SS1.p4.4.m4.2.3.3.1.cmml" xref="S2.SS1.p4.4.m4.2.3.3.2"><ci id="S2.SS1.p4.4.m4.1.1.cmml" xref="S2.SS1.p4.4.m4.1.1">𝒞</ci><ci id="S2.SS1.p4.4.m4.2.2.cmml" xref="S2.SS1.p4.4.m4.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.4.m4.2c">C^{*}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.4.m4.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> where for each <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S2.SS1.p4.5.m5.1"><semantics id="S2.SS1.p4.5.m5.1a"><mrow id="S2.SS1.p4.5.m5.1.1" xref="S2.SS1.p4.5.m5.1.1.cmml"><mi id="S2.SS1.p4.5.m5.1.1.2" xref="S2.SS1.p4.5.m5.1.1.2.cmml">n</mi><mo id="S2.SS1.p4.5.m5.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.cmml">≥</mo><mn id="S2.SS1.p4.5.m5.1.1.3" xref="S2.SS1.p4.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.5.m5.1b"><apply id="S2.SS1.p4.5.m5.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1"><geq id="S2.SS1.p4.5.m5.1.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1.1"></geq><ci id="S2.SS1.p4.5.m5.1.1.2.cmml" xref="S2.SS1.p4.5.m5.1.1.2">𝑛</ci><cn id="S2.SS1.p4.5.m5.1.1.3.cmml" type="integer" xref="S2.SS1.p4.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.5.m5.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.5.m5.1d">italic_n ≥ 0</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C^{n}(\mathcal{C};M)=\{f:N\mathcal{C}_{n}\to\coprod_{c\in\mathcal{C}}M(c)\,|\ % f(c_{0}\to\cdots\to c_{n})\in M(c_{n})\}" class="ltx_Math" display="block" id="S2.Ex16.m1.5"><semantics id="S2.Ex16.m1.5a"><mrow id="S2.Ex16.m1.5.5" xref="S2.Ex16.m1.5.5.cmml"><mrow id="S2.Ex16.m1.5.5.3" xref="S2.Ex16.m1.5.5.3.cmml"><msup id="S2.Ex16.m1.5.5.3.2" xref="S2.Ex16.m1.5.5.3.2.cmml"><mi id="S2.Ex16.m1.5.5.3.2.2" xref="S2.Ex16.m1.5.5.3.2.2.cmml">C</mi><mi id="S2.Ex16.m1.5.5.3.2.3" xref="S2.Ex16.m1.5.5.3.2.3.cmml">n</mi></msup><mo id="S2.Ex16.m1.5.5.3.1" xref="S2.Ex16.m1.5.5.3.1.cmml">⁢</mo><mrow id="S2.Ex16.m1.5.5.3.3.2" xref="S2.Ex16.m1.5.5.3.3.1.cmml"><mo id="S2.Ex16.m1.5.5.3.3.2.1" stretchy="false" xref="S2.Ex16.m1.5.5.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex16.m1.1.1" xref="S2.Ex16.m1.1.1.cmml">𝒞</mi><mo id="S2.Ex16.m1.5.5.3.3.2.2" xref="S2.Ex16.m1.5.5.3.3.1.cmml">;</mo><mi id="S2.Ex16.m1.2.2" xref="S2.Ex16.m1.2.2.cmml">M</mi><mo id="S2.Ex16.m1.5.5.3.3.2.3" stretchy="false" xref="S2.Ex16.m1.5.5.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex16.m1.5.5.2" xref="S2.Ex16.m1.5.5.2.cmml">=</mo><mrow id="S2.Ex16.m1.5.5.1.1" xref="S2.Ex16.m1.5.5.1.2.cmml"><mo id="S2.Ex16.m1.5.5.1.1.2" stretchy="false" xref="S2.Ex16.m1.5.5.1.2.1.cmml">{</mo><mi id="S2.Ex16.m1.4.4" xref="S2.Ex16.m1.4.4.cmml">f</mi><mo id="S2.Ex16.m1.5.5.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.Ex16.m1.5.5.1.2.1.cmml">:</mo><mrow id="S2.Ex16.m1.5.5.1.1.1" xref="S2.Ex16.m1.5.5.1.1.1.cmml"><mrow id="S2.Ex16.m1.5.5.1.1.1.4" xref="S2.Ex16.m1.5.5.1.1.1.4.cmml"><mi id="S2.Ex16.m1.5.5.1.1.1.4.2" xref="S2.Ex16.m1.5.5.1.1.1.4.2.cmml">N</mi><mo id="S2.Ex16.m1.5.5.1.1.1.4.1" xref="S2.Ex16.m1.5.5.1.1.1.4.1.cmml">⁢</mo><msub id="S2.Ex16.m1.5.5.1.1.1.4.3" xref="S2.Ex16.m1.5.5.1.1.1.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex16.m1.5.5.1.1.1.4.3.2" xref="S2.Ex16.m1.5.5.1.1.1.4.3.2.cmml">𝒞</mi><mi id="S2.Ex16.m1.5.5.1.1.1.4.3.3" xref="S2.Ex16.m1.5.5.1.1.1.4.3.3.cmml">n</mi></msub></mrow><mo id="S2.Ex16.m1.5.5.1.1.1.5" rspace="0.111em" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.5.cmml">→</mo><mrow id="S2.Ex16.m1.5.5.1.1.1.1" xref="S2.Ex16.m1.5.5.1.1.1.1.cmml"><mrow id="S2.Ex16.m1.5.5.1.1.1.1.3" xref="S2.Ex16.m1.5.5.1.1.1.1.3.cmml"><munder id="S2.Ex16.m1.5.5.1.1.1.1.3.1" xref="S2.Ex16.m1.5.5.1.1.1.1.3.1.cmml"><mo id="S2.Ex16.m1.5.5.1.1.1.1.3.1.2" movablelimits="false" xref="S2.Ex16.m1.5.5.1.1.1.1.3.1.2.cmml">∐</mo><mrow id="S2.Ex16.m1.5.5.1.1.1.1.3.1.3" xref="S2.Ex16.m1.5.5.1.1.1.1.3.1.3.cmml"><mi id="S2.Ex16.m1.5.5.1.1.1.1.3.1.3.2" xref="S2.Ex16.m1.5.5.1.1.1.1.3.1.3.2.cmml">c</mi><mo id="S2.Ex16.m1.5.5.1.1.1.1.3.1.3.1" xref="S2.Ex16.m1.5.5.1.1.1.1.3.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex16.m1.5.5.1.1.1.1.3.1.3.3" xref="S2.Ex16.m1.5.5.1.1.1.1.3.1.3.3.cmml">𝒞</mi></mrow></munder><mrow id="S2.Ex16.m1.5.5.1.1.1.1.3.2" xref="S2.Ex16.m1.5.5.1.1.1.1.3.2.cmml"><mi id="S2.Ex16.m1.5.5.1.1.1.1.3.2.2" xref="S2.Ex16.m1.5.5.1.1.1.1.3.2.2.cmml">M</mi><mo id="S2.Ex16.m1.5.5.1.1.1.1.3.2.1" xref="S2.Ex16.m1.5.5.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex16.m1.5.5.1.1.1.1.3.2.3.2" xref="S2.Ex16.m1.5.5.1.1.1.1.3.2.cmml"><mo id="S2.Ex16.m1.5.5.1.1.1.1.3.2.3.2.1" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.1.3.2.cmml">(</mo><mi id="S2.Ex16.m1.3.3" xref="S2.Ex16.m1.3.3.cmml">c</mi><mo id="S2.Ex16.m1.5.5.1.1.1.1.3.2.3.2.2" rspace="0.170em" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.1.3.2.cmml">)</mo></mrow></mrow></mrow><mo fence="false" id="S2.Ex16.m1.5.5.1.1.1.1.2" rspace="0.778em" xref="S2.Ex16.m1.5.5.1.1.1.1.2.cmml">|</mo><mrow id="S2.Ex16.m1.5.5.1.1.1.1.1" xref="S2.Ex16.m1.5.5.1.1.1.1.1.cmml"><mi id="S2.Ex16.m1.5.5.1.1.1.1.1.3" xref="S2.Ex16.m1.5.5.1.1.1.1.1.3.cmml">f</mi><mo id="S2.Ex16.m1.5.5.1.1.1.1.1.2" xref="S2.Ex16.m1.5.5.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.cmml"><msub id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.2" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.2.2" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.2.2.cmml">c</mi><mn id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.2.3" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.3.cmml">→</mo><mi id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.4" mathvariant="normal" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.4.cmml">⋯</mi><mo id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.5" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.5.cmml">→</mo><msub id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.6" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.6.cmml"><mi id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.6.2" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.6.2.cmml">c</mi><mi id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.6.3" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.6.3.cmml">n</mi></msub></mrow><mo id="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex16.m1.5.5.1.1.1.6" xref="S2.Ex16.m1.5.5.1.1.1.6.cmml">∈</mo><mrow id="S2.Ex16.m1.5.5.1.1.1.2" xref="S2.Ex16.m1.5.5.1.1.1.2.cmml"><mi id="S2.Ex16.m1.5.5.1.1.1.2.3" xref="S2.Ex16.m1.5.5.1.1.1.2.3.cmml">M</mi><mo id="S2.Ex16.m1.5.5.1.1.1.2.2" xref="S2.Ex16.m1.5.5.1.1.1.2.2.cmml">⁢</mo><mrow id="S2.Ex16.m1.5.5.1.1.1.2.1.1" xref="S2.Ex16.m1.5.5.1.1.1.2.1.1.1.cmml"><mo id="S2.Ex16.m1.5.5.1.1.1.2.1.1.2" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.2.1.1.1.cmml">(</mo><msub id="S2.Ex16.m1.5.5.1.1.1.2.1.1.1" xref="S2.Ex16.m1.5.5.1.1.1.2.1.1.1.cmml"><mi id="S2.Ex16.m1.5.5.1.1.1.2.1.1.1.2" xref="S2.Ex16.m1.5.5.1.1.1.2.1.1.1.2.cmml">c</mi><mi id="S2.Ex16.m1.5.5.1.1.1.2.1.1.1.3" xref="S2.Ex16.m1.5.5.1.1.1.2.1.1.1.3.cmml">n</mi></msub><mo id="S2.Ex16.m1.5.5.1.1.1.2.1.1.3" stretchy="false" xref="S2.Ex16.m1.5.5.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex16.m1.5.5.1.1.4" stretchy="false" xref="S2.Ex16.m1.5.5.1.2.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex16.m1.5b"><apply id="S2.Ex16.m1.5.5.cmml" xref="S2.Ex16.m1.5.5"><eq id="S2.Ex16.m1.5.5.2.cmml" xref="S2.Ex16.m1.5.5.2"></eq><apply id="S2.Ex16.m1.5.5.3.cmml" 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id="S2.Ex16.m1.5c">C^{n}(\mathcal{C};M)=\{f:N\mathcal{C}_{n}\to\coprod_{c\in\mathcal{C}}M(c)\,|\ % f(c_{0}\to\cdots\to c_{n})\in M(c_{n})\}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex16.m1.5d">italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M ) = { italic_f : italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → ∐ start_POSTSUBSCRIPT italic_c ∈ caligraphic_C end_POSTSUBSCRIPT italic_M ( italic_c ) | italic_f ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ⋯ → italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ italic_M ( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p4.6">and the coboundary map <math alttext="\delta^{n-1}:C^{n-1}(\mathcal{C};M)\to C^{n}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.SS1.p4.6.m1.4"><semantics 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id="S2.SS1.p4.6.m1.4.5.3.2.1.cmml" xref="S2.SS1.p4.6.m1.4.5.3.2.1"></times><apply id="S2.SS1.p4.6.m1.4.5.3.2.2.cmml" xref="S2.SS1.p4.6.m1.4.5.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p4.6.m1.4.5.3.2.2.1.cmml" xref="S2.SS1.p4.6.m1.4.5.3.2.2">superscript</csymbol><ci id="S2.SS1.p4.6.m1.4.5.3.2.2.2.cmml" xref="S2.SS1.p4.6.m1.4.5.3.2.2.2">𝐶</ci><apply id="S2.SS1.p4.6.m1.4.5.3.2.2.3.cmml" xref="S2.SS1.p4.6.m1.4.5.3.2.2.3"><minus id="S2.SS1.p4.6.m1.4.5.3.2.2.3.1.cmml" xref="S2.SS1.p4.6.m1.4.5.3.2.2.3.1"></minus><ci id="S2.SS1.p4.6.m1.4.5.3.2.2.3.2.cmml" xref="S2.SS1.p4.6.m1.4.5.3.2.2.3.2">𝑛</ci><cn id="S2.SS1.p4.6.m1.4.5.3.2.2.3.3.cmml" type="integer" xref="S2.SS1.p4.6.m1.4.5.3.2.2.3.3">1</cn></apply></apply><list id="S2.SS1.p4.6.m1.4.5.3.2.3.1.cmml" xref="S2.SS1.p4.6.m1.4.5.3.2.3.2"><ci id="S2.SS1.p4.6.m1.1.1.cmml" xref="S2.SS1.p4.6.m1.1.1">𝒞</ci><ci id="S2.SS1.p4.6.m1.2.2.cmml" xref="S2.SS1.p4.6.m1.2.2">𝑀</ci></list></apply><apply id="S2.SS1.p4.6.m1.4.5.3.3.cmml" xref="S2.SS1.p4.6.m1.4.5.3.3"><times id="S2.SS1.p4.6.m1.4.5.3.3.1.cmml" xref="S2.SS1.p4.6.m1.4.5.3.3.1"></times><apply id="S2.SS1.p4.6.m1.4.5.3.3.2.cmml" xref="S2.SS1.p4.6.m1.4.5.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.6.m1.4.5.3.3.2.1.cmml" xref="S2.SS1.p4.6.m1.4.5.3.3.2">superscript</csymbol><ci id="S2.SS1.p4.6.m1.4.5.3.3.2.2.cmml" xref="S2.SS1.p4.6.m1.4.5.3.3.2.2">𝐶</ci><ci id="S2.SS1.p4.6.m1.4.5.3.3.2.3.cmml" xref="S2.SS1.p4.6.m1.4.5.3.3.2.3">𝑛</ci></apply><list id="S2.SS1.p4.6.m1.4.5.3.3.3.1.cmml" xref="S2.SS1.p4.6.m1.4.5.3.3.3.2"><ci id="S2.SS1.p4.6.m1.3.3.cmml" xref="S2.SS1.p4.6.m1.3.3">𝒞</ci><ci id="S2.SS1.p4.6.m1.4.4.cmml" xref="S2.SS1.p4.6.m1.4.4">𝑀</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.6.m1.4c">\delta^{n-1}:C^{n-1}(\mathcal{C};M)\to C^{n}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.6.m1.4d">italic_δ start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT : italic_C start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M ) → italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(\delta^{n-1}f)(\sigma)=\sum_{i=0}^{n}(-1)^{i}f(d_{i}\sigma)+(-1)^{n}M(\alpha_% {n})f(d_{n}\sigma)" class="ltx_Math" display="block" id="S2.Ex17.m1.7"><semantics id="S2.Ex17.m1.7a"><mrow id="S2.Ex17.m1.7.7" xref="S2.Ex17.m1.7.7.cmml"><mrow id="S2.Ex17.m1.2.2.1" xref="S2.Ex17.m1.2.2.1.cmml"><mrow id="S2.Ex17.m1.2.2.1.1.1" xref="S2.Ex17.m1.2.2.1.1.1.1.cmml"><mo id="S2.Ex17.m1.2.2.1.1.1.2" stretchy="false" xref="S2.Ex17.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.Ex17.m1.2.2.1.1.1.1" xref="S2.Ex17.m1.2.2.1.1.1.1.cmml"><msup id="S2.Ex17.m1.2.2.1.1.1.1.2" xref="S2.Ex17.m1.2.2.1.1.1.1.2.cmml"><mi id="S2.Ex17.m1.2.2.1.1.1.1.2.2" xref="S2.Ex17.m1.2.2.1.1.1.1.2.2.cmml">δ</mi><mrow id="S2.Ex17.m1.2.2.1.1.1.1.2.3" xref="S2.Ex17.m1.2.2.1.1.1.1.2.3.cmml"><mi id="S2.Ex17.m1.2.2.1.1.1.1.2.3.2" xref="S2.Ex17.m1.2.2.1.1.1.1.2.3.2.cmml">n</mi><mo id="S2.Ex17.m1.2.2.1.1.1.1.2.3.1" xref="S2.Ex17.m1.2.2.1.1.1.1.2.3.1.cmml">−</mo><mn id="S2.Ex17.m1.2.2.1.1.1.1.2.3.3" xref="S2.Ex17.m1.2.2.1.1.1.1.2.3.3.cmml">1</mn></mrow></msup><mo id="S2.Ex17.m1.2.2.1.1.1.1.1" xref="S2.Ex17.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S2.Ex17.m1.2.2.1.1.1.1.3" xref="S2.Ex17.m1.2.2.1.1.1.1.3.cmml">f</mi></mrow><mo id="S2.Ex17.m1.2.2.1.1.1.3" stretchy="false" xref="S2.Ex17.m1.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex17.m1.2.2.1.2" xref="S2.Ex17.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.Ex17.m1.2.2.1.3.2" xref="S2.Ex17.m1.2.2.1.cmml"><mo id="S2.Ex17.m1.2.2.1.3.2.1" stretchy="false" xref="S2.Ex17.m1.2.2.1.cmml">(</mo><mi id="S2.Ex17.m1.1.1" xref="S2.Ex17.m1.1.1.cmml">σ</mi><mo id="S2.Ex17.m1.2.2.1.3.2.2" stretchy="false" xref="S2.Ex17.m1.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex17.m1.7.7.7" rspace="0.111em" xref="S2.Ex17.m1.7.7.7.cmml">=</mo><mrow id="S2.Ex17.m1.7.7.6" xref="S2.Ex17.m1.7.7.6.cmml"><mrow id="S2.Ex17.m1.4.4.3.2" xref="S2.Ex17.m1.4.4.3.2.cmml"><munderover id="S2.Ex17.m1.4.4.3.2.3" xref="S2.Ex17.m1.4.4.3.2.3.cmml"><mo id="S2.Ex17.m1.4.4.3.2.3.2.2" movablelimits="false" rspace="0em" xref="S2.Ex17.m1.4.4.3.2.3.2.2.cmml">∑</mo><mrow id="S2.Ex17.m1.4.4.3.2.3.2.3" xref="S2.Ex17.m1.4.4.3.2.3.2.3.cmml"><mi id="S2.Ex17.m1.4.4.3.2.3.2.3.2" xref="S2.Ex17.m1.4.4.3.2.3.2.3.2.cmml">i</mi><mo id="S2.Ex17.m1.4.4.3.2.3.2.3.1" xref="S2.Ex17.m1.4.4.3.2.3.2.3.1.cmml">=</mo><mn id="S2.Ex17.m1.4.4.3.2.3.2.3.3" xref="S2.Ex17.m1.4.4.3.2.3.2.3.3.cmml">0</mn></mrow><mi id="S2.Ex17.m1.4.4.3.2.3.3" xref="S2.Ex17.m1.4.4.3.2.3.3.cmml">n</mi></munderover><mrow id="S2.Ex17.m1.4.4.3.2.2" xref="S2.Ex17.m1.4.4.3.2.2.cmml"><msup 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id="S2.Ex17.m1.4.4.3.2.2.2.1.2" stretchy="false" xref="S2.Ex17.m1.4.4.3.2.2.2.1.1.cmml">(</mo><mrow id="S2.Ex17.m1.4.4.3.2.2.2.1.1" xref="S2.Ex17.m1.4.4.3.2.2.2.1.1.cmml"><msub id="S2.Ex17.m1.4.4.3.2.2.2.1.1.2" xref="S2.Ex17.m1.4.4.3.2.2.2.1.1.2.cmml"><mi id="S2.Ex17.m1.4.4.3.2.2.2.1.1.2.2" xref="S2.Ex17.m1.4.4.3.2.2.2.1.1.2.2.cmml">d</mi><mi id="S2.Ex17.m1.4.4.3.2.2.2.1.1.2.3" xref="S2.Ex17.m1.4.4.3.2.2.2.1.1.2.3.cmml">i</mi></msub><mo id="S2.Ex17.m1.4.4.3.2.2.2.1.1.1" xref="S2.Ex17.m1.4.4.3.2.2.2.1.1.1.cmml">⁢</mo><mi id="S2.Ex17.m1.4.4.3.2.2.2.1.1.3" xref="S2.Ex17.m1.4.4.3.2.2.2.1.1.3.cmml">σ</mi></mrow><mo id="S2.Ex17.m1.4.4.3.2.2.2.1.3" stretchy="false" xref="S2.Ex17.m1.4.4.3.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex17.m1.7.7.6.6" xref="S2.Ex17.m1.7.7.6.6.cmml">+</mo><mrow id="S2.Ex17.m1.7.7.6.5" xref="S2.Ex17.m1.7.7.6.5.cmml"><msup id="S2.Ex17.m1.5.5.4.3.1" xref="S2.Ex17.m1.5.5.4.3.1.cmml"><mrow id="S2.Ex17.m1.5.5.4.3.1.1.1" 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id="S2.Ex17.m1.7c">(\delta^{n-1}f)(\sigma)=\sum_{i=0}^{n}(-1)^{i}f(d_{i}\sigma)+(-1)^{n}M(\alpha_% {n})f(d_{n}\sigma)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex17.m1.7d">( italic_δ start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_f ) ( italic_σ ) = ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_f ( italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ ) + ( - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_M ( italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) italic_f ( italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_σ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p4.8">for every <math alttext="\sigma=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% 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xref="S2.SS1.p4.7.m1.1.1.1.1.1.3.2.3.2.2">𝑐</ci><ci id="S2.SS1.p4.7.m1.1.1.1.1.1.3.2.3.2.3.cmml" xref="S2.SS1.p4.7.m1.1.1.1.1.1.3.2.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.7.m1.1c">\sigma=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.7.m1.1d">italic_σ = ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> in <math alttext="N\mathcal{C}_{n}" class="ltx_Math" display="inline" id="S2.SS1.p4.8.m2.1"><semantics id="S2.SS1.p4.8.m2.1a"><mrow id="S2.SS1.p4.8.m2.1.1" xref="S2.SS1.p4.8.m2.1.1.cmml"><mi id="S2.SS1.p4.8.m2.1.1.2" xref="S2.SS1.p4.8.m2.1.1.2.cmml">N</mi><mo id="S2.SS1.p4.8.m2.1.1.1" xref="S2.SS1.p4.8.m2.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p4.8.m2.1.1.3" xref="S2.SS1.p4.8.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.8.m2.1.1.3.2" xref="S2.SS1.p4.8.m2.1.1.3.2.cmml">𝒞</mi><mi id="S2.SS1.p4.8.m2.1.1.3.3" xref="S2.SS1.p4.8.m2.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.8.m2.1b"><apply id="S2.SS1.p4.8.m2.1.1.cmml" xref="S2.SS1.p4.8.m2.1.1"><times id="S2.SS1.p4.8.m2.1.1.1.cmml" xref="S2.SS1.p4.8.m2.1.1.1"></times><ci id="S2.SS1.p4.8.m2.1.1.2.cmml" xref="S2.SS1.p4.8.m2.1.1.2">𝑁</ci><apply id="S2.SS1.p4.8.m2.1.1.3.cmml" xref="S2.SS1.p4.8.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.8.m2.1.1.3.1.cmml" xref="S2.SS1.p4.8.m2.1.1.3">subscript</csymbol><ci id="S2.SS1.p4.8.m2.1.1.3.2.cmml" xref="S2.SS1.p4.8.m2.1.1.3.2">𝒞</ci><ci id="S2.SS1.p4.8.m2.1.1.3.3.cmml" xref="S2.SS1.p4.8.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.8.m2.1c">N\mathcal{C}_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.8.m2.1d">italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. For more details on cohomology of categories we refer the reader to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib16" title="">16</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib17" title="">17</a>]</cite>, and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib21" title="">21</a>]</cite>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2. </span>Baues-Wirsching Cohomology</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.3">The Baues-Wirsching cohomology <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib1" title="">1</a>]</cite> of a small category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><ci id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">caligraphic_C</annotation></semantics></math> is a cohomology theory for <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><ci id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">caligraphic_C</annotation></semantics></math> with more general coefficients, called natural systems. Throughout the section, <math alttext="R" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m3.1"><semantics id="S2.SS2.p1.3.m3.1a"><mi id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m3.1b"><ci id="S2.SS2.p1.3.m3.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m3.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m3.1d">italic_R</annotation></semantics></math> denotes a commutative ring with unity.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.1.1.1">Definition 2.5</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem5.p1"> <p class="ltx_p" id="S2.Thmtheorem5.p1.6">The <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem5.p1.6.1">category of factorizations</em> of a small category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.1.m1.1"><semantics id="S2.Thmtheorem5.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem5.p1.1.m1.1.1" xref="S2.Thmtheorem5.p1.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.1.m1.1b"><ci id="S2.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem5.p1.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.1.m1.1d">caligraphic_C</annotation></semantics></math> is the category <math alttext="\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.2.m2.1"><semantics id="S2.Thmtheorem5.p1.2.m2.1a"><mrow id="S2.Thmtheorem5.p1.2.m2.1.1" xref="S2.Thmtheorem5.p1.2.m2.1.1.cmml"><mi id="S2.Thmtheorem5.p1.2.m2.1.1.2" xref="S2.Thmtheorem5.p1.2.m2.1.1.2.cmml">𝔉</mi><mo id="S2.Thmtheorem5.p1.2.m2.1.1.1" xref="S2.Thmtheorem5.p1.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem5.p1.2.m2.1.1.3" xref="S2.Thmtheorem5.p1.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.2.m2.1b"><apply id="S2.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem5.p1.2.m2.1.1"><times id="S2.Thmtheorem5.p1.2.m2.1.1.1.cmml" xref="S2.Thmtheorem5.p1.2.m2.1.1.1"></times><ci id="S2.Thmtheorem5.p1.2.m2.1.1.2.cmml" xref="S2.Thmtheorem5.p1.2.m2.1.1.2">𝔉</ci><ci id="S2.Thmtheorem5.p1.2.m2.1.1.3.cmml" xref="S2.Thmtheorem5.p1.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.2.m2.1c">\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.2.m2.1d">fraktur_F caligraphic_C</annotation></semantics></math> whose objects are the morphisms <math alttext="\alpha:x\to y" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.3.m3.1"><semantics id="S2.Thmtheorem5.p1.3.m3.1a"><mrow id="S2.Thmtheorem5.p1.3.m3.1.1" xref="S2.Thmtheorem5.p1.3.m3.1.1.cmml"><mi id="S2.Thmtheorem5.p1.3.m3.1.1.2" xref="S2.Thmtheorem5.p1.3.m3.1.1.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem5.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S2.Thmtheorem5.p1.3.m3.1.1.3" xref="S2.Thmtheorem5.p1.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem5.p1.3.m3.1.1.3.2" xref="S2.Thmtheorem5.p1.3.m3.1.1.3.2.cmml">x</mi><mo id="S2.Thmtheorem5.p1.3.m3.1.1.3.1" stretchy="false" xref="S2.Thmtheorem5.p1.3.m3.1.1.3.1.cmml">→</mo><mi id="S2.Thmtheorem5.p1.3.m3.1.1.3.3" xref="S2.Thmtheorem5.p1.3.m3.1.1.3.3.cmml">y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.3.m3.1b"><apply id="S2.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem5.p1.3.m3.1.1"><ci id="S2.Thmtheorem5.p1.3.m3.1.1.1.cmml" xref="S2.Thmtheorem5.p1.3.m3.1.1.1">:</ci><ci id="S2.Thmtheorem5.p1.3.m3.1.1.2.cmml" xref="S2.Thmtheorem5.p1.3.m3.1.1.2">𝛼</ci><apply id="S2.Thmtheorem5.p1.3.m3.1.1.3.cmml" xref="S2.Thmtheorem5.p1.3.m3.1.1.3"><ci id="S2.Thmtheorem5.p1.3.m3.1.1.3.1.cmml" xref="S2.Thmtheorem5.p1.3.m3.1.1.3.1">→</ci><ci id="S2.Thmtheorem5.p1.3.m3.1.1.3.2.cmml" xref="S2.Thmtheorem5.p1.3.m3.1.1.3.2">𝑥</ci><ci id="S2.Thmtheorem5.p1.3.m3.1.1.3.3.cmml" xref="S2.Thmtheorem5.p1.3.m3.1.1.3.3">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.3.m3.1c">\alpha:x\to y</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.3.m3.1d">italic_α : italic_x → italic_y</annotation></semantics></math> in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.4.m4.1"><semantics id="S2.Thmtheorem5.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem5.p1.4.m4.1.1" xref="S2.Thmtheorem5.p1.4.m4.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.4.m4.1b"><ci id="S2.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem5.p1.4.m4.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.4.m4.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.4.m4.1d">caligraphic_C</annotation></semantics></math>, and whose morphisms <math alttext="(u,v):\alpha\to\alpha^{\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.5.m5.2"><semantics id="S2.Thmtheorem5.p1.5.m5.2a"><mrow id="S2.Thmtheorem5.p1.5.m5.2.3" xref="S2.Thmtheorem5.p1.5.m5.2.3.cmml"><mrow id="S2.Thmtheorem5.p1.5.m5.2.3.2.2" xref="S2.Thmtheorem5.p1.5.m5.2.3.2.1.cmml"><mo id="S2.Thmtheorem5.p1.5.m5.2.3.2.2.1" stretchy="false" xref="S2.Thmtheorem5.p1.5.m5.2.3.2.1.cmml">(</mo><mi id="S2.Thmtheorem5.p1.5.m5.1.1" xref="S2.Thmtheorem5.p1.5.m5.1.1.cmml">u</mi><mo id="S2.Thmtheorem5.p1.5.m5.2.3.2.2.2" xref="S2.Thmtheorem5.p1.5.m5.2.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem5.p1.5.m5.2.2" xref="S2.Thmtheorem5.p1.5.m5.2.2.cmml">v</mi><mo id="S2.Thmtheorem5.p1.5.m5.2.3.2.2.3" rspace="0.278em" stretchy="false" xref="S2.Thmtheorem5.p1.5.m5.2.3.2.1.cmml">)</mo></mrow><mo id="S2.Thmtheorem5.p1.5.m5.2.3.1" rspace="0.278em" xref="S2.Thmtheorem5.p1.5.m5.2.3.1.cmml">:</mo><mrow id="S2.Thmtheorem5.p1.5.m5.2.3.3" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.cmml"><mi id="S2.Thmtheorem5.p1.5.m5.2.3.3.2" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.5.m5.2.3.3.1" stretchy="false" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.1.cmml">→</mo><msup id="S2.Thmtheorem5.p1.5.m5.2.3.3.3" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.3.cmml"><mi id="S2.Thmtheorem5.p1.5.m5.2.3.3.3.2" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.3.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.5.m5.2.3.3.3.3" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.5.m5.2b"><apply id="S2.Thmtheorem5.p1.5.m5.2.3.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3"><ci id="S2.Thmtheorem5.p1.5.m5.2.3.1.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.1">:</ci><interval closure="open" id="S2.Thmtheorem5.p1.5.m5.2.3.2.1.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.2.2"><ci id="S2.Thmtheorem5.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem5.p1.5.m5.1.1">𝑢</ci><ci id="S2.Thmtheorem5.p1.5.m5.2.2.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.2">𝑣</ci></interval><apply id="S2.Thmtheorem5.p1.5.m5.2.3.3.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.3"><ci id="S2.Thmtheorem5.p1.5.m5.2.3.3.1.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.1">→</ci><ci id="S2.Thmtheorem5.p1.5.m5.2.3.3.2.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.2">𝛼</ci><apply id="S2.Thmtheorem5.p1.5.m5.2.3.3.3.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.5.m5.2.3.3.3.1.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.3">superscript</csymbol><ci id="S2.Thmtheorem5.p1.5.m5.2.3.3.3.2.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.3.2">𝛼</ci><ci id="S2.Thmtheorem5.p1.5.m5.2.3.3.3.3.cmml" xref="S2.Thmtheorem5.p1.5.m5.2.3.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.5.m5.2c">(u,v):\alpha\to\alpha^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.5.m5.2d">( italic_u , italic_v ) : italic_α → italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are given by a pair of morphisms in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.6.m6.1"><semantics id="S2.Thmtheorem5.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem5.p1.6.m6.1.1" xref="S2.Thmtheorem5.p1.6.m6.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.6.m6.1b"><ci id="S2.Thmtheorem5.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem5.p1.6.m6.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.6.m6.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.6.m6.1d">caligraphic_C</annotation></semantics></math> such that the following diagram commutes:</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><svg class="ltx_picture ltx_markedasmath" height="70.55" id="S2.Ex18.m1.1.1.pic1" overflow="visible" version="1.1" width="78.43"><g transform="matrix(1 0 0 -1 13.34 15.93) translate(13.34,0)"><g transform="translate(-8.11,0) translate(4.15,0)"><foreignobject height="5.96" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="7.91"><math alttext="\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}" class="ltx_Math" display="inline" id="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex18.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_x</annotation></semantics></math></foreignobject></g><path class="droprule" d="M 8.11 -0.28 L 8.11 0.28" fill="none" stroke="#000000"></path><g transform="translate(18.37,0) translate(0,6.24) translate(4.15,0) translate(4.15,0) translate(0,-2.09)"><foreignobject height="4.17" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.2"><math alttext="\scriptstyle{\alpha}" class="ltx_Math" display="inline" id="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" mathsize="70%" xref="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\scriptstyle{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex18.m1.1.1.pic1.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_α</annotation></semantics></math></foreignobject></g><g transform="translate(43.44,0)"><path d="M 0 0 A 13.84 13.84 45 0 0 -6.92 2.77" fill="none" stroke="#000000"></path><path d="M 0 0 A 13.84 13.84 45 0 1 -6.92 -2.77" fill="none" stroke="#000000"></path></g><path class="droprule" d="M 8.11 -0.28 L 43.44 0.28" fill="none" stroke="#000000"></path><path class="droprule" d="M 43.44 -0.28 L 43.44 0.28" fill="none" stroke="#000000"></path><g transform="translate(43.44,0) translate(4.15,0)"><foreignobject height="8.65" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="7.28"><math alttext="\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}" class="ltx_Math" display="inline" id="S2.Ex18.m1.1.1.pic1.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.m1.1"><semantics 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stroke="#000000"></path></g><path class="droprule" d="M 9.17 -51.51 L 42.38 -50.96" fill="none" stroke="#000000"></path><path class="droprule" d="M 42.38 -51.51 L 42.38 -50.96" fill="none" stroke="#000000"></path><g transform="translate(42.38,0) translate(0,-51.24) translate(4.15,0)"><foreignobject height="10.8" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="9.41"><math alttext="\textstyle{y^{\prime}}" class="ltx_Math" display="inline" id="S2.Ex18.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S2.Ex18.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1a"><msup id="S2.Ex18.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S2.Ex18.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S2.Ex18.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S2.Ex18.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">y</mi><mo id="S2.Ex18.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.3" 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encoding="application/x-llamapun" id="S2.Ex18.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1d">italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math></foreignobject></g></g></svg></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem5.p1.14">i.e., <math alttext="\alpha^{\prime}=u\alpha v" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.7.m1.1"><semantics id="S2.Thmtheorem5.p1.7.m1.1a"><mrow id="S2.Thmtheorem5.p1.7.m1.1.1" xref="S2.Thmtheorem5.p1.7.m1.1.1.cmml"><msup id="S2.Thmtheorem5.p1.7.m1.1.1.2" xref="S2.Thmtheorem5.p1.7.m1.1.1.2.cmml"><mi id="S2.Thmtheorem5.p1.7.m1.1.1.2.2" xref="S2.Thmtheorem5.p1.7.m1.1.1.2.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.7.m1.1.1.2.3" xref="S2.Thmtheorem5.p1.7.m1.1.1.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem5.p1.7.m1.1.1.1" xref="S2.Thmtheorem5.p1.7.m1.1.1.1.cmml">=</mo><mrow id="S2.Thmtheorem5.p1.7.m1.1.1.3" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.cmml"><mi id="S2.Thmtheorem5.p1.7.m1.1.1.3.2" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.2.cmml">u</mi><mo id="S2.Thmtheorem5.p1.7.m1.1.1.3.1" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.Thmtheorem5.p1.7.m1.1.1.3.3" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.3.cmml">α</mi><mo id="S2.Thmtheorem5.p1.7.m1.1.1.3.1a" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.Thmtheorem5.p1.7.m1.1.1.3.4" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.4.cmml">v</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.7.m1.1b"><apply id="S2.Thmtheorem5.p1.7.m1.1.1.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1"><eq id="S2.Thmtheorem5.p1.7.m1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.1"></eq><apply id="S2.Thmtheorem5.p1.7.m1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.7.m1.1.1.2.1.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem5.p1.7.m1.1.1.2.2.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.2.2">𝛼</ci><ci id="S2.Thmtheorem5.p1.7.m1.1.1.2.3.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.2.3">′</ci></apply><apply id="S2.Thmtheorem5.p1.7.m1.1.1.3.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.3"><times id="S2.Thmtheorem5.p1.7.m1.1.1.3.1.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.1"></times><ci id="S2.Thmtheorem5.p1.7.m1.1.1.3.2.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.2">𝑢</ci><ci id="S2.Thmtheorem5.p1.7.m1.1.1.3.3.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.3">𝛼</ci><ci id="S2.Thmtheorem5.p1.7.m1.1.1.3.4.cmml" xref="S2.Thmtheorem5.p1.7.m1.1.1.3.4">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.7.m1.1c">\alpha^{\prime}=u\alpha v</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.7.m1.1d">italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_u italic_α italic_v</annotation></semantics></math>. The composition <math alttext="(u^{\prime},v^{\prime})\circ(u,v)" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.8.m2.4"><semantics id="S2.Thmtheorem5.p1.8.m2.4a"><mrow id="S2.Thmtheorem5.p1.8.m2.4.4" xref="S2.Thmtheorem5.p1.8.m2.4.4.cmml"><mrow id="S2.Thmtheorem5.p1.8.m2.4.4.2.2" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.3.cmml"><mo id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.3" stretchy="false" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.3.cmml">(</mo><msup id="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1" xref="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.cmml"><mi id="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.2" xref="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.3" xref="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.4" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.3.cmml">,</mo><msup id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.cmml"><mi id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.2" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.2.cmml">v</mi><mo id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.3" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.5" rspace="0.055em" stretchy="false" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.3.cmml">)</mo></mrow><mo id="S2.Thmtheorem5.p1.8.m2.4.4.3" rspace="0.222em" xref="S2.Thmtheorem5.p1.8.m2.4.4.3.cmml">∘</mo><mrow id="S2.Thmtheorem5.p1.8.m2.4.4.4.2" xref="S2.Thmtheorem5.p1.8.m2.4.4.4.1.cmml"><mo id="S2.Thmtheorem5.p1.8.m2.4.4.4.2.1" stretchy="false" xref="S2.Thmtheorem5.p1.8.m2.4.4.4.1.cmml">(</mo><mi id="S2.Thmtheorem5.p1.8.m2.1.1" xref="S2.Thmtheorem5.p1.8.m2.1.1.cmml">u</mi><mo id="S2.Thmtheorem5.p1.8.m2.4.4.4.2.2" xref="S2.Thmtheorem5.p1.8.m2.4.4.4.1.cmml">,</mo><mi id="S2.Thmtheorem5.p1.8.m2.2.2" xref="S2.Thmtheorem5.p1.8.m2.2.2.cmml">v</mi><mo id="S2.Thmtheorem5.p1.8.m2.4.4.4.2.3" stretchy="false" xref="S2.Thmtheorem5.p1.8.m2.4.4.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.8.m2.4b"><apply id="S2.Thmtheorem5.p1.8.m2.4.4.cmml" xref="S2.Thmtheorem5.p1.8.m2.4.4"><compose id="S2.Thmtheorem5.p1.8.m2.4.4.3.cmml" xref="S2.Thmtheorem5.p1.8.m2.4.4.3"></compose><interval closure="open" id="S2.Thmtheorem5.p1.8.m2.4.4.2.3.cmml" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.2"><apply id="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.cmml" xref="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.3.cmml" xref="S2.Thmtheorem5.p1.8.m2.3.3.1.1.1.3">′</ci></apply><apply id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.cmml" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.1.cmml" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2">superscript</csymbol><ci id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.2.cmml" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.2">𝑣</ci><ci id="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.3.cmml" xref="S2.Thmtheorem5.p1.8.m2.4.4.2.2.2.3">′</ci></apply></interval><interval closure="open" id="S2.Thmtheorem5.p1.8.m2.4.4.4.1.cmml" xref="S2.Thmtheorem5.p1.8.m2.4.4.4.2"><ci id="S2.Thmtheorem5.p1.8.m2.1.1.cmml" xref="S2.Thmtheorem5.p1.8.m2.1.1">𝑢</ci><ci id="S2.Thmtheorem5.p1.8.m2.2.2.cmml" xref="S2.Thmtheorem5.p1.8.m2.2.2">𝑣</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.8.m2.4c">(u^{\prime},v^{\prime})\circ(u,v)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.8.m2.4d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∘ ( italic_u , italic_v )</annotation></semantics></math> of two morphisms <math alttext="(u,v):\alpha\to\alpha^{\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.9.m3.2"><semantics id="S2.Thmtheorem5.p1.9.m3.2a"><mrow id="S2.Thmtheorem5.p1.9.m3.2.3" xref="S2.Thmtheorem5.p1.9.m3.2.3.cmml"><mrow id="S2.Thmtheorem5.p1.9.m3.2.3.2.2" xref="S2.Thmtheorem5.p1.9.m3.2.3.2.1.cmml"><mo id="S2.Thmtheorem5.p1.9.m3.2.3.2.2.1" stretchy="false" xref="S2.Thmtheorem5.p1.9.m3.2.3.2.1.cmml">(</mo><mi id="S2.Thmtheorem5.p1.9.m3.1.1" xref="S2.Thmtheorem5.p1.9.m3.1.1.cmml">u</mi><mo id="S2.Thmtheorem5.p1.9.m3.2.3.2.2.2" xref="S2.Thmtheorem5.p1.9.m3.2.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem5.p1.9.m3.2.2" xref="S2.Thmtheorem5.p1.9.m3.2.2.cmml">v</mi><mo id="S2.Thmtheorem5.p1.9.m3.2.3.2.2.3" rspace="0.278em" stretchy="false" xref="S2.Thmtheorem5.p1.9.m3.2.3.2.1.cmml">)</mo></mrow><mo id="S2.Thmtheorem5.p1.9.m3.2.3.1" rspace="0.278em" xref="S2.Thmtheorem5.p1.9.m3.2.3.1.cmml">:</mo><mrow id="S2.Thmtheorem5.p1.9.m3.2.3.3" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.cmml"><mi id="S2.Thmtheorem5.p1.9.m3.2.3.3.2" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.9.m3.2.3.3.1" stretchy="false" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.1.cmml">→</mo><msup id="S2.Thmtheorem5.p1.9.m3.2.3.3.3" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.3.cmml"><mi id="S2.Thmtheorem5.p1.9.m3.2.3.3.3.2" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.3.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.9.m3.2.3.3.3.3" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.9.m3.2b"><apply id="S2.Thmtheorem5.p1.9.m3.2.3.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3"><ci id="S2.Thmtheorem5.p1.9.m3.2.3.1.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.1">:</ci><interval closure="open" id="S2.Thmtheorem5.p1.9.m3.2.3.2.1.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.2.2"><ci id="S2.Thmtheorem5.p1.9.m3.1.1.cmml" xref="S2.Thmtheorem5.p1.9.m3.1.1">𝑢</ci><ci id="S2.Thmtheorem5.p1.9.m3.2.2.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.2">𝑣</ci></interval><apply id="S2.Thmtheorem5.p1.9.m3.2.3.3.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.3"><ci id="S2.Thmtheorem5.p1.9.m3.2.3.3.1.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.1">→</ci><ci id="S2.Thmtheorem5.p1.9.m3.2.3.3.2.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.2">𝛼</ci><apply id="S2.Thmtheorem5.p1.9.m3.2.3.3.3.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.9.m3.2.3.3.3.1.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.3">superscript</csymbol><ci id="S2.Thmtheorem5.p1.9.m3.2.3.3.3.2.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.3.2">𝛼</ci><ci id="S2.Thmtheorem5.p1.9.m3.2.3.3.3.3.cmml" xref="S2.Thmtheorem5.p1.9.m3.2.3.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.9.m3.2c">(u,v):\alpha\to\alpha^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.9.m3.2d">( italic_u , italic_v ) : italic_α → italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="(u^{\prime},v^{\prime}):\alpha^{\prime}\to\alpha^{\prime\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.10.m4.2"><semantics id="S2.Thmtheorem5.p1.10.m4.2a"><mrow id="S2.Thmtheorem5.p1.10.m4.2.2" xref="S2.Thmtheorem5.p1.10.m4.2.2.cmml"><mrow id="S2.Thmtheorem5.p1.10.m4.2.2.2.2" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.3.cmml"><mo id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.3" stretchy="false" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.3.cmml">(</mo><msup id="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1" xref="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.2" xref="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.3" xref="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.4" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.3.cmml">,</mo><msup id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.cmml"><mi id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.2" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.2.cmml">v</mi><mo id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.3" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.5" rspace="0.278em" stretchy="false" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.3.cmml">)</mo></mrow><mo id="S2.Thmtheorem5.p1.10.m4.2.2.3" rspace="0.278em" xref="S2.Thmtheorem5.p1.10.m4.2.2.3.cmml">:</mo><mrow id="S2.Thmtheorem5.p1.10.m4.2.2.4" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.cmml"><msup id="S2.Thmtheorem5.p1.10.m4.2.2.4.2" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.2.cmml"><mi id="S2.Thmtheorem5.p1.10.m4.2.2.4.2.2" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.2.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.10.m4.2.2.4.2.3" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem5.p1.10.m4.2.2.4.1" stretchy="false" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.1.cmml">→</mo><msup id="S2.Thmtheorem5.p1.10.m4.2.2.4.3" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.3.cmml"><mi id="S2.Thmtheorem5.p1.10.m4.2.2.4.3.2" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.3.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.10.m4.2.2.4.3.3" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.3.3.cmml">′′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.10.m4.2b"><apply id="S2.Thmtheorem5.p1.10.m4.2.2.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2"><ci id="S2.Thmtheorem5.p1.10.m4.2.2.3.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.3">:</ci><interval closure="open" id="S2.Thmtheorem5.p1.10.m4.2.2.2.3.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.2"><apply id="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem5.p1.10.m4.1.1.1.1.1.3">′</ci></apply><apply id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2">superscript</csymbol><ci id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.2">𝑣</ci><ci id="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.3.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.2.2.2.3">′</ci></apply></interval><apply id="S2.Thmtheorem5.p1.10.m4.2.2.4.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4"><ci id="S2.Thmtheorem5.p1.10.m4.2.2.4.1.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.1">→</ci><apply id="S2.Thmtheorem5.p1.10.m4.2.2.4.2.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.2"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.10.m4.2.2.4.2.1.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.2">superscript</csymbol><ci id="S2.Thmtheorem5.p1.10.m4.2.2.4.2.2.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.2.2">𝛼</ci><ci id="S2.Thmtheorem5.p1.10.m4.2.2.4.2.3.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.2.3">′</ci></apply><apply id="S2.Thmtheorem5.p1.10.m4.2.2.4.3.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.3"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.10.m4.2.2.4.3.1.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.3">superscript</csymbol><ci id="S2.Thmtheorem5.p1.10.m4.2.2.4.3.2.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.3.2">𝛼</ci><ci id="S2.Thmtheorem5.p1.10.m4.2.2.4.3.3.cmml" xref="S2.Thmtheorem5.p1.10.m4.2.2.4.3.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.10.m4.2c">(u^{\prime},v^{\prime}):\alpha^{\prime}\to\alpha^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.10.m4.2d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) : italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_α start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is defined by <math alttext="(u^{\prime}u,vv^{\prime}):\alpha\to\alpha^{\prime\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.11.m5.2"><semantics id="S2.Thmtheorem5.p1.11.m5.2a"><mrow id="S2.Thmtheorem5.p1.11.m5.2.2" xref="S2.Thmtheorem5.p1.11.m5.2.2.cmml"><mrow id="S2.Thmtheorem5.p1.11.m5.2.2.2.2" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.3.cmml"><mo id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.3" stretchy="false" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.3.cmml">(</mo><mrow id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.cmml"><msup id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.cmml"><mi id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.2" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.2.cmml">u</mi><mo id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.3" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.1" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.3" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.4" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.3.cmml">,</mo><mrow id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.cmml"><mi id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.2" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.2.cmml">v</mi><mo id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.1" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.1.cmml">⁢</mo><msup id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.cmml"><mi id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.2" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.2.cmml">v</mi><mo id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.3" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.3.cmml">′</mo></msup></mrow><mo id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.5" rspace="0.278em" stretchy="false" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.3.cmml">)</mo></mrow><mo id="S2.Thmtheorem5.p1.11.m5.2.2.3" rspace="0.278em" xref="S2.Thmtheorem5.p1.11.m5.2.2.3.cmml">:</mo><mrow id="S2.Thmtheorem5.p1.11.m5.2.2.4" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.cmml"><mi id="S2.Thmtheorem5.p1.11.m5.2.2.4.2" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.11.m5.2.2.4.1" stretchy="false" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.1.cmml">→</mo><msup id="S2.Thmtheorem5.p1.11.m5.2.2.4.3" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.3.cmml"><mi id="S2.Thmtheorem5.p1.11.m5.2.2.4.3.2" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.3.2.cmml">α</mi><mo id="S2.Thmtheorem5.p1.11.m5.2.2.4.3.3" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.3.3.cmml">′′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.11.m5.2b"><apply id="S2.Thmtheorem5.p1.11.m5.2.2.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2"><ci id="S2.Thmtheorem5.p1.11.m5.2.2.3.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.3">:</ci><interval closure="open" id="S2.Thmtheorem5.p1.11.m5.2.2.2.3.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2"><apply id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1"><times id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.1"></times><apply id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.2.cmml" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.2">𝑢</ci><ci id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.3.cmml" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.2.3">′</ci></apply><ci id="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem5.p1.11.m5.1.1.1.1.1.3">𝑢</ci></apply><apply id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2"><times id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.1"></times><ci id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.2">𝑣</ci><apply id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.1.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3">superscript</csymbol><ci id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.2.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.2">𝑣</ci><ci id="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.3.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.2.2.2.3.3">′</ci></apply></apply></interval><apply id="S2.Thmtheorem5.p1.11.m5.2.2.4.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.4"><ci id="S2.Thmtheorem5.p1.11.m5.2.2.4.1.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.1">→</ci><ci id="S2.Thmtheorem5.p1.11.m5.2.2.4.2.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.2">𝛼</ci><apply id="S2.Thmtheorem5.p1.11.m5.2.2.4.3.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.3"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.11.m5.2.2.4.3.1.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.3">superscript</csymbol><ci id="S2.Thmtheorem5.p1.11.m5.2.2.4.3.2.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.3.2">𝛼</ci><ci id="S2.Thmtheorem5.p1.11.m5.2.2.4.3.3.cmml" xref="S2.Thmtheorem5.p1.11.m5.2.2.4.3.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.11.m5.2c">(u^{\prime}u,vv^{\prime}):\alpha\to\alpha^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.11.m5.2d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_u , italic_v italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) : italic_α → italic_α start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. A functor <math alttext="M:\mathfrak{F}\mathcal{C}\to R" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.12.m6.1"><semantics id="S2.Thmtheorem5.p1.12.m6.1a"><mrow id="S2.Thmtheorem5.p1.12.m6.1.1" xref="S2.Thmtheorem5.p1.12.m6.1.1.cmml"><mi id="S2.Thmtheorem5.p1.12.m6.1.1.2" xref="S2.Thmtheorem5.p1.12.m6.1.1.2.cmml">M</mi><mo id="S2.Thmtheorem5.p1.12.m6.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem5.p1.12.m6.1.1.1.cmml">:</mo><mrow id="S2.Thmtheorem5.p1.12.m6.1.1.3" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.cmml"><mrow id="S2.Thmtheorem5.p1.12.m6.1.1.3.2" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.2.cmml"><mi id="S2.Thmtheorem5.p1.12.m6.1.1.3.2.2" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.2.2.cmml">𝔉</mi><mo id="S2.Thmtheorem5.p1.12.m6.1.1.3.2.1" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem5.p1.12.m6.1.1.3.2.3" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.2.3.cmml">𝒞</mi></mrow><mo id="S2.Thmtheorem5.p1.12.m6.1.1.3.1" stretchy="false" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.1.cmml">→</mo><mi id="S2.Thmtheorem5.p1.12.m6.1.1.3.3" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.12.m6.1b"><apply id="S2.Thmtheorem5.p1.12.m6.1.1.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1"><ci id="S2.Thmtheorem5.p1.12.m6.1.1.1.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.1">:</ci><ci id="S2.Thmtheorem5.p1.12.m6.1.1.2.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.2">𝑀</ci><apply id="S2.Thmtheorem5.p1.12.m6.1.1.3.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.3"><ci id="S2.Thmtheorem5.p1.12.m6.1.1.3.1.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.1">→</ci><apply id="S2.Thmtheorem5.p1.12.m6.1.1.3.2.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.2"><times id="S2.Thmtheorem5.p1.12.m6.1.1.3.2.1.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.2.1"></times><ci id="S2.Thmtheorem5.p1.12.m6.1.1.3.2.2.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.2.2">𝔉</ci><ci id="S2.Thmtheorem5.p1.12.m6.1.1.3.2.3.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.2.3">𝒞</ci></apply><ci id="S2.Thmtheorem5.p1.12.m6.1.1.3.3.cmml" xref="S2.Thmtheorem5.p1.12.m6.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.12.m6.1c">M:\mathfrak{F}\mathcal{C}\to R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.12.m6.1d">italic_M : fraktur_F caligraphic_C → italic_R</annotation></semantics></math>-Mod is called a <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem5.p1.14.1">natural system</em> for <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.13.m7.1"><semantics id="S2.Thmtheorem5.p1.13.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem5.p1.13.m7.1.1" xref="S2.Thmtheorem5.p1.13.m7.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.13.m7.1b"><ci id="S2.Thmtheorem5.p1.13.m7.1.1.cmml" xref="S2.Thmtheorem5.p1.13.m7.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.13.m7.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.13.m7.1d">caligraphic_C</annotation></semantics></math> over <math alttext="R" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.14.m8.1"><semantics id="S2.Thmtheorem5.p1.14.m8.1a"><mi id="S2.Thmtheorem5.p1.14.m8.1.1" xref="S2.Thmtheorem5.p1.14.m8.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.14.m8.1b"><ci id="S2.Thmtheorem5.p1.14.m8.1.1.cmml" xref="S2.Thmtheorem5.p1.14.m8.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.14.m8.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.14.m8.1d">italic_R</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.11">For the morphism <math alttext="(u,1):\alpha\to u\alpha" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.2"><semantics id="S2.SS2.p2.1.m1.2a"><mrow id="S2.SS2.p2.1.m1.2.3" xref="S2.SS2.p2.1.m1.2.3.cmml"><mrow id="S2.SS2.p2.1.m1.2.3.2.2" xref="S2.SS2.p2.1.m1.2.3.2.1.cmml"><mo id="S2.SS2.p2.1.m1.2.3.2.2.1" stretchy="false" xref="S2.SS2.p2.1.m1.2.3.2.1.cmml">(</mo><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">u</mi><mo id="S2.SS2.p2.1.m1.2.3.2.2.2" xref="S2.SS2.p2.1.m1.2.3.2.1.cmml">,</mo><mn id="S2.SS2.p2.1.m1.2.2" xref="S2.SS2.p2.1.m1.2.2.cmml">1</mn><mo id="S2.SS2.p2.1.m1.2.3.2.2.3" rspace="0.278em" stretchy="false" xref="S2.SS2.p2.1.m1.2.3.2.1.cmml">)</mo></mrow><mo id="S2.SS2.p2.1.m1.2.3.1" rspace="0.278em" xref="S2.SS2.p2.1.m1.2.3.1.cmml">:</mo><mrow id="S2.SS2.p2.1.m1.2.3.3" xref="S2.SS2.p2.1.m1.2.3.3.cmml"><mi id="S2.SS2.p2.1.m1.2.3.3.2" xref="S2.SS2.p2.1.m1.2.3.3.2.cmml">α</mi><mo id="S2.SS2.p2.1.m1.2.3.3.1" stretchy="false" xref="S2.SS2.p2.1.m1.2.3.3.1.cmml">→</mo><mrow id="S2.SS2.p2.1.m1.2.3.3.3" xref="S2.SS2.p2.1.m1.2.3.3.3.cmml"><mi id="S2.SS2.p2.1.m1.2.3.3.3.2" xref="S2.SS2.p2.1.m1.2.3.3.3.2.cmml">u</mi><mo id="S2.SS2.p2.1.m1.2.3.3.3.1" xref="S2.SS2.p2.1.m1.2.3.3.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.1.m1.2.3.3.3.3" xref="S2.SS2.p2.1.m1.2.3.3.3.3.cmml">α</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.2b"><apply id="S2.SS2.p2.1.m1.2.3.cmml" xref="S2.SS2.p2.1.m1.2.3"><ci id="S2.SS2.p2.1.m1.2.3.1.cmml" xref="S2.SS2.p2.1.m1.2.3.1">:</ci><interval closure="open" id="S2.SS2.p2.1.m1.2.3.2.1.cmml" xref="S2.SS2.p2.1.m1.2.3.2.2"><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝑢</ci><cn id="S2.SS2.p2.1.m1.2.2.cmml" type="integer" xref="S2.SS2.p2.1.m1.2.2">1</cn></interval><apply id="S2.SS2.p2.1.m1.2.3.3.cmml" xref="S2.SS2.p2.1.m1.2.3.3"><ci id="S2.SS2.p2.1.m1.2.3.3.1.cmml" xref="S2.SS2.p2.1.m1.2.3.3.1">→</ci><ci id="S2.SS2.p2.1.m1.2.3.3.2.cmml" xref="S2.SS2.p2.1.m1.2.3.3.2">𝛼</ci><apply id="S2.SS2.p2.1.m1.2.3.3.3.cmml" xref="S2.SS2.p2.1.m1.2.3.3.3"><times id="S2.SS2.p2.1.m1.2.3.3.3.1.cmml" xref="S2.SS2.p2.1.m1.2.3.3.3.1"></times><ci id="S2.SS2.p2.1.m1.2.3.3.3.2.cmml" xref="S2.SS2.p2.1.m1.2.3.3.3.2">𝑢</ci><ci id="S2.SS2.p2.1.m1.2.3.3.3.3.cmml" xref="S2.SS2.p2.1.m1.2.3.3.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.2c">(u,1):\alpha\to u\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.2d">( italic_u , 1 ) : italic_α → italic_u italic_α</annotation></semantics></math> in <math alttext="\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.1"><semantics id="S2.SS2.p2.2.m2.1a"><mrow id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml"><mi id="S2.SS2.p2.2.m2.1.1.2" xref="S2.SS2.p2.2.m2.1.1.2.cmml">𝔉</mi><mo id="S2.SS2.p2.2.m2.1.1.1" xref="S2.SS2.p2.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.2.m2.1.1.3" xref="S2.SS2.p2.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.1b"><apply id="S2.SS2.p2.2.m2.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1"><times id="S2.SS2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1.1"></times><ci id="S2.SS2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.p2.2.m2.1.1.2">𝔉</ci><ci id="S2.SS2.p2.2.m2.1.1.3.cmml" xref="S2.SS2.p2.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">fraktur_F caligraphic_C</annotation></semantics></math>, the induced map <math alttext="M(u,1):M(\alpha)\to M(u\alpha)" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.4"><semantics id="S2.SS2.p2.3.m3.4a"><mrow id="S2.SS2.p2.3.m3.4.4" xref="S2.SS2.p2.3.m3.4.4.cmml"><mrow id="S2.SS2.p2.3.m3.4.4.3" xref="S2.SS2.p2.3.m3.4.4.3.cmml"><mi id="S2.SS2.p2.3.m3.4.4.3.2" xref="S2.SS2.p2.3.m3.4.4.3.2.cmml">M</mi><mo id="S2.SS2.p2.3.m3.4.4.3.1" xref="S2.SS2.p2.3.m3.4.4.3.1.cmml">⁢</mo><mrow id="S2.SS2.p2.3.m3.4.4.3.3.2" xref="S2.SS2.p2.3.m3.4.4.3.3.1.cmml"><mo id="S2.SS2.p2.3.m3.4.4.3.3.2.1" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.3.3.1.cmml">(</mo><mi id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml">u</mi><mo id="S2.SS2.p2.3.m3.4.4.3.3.2.2" xref="S2.SS2.p2.3.m3.4.4.3.3.1.cmml">,</mo><mn id="S2.SS2.p2.3.m3.2.2" xref="S2.SS2.p2.3.m3.2.2.cmml">1</mn><mo id="S2.SS2.p2.3.m3.4.4.3.3.2.3" rspace="0.278em" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.3.m3.4.4.2" rspace="0.278em" xref="S2.SS2.p2.3.m3.4.4.2.cmml">:</mo><mrow id="S2.SS2.p2.3.m3.4.4.1" xref="S2.SS2.p2.3.m3.4.4.1.cmml"><mrow id="S2.SS2.p2.3.m3.4.4.1.3" xref="S2.SS2.p2.3.m3.4.4.1.3.cmml"><mi id="S2.SS2.p2.3.m3.4.4.1.3.2" xref="S2.SS2.p2.3.m3.4.4.1.3.2.cmml">M</mi><mo id="S2.SS2.p2.3.m3.4.4.1.3.1" xref="S2.SS2.p2.3.m3.4.4.1.3.1.cmml">⁢</mo><mrow id="S2.SS2.p2.3.m3.4.4.1.3.3.2" xref="S2.SS2.p2.3.m3.4.4.1.3.cmml"><mo id="S2.SS2.p2.3.m3.4.4.1.3.3.2.1" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.1.3.cmml">(</mo><mi id="S2.SS2.p2.3.m3.3.3" xref="S2.SS2.p2.3.m3.3.3.cmml">α</mi><mo id="S2.SS2.p2.3.m3.4.4.1.3.3.2.2" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.1.3.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.3.m3.4.4.1.2" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.1.2.cmml">→</mo><mrow id="S2.SS2.p2.3.m3.4.4.1.1" xref="S2.SS2.p2.3.m3.4.4.1.1.cmml"><mi id="S2.SS2.p2.3.m3.4.4.1.1.3" xref="S2.SS2.p2.3.m3.4.4.1.1.3.cmml">M</mi><mo id="S2.SS2.p2.3.m3.4.4.1.1.2" xref="S2.SS2.p2.3.m3.4.4.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.3.m3.4.4.1.1.1.1" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.cmml"><mo id="S2.SS2.p2.3.m3.4.4.1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p2.3.m3.4.4.1.1.1.1.1" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.cmml"><mi id="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.2" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.2.cmml">u</mi><mo id="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.1" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.3" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.3.cmml">α</mi></mrow><mo id="S2.SS2.p2.3.m3.4.4.1.1.1.1.3" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m3.4b"><apply id="S2.SS2.p2.3.m3.4.4.cmml" xref="S2.SS2.p2.3.m3.4.4"><ci id="S2.SS2.p2.3.m3.4.4.2.cmml" xref="S2.SS2.p2.3.m3.4.4.2">:</ci><apply id="S2.SS2.p2.3.m3.4.4.3.cmml" xref="S2.SS2.p2.3.m3.4.4.3"><times id="S2.SS2.p2.3.m3.4.4.3.1.cmml" xref="S2.SS2.p2.3.m3.4.4.3.1"></times><ci id="S2.SS2.p2.3.m3.4.4.3.2.cmml" xref="S2.SS2.p2.3.m3.4.4.3.2">𝑀</ci><interval closure="open" id="S2.SS2.p2.3.m3.4.4.3.3.1.cmml" xref="S2.SS2.p2.3.m3.4.4.3.3.2"><ci id="S2.SS2.p2.3.m3.1.1.cmml" xref="S2.SS2.p2.3.m3.1.1">𝑢</ci><cn id="S2.SS2.p2.3.m3.2.2.cmml" type="integer" xref="S2.SS2.p2.3.m3.2.2">1</cn></interval></apply><apply id="S2.SS2.p2.3.m3.4.4.1.cmml" xref="S2.SS2.p2.3.m3.4.4.1"><ci id="S2.SS2.p2.3.m3.4.4.1.2.cmml" xref="S2.SS2.p2.3.m3.4.4.1.2">→</ci><apply id="S2.SS2.p2.3.m3.4.4.1.3.cmml" xref="S2.SS2.p2.3.m3.4.4.1.3"><times id="S2.SS2.p2.3.m3.4.4.1.3.1.cmml" xref="S2.SS2.p2.3.m3.4.4.1.3.1"></times><ci id="S2.SS2.p2.3.m3.4.4.1.3.2.cmml" xref="S2.SS2.p2.3.m3.4.4.1.3.2">𝑀</ci><ci id="S2.SS2.p2.3.m3.3.3.cmml" xref="S2.SS2.p2.3.m3.3.3">𝛼</ci></apply><apply id="S2.SS2.p2.3.m3.4.4.1.1.cmml" xref="S2.SS2.p2.3.m3.4.4.1.1"><times id="S2.SS2.p2.3.m3.4.4.1.1.2.cmml" xref="S2.SS2.p2.3.m3.4.4.1.1.2"></times><ci id="S2.SS2.p2.3.m3.4.4.1.1.3.cmml" xref="S2.SS2.p2.3.m3.4.4.1.1.3">𝑀</ci><apply id="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.cmml" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1"><times id="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.1"></times><ci id="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.2.cmml" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.2">𝑢</ci><ci id="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.3.cmml" xref="S2.SS2.p2.3.m3.4.4.1.1.1.1.1.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.4c">M(u,1):M(\alpha)\to M(u\alpha)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.4d">italic_M ( italic_u , 1 ) : italic_M ( italic_α ) → italic_M ( italic_u italic_α )</annotation></semantics></math> is denoted <math alttext="u_{*}" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m4.1"><semantics id="S2.SS2.p2.4.m4.1a"><msub id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml"><mi id="S2.SS2.p2.4.m4.1.1.2" xref="S2.SS2.p2.4.m4.1.1.2.cmml">u</mi><mo id="S2.SS2.p2.4.m4.1.1.3" xref="S2.SS2.p2.4.m4.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m4.1b"><apply id="S2.SS2.p2.4.m4.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m4.1.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.p2.4.m4.1.1.2.cmml" xref="S2.SS2.p2.4.m4.1.1.2">𝑢</ci><times id="S2.SS2.p2.4.m4.1.1.3.cmml" xref="S2.SS2.p2.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.1c">u_{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.1d">italic_u start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>, and for <math alttext="(1,v):\alpha\to\alpha v" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m5.2"><semantics id="S2.SS2.p2.5.m5.2a"><mrow id="S2.SS2.p2.5.m5.2.3" xref="S2.SS2.p2.5.m5.2.3.cmml"><mrow id="S2.SS2.p2.5.m5.2.3.2.2" xref="S2.SS2.p2.5.m5.2.3.2.1.cmml"><mo id="S2.SS2.p2.5.m5.2.3.2.2.1" stretchy="false" xref="S2.SS2.p2.5.m5.2.3.2.1.cmml">(</mo><mn id="S2.SS2.p2.5.m5.1.1" xref="S2.SS2.p2.5.m5.1.1.cmml">1</mn><mo id="S2.SS2.p2.5.m5.2.3.2.2.2" xref="S2.SS2.p2.5.m5.2.3.2.1.cmml">,</mo><mi id="S2.SS2.p2.5.m5.2.2" xref="S2.SS2.p2.5.m5.2.2.cmml">v</mi><mo id="S2.SS2.p2.5.m5.2.3.2.2.3" rspace="0.278em" stretchy="false" xref="S2.SS2.p2.5.m5.2.3.2.1.cmml">)</mo></mrow><mo id="S2.SS2.p2.5.m5.2.3.1" rspace="0.278em" xref="S2.SS2.p2.5.m5.2.3.1.cmml">:</mo><mrow id="S2.SS2.p2.5.m5.2.3.3" xref="S2.SS2.p2.5.m5.2.3.3.cmml"><mi id="S2.SS2.p2.5.m5.2.3.3.2" xref="S2.SS2.p2.5.m5.2.3.3.2.cmml">α</mi><mo id="S2.SS2.p2.5.m5.2.3.3.1" stretchy="false" xref="S2.SS2.p2.5.m5.2.3.3.1.cmml">→</mo><mrow id="S2.SS2.p2.5.m5.2.3.3.3" xref="S2.SS2.p2.5.m5.2.3.3.3.cmml"><mi id="S2.SS2.p2.5.m5.2.3.3.3.2" xref="S2.SS2.p2.5.m5.2.3.3.3.2.cmml">α</mi><mo id="S2.SS2.p2.5.m5.2.3.3.3.1" xref="S2.SS2.p2.5.m5.2.3.3.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.5.m5.2.3.3.3.3" xref="S2.SS2.p2.5.m5.2.3.3.3.3.cmml">v</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m5.2b"><apply id="S2.SS2.p2.5.m5.2.3.cmml" xref="S2.SS2.p2.5.m5.2.3"><ci id="S2.SS2.p2.5.m5.2.3.1.cmml" xref="S2.SS2.p2.5.m5.2.3.1">:</ci><interval closure="open" id="S2.SS2.p2.5.m5.2.3.2.1.cmml" xref="S2.SS2.p2.5.m5.2.3.2.2"><cn id="S2.SS2.p2.5.m5.1.1.cmml" type="integer" xref="S2.SS2.p2.5.m5.1.1">1</cn><ci id="S2.SS2.p2.5.m5.2.2.cmml" xref="S2.SS2.p2.5.m5.2.2">𝑣</ci></interval><apply id="S2.SS2.p2.5.m5.2.3.3.cmml" xref="S2.SS2.p2.5.m5.2.3.3"><ci id="S2.SS2.p2.5.m5.2.3.3.1.cmml" xref="S2.SS2.p2.5.m5.2.3.3.1">→</ci><ci id="S2.SS2.p2.5.m5.2.3.3.2.cmml" xref="S2.SS2.p2.5.m5.2.3.3.2">𝛼</ci><apply id="S2.SS2.p2.5.m5.2.3.3.3.cmml" xref="S2.SS2.p2.5.m5.2.3.3.3"><times id="S2.SS2.p2.5.m5.2.3.3.3.1.cmml" xref="S2.SS2.p2.5.m5.2.3.3.3.1"></times><ci id="S2.SS2.p2.5.m5.2.3.3.3.2.cmml" xref="S2.SS2.p2.5.m5.2.3.3.3.2">𝛼</ci><ci id="S2.SS2.p2.5.m5.2.3.3.3.3.cmml" xref="S2.SS2.p2.5.m5.2.3.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m5.2c">(1,v):\alpha\to\alpha v</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m5.2d">( 1 , italic_v ) : italic_α → italic_α italic_v</annotation></semantics></math> in <math alttext="\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m6.1"><semantics id="S2.SS2.p2.6.m6.1a"><mrow id="S2.SS2.p2.6.m6.1.1" xref="S2.SS2.p2.6.m6.1.1.cmml"><mi id="S2.SS2.p2.6.m6.1.1.2" xref="S2.SS2.p2.6.m6.1.1.2.cmml">𝔉</mi><mo id="S2.SS2.p2.6.m6.1.1.1" xref="S2.SS2.p2.6.m6.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.6.m6.1.1.3" xref="S2.SS2.p2.6.m6.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m6.1b"><apply id="S2.SS2.p2.6.m6.1.1.cmml" xref="S2.SS2.p2.6.m6.1.1"><times id="S2.SS2.p2.6.m6.1.1.1.cmml" xref="S2.SS2.p2.6.m6.1.1.1"></times><ci id="S2.SS2.p2.6.m6.1.1.2.cmml" xref="S2.SS2.p2.6.m6.1.1.2">𝔉</ci><ci id="S2.SS2.p2.6.m6.1.1.3.cmml" xref="S2.SS2.p2.6.m6.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m6.1c">\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m6.1d">fraktur_F caligraphic_C</annotation></semantics></math>, the induced map <math alttext="M(1,v):M(\alpha)\to M(\alpha v)" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m7.4"><semantics id="S2.SS2.p2.7.m7.4a"><mrow id="S2.SS2.p2.7.m7.4.4" xref="S2.SS2.p2.7.m7.4.4.cmml"><mrow id="S2.SS2.p2.7.m7.4.4.3" xref="S2.SS2.p2.7.m7.4.4.3.cmml"><mi id="S2.SS2.p2.7.m7.4.4.3.2" xref="S2.SS2.p2.7.m7.4.4.3.2.cmml">M</mi><mo id="S2.SS2.p2.7.m7.4.4.3.1" xref="S2.SS2.p2.7.m7.4.4.3.1.cmml">⁢</mo><mrow id="S2.SS2.p2.7.m7.4.4.3.3.2" xref="S2.SS2.p2.7.m7.4.4.3.3.1.cmml"><mo id="S2.SS2.p2.7.m7.4.4.3.3.2.1" stretchy="false" xref="S2.SS2.p2.7.m7.4.4.3.3.1.cmml">(</mo><mn id="S2.SS2.p2.7.m7.1.1" xref="S2.SS2.p2.7.m7.1.1.cmml">1</mn><mo id="S2.SS2.p2.7.m7.4.4.3.3.2.2" xref="S2.SS2.p2.7.m7.4.4.3.3.1.cmml">,</mo><mi id="S2.SS2.p2.7.m7.2.2" xref="S2.SS2.p2.7.m7.2.2.cmml">v</mi><mo id="S2.SS2.p2.7.m7.4.4.3.3.2.3" rspace="0.278em" stretchy="false" xref="S2.SS2.p2.7.m7.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.7.m7.4.4.2" rspace="0.278em" xref="S2.SS2.p2.7.m7.4.4.2.cmml">:</mo><mrow id="S2.SS2.p2.7.m7.4.4.1" xref="S2.SS2.p2.7.m7.4.4.1.cmml"><mrow id="S2.SS2.p2.7.m7.4.4.1.3" xref="S2.SS2.p2.7.m7.4.4.1.3.cmml"><mi id="S2.SS2.p2.7.m7.4.4.1.3.2" xref="S2.SS2.p2.7.m7.4.4.1.3.2.cmml">M</mi><mo id="S2.SS2.p2.7.m7.4.4.1.3.1" xref="S2.SS2.p2.7.m7.4.4.1.3.1.cmml">⁢</mo><mrow id="S2.SS2.p2.7.m7.4.4.1.3.3.2" xref="S2.SS2.p2.7.m7.4.4.1.3.cmml"><mo id="S2.SS2.p2.7.m7.4.4.1.3.3.2.1" stretchy="false" xref="S2.SS2.p2.7.m7.4.4.1.3.cmml">(</mo><mi id="S2.SS2.p2.7.m7.3.3" xref="S2.SS2.p2.7.m7.3.3.cmml">α</mi><mo id="S2.SS2.p2.7.m7.4.4.1.3.3.2.2" stretchy="false" xref="S2.SS2.p2.7.m7.4.4.1.3.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.7.m7.4.4.1.2" stretchy="false" xref="S2.SS2.p2.7.m7.4.4.1.2.cmml">→</mo><mrow id="S2.SS2.p2.7.m7.4.4.1.1" xref="S2.SS2.p2.7.m7.4.4.1.1.cmml"><mi id="S2.SS2.p2.7.m7.4.4.1.1.3" xref="S2.SS2.p2.7.m7.4.4.1.1.3.cmml">M</mi><mo id="S2.SS2.p2.7.m7.4.4.1.1.2" xref="S2.SS2.p2.7.m7.4.4.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.7.m7.4.4.1.1.1.1" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.cmml"><mo id="S2.SS2.p2.7.m7.4.4.1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p2.7.m7.4.4.1.1.1.1.1" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.cmml"><mi id="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.2" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.2.cmml">α</mi><mo id="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.1" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.3" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.SS2.p2.7.m7.4.4.1.1.1.1.3" stretchy="false" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m7.4b"><apply id="S2.SS2.p2.7.m7.4.4.cmml" xref="S2.SS2.p2.7.m7.4.4"><ci id="S2.SS2.p2.7.m7.4.4.2.cmml" xref="S2.SS2.p2.7.m7.4.4.2">:</ci><apply id="S2.SS2.p2.7.m7.4.4.3.cmml" xref="S2.SS2.p2.7.m7.4.4.3"><times id="S2.SS2.p2.7.m7.4.4.3.1.cmml" xref="S2.SS2.p2.7.m7.4.4.3.1"></times><ci id="S2.SS2.p2.7.m7.4.4.3.2.cmml" xref="S2.SS2.p2.7.m7.4.4.3.2">𝑀</ci><interval closure="open" id="S2.SS2.p2.7.m7.4.4.3.3.1.cmml" xref="S2.SS2.p2.7.m7.4.4.3.3.2"><cn id="S2.SS2.p2.7.m7.1.1.cmml" type="integer" xref="S2.SS2.p2.7.m7.1.1">1</cn><ci id="S2.SS2.p2.7.m7.2.2.cmml" xref="S2.SS2.p2.7.m7.2.2">𝑣</ci></interval></apply><apply id="S2.SS2.p2.7.m7.4.4.1.cmml" xref="S2.SS2.p2.7.m7.4.4.1"><ci id="S2.SS2.p2.7.m7.4.4.1.2.cmml" xref="S2.SS2.p2.7.m7.4.4.1.2">→</ci><apply id="S2.SS2.p2.7.m7.4.4.1.3.cmml" xref="S2.SS2.p2.7.m7.4.4.1.3"><times id="S2.SS2.p2.7.m7.4.4.1.3.1.cmml" xref="S2.SS2.p2.7.m7.4.4.1.3.1"></times><ci id="S2.SS2.p2.7.m7.4.4.1.3.2.cmml" xref="S2.SS2.p2.7.m7.4.4.1.3.2">𝑀</ci><ci id="S2.SS2.p2.7.m7.3.3.cmml" xref="S2.SS2.p2.7.m7.3.3">𝛼</ci></apply><apply id="S2.SS2.p2.7.m7.4.4.1.1.cmml" xref="S2.SS2.p2.7.m7.4.4.1.1"><times id="S2.SS2.p2.7.m7.4.4.1.1.2.cmml" xref="S2.SS2.p2.7.m7.4.4.1.1.2"></times><ci id="S2.SS2.p2.7.m7.4.4.1.1.3.cmml" xref="S2.SS2.p2.7.m7.4.4.1.1.3">𝑀</ci><apply id="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.cmml" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1"><times id="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.1"></times><ci id="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.2.cmml" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.2">𝛼</ci><ci id="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.3.cmml" xref="S2.SS2.p2.7.m7.4.4.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m7.4c">M(1,v):M(\alpha)\to M(\alpha v)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m7.4d">italic_M ( 1 , italic_v ) : italic_M ( italic_α ) → italic_M ( italic_α italic_v )</annotation></semantics></math> is denoted <math alttext="v^{*}" class="ltx_Math" display="inline" id="S2.SS2.p2.8.m8.1"><semantics id="S2.SS2.p2.8.m8.1a"><msup id="S2.SS2.p2.8.m8.1.1" xref="S2.SS2.p2.8.m8.1.1.cmml"><mi id="S2.SS2.p2.8.m8.1.1.2" xref="S2.SS2.p2.8.m8.1.1.2.cmml">v</mi><mo id="S2.SS2.p2.8.m8.1.1.3" xref="S2.SS2.p2.8.m8.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.8.m8.1b"><apply id="S2.SS2.p2.8.m8.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.8.m8.1.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1">superscript</csymbol><ci id="S2.SS2.p2.8.m8.1.1.2.cmml" xref="S2.SS2.p2.8.m8.1.1.2">𝑣</ci><times id="S2.SS2.p2.8.m8.1.1.3.cmml" xref="S2.SS2.p2.8.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.8.m8.1c">v^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.8.m8.1d">italic_v start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. So, for each morphism <math alttext="(u,v)" class="ltx_Math" display="inline" id="S2.SS2.p2.9.m9.2"><semantics id="S2.SS2.p2.9.m9.2a"><mrow id="S2.SS2.p2.9.m9.2.3.2" xref="S2.SS2.p2.9.m9.2.3.1.cmml"><mo id="S2.SS2.p2.9.m9.2.3.2.1" stretchy="false" xref="S2.SS2.p2.9.m9.2.3.1.cmml">(</mo><mi id="S2.SS2.p2.9.m9.1.1" xref="S2.SS2.p2.9.m9.1.1.cmml">u</mi><mo id="S2.SS2.p2.9.m9.2.3.2.2" xref="S2.SS2.p2.9.m9.2.3.1.cmml">,</mo><mi id="S2.SS2.p2.9.m9.2.2" xref="S2.SS2.p2.9.m9.2.2.cmml">v</mi><mo id="S2.SS2.p2.9.m9.2.3.2.3" stretchy="false" xref="S2.SS2.p2.9.m9.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.9.m9.2b"><interval closure="open" id="S2.SS2.p2.9.m9.2.3.1.cmml" xref="S2.SS2.p2.9.m9.2.3.2"><ci id="S2.SS2.p2.9.m9.1.1.cmml" xref="S2.SS2.p2.9.m9.1.1">𝑢</ci><ci id="S2.SS2.p2.9.m9.2.2.cmml" xref="S2.SS2.p2.9.m9.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.9.m9.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.9.m9.2d">( italic_u , italic_v )</annotation></semantics></math> in <math alttext="\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p2.10.m10.1"><semantics id="S2.SS2.p2.10.m10.1a"><mrow id="S2.SS2.p2.10.m10.1.1" xref="S2.SS2.p2.10.m10.1.1.cmml"><mi id="S2.SS2.p2.10.m10.1.1.2" xref="S2.SS2.p2.10.m10.1.1.2.cmml">𝔉</mi><mo id="S2.SS2.p2.10.m10.1.1.1" xref="S2.SS2.p2.10.m10.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.10.m10.1.1.3" xref="S2.SS2.p2.10.m10.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.10.m10.1b"><apply id="S2.SS2.p2.10.m10.1.1.cmml" xref="S2.SS2.p2.10.m10.1.1"><times id="S2.SS2.p2.10.m10.1.1.1.cmml" xref="S2.SS2.p2.10.m10.1.1.1"></times><ci id="S2.SS2.p2.10.m10.1.1.2.cmml" xref="S2.SS2.p2.10.m10.1.1.2">𝔉</ci><ci id="S2.SS2.p2.10.m10.1.1.3.cmml" xref="S2.SS2.p2.10.m10.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.10.m10.1c">\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.10.m10.1d">fraktur_F caligraphic_C</annotation></semantics></math>, we have <math alttext="M(u,v)=u_{*}v^{*}=v^{*}u_{*}" class="ltx_Math" display="inline" id="S2.SS2.p2.11.m11.2"><semantics id="S2.SS2.p2.11.m11.2a"><mrow id="S2.SS2.p2.11.m11.2.3" xref="S2.SS2.p2.11.m11.2.3.cmml"><mrow id="S2.SS2.p2.11.m11.2.3.2" xref="S2.SS2.p2.11.m11.2.3.2.cmml"><mi id="S2.SS2.p2.11.m11.2.3.2.2" xref="S2.SS2.p2.11.m11.2.3.2.2.cmml">M</mi><mo id="S2.SS2.p2.11.m11.2.3.2.1" xref="S2.SS2.p2.11.m11.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.11.m11.2.3.2.3.2" xref="S2.SS2.p2.11.m11.2.3.2.3.1.cmml"><mo id="S2.SS2.p2.11.m11.2.3.2.3.2.1" stretchy="false" xref="S2.SS2.p2.11.m11.2.3.2.3.1.cmml">(</mo><mi id="S2.SS2.p2.11.m11.1.1" xref="S2.SS2.p2.11.m11.1.1.cmml">u</mi><mo id="S2.SS2.p2.11.m11.2.3.2.3.2.2" xref="S2.SS2.p2.11.m11.2.3.2.3.1.cmml">,</mo><mi id="S2.SS2.p2.11.m11.2.2" xref="S2.SS2.p2.11.m11.2.2.cmml">v</mi><mo id="S2.SS2.p2.11.m11.2.3.2.3.2.3" stretchy="false" xref="S2.SS2.p2.11.m11.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.11.m11.2.3.3" xref="S2.SS2.p2.11.m11.2.3.3.cmml">=</mo><mrow id="S2.SS2.p2.11.m11.2.3.4" xref="S2.SS2.p2.11.m11.2.3.4.cmml"><msub id="S2.SS2.p2.11.m11.2.3.4.2" xref="S2.SS2.p2.11.m11.2.3.4.2.cmml"><mi id="S2.SS2.p2.11.m11.2.3.4.2.2" xref="S2.SS2.p2.11.m11.2.3.4.2.2.cmml">u</mi><mo id="S2.SS2.p2.11.m11.2.3.4.2.3" xref="S2.SS2.p2.11.m11.2.3.4.2.3.cmml">∗</mo></msub><mo id="S2.SS2.p2.11.m11.2.3.4.1" xref="S2.SS2.p2.11.m11.2.3.4.1.cmml">⁢</mo><msup id="S2.SS2.p2.11.m11.2.3.4.3" xref="S2.SS2.p2.11.m11.2.3.4.3.cmml"><mi id="S2.SS2.p2.11.m11.2.3.4.3.2" xref="S2.SS2.p2.11.m11.2.3.4.3.2.cmml">v</mi><mo id="S2.SS2.p2.11.m11.2.3.4.3.3" xref="S2.SS2.p2.11.m11.2.3.4.3.3.cmml">∗</mo></msup></mrow><mo id="S2.SS2.p2.11.m11.2.3.5" xref="S2.SS2.p2.11.m11.2.3.5.cmml">=</mo><mrow id="S2.SS2.p2.11.m11.2.3.6" xref="S2.SS2.p2.11.m11.2.3.6.cmml"><msup id="S2.SS2.p2.11.m11.2.3.6.2" xref="S2.SS2.p2.11.m11.2.3.6.2.cmml"><mi id="S2.SS2.p2.11.m11.2.3.6.2.2" xref="S2.SS2.p2.11.m11.2.3.6.2.2.cmml">v</mi><mo id="S2.SS2.p2.11.m11.2.3.6.2.3" xref="S2.SS2.p2.11.m11.2.3.6.2.3.cmml">∗</mo></msup><mo id="S2.SS2.p2.11.m11.2.3.6.1" xref="S2.SS2.p2.11.m11.2.3.6.1.cmml">⁢</mo><msub id="S2.SS2.p2.11.m11.2.3.6.3" xref="S2.SS2.p2.11.m11.2.3.6.3.cmml"><mi id="S2.SS2.p2.11.m11.2.3.6.3.2" xref="S2.SS2.p2.11.m11.2.3.6.3.2.cmml">u</mi><mo id="S2.SS2.p2.11.m11.2.3.6.3.3" xref="S2.SS2.p2.11.m11.2.3.6.3.3.cmml">∗</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.11.m11.2b"><apply id="S2.SS2.p2.11.m11.2.3.cmml" xref="S2.SS2.p2.11.m11.2.3"><and id="S2.SS2.p2.11.m11.2.3a.cmml" xref="S2.SS2.p2.11.m11.2.3"></and><apply id="S2.SS2.p2.11.m11.2.3b.cmml" xref="S2.SS2.p2.11.m11.2.3"><eq id="S2.SS2.p2.11.m11.2.3.3.cmml" xref="S2.SS2.p2.11.m11.2.3.3"></eq><apply id="S2.SS2.p2.11.m11.2.3.2.cmml" xref="S2.SS2.p2.11.m11.2.3.2"><times id="S2.SS2.p2.11.m11.2.3.2.1.cmml" xref="S2.SS2.p2.11.m11.2.3.2.1"></times><ci id="S2.SS2.p2.11.m11.2.3.2.2.cmml" xref="S2.SS2.p2.11.m11.2.3.2.2">𝑀</ci><interval closure="open" id="S2.SS2.p2.11.m11.2.3.2.3.1.cmml" xref="S2.SS2.p2.11.m11.2.3.2.3.2"><ci id="S2.SS2.p2.11.m11.1.1.cmml" xref="S2.SS2.p2.11.m11.1.1">𝑢</ci><ci id="S2.SS2.p2.11.m11.2.2.cmml" xref="S2.SS2.p2.11.m11.2.2">𝑣</ci></interval></apply><apply id="S2.SS2.p2.11.m11.2.3.4.cmml" xref="S2.SS2.p2.11.m11.2.3.4"><times id="S2.SS2.p2.11.m11.2.3.4.1.cmml" xref="S2.SS2.p2.11.m11.2.3.4.1"></times><apply id="S2.SS2.p2.11.m11.2.3.4.2.cmml" xref="S2.SS2.p2.11.m11.2.3.4.2"><csymbol cd="ambiguous" id="S2.SS2.p2.11.m11.2.3.4.2.1.cmml" xref="S2.SS2.p2.11.m11.2.3.4.2">subscript</csymbol><ci id="S2.SS2.p2.11.m11.2.3.4.2.2.cmml" xref="S2.SS2.p2.11.m11.2.3.4.2.2">𝑢</ci><times id="S2.SS2.p2.11.m11.2.3.4.2.3.cmml" xref="S2.SS2.p2.11.m11.2.3.4.2.3"></times></apply><apply id="S2.SS2.p2.11.m11.2.3.4.3.cmml" xref="S2.SS2.p2.11.m11.2.3.4.3"><csymbol cd="ambiguous" id="S2.SS2.p2.11.m11.2.3.4.3.1.cmml" xref="S2.SS2.p2.11.m11.2.3.4.3">superscript</csymbol><ci id="S2.SS2.p2.11.m11.2.3.4.3.2.cmml" xref="S2.SS2.p2.11.m11.2.3.4.3.2">𝑣</ci><times id="S2.SS2.p2.11.m11.2.3.4.3.3.cmml" xref="S2.SS2.p2.11.m11.2.3.4.3.3"></times></apply></apply></apply><apply id="S2.SS2.p2.11.m11.2.3c.cmml" xref="S2.SS2.p2.11.m11.2.3"><eq id="S2.SS2.p2.11.m11.2.3.5.cmml" xref="S2.SS2.p2.11.m11.2.3.5"></eq><share href="https://arxiv.org/html/2503.14659v1#S2.SS2.p2.11.m11.2.3.4.cmml" id="S2.SS2.p2.11.m11.2.3d.cmml" xref="S2.SS2.p2.11.m11.2.3"></share><apply id="S2.SS2.p2.11.m11.2.3.6.cmml" xref="S2.SS2.p2.11.m11.2.3.6"><times id="S2.SS2.p2.11.m11.2.3.6.1.cmml" xref="S2.SS2.p2.11.m11.2.3.6.1"></times><apply id="S2.SS2.p2.11.m11.2.3.6.2.cmml" xref="S2.SS2.p2.11.m11.2.3.6.2"><csymbol cd="ambiguous" id="S2.SS2.p2.11.m11.2.3.6.2.1.cmml" xref="S2.SS2.p2.11.m11.2.3.6.2">superscript</csymbol><ci id="S2.SS2.p2.11.m11.2.3.6.2.2.cmml" xref="S2.SS2.p2.11.m11.2.3.6.2.2">𝑣</ci><times id="S2.SS2.p2.11.m11.2.3.6.2.3.cmml" xref="S2.SS2.p2.11.m11.2.3.6.2.3"></times></apply><apply id="S2.SS2.p2.11.m11.2.3.6.3.cmml" xref="S2.SS2.p2.11.m11.2.3.6.3"><csymbol cd="ambiguous" id="S2.SS2.p2.11.m11.2.3.6.3.1.cmml" xref="S2.SS2.p2.11.m11.2.3.6.3">subscript</csymbol><ci id="S2.SS2.p2.11.m11.2.3.6.3.2.cmml" xref="S2.SS2.p2.11.m11.2.3.6.3.2">𝑢</ci><times id="S2.SS2.p2.11.m11.2.3.6.3.3.cmml" xref="S2.SS2.p2.11.m11.2.3.6.3.3"></times></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.11.m11.2c">M(u,v)=u_{*}v^{*}=v^{*}u_{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.11.m11.2d">italic_M ( italic_u , italic_v ) = italic_u start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_v start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.1.1.1">Definition 2.6</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem6.p1"> <p class="ltx_p" id="S2.Thmtheorem6.p1.10">Let <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.1.m1.1"><semantics id="S2.Thmtheorem6.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem6.p1.1.m1.1.1" xref="S2.Thmtheorem6.p1.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.1.m1.1b"><ci id="S2.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.1.m1.1d">caligraphic_C</annotation></semantics></math> be a small category and <math alttext="M" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.2.m2.1"><semantics id="S2.Thmtheorem6.p1.2.m2.1a"><mi id="S2.Thmtheorem6.p1.2.m2.1.1" xref="S2.Thmtheorem6.p1.2.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.2.m2.1b"><ci id="S2.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem6.p1.2.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.2.m2.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.2.m2.1d">italic_M</annotation></semantics></math> be a natural system for <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.3.m3.1"><semantics id="S2.Thmtheorem6.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem6.p1.3.m3.1.1" xref="S2.Thmtheorem6.p1.3.m3.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.3.m3.1b"><ci id="S2.Thmtheorem6.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem6.p1.3.m3.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.3.m3.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.3.m3.1d">caligraphic_C</annotation></semantics></math> over <math alttext="R" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.4.m4.1"><semantics id="S2.Thmtheorem6.p1.4.m4.1a"><mi id="S2.Thmtheorem6.p1.4.m4.1.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.4.m4.1b"><ci id="S2.Thmtheorem6.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.4.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.4.m4.1d">italic_R</annotation></semantics></math> . The <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem6.p1.10.1">Baues-Wirsching cohomology</em> <math alttext="H^{*}_{BW}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.5.m5.2"><semantics id="S2.Thmtheorem6.p1.5.m5.2a"><mrow id="S2.Thmtheorem6.p1.5.m5.2.3" xref="S2.Thmtheorem6.p1.5.m5.2.3.cmml"><msubsup id="S2.Thmtheorem6.p1.5.m5.2.3.2" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.cmml"><mi id="S2.Thmtheorem6.p1.5.m5.2.3.2.2.2" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.2.2.cmml">H</mi><mrow id="S2.Thmtheorem6.p1.5.m5.2.3.2.3" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.3.cmml"><mi id="S2.Thmtheorem6.p1.5.m5.2.3.2.3.2" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.3.2.cmml">B</mi><mo id="S2.Thmtheorem6.p1.5.m5.2.3.2.3.1" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.3.1.cmml">⁢</mo><mi id="S2.Thmtheorem6.p1.5.m5.2.3.2.3.3" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.3.3.cmml">W</mi></mrow><mo id="S2.Thmtheorem6.p1.5.m5.2.3.2.2.3" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S2.Thmtheorem6.p1.5.m5.2.3.1" xref="S2.Thmtheorem6.p1.5.m5.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.5.m5.2.3.3.2" xref="S2.Thmtheorem6.p1.5.m5.2.3.3.1.cmml"><mo id="S2.Thmtheorem6.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.5.m5.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem6.p1.5.m5.1.1" xref="S2.Thmtheorem6.p1.5.m5.1.1.cmml">𝒞</mi><mo id="S2.Thmtheorem6.p1.5.m5.2.3.3.2.2" xref="S2.Thmtheorem6.p1.5.m5.2.3.3.1.cmml">;</mo><mi id="S2.Thmtheorem6.p1.5.m5.2.2" xref="S2.Thmtheorem6.p1.5.m5.2.2.cmml">M</mi><mo id="S2.Thmtheorem6.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem6.p1.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.5.m5.2b"><apply id="S2.Thmtheorem6.p1.5.m5.2.3.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3"><times id="S2.Thmtheorem6.p1.5.m5.2.3.1.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.1"></times><apply id="S2.Thmtheorem6.p1.5.m5.2.3.2.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.5.m5.2.3.2.1.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2">subscript</csymbol><apply id="S2.Thmtheorem6.p1.5.m5.2.3.2.2.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.5.m5.2.3.2.2.1.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem6.p1.5.m5.2.3.2.2.2.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.2.2">𝐻</ci><times id="S2.Thmtheorem6.p1.5.m5.2.3.2.2.3.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.2.3"></times></apply><apply id="S2.Thmtheorem6.p1.5.m5.2.3.2.3.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.3"><times id="S2.Thmtheorem6.p1.5.m5.2.3.2.3.1.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.3.1"></times><ci id="S2.Thmtheorem6.p1.5.m5.2.3.2.3.2.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.3.2">𝐵</ci><ci id="S2.Thmtheorem6.p1.5.m5.2.3.2.3.3.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.2.3.3">𝑊</ci></apply></apply><list id="S2.Thmtheorem6.p1.5.m5.2.3.3.1.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.3.3.2"><ci id="S2.Thmtheorem6.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem6.p1.5.m5.1.1">𝒞</ci><ci id="S2.Thmtheorem6.p1.5.m5.2.2.cmml" xref="S2.Thmtheorem6.p1.5.m5.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.5.m5.2c">H^{*}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.5.m5.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> of <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.6.m6.1"><semantics id="S2.Thmtheorem6.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem6.p1.6.m6.1.1" xref="S2.Thmtheorem6.p1.6.m6.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.6.m6.1b"><ci id="S2.Thmtheorem6.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem6.p1.6.m6.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.6.m6.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.6.m6.1d">caligraphic_C</annotation></semantics></math> with coefficients in <math alttext="M" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.7.m7.1"><semantics id="S2.Thmtheorem6.p1.7.m7.1a"><mi id="S2.Thmtheorem6.p1.7.m7.1.1" xref="S2.Thmtheorem6.p1.7.m7.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.7.m7.1b"><ci id="S2.Thmtheorem6.p1.7.m7.1.1.cmml" xref="S2.Thmtheorem6.p1.7.m7.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.7.m7.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.7.m7.1d">italic_M</annotation></semantics></math> is the cohomology of a cochain complex <math alttext="C^{*}_{BW}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.8.m8.2"><semantics id="S2.Thmtheorem6.p1.8.m8.2a"><mrow id="S2.Thmtheorem6.p1.8.m8.2.3" xref="S2.Thmtheorem6.p1.8.m8.2.3.cmml"><msubsup id="S2.Thmtheorem6.p1.8.m8.2.3.2" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.cmml"><mi id="S2.Thmtheorem6.p1.8.m8.2.3.2.2.2" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.2.2.cmml">C</mi><mrow id="S2.Thmtheorem6.p1.8.m8.2.3.2.3" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.3.cmml"><mi id="S2.Thmtheorem6.p1.8.m8.2.3.2.3.2" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.3.2.cmml">B</mi><mo id="S2.Thmtheorem6.p1.8.m8.2.3.2.3.1" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.3.1.cmml">⁢</mo><mi id="S2.Thmtheorem6.p1.8.m8.2.3.2.3.3" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.3.3.cmml">W</mi></mrow><mo id="S2.Thmtheorem6.p1.8.m8.2.3.2.2.3" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S2.Thmtheorem6.p1.8.m8.2.3.1" xref="S2.Thmtheorem6.p1.8.m8.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.8.m8.2.3.3.2" xref="S2.Thmtheorem6.p1.8.m8.2.3.3.1.cmml"><mo id="S2.Thmtheorem6.p1.8.m8.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.8.m8.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem6.p1.8.m8.1.1" xref="S2.Thmtheorem6.p1.8.m8.1.1.cmml">𝒞</mi><mo id="S2.Thmtheorem6.p1.8.m8.2.3.3.2.2" xref="S2.Thmtheorem6.p1.8.m8.2.3.3.1.cmml">;</mo><mi id="S2.Thmtheorem6.p1.8.m8.2.2" xref="S2.Thmtheorem6.p1.8.m8.2.2.cmml">M</mi><mo id="S2.Thmtheorem6.p1.8.m8.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem6.p1.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.8.m8.2b"><apply id="S2.Thmtheorem6.p1.8.m8.2.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3"><times id="S2.Thmtheorem6.p1.8.m8.2.3.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.1"></times><apply id="S2.Thmtheorem6.p1.8.m8.2.3.2.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.8.m8.2.3.2.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2">subscript</csymbol><apply id="S2.Thmtheorem6.p1.8.m8.2.3.2.2.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.8.m8.2.3.2.2.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem6.p1.8.m8.2.3.2.2.2.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.2.2">𝐶</ci><times id="S2.Thmtheorem6.p1.8.m8.2.3.2.2.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.2.3"></times></apply><apply id="S2.Thmtheorem6.p1.8.m8.2.3.2.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.3"><times id="S2.Thmtheorem6.p1.8.m8.2.3.2.3.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.3.1"></times><ci id="S2.Thmtheorem6.p1.8.m8.2.3.2.3.2.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.3.2">𝐵</ci><ci id="S2.Thmtheorem6.p1.8.m8.2.3.2.3.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.2.3.3">𝑊</ci></apply></apply><list id="S2.Thmtheorem6.p1.8.m8.2.3.3.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.3.3.2"><ci id="S2.Thmtheorem6.p1.8.m8.1.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1">𝒞</ci><ci id="S2.Thmtheorem6.p1.8.m8.2.2.cmml" xref="S2.Thmtheorem6.p1.8.m8.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.8.m8.2c">C^{*}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.8.m8.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> where for all <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.9.m9.1"><semantics id="S2.Thmtheorem6.p1.9.m9.1a"><mrow id="S2.Thmtheorem6.p1.9.m9.1.1" xref="S2.Thmtheorem6.p1.9.m9.1.1.cmml"><mi id="S2.Thmtheorem6.p1.9.m9.1.1.2" xref="S2.Thmtheorem6.p1.9.m9.1.1.2.cmml">n</mi><mo id="S2.Thmtheorem6.p1.9.m9.1.1.1" xref="S2.Thmtheorem6.p1.9.m9.1.1.1.cmml">≥</mo><mn id="S2.Thmtheorem6.p1.9.m9.1.1.3" xref="S2.Thmtheorem6.p1.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.9.m9.1b"><apply id="S2.Thmtheorem6.p1.9.m9.1.1.cmml" xref="S2.Thmtheorem6.p1.9.m9.1.1"><geq id="S2.Thmtheorem6.p1.9.m9.1.1.1.cmml" xref="S2.Thmtheorem6.p1.9.m9.1.1.1"></geq><ci id="S2.Thmtheorem6.p1.9.m9.1.1.2.cmml" xref="S2.Thmtheorem6.p1.9.m9.1.1.2">𝑛</ci><cn id="S2.Thmtheorem6.p1.9.m9.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.9.m9.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.9.m9.1d">italic_n ≥ 0</annotation></semantics></math>, <math alttext="C^{n}_{BW}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.10.m10.2"><semantics id="S2.Thmtheorem6.p1.10.m10.2a"><mrow id="S2.Thmtheorem6.p1.10.m10.2.3" xref="S2.Thmtheorem6.p1.10.m10.2.3.cmml"><msubsup id="S2.Thmtheorem6.p1.10.m10.2.3.2" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.cmml"><mi id="S2.Thmtheorem6.p1.10.m10.2.3.2.2.2" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.2.2.cmml">C</mi><mrow id="S2.Thmtheorem6.p1.10.m10.2.3.2.3" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.3.cmml"><mi id="S2.Thmtheorem6.p1.10.m10.2.3.2.3.2" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.3.2.cmml">B</mi><mo id="S2.Thmtheorem6.p1.10.m10.2.3.2.3.1" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.3.1.cmml">⁢</mo><mi id="S2.Thmtheorem6.p1.10.m10.2.3.2.3.3" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.3.3.cmml">W</mi></mrow><mi id="S2.Thmtheorem6.p1.10.m10.2.3.2.2.3" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.2.3.cmml">n</mi></msubsup><mo id="S2.Thmtheorem6.p1.10.m10.2.3.1" xref="S2.Thmtheorem6.p1.10.m10.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.10.m10.2.3.3.2" xref="S2.Thmtheorem6.p1.10.m10.2.3.3.1.cmml"><mo id="S2.Thmtheorem6.p1.10.m10.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.10.m10.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem6.p1.10.m10.1.1" xref="S2.Thmtheorem6.p1.10.m10.1.1.cmml">𝒞</mi><mo id="S2.Thmtheorem6.p1.10.m10.2.3.3.2.2" xref="S2.Thmtheorem6.p1.10.m10.2.3.3.1.cmml">;</mo><mi id="S2.Thmtheorem6.p1.10.m10.2.2" xref="S2.Thmtheorem6.p1.10.m10.2.2.cmml">M</mi><mo id="S2.Thmtheorem6.p1.10.m10.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem6.p1.10.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.10.m10.2b"><apply id="S2.Thmtheorem6.p1.10.m10.2.3.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3"><times id="S2.Thmtheorem6.p1.10.m10.2.3.1.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.1"></times><apply id="S2.Thmtheorem6.p1.10.m10.2.3.2.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.10.m10.2.3.2.1.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2">subscript</csymbol><apply id="S2.Thmtheorem6.p1.10.m10.2.3.2.2.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.10.m10.2.3.2.2.1.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem6.p1.10.m10.2.3.2.2.2.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.2.2">𝐶</ci><ci id="S2.Thmtheorem6.p1.10.m10.2.3.2.2.3.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.2.3">𝑛</ci></apply><apply id="S2.Thmtheorem6.p1.10.m10.2.3.2.3.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.3"><times id="S2.Thmtheorem6.p1.10.m10.2.3.2.3.1.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.3.1"></times><ci id="S2.Thmtheorem6.p1.10.m10.2.3.2.3.2.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.3.2">𝐵</ci><ci id="S2.Thmtheorem6.p1.10.m10.2.3.2.3.3.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.2.3.3">𝑊</ci></apply></apply><list id="S2.Thmtheorem6.p1.10.m10.2.3.3.1.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.3.3.2"><ci id="S2.Thmtheorem6.p1.10.m10.1.1.cmml" xref="S2.Thmtheorem6.p1.10.m10.1.1">𝒞</ci><ci id="S2.Thmtheorem6.p1.10.m10.2.2.cmml" xref="S2.Thmtheorem6.p1.10.m10.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.10.m10.2c">C^{n}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.10.m10.2d">italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> is the abelian group of all functions</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f:N\mathcal{C}_{n}\to\coprod_{\alpha\in\mathrm{Mor}(\mathcal{C})}M(\alpha)" class="ltx_Math" display="block" id="S2.Ex19.m1.2"><semantics id="S2.Ex19.m1.2a"><mrow id="S2.Ex19.m1.2.3" xref="S2.Ex19.m1.2.3.cmml"><mi id="S2.Ex19.m1.2.3.2" xref="S2.Ex19.m1.2.3.2.cmml">f</mi><mo id="S2.Ex19.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex19.m1.2.3.1.cmml">:</mo><mrow id="S2.Ex19.m1.2.3.3" xref="S2.Ex19.m1.2.3.3.cmml"><mrow id="S2.Ex19.m1.2.3.3.2" xref="S2.Ex19.m1.2.3.3.2.cmml"><mi id="S2.Ex19.m1.2.3.3.2.2" xref="S2.Ex19.m1.2.3.3.2.2.cmml">N</mi><mo id="S2.Ex19.m1.2.3.3.2.1" xref="S2.Ex19.m1.2.3.3.2.1.cmml">⁢</mo><msub id="S2.Ex19.m1.2.3.3.2.3" xref="S2.Ex19.m1.2.3.3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex19.m1.2.3.3.2.3.2" xref="S2.Ex19.m1.2.3.3.2.3.2.cmml">𝒞</mi><mi id="S2.Ex19.m1.2.3.3.2.3.3" xref="S2.Ex19.m1.2.3.3.2.3.3.cmml">n</mi></msub></mrow><mo id="S2.Ex19.m1.2.3.3.1" rspace="0.111em" stretchy="false" xref="S2.Ex19.m1.2.3.3.1.cmml">→</mo><mrow id="S2.Ex19.m1.2.3.3.3" xref="S2.Ex19.m1.2.3.3.3.cmml"><munder id="S2.Ex19.m1.2.3.3.3.1" xref="S2.Ex19.m1.2.3.3.3.1.cmml"><mo id="S2.Ex19.m1.2.3.3.3.1.2" movablelimits="false" xref="S2.Ex19.m1.2.3.3.3.1.2.cmml">∐</mo><mrow id="S2.Ex19.m1.1.1.1" xref="S2.Ex19.m1.1.1.1.cmml"><mi id="S2.Ex19.m1.1.1.1.3" xref="S2.Ex19.m1.1.1.1.3.cmml">α</mi><mo id="S2.Ex19.m1.1.1.1.2" xref="S2.Ex19.m1.1.1.1.2.cmml">∈</mo><mrow id="S2.Ex19.m1.1.1.1.4" xref="S2.Ex19.m1.1.1.1.4.cmml"><mi id="S2.Ex19.m1.1.1.1.4.2" xref="S2.Ex19.m1.1.1.1.4.2.cmml">Mor</mi><mo id="S2.Ex19.m1.1.1.1.4.1" xref="S2.Ex19.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.Ex19.m1.1.1.1.4.3.2" xref="S2.Ex19.m1.1.1.1.4.cmml"><mo id="S2.Ex19.m1.1.1.1.4.3.2.1" stretchy="false" xref="S2.Ex19.m1.1.1.1.4.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex19.m1.1.1.1.1" xref="S2.Ex19.m1.1.1.1.1.cmml">𝒞</mi><mo id="S2.Ex19.m1.1.1.1.4.3.2.2" stretchy="false" xref="S2.Ex19.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.Ex19.m1.2.3.3.3.2" xref="S2.Ex19.m1.2.3.3.3.2.cmml"><mi id="S2.Ex19.m1.2.3.3.3.2.2" xref="S2.Ex19.m1.2.3.3.3.2.2.cmml">M</mi><mo id="S2.Ex19.m1.2.3.3.3.2.1" xref="S2.Ex19.m1.2.3.3.3.2.1.cmml">⁢</mo><mrow id="S2.Ex19.m1.2.3.3.3.2.3.2" xref="S2.Ex19.m1.2.3.3.3.2.cmml"><mo id="S2.Ex19.m1.2.3.3.3.2.3.2.1" stretchy="false" xref="S2.Ex19.m1.2.3.3.3.2.cmml">(</mo><mi id="S2.Ex19.m1.2.2" xref="S2.Ex19.m1.2.2.cmml">α</mi><mo id="S2.Ex19.m1.2.3.3.3.2.3.2.2" stretchy="false" xref="S2.Ex19.m1.2.3.3.3.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex19.m1.2b"><apply id="S2.Ex19.m1.2.3.cmml" xref="S2.Ex19.m1.2.3"><ci id="S2.Ex19.m1.2.3.1.cmml" xref="S2.Ex19.m1.2.3.1">:</ci><ci id="S2.Ex19.m1.2.3.2.cmml" xref="S2.Ex19.m1.2.3.2">𝑓</ci><apply id="S2.Ex19.m1.2.3.3.cmml" xref="S2.Ex19.m1.2.3.3"><ci id="S2.Ex19.m1.2.3.3.1.cmml" xref="S2.Ex19.m1.2.3.3.1">→</ci><apply id="S2.Ex19.m1.2.3.3.2.cmml" xref="S2.Ex19.m1.2.3.3.2"><times id="S2.Ex19.m1.2.3.3.2.1.cmml" xref="S2.Ex19.m1.2.3.3.2.1"></times><ci id="S2.Ex19.m1.2.3.3.2.2.cmml" xref="S2.Ex19.m1.2.3.3.2.2">𝑁</ci><apply id="S2.Ex19.m1.2.3.3.2.3.cmml" xref="S2.Ex19.m1.2.3.3.2.3"><csymbol cd="ambiguous" id="S2.Ex19.m1.2.3.3.2.3.1.cmml" xref="S2.Ex19.m1.2.3.3.2.3">subscript</csymbol><ci id="S2.Ex19.m1.2.3.3.2.3.2.cmml" xref="S2.Ex19.m1.2.3.3.2.3.2">𝒞</ci><ci id="S2.Ex19.m1.2.3.3.2.3.3.cmml" xref="S2.Ex19.m1.2.3.3.2.3.3">𝑛</ci></apply></apply><apply id="S2.Ex19.m1.2.3.3.3.cmml" xref="S2.Ex19.m1.2.3.3.3"><apply id="S2.Ex19.m1.2.3.3.3.1.cmml" xref="S2.Ex19.m1.2.3.3.3.1"><csymbol cd="ambiguous" id="S2.Ex19.m1.2.3.3.3.1.1.cmml" xref="S2.Ex19.m1.2.3.3.3.1">subscript</csymbol><csymbol cd="latexml" id="S2.Ex19.m1.2.3.3.3.1.2.cmml" xref="S2.Ex19.m1.2.3.3.3.1.2">coproduct</csymbol><apply id="S2.Ex19.m1.1.1.1.cmml" xref="S2.Ex19.m1.1.1.1"><in id="S2.Ex19.m1.1.1.1.2.cmml" xref="S2.Ex19.m1.1.1.1.2"></in><ci id="S2.Ex19.m1.1.1.1.3.cmml" xref="S2.Ex19.m1.1.1.1.3">𝛼</ci><apply id="S2.Ex19.m1.1.1.1.4.cmml" xref="S2.Ex19.m1.1.1.1.4"><times id="S2.Ex19.m1.1.1.1.4.1.cmml" xref="S2.Ex19.m1.1.1.1.4.1"></times><ci id="S2.Ex19.m1.1.1.1.4.2.cmml" xref="S2.Ex19.m1.1.1.1.4.2">Mor</ci><ci id="S2.Ex19.m1.1.1.1.1.cmml" xref="S2.Ex19.m1.1.1.1.1">𝒞</ci></apply></apply></apply><apply id="S2.Ex19.m1.2.3.3.3.2.cmml" xref="S2.Ex19.m1.2.3.3.3.2"><times id="S2.Ex19.m1.2.3.3.3.2.1.cmml" xref="S2.Ex19.m1.2.3.3.3.2.1"></times><ci id="S2.Ex19.m1.2.3.3.3.2.2.cmml" xref="S2.Ex19.m1.2.3.3.3.2.2">𝑀</ci><ci id="S2.Ex19.m1.2.2.cmml" xref="S2.Ex19.m1.2.2">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex19.m1.2c">f:N\mathcal{C}_{n}\to\coprod_{\alpha\in\mathrm{Mor}(\mathcal{C})}M(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex19.m1.2d">italic_f : italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → ∐ start_POSTSUBSCRIPT italic_α ∈ roman_Mor ( caligraphic_C ) end_POSTSUBSCRIPT italic_M ( italic_α )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem6.p1.16">such that <math alttext="f(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})\in M(\alpha_{n}\cdots% \alpha_{1})" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.11.m1.2"><semantics id="S2.Thmtheorem6.p1.11.m1.2a"><mrow id="S2.Thmtheorem6.p1.11.m1.2.2" xref="S2.Thmtheorem6.p1.11.m1.2.2.cmml"><mrow id="S2.Thmtheorem6.p1.11.m1.1.1.1" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.11.m1.1.1.1.3" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.3.cmml">f</mi><mo id="S2.Thmtheorem6.p1.11.m1.1.1.1.2" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.cmml"><msub id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.cmml"><mi id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.2" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.2.cmml">c</mi><mn id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.3" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.1" lspace="0.167em" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.cmml"><mover id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.cmml"><mo id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.2.cmml">⟶</mo><msub id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.cmml"><mi id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.2" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.2.cmml">α</mi><mn id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.3" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.3.cmml">1</mn></msub></mover><mrow id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.2" mathvariant="normal" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.2.cmml">⋯</mi><mo id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.1" lspace="0.337em" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.cmml"><mover id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.cmml"><mo id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.cmml"><mi id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.2" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.2.cmml">α</mi><mi id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.3" 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xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.cmml"><msub id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.cmml"><mi id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.2" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.2.cmml">α</mi><mi id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.3" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.3.cmml">n</mi></msub><mo id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.1" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.3" mathvariant="normal" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.3.cmml">⋯</mi><mo id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.1a" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.1.cmml">⁢</mo><msub id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.cmml"><mi id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.2" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.2.cmml">α</mi><mn id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.3" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.3.cmml">1</mn></msub></mrow><mo id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.11.m1.2b"><apply id="S2.Thmtheorem6.p1.11.m1.2.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2"><in id="S2.Thmtheorem6.p1.11.m1.2.2.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.3"></in><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1"><times id="S2.Thmtheorem6.p1.11.m1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.2"></times><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.3">𝑓</ci><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1"><times id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.1"></times><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.2">𝑐</ci><cn id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.2.3">0</cn></apply><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3"><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1">superscript</csymbol><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.2">⟶</ci><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3">subscript</csymbol><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.2">𝛼</ci><cn id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.1.3.3">1</cn></apply></apply><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2"><times id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.1"></times><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.2">⋯</ci><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3"><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1">superscript</csymbol><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.2">⟶</ci><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3">subscript</csymbol><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.2">𝛼</ci><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.1.3.3">𝑛</ci></apply></apply><apply id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.2.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.2">subscript</csymbol><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.2.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.2.2">𝑐</ci><ci id="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.2.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.1.1.1.1.1.1.3.2.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply><apply id="S2.Thmtheorem6.p1.11.m1.2.2.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2"><times id="S2.Thmtheorem6.p1.11.m1.2.2.2.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.2"></times><ci id="S2.Thmtheorem6.p1.11.m1.2.2.2.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.3">𝑀</ci><apply id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1"><times id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.1"></times><apply id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.2">𝛼</ci><ci id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.2.3">𝑛</ci></apply><ci id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.3">⋯</ci><apply id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.1.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4">subscript</csymbol><ci id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.2.cmml" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.2">𝛼</ci><cn id="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.11.m1.2.2.2.1.1.1.4.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.11.m1.2c">f(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots\smash{\,% \mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})\in M(\alpha_{n}\cdots% \alpha_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.11.m1.2d">italic_f ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ italic_M ( italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> if <math alttext="n>0" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.12.m2.1"><semantics id="S2.Thmtheorem6.p1.12.m2.1a"><mrow id="S2.Thmtheorem6.p1.12.m2.1.1" xref="S2.Thmtheorem6.p1.12.m2.1.1.cmml"><mi id="S2.Thmtheorem6.p1.12.m2.1.1.2" xref="S2.Thmtheorem6.p1.12.m2.1.1.2.cmml">n</mi><mo id="S2.Thmtheorem6.p1.12.m2.1.1.1" xref="S2.Thmtheorem6.p1.12.m2.1.1.1.cmml">></mo><mn id="S2.Thmtheorem6.p1.12.m2.1.1.3" xref="S2.Thmtheorem6.p1.12.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.12.m2.1b"><apply id="S2.Thmtheorem6.p1.12.m2.1.1.cmml" xref="S2.Thmtheorem6.p1.12.m2.1.1"><gt id="S2.Thmtheorem6.p1.12.m2.1.1.1.cmml" xref="S2.Thmtheorem6.p1.12.m2.1.1.1"></gt><ci id="S2.Thmtheorem6.p1.12.m2.1.1.2.cmml" xref="S2.Thmtheorem6.p1.12.m2.1.1.2">𝑛</ci><cn id="S2.Thmtheorem6.p1.12.m2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.12.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.12.m2.1c">n>0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.12.m2.1d">italic_n > 0</annotation></semantics></math>. For <math alttext="n=0" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.13.m3.1"><semantics id="S2.Thmtheorem6.p1.13.m3.1a"><mrow id="S2.Thmtheorem6.p1.13.m3.1.1" xref="S2.Thmtheorem6.p1.13.m3.1.1.cmml"><mi id="S2.Thmtheorem6.p1.13.m3.1.1.2" xref="S2.Thmtheorem6.p1.13.m3.1.1.2.cmml">n</mi><mo id="S2.Thmtheorem6.p1.13.m3.1.1.1" xref="S2.Thmtheorem6.p1.13.m3.1.1.1.cmml">=</mo><mn id="S2.Thmtheorem6.p1.13.m3.1.1.3" xref="S2.Thmtheorem6.p1.13.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.13.m3.1b"><apply id="S2.Thmtheorem6.p1.13.m3.1.1.cmml" xref="S2.Thmtheorem6.p1.13.m3.1.1"><eq id="S2.Thmtheorem6.p1.13.m3.1.1.1.cmml" xref="S2.Thmtheorem6.p1.13.m3.1.1.1"></eq><ci id="S2.Thmtheorem6.p1.13.m3.1.1.2.cmml" xref="S2.Thmtheorem6.p1.13.m3.1.1.2">𝑛</ci><cn id="S2.Thmtheorem6.p1.13.m3.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.13.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.13.m3.1c">n=0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.13.m3.1d">italic_n = 0</annotation></semantics></math>, we require <math alttext="f(c)\in M(\mathrm{id}_{c})" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.14.m4.2"><semantics id="S2.Thmtheorem6.p1.14.m4.2a"><mrow id="S2.Thmtheorem6.p1.14.m4.2.2" xref="S2.Thmtheorem6.p1.14.m4.2.2.cmml"><mrow id="S2.Thmtheorem6.p1.14.m4.2.2.3" xref="S2.Thmtheorem6.p1.14.m4.2.2.3.cmml"><mi id="S2.Thmtheorem6.p1.14.m4.2.2.3.2" xref="S2.Thmtheorem6.p1.14.m4.2.2.3.2.cmml">f</mi><mo id="S2.Thmtheorem6.p1.14.m4.2.2.3.1" xref="S2.Thmtheorem6.p1.14.m4.2.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.14.m4.2.2.3.3.2" xref="S2.Thmtheorem6.p1.14.m4.2.2.3.cmml"><mo id="S2.Thmtheorem6.p1.14.m4.2.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.14.m4.2.2.3.cmml">(</mo><mi id="S2.Thmtheorem6.p1.14.m4.1.1" xref="S2.Thmtheorem6.p1.14.m4.1.1.cmml">c</mi><mo id="S2.Thmtheorem6.p1.14.m4.2.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem6.p1.14.m4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem6.p1.14.m4.2.2.2" xref="S2.Thmtheorem6.p1.14.m4.2.2.2.cmml">∈</mo><mrow id="S2.Thmtheorem6.p1.14.m4.2.2.1" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.cmml"><mi id="S2.Thmtheorem6.p1.14.m4.2.2.1.3" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.3.cmml">M</mi><mo id="S2.Thmtheorem6.p1.14.m4.2.2.1.2" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.cmml">(</mo><msub id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.2" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.2.cmml">id</mi><mi id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.3" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.3.cmml">c</mi></msub><mo id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.14.m4.2b"><apply id="S2.Thmtheorem6.p1.14.m4.2.2.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2"><in id="S2.Thmtheorem6.p1.14.m4.2.2.2.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.2"></in><apply id="S2.Thmtheorem6.p1.14.m4.2.2.3.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.3"><times id="S2.Thmtheorem6.p1.14.m4.2.2.3.1.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.3.1"></times><ci id="S2.Thmtheorem6.p1.14.m4.2.2.3.2.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.3.2">𝑓</ci><ci id="S2.Thmtheorem6.p1.14.m4.1.1.cmml" xref="S2.Thmtheorem6.p1.14.m4.1.1">𝑐</ci></apply><apply id="S2.Thmtheorem6.p1.14.m4.2.2.1.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.1"><times id="S2.Thmtheorem6.p1.14.m4.2.2.1.2.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.2"></times><ci id="S2.Thmtheorem6.p1.14.m4.2.2.1.3.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.3">𝑀</ci><apply id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.2">id</ci><ci id="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.14.m4.2.2.1.1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.14.m4.2c">f(c)\in M(\mathrm{id}_{c})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.14.m4.2d">italic_f ( italic_c ) ∈ italic_M ( roman_id start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math> for all <math alttext="c\in\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.15.m5.1"><semantics id="S2.Thmtheorem6.p1.15.m5.1a"><mrow id="S2.Thmtheorem6.p1.15.m5.1.1" xref="S2.Thmtheorem6.p1.15.m5.1.1.cmml"><mi id="S2.Thmtheorem6.p1.15.m5.1.1.2" xref="S2.Thmtheorem6.p1.15.m5.1.1.2.cmml">c</mi><mo id="S2.Thmtheorem6.p1.15.m5.1.1.1" xref="S2.Thmtheorem6.p1.15.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem6.p1.15.m5.1.1.3" xref="S2.Thmtheorem6.p1.15.m5.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.15.m5.1b"><apply id="S2.Thmtheorem6.p1.15.m5.1.1.cmml" xref="S2.Thmtheorem6.p1.15.m5.1.1"><in id="S2.Thmtheorem6.p1.15.m5.1.1.1.cmml" xref="S2.Thmtheorem6.p1.15.m5.1.1.1"></in><ci id="S2.Thmtheorem6.p1.15.m5.1.1.2.cmml" xref="S2.Thmtheorem6.p1.15.m5.1.1.2">𝑐</ci><ci id="S2.Thmtheorem6.p1.15.m5.1.1.3.cmml" xref="S2.Thmtheorem6.p1.15.m5.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.15.m5.1c">c\in\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.15.m5.1d">italic_c ∈ caligraphic_C</annotation></semantics></math>. For <math alttext="n>1" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.16.m6.1"><semantics id="S2.Thmtheorem6.p1.16.m6.1a"><mrow id="S2.Thmtheorem6.p1.16.m6.1.1" xref="S2.Thmtheorem6.p1.16.m6.1.1.cmml"><mi id="S2.Thmtheorem6.p1.16.m6.1.1.2" xref="S2.Thmtheorem6.p1.16.m6.1.1.2.cmml">n</mi><mo id="S2.Thmtheorem6.p1.16.m6.1.1.1" xref="S2.Thmtheorem6.p1.16.m6.1.1.1.cmml">></mo><mn id="S2.Thmtheorem6.p1.16.m6.1.1.3" xref="S2.Thmtheorem6.p1.16.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.16.m6.1b"><apply id="S2.Thmtheorem6.p1.16.m6.1.1.cmml" xref="S2.Thmtheorem6.p1.16.m6.1.1"><gt id="S2.Thmtheorem6.p1.16.m6.1.1.1.cmml" xref="S2.Thmtheorem6.p1.16.m6.1.1.1"></gt><ci id="S2.Thmtheorem6.p1.16.m6.1.1.2.cmml" xref="S2.Thmtheorem6.p1.16.m6.1.1.2">𝑛</ci><cn id="S2.Thmtheorem6.p1.16.m6.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.16.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.16.m6.1c">n>1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.16.m6.1d">italic_n > 1</annotation></semantics></math>, the coboundary map</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\delta^{n-1}:C^{n-1}_{BW}(\mathcal{C};M)\to C^{n}_{BW}(\mathcal{C};M)" class="ltx_Math" display="block" id="S2.Ex20.m1.4"><semantics id="S2.Ex20.m1.4a"><mrow id="S2.Ex20.m1.4.5" xref="S2.Ex20.m1.4.5.cmml"><msup id="S2.Ex20.m1.4.5.2" xref="S2.Ex20.m1.4.5.2.cmml"><mi id="S2.Ex20.m1.4.5.2.2" xref="S2.Ex20.m1.4.5.2.2.cmml">δ</mi><mrow id="S2.Ex20.m1.4.5.2.3" xref="S2.Ex20.m1.4.5.2.3.cmml"><mi id="S2.Ex20.m1.4.5.2.3.2" xref="S2.Ex20.m1.4.5.2.3.2.cmml">n</mi><mo id="S2.Ex20.m1.4.5.2.3.1" xref="S2.Ex20.m1.4.5.2.3.1.cmml">−</mo><mn id="S2.Ex20.m1.4.5.2.3.3" xref="S2.Ex20.m1.4.5.2.3.3.cmml">1</mn></mrow></msup><mo id="S2.Ex20.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex20.m1.4.5.1.cmml">:</mo><mrow id="S2.Ex20.m1.4.5.3" xref="S2.Ex20.m1.4.5.3.cmml"><mrow id="S2.Ex20.m1.4.5.3.2" xref="S2.Ex20.m1.4.5.3.2.cmml"><msubsup id="S2.Ex20.m1.4.5.3.2.2" xref="S2.Ex20.m1.4.5.3.2.2.cmml"><mi id="S2.Ex20.m1.4.5.3.2.2.2.2" xref="S2.Ex20.m1.4.5.3.2.2.2.2.cmml">C</mi><mrow id="S2.Ex20.m1.4.5.3.2.2.3" xref="S2.Ex20.m1.4.5.3.2.2.3.cmml"><mi id="S2.Ex20.m1.4.5.3.2.2.3.2" xref="S2.Ex20.m1.4.5.3.2.2.3.2.cmml">B</mi><mo id="S2.Ex20.m1.4.5.3.2.2.3.1" xref="S2.Ex20.m1.4.5.3.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex20.m1.4.5.3.2.2.3.3" xref="S2.Ex20.m1.4.5.3.2.2.3.3.cmml">W</mi></mrow><mrow id="S2.Ex20.m1.4.5.3.2.2.2.3" xref="S2.Ex20.m1.4.5.3.2.2.2.3.cmml"><mi id="S2.Ex20.m1.4.5.3.2.2.2.3.2" xref="S2.Ex20.m1.4.5.3.2.2.2.3.2.cmml">n</mi><mo id="S2.Ex20.m1.4.5.3.2.2.2.3.1" xref="S2.Ex20.m1.4.5.3.2.2.2.3.1.cmml">−</mo><mn id="S2.Ex20.m1.4.5.3.2.2.2.3.3" xref="S2.Ex20.m1.4.5.3.2.2.2.3.3.cmml">1</mn></mrow></msubsup><mo id="S2.Ex20.m1.4.5.3.2.1" xref="S2.Ex20.m1.4.5.3.2.1.cmml">⁢</mo><mrow id="S2.Ex20.m1.4.5.3.2.3.2" xref="S2.Ex20.m1.4.5.3.2.3.1.cmml"><mo id="S2.Ex20.m1.4.5.3.2.3.2.1" stretchy="false" xref="S2.Ex20.m1.4.5.3.2.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex20.m1.1.1" xref="S2.Ex20.m1.1.1.cmml">𝒞</mi><mo id="S2.Ex20.m1.4.5.3.2.3.2.2" xref="S2.Ex20.m1.4.5.3.2.3.1.cmml">;</mo><mi id="S2.Ex20.m1.2.2" xref="S2.Ex20.m1.2.2.cmml">M</mi><mo id="S2.Ex20.m1.4.5.3.2.3.2.3" stretchy="false" xref="S2.Ex20.m1.4.5.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex20.m1.4.5.3.1" stretchy="false" xref="S2.Ex20.m1.4.5.3.1.cmml">→</mo><mrow id="S2.Ex20.m1.4.5.3.3" xref="S2.Ex20.m1.4.5.3.3.cmml"><msubsup id="S2.Ex20.m1.4.5.3.3.2" xref="S2.Ex20.m1.4.5.3.3.2.cmml"><mi id="S2.Ex20.m1.4.5.3.3.2.2.2" xref="S2.Ex20.m1.4.5.3.3.2.2.2.cmml">C</mi><mrow id="S2.Ex20.m1.4.5.3.3.2.3" xref="S2.Ex20.m1.4.5.3.3.2.3.cmml"><mi id="S2.Ex20.m1.4.5.3.3.2.3.2" xref="S2.Ex20.m1.4.5.3.3.2.3.2.cmml">B</mi><mo id="S2.Ex20.m1.4.5.3.3.2.3.1" xref="S2.Ex20.m1.4.5.3.3.2.3.1.cmml">⁢</mo><mi id="S2.Ex20.m1.4.5.3.3.2.3.3" xref="S2.Ex20.m1.4.5.3.3.2.3.3.cmml">W</mi></mrow><mi id="S2.Ex20.m1.4.5.3.3.2.2.3" xref="S2.Ex20.m1.4.5.3.3.2.2.3.cmml">n</mi></msubsup><mo id="S2.Ex20.m1.4.5.3.3.1" xref="S2.Ex20.m1.4.5.3.3.1.cmml">⁢</mo><mrow id="S2.Ex20.m1.4.5.3.3.3.2" xref="S2.Ex20.m1.4.5.3.3.3.1.cmml"><mo id="S2.Ex20.m1.4.5.3.3.3.2.1" stretchy="false" xref="S2.Ex20.m1.4.5.3.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex20.m1.3.3" xref="S2.Ex20.m1.3.3.cmml">𝒞</mi><mo id="S2.Ex20.m1.4.5.3.3.3.2.2" xref="S2.Ex20.m1.4.5.3.3.3.1.cmml">;</mo><mi id="S2.Ex20.m1.4.4" xref="S2.Ex20.m1.4.4.cmml">M</mi><mo id="S2.Ex20.m1.4.5.3.3.3.2.3" stretchy="false" xref="S2.Ex20.m1.4.5.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex20.m1.4b"><apply id="S2.Ex20.m1.4.5.cmml" xref="S2.Ex20.m1.4.5"><ci id="S2.Ex20.m1.4.5.1.cmml" xref="S2.Ex20.m1.4.5.1">:</ci><apply id="S2.Ex20.m1.4.5.2.cmml" xref="S2.Ex20.m1.4.5.2"><csymbol cd="ambiguous" id="S2.Ex20.m1.4.5.2.1.cmml" xref="S2.Ex20.m1.4.5.2">superscript</csymbol><ci id="S2.Ex20.m1.4.5.2.2.cmml" xref="S2.Ex20.m1.4.5.2.2">𝛿</ci><apply id="S2.Ex20.m1.4.5.2.3.cmml" xref="S2.Ex20.m1.4.5.2.3"><minus id="S2.Ex20.m1.4.5.2.3.1.cmml" xref="S2.Ex20.m1.4.5.2.3.1"></minus><ci id="S2.Ex20.m1.4.5.2.3.2.cmml" xref="S2.Ex20.m1.4.5.2.3.2">𝑛</ci><cn id="S2.Ex20.m1.4.5.2.3.3.cmml" type="integer" xref="S2.Ex20.m1.4.5.2.3.3">1</cn></apply></apply><apply id="S2.Ex20.m1.4.5.3.cmml" xref="S2.Ex20.m1.4.5.3"><ci id="S2.Ex20.m1.4.5.3.1.cmml" xref="S2.Ex20.m1.4.5.3.1">→</ci><apply id="S2.Ex20.m1.4.5.3.2.cmml" xref="S2.Ex20.m1.4.5.3.2"><times id="S2.Ex20.m1.4.5.3.2.1.cmml" xref="S2.Ex20.m1.4.5.3.2.1"></times><apply id="S2.Ex20.m1.4.5.3.2.2.cmml" xref="S2.Ex20.m1.4.5.3.2.2"><csymbol cd="ambiguous" id="S2.Ex20.m1.4.5.3.2.2.1.cmml" xref="S2.Ex20.m1.4.5.3.2.2">subscript</csymbol><apply id="S2.Ex20.m1.4.5.3.2.2.2.cmml" xref="S2.Ex20.m1.4.5.3.2.2"><csymbol cd="ambiguous" id="S2.Ex20.m1.4.5.3.2.2.2.1.cmml" xref="S2.Ex20.m1.4.5.3.2.2">superscript</csymbol><ci id="S2.Ex20.m1.4.5.3.2.2.2.2.cmml" xref="S2.Ex20.m1.4.5.3.2.2.2.2">𝐶</ci><apply id="S2.Ex20.m1.4.5.3.2.2.2.3.cmml" xref="S2.Ex20.m1.4.5.3.2.2.2.3"><minus id="S2.Ex20.m1.4.5.3.2.2.2.3.1.cmml" xref="S2.Ex20.m1.4.5.3.2.2.2.3.1"></minus><ci id="S2.Ex20.m1.4.5.3.2.2.2.3.2.cmml" xref="S2.Ex20.m1.4.5.3.2.2.2.3.2">𝑛</ci><cn id="S2.Ex20.m1.4.5.3.2.2.2.3.3.cmml" type="integer" xref="S2.Ex20.m1.4.5.3.2.2.2.3.3">1</cn></apply></apply><apply id="S2.Ex20.m1.4.5.3.2.2.3.cmml" xref="S2.Ex20.m1.4.5.3.2.2.3"><times id="S2.Ex20.m1.4.5.3.2.2.3.1.cmml" xref="S2.Ex20.m1.4.5.3.2.2.3.1"></times><ci id="S2.Ex20.m1.4.5.3.2.2.3.2.cmml" xref="S2.Ex20.m1.4.5.3.2.2.3.2">𝐵</ci><ci id="S2.Ex20.m1.4.5.3.2.2.3.3.cmml" xref="S2.Ex20.m1.4.5.3.2.2.3.3">𝑊</ci></apply></apply><list id="S2.Ex20.m1.4.5.3.2.3.1.cmml" xref="S2.Ex20.m1.4.5.3.2.3.2"><ci id="S2.Ex20.m1.1.1.cmml" xref="S2.Ex20.m1.1.1">𝒞</ci><ci id="S2.Ex20.m1.2.2.cmml" xref="S2.Ex20.m1.2.2">𝑀</ci></list></apply><apply id="S2.Ex20.m1.4.5.3.3.cmml" xref="S2.Ex20.m1.4.5.3.3"><times id="S2.Ex20.m1.4.5.3.3.1.cmml" xref="S2.Ex20.m1.4.5.3.3.1"></times><apply id="S2.Ex20.m1.4.5.3.3.2.cmml" xref="S2.Ex20.m1.4.5.3.3.2"><csymbol cd="ambiguous" id="S2.Ex20.m1.4.5.3.3.2.1.cmml" xref="S2.Ex20.m1.4.5.3.3.2">subscript</csymbol><apply id="S2.Ex20.m1.4.5.3.3.2.2.cmml" xref="S2.Ex20.m1.4.5.3.3.2"><csymbol cd="ambiguous" id="S2.Ex20.m1.4.5.3.3.2.2.1.cmml" xref="S2.Ex20.m1.4.5.3.3.2">superscript</csymbol><ci id="S2.Ex20.m1.4.5.3.3.2.2.2.cmml" xref="S2.Ex20.m1.4.5.3.3.2.2.2">𝐶</ci><ci id="S2.Ex20.m1.4.5.3.3.2.2.3.cmml" xref="S2.Ex20.m1.4.5.3.3.2.2.3">𝑛</ci></apply><apply id="S2.Ex20.m1.4.5.3.3.2.3.cmml" xref="S2.Ex20.m1.4.5.3.3.2.3"><times id="S2.Ex20.m1.4.5.3.3.2.3.1.cmml" xref="S2.Ex20.m1.4.5.3.3.2.3.1"></times><ci id="S2.Ex20.m1.4.5.3.3.2.3.2.cmml" xref="S2.Ex20.m1.4.5.3.3.2.3.2">𝐵</ci><ci id="S2.Ex20.m1.4.5.3.3.2.3.3.cmml" xref="S2.Ex20.m1.4.5.3.3.2.3.3">𝑊</ci></apply></apply><list id="S2.Ex20.m1.4.5.3.3.3.1.cmml" xref="S2.Ex20.m1.4.5.3.3.3.2"><ci id="S2.Ex20.m1.3.3.cmml" xref="S2.Ex20.m1.3.3">𝒞</ci><ci id="S2.Ex20.m1.4.4.cmml" xref="S2.Ex20.m1.4.4">𝑀</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex20.m1.4c">\delta^{n-1}:C^{n-1}_{BW}(\mathcal{C};M)\to C^{n}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex20.m1.4d">italic_δ start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT : italic_C start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M ) → italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem6.p1.19">is defined by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S7.EGx2"> <tbody id="S2.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\delta^{n}(f)(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{% \alpha_{1}}\,}\cdots\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_% {n})" class="ltx_Math" display="inline" id="S2.Ex21.m1.2"><semantics id="S2.Ex21.m1.2a"><mrow id="S2.Ex21.m1.2.2" xref="S2.Ex21.m1.2.2.cmml"><msup id="S2.Ex21.m1.2.2.3" xref="S2.Ex21.m1.2.2.3.cmml"><mi id="S2.Ex21.m1.2.2.3.2" xref="S2.Ex21.m1.2.2.3.2.cmml">δ</mi><mi id="S2.Ex21.m1.2.2.3.3" xref="S2.Ex21.m1.2.2.3.3.cmml">n</mi></msup><mo id="S2.Ex21.m1.2.2.2" xref="S2.Ex21.m1.2.2.2.cmml">⁢</mo><mrow id="S2.Ex21.m1.2.2.4.2" xref="S2.Ex21.m1.2.2.cmml"><mo id="S2.Ex21.m1.2.2.4.2.1" stretchy="false" xref="S2.Ex21.m1.2.2.cmml">(</mo><mi id="S2.Ex21.m1.1.1" xref="S2.Ex21.m1.1.1.cmml">f</mi><mo id="S2.Ex21.m1.2.2.4.2.2" stretchy="false" xref="S2.Ex21.m1.2.2.cmml">)</mo></mrow><mo id="S2.Ex21.m1.2.2.2a" xref="S2.Ex21.m1.2.2.2.cmml">⁢</mo><mrow id="S2.Ex21.m1.2.2.1.1" xref="S2.Ex21.m1.2.2.1.1.1.cmml"><mo id="S2.Ex21.m1.2.2.1.1.2" stretchy="false" xref="S2.Ex21.m1.2.2.1.1.1.cmml">(</mo><mrow id="S2.Ex21.m1.2.2.1.1.1" xref="S2.Ex21.m1.2.2.1.1.1.cmml"><msub id="S2.Ex21.m1.2.2.1.1.1.2" xref="S2.Ex21.m1.2.2.1.1.1.2.cmml"><mi id="S2.Ex21.m1.2.2.1.1.1.2.2" xref="S2.Ex21.m1.2.2.1.1.1.2.2.cmml">c</mi><mn id="S2.Ex21.m1.2.2.1.1.1.2.3" xref="S2.Ex21.m1.2.2.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Ex21.m1.2.2.1.1.1.1" lspace="0.167em" xref="S2.Ex21.m1.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex21.m1.2.2.1.1.1.3" xref="S2.Ex21.m1.2.2.1.1.1.3.cmml"><mover 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xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.cmml"><mi id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.2" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.2.cmml">α</mi><mi id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.3" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.3.cmml">n</mi></msub></mover><msub id="S2.Ex21.m1.2.2.1.1.1.3.2.3.2" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.cmml"><mi id="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.2" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.2.cmml">c</mi><mi id="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.3" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.3.cmml">n</mi></msub></mrow></mrow></mrow></mrow><mo id="S2.Ex21.m1.2.2.1.1.3" stretchy="false" xref="S2.Ex21.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex21.m1.2b"><apply id="S2.Ex21.m1.2.2.cmml" xref="S2.Ex21.m1.2.2"><times id="S2.Ex21.m1.2.2.2.cmml" xref="S2.Ex21.m1.2.2.2"></times><apply id="S2.Ex21.m1.2.2.3.cmml" xref="S2.Ex21.m1.2.2.3"><csymbol cd="ambiguous" 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xref="S2.Ex21.m1.2.2.1.1.1.3.1">superscript</csymbol><ci id="S2.Ex21.m1.2.2.1.1.1.3.1.2.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.1.2">⟶</ci><apply id="S2.Ex21.m1.2.2.1.1.1.3.1.3.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.1.3"><csymbol cd="ambiguous" id="S2.Ex21.m1.2.2.1.1.1.3.1.3.1.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.1.3">subscript</csymbol><ci id="S2.Ex21.m1.2.2.1.1.1.3.1.3.2.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.1.3.2">𝛼</ci><cn id="S2.Ex21.m1.2.2.1.1.1.3.1.3.3.cmml" type="integer" xref="S2.Ex21.m1.2.2.1.1.1.3.1.3.3">1</cn></apply></apply><apply id="S2.Ex21.m1.2.2.1.1.1.3.2.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2"><times id="S2.Ex21.m1.2.2.1.1.1.3.2.1.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.1"></times><ci id="S2.Ex21.m1.2.2.1.1.1.3.2.2.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.2">⋯</ci><apply id="S2.Ex21.m1.2.2.1.1.1.3.2.3.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3"><apply id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.1.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1">superscript</csymbol><ci id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.2.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.2">⟶</ci><apply id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3"><csymbol cd="ambiguous" id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.1.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3">subscript</csymbol><ci id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.2.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.2">𝛼</ci><ci id="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.3.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.1.3.3">𝑛</ci></apply></apply><apply id="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.1.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.2">subscript</csymbol><ci id="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.2.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.2">𝑐</ci><ci id="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.3.cmml" xref="S2.Ex21.m1.2.2.1.1.1.3.2.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex21.m1.2c">\displaystyle\delta^{n}(f)(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{% \alpha_{1}}\,}\cdots\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_% {n})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex21.m1.2d">italic_δ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_f ) ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\alpha_{1}^{*}f(c_{1}\smash{\,\mathop{\longrightarrow}\limits^{% \alpha_{2}}\,}\cdots\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_% {n})" class="ltx_Math" display="inline" id="S2.Ex21.m2.1"><semantics id="S2.Ex21.m2.1a"><mrow id="S2.Ex21.m2.1.1" xref="S2.Ex21.m2.1.1.cmml"><mi id="S2.Ex21.m2.1.1.3" xref="S2.Ex21.m2.1.1.3.cmml"></mi><mo id="S2.Ex21.m2.1.1.2" xref="S2.Ex21.m2.1.1.2.cmml">=</mo><mrow id="S2.Ex21.m2.1.1.1" xref="S2.Ex21.m2.1.1.1.cmml"><msubsup id="S2.Ex21.m2.1.1.1.3" xref="S2.Ex21.m2.1.1.1.3.cmml"><mi id="S2.Ex21.m2.1.1.1.3.2.2" xref="S2.Ex21.m2.1.1.1.3.2.2.cmml">α</mi><mn id="S2.Ex21.m2.1.1.1.3.2.3" xref="S2.Ex21.m2.1.1.1.3.2.3.cmml">1</mn><mo id="S2.Ex21.m2.1.1.1.3.3" xref="S2.Ex21.m2.1.1.1.3.3.cmml">∗</mo></msubsup><mo id="S2.Ex21.m2.1.1.1.2" xref="S2.Ex21.m2.1.1.1.2.cmml">⁢</mo><mi id="S2.Ex21.m2.1.1.1.4" xref="S2.Ex21.m2.1.1.1.4.cmml">f</mi><mo id="S2.Ex21.m2.1.1.1.2a" xref="S2.Ex21.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex21.m2.1.1.1.1.1" xref="S2.Ex21.m2.1.1.1.1.1.1.cmml"><mo id="S2.Ex21.m2.1.1.1.1.1.2" stretchy="false" 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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\sum_{i=1}^{n-1}(-1)^{i}f(c_{0}\smash{\,\mathop{\longrightarrow}% \limits^{\alpha_{1}}\,}\cdots c_{i-1}\smash{\,\mathop{\longrightarrow}\limits^% {\alpha_{i+1}\alpha_{i}}\,}c_{i+1}\cdots\smash{\,\mathop{\longrightarrow}% \limits^{\alpha_{n}}\,}c_{n})" class="ltx_Math" display="inline" id="S2.Ex22.m1.2"><semantics id="S2.Ex22.m1.2a"><mrow id="S2.Ex22.m1.2.2" xref="S2.Ex22.m1.2.2.cmml"><mo id="S2.Ex22.m1.2.2a" xref="S2.Ex22.m1.2.2.cmml">+</mo><mrow id="S2.Ex22.m1.2.2.2" xref="S2.Ex22.m1.2.2.2.cmml"><mstyle displaystyle="true" id="S2.Ex22.m1.2.2.2.3" xref="S2.Ex22.m1.2.2.2.3.cmml"><munderover id="S2.Ex22.m1.2.2.2.3a" xref="S2.Ex22.m1.2.2.2.3.cmml"><mo id="S2.Ex22.m1.2.2.2.3.2.2" 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xref="S2.Ex22.m1.2.2.2.2.2.1.1.3.2.4.2.4.2"><csymbol cd="ambiguous" id="S2.Ex22.m1.2.2.2.2.2.1.1.3.2.4.2.4.2.1.cmml" xref="S2.Ex22.m1.2.2.2.2.2.1.1.3.2.4.2.4.2">subscript</csymbol><ci id="S2.Ex22.m1.2.2.2.2.2.1.1.3.2.4.2.4.2.2.cmml" xref="S2.Ex22.m1.2.2.2.2.2.1.1.3.2.4.2.4.2.2">𝑐</ci><ci id="S2.Ex22.m1.2.2.2.2.2.1.1.3.2.4.2.4.2.3.cmml" xref="S2.Ex22.m1.2.2.2.2.2.1.1.3.2.4.2.4.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex22.m1.2c">\displaystyle+\sum_{i=1}^{n-1}(-1)^{i}f(c_{0}\smash{\,\mathop{\longrightarrow}% \limits^{\alpha_{1}}\,}\cdots c_{i-1}\smash{\,\mathop{\longrightarrow}\limits^% {\alpha_{i+1}\alpha_{i}}\,}c_{i+1}\cdots\smash{\,\mathop{\longrightarrow}% \limits^{\alpha_{n}}\,}c_{n})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex22.m1.2d">+ ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_f ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ italic_c start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.1.3.3.1.cmml" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.1.3.3.1"></minus><ci id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.1.3.3.2.cmml" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.1.3.3.2">𝑛</ci><cn id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.1.3.3.3.cmml" type="integer" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.1.3.3.3">1</cn></apply></apply></apply><apply id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.cmml" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.1.cmml" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2">subscript</csymbol><ci id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.2.cmml" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.2">𝑐</ci><apply id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.3.cmml" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.3"><minus id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.3.1.cmml" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.3.1"></minus><ci id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.3.2.cmml" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.3.2">𝑛</ci><cn id="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.3.3.cmml" type="integer" xref="S2.Ex23.m1.1.1.1.1.3.3.1.1.3.2.3.2.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex23.m1.1c">\displaystyle+(-1)^{n}(\alpha_{n})_{*}f(c_{0}\smash{\,\mathop{\longrightarrow}% \limits^{\alpha_{1}}\,}\cdots\smash{\,\mathop{\longrightarrow}\limits^{\alpha_% {n-1}}\,}c_{n-1}).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex23.m1.1d">+ ( - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT italic_f ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem6.p1.18">For <math alttext="n=1" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.17.m1.1"><semantics id="S2.Thmtheorem6.p1.17.m1.1a"><mrow id="S2.Thmtheorem6.p1.17.m1.1.1" xref="S2.Thmtheorem6.p1.17.m1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.17.m1.1.1.2" xref="S2.Thmtheorem6.p1.17.m1.1.1.2.cmml">n</mi><mo id="S2.Thmtheorem6.p1.17.m1.1.1.1" xref="S2.Thmtheorem6.p1.17.m1.1.1.1.cmml">=</mo><mn id="S2.Thmtheorem6.p1.17.m1.1.1.3" xref="S2.Thmtheorem6.p1.17.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.17.m1.1b"><apply id="S2.Thmtheorem6.p1.17.m1.1.1.cmml" xref="S2.Thmtheorem6.p1.17.m1.1.1"><eq id="S2.Thmtheorem6.p1.17.m1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.17.m1.1.1.1"></eq><ci id="S2.Thmtheorem6.p1.17.m1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.17.m1.1.1.2">𝑛</ci><cn id="S2.Thmtheorem6.p1.17.m1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.17.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.17.m1.1c">n=1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.17.m1.1d">italic_n = 1</annotation></semantics></math>, we have <math alttext="\delta^{0}(f)(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha}\,}c_{1})=% \alpha^{*}f(c_{1})-\alpha_{*}f(c_{0})" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.18.m2.4"><semantics id="S2.Thmtheorem6.p1.18.m2.4a"><mrow id="S2.Thmtheorem6.p1.18.m2.4.4" xref="S2.Thmtheorem6.p1.18.m2.4.4.cmml"><mrow id="S2.Thmtheorem6.p1.18.m2.2.2.1" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.cmml"><msup id="S2.Thmtheorem6.p1.18.m2.2.2.1.3" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.3.cmml"><mi id="S2.Thmtheorem6.p1.18.m2.2.2.1.3.2" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.3.2.cmml">δ</mi><mn id="S2.Thmtheorem6.p1.18.m2.2.2.1.3.3" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.3.3.cmml">0</mn></msup><mo id="S2.Thmtheorem6.p1.18.m2.2.2.1.2" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.18.m2.2.2.1.4.2" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.cmml"><mo id="S2.Thmtheorem6.p1.18.m2.2.2.1.4.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.cmml">(</mo><mi id="S2.Thmtheorem6.p1.18.m2.1.1" xref="S2.Thmtheorem6.p1.18.m2.1.1.cmml">f</mi><mo id="S2.Thmtheorem6.p1.18.m2.2.2.1.4.2.2" stretchy="false" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.cmml">)</mo></mrow><mo id="S2.Thmtheorem6.p1.18.m2.2.2.1.2a" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.cmml"><msub id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.cmml"><mi id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.2" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.2.cmml">c</mi><mn id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.3" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.1" lspace="0.167em" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.cmml"><mover id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.1" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.1.cmml"><mo id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.1.2.cmml">⟶</mo><mi id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.1.3" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.1.3.cmml">α</mi></mover><msub id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.2" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.2.cmml"><mi id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.2.2" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.2.2.cmml">c</mi><mn id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.2.3" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.3.2.3.cmml">1</mn></msub></mrow></mrow><mo id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem6.p1.18.m2.4.4.4" xref="S2.Thmtheorem6.p1.18.m2.4.4.4.cmml">=</mo><mrow id="S2.Thmtheorem6.p1.18.m2.4.4.3" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.cmml"><mrow id="S2.Thmtheorem6.p1.18.m2.3.3.2.1" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.cmml"><msup id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.3" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.3.cmml"><mi id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.3.2" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.3.2.cmml">α</mi><mo id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.3.3" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.3.3.cmml">∗</mo></msup><mo id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.2" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.2.cmml">⁢</mo><mi id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.4" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.4.cmml">f</mi><mo id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.2a" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1.cmml">(</mo><msub id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1.2" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1.2.cmml">c</mi><mn id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1.3" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.18.m2.3.3.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem6.p1.18.m2.4.4.3.3" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.3.cmml">−</mo><mrow id="S2.Thmtheorem6.p1.18.m2.4.4.3.2" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.cmml"><msub id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.cmml"><mi id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.2" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.2.cmml">α</mi><mo id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.3" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.3.cmml">∗</mo></msub><mo id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.2" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.2.cmml">⁢</mo><mi id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.4" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.4.cmml">f</mi><mo id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.2a" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.cmml">(</mo><msub id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.2" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.2.cmml">c</mi><mn id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.3" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.3.cmml">0</mn></msub><mo id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.18.m2.4b"><apply id="S2.Thmtheorem6.p1.18.m2.4.4.cmml" xref="S2.Thmtheorem6.p1.18.m2.4.4"><eq id="S2.Thmtheorem6.p1.18.m2.4.4.4.cmml" xref="S2.Thmtheorem6.p1.18.m2.4.4.4"></eq><apply id="S2.Thmtheorem6.p1.18.m2.2.2.1.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1"><times id="S2.Thmtheorem6.p1.18.m2.2.2.1.2.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.2"></times><apply id="S2.Thmtheorem6.p1.18.m2.2.2.1.3.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.18.m2.2.2.1.3.1.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.3">superscript</csymbol><ci id="S2.Thmtheorem6.p1.18.m2.2.2.1.3.2.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.3.2">𝛿</ci><cn id="S2.Thmtheorem6.p1.18.m2.2.2.1.3.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.3.3">0</cn></apply><ci id="S2.Thmtheorem6.p1.18.m2.1.1.cmml" xref="S2.Thmtheorem6.p1.18.m2.1.1">𝑓</ci><apply id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1"><times id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.1"></times><apply id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.2.cmml" xref="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.2">𝑐</ci><cn id="S2.Thmtheorem6.p1.18.m2.2.2.1.1.1.1.2.3.cmml" type="integer" 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id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.2.cmml" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.2">𝛼</ci><times id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.3.cmml" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.3.3"></times></apply><ci id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.4.cmml" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.4">𝑓</ci><apply id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.cmml" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1">subscript</csymbol><ci id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.2">𝑐</ci><cn id="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.18.m2.4.4.3.2.1.1.1.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.18.m2.4c">\delta^{0}(f)(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha}\,}c_{1})=% \alpha^{*}f(c_{1})-\alpha_{*}f(c_{0})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.18.m2.4d">italic_δ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_f ) ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_α start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_f ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_α start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT italic_f ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.1">The Baues-Wirsching cohomology satisfies many properties that one would expect to hold for a cohomology theory. For example, if <math alttext="\varphi:\mathcal{C}\to\mathcal{D}" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><mrow id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml">φ</mi><mo id="S2.SS2.p3.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p3.1.m1.1.1.1.cmml">:</mo><mrow id="S2.SS2.p3.1.m1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.1.m1.1.1.3.2" xref="S2.SS2.p3.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="S2.SS2.p3.1.m1.1.1.3.1" stretchy="false" xref="S2.SS2.p3.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.1.m1.1.1.3.3" xref="S2.SS2.p3.1.m1.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><apply id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"><ci id="S2.SS2.p3.1.m1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1.1">:</ci><ci id="S2.SS2.p3.1.m1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1.2">𝜑</ci><apply id="S2.SS2.p3.1.m1.1.1.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3"><ci id="S2.SS2.p3.1.m1.1.1.3.1.cmml" xref="S2.SS2.p3.1.m1.1.1.3.1">→</ci><ci id="S2.SS2.p3.1.m1.1.1.3.2.cmml" xref="S2.SS2.p3.1.m1.1.1.3.2">𝒞</ci><ci id="S2.SS2.p3.1.m1.1.1.3.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">\varphi:\mathcal{C}\to\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">italic_φ : caligraphic_C → caligraphic_D</annotation></semantics></math> is an equivalence of categories then the induced map</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\varphi^{*}:H^{*}_{BW}(\mathcal{D};M)\to H^{*}_{BW}(\mathcal{C};\varphi^{*}M)" class="ltx_Math" display="block" id="S2.Ex24.m1.4"><semantics id="S2.Ex24.m1.4a"><mrow id="S2.Ex24.m1.4.4" xref="S2.Ex24.m1.4.4.cmml"><msup id="S2.Ex24.m1.4.4.3" xref="S2.Ex24.m1.4.4.3.cmml"><mi id="S2.Ex24.m1.4.4.3.2" xref="S2.Ex24.m1.4.4.3.2.cmml">φ</mi><mo id="S2.Ex24.m1.4.4.3.3" xref="S2.Ex24.m1.4.4.3.3.cmml">∗</mo></msup><mo id="S2.Ex24.m1.4.4.2" lspace="0.278em" rspace="0.278em" xref="S2.Ex24.m1.4.4.2.cmml">:</mo><mrow id="S2.Ex24.m1.4.4.1" xref="S2.Ex24.m1.4.4.1.cmml"><mrow id="S2.Ex24.m1.4.4.1.3" xref="S2.Ex24.m1.4.4.1.3.cmml"><msubsup id="S2.Ex24.m1.4.4.1.3.2" xref="S2.Ex24.m1.4.4.1.3.2.cmml"><mi id="S2.Ex24.m1.4.4.1.3.2.2.2" xref="S2.Ex24.m1.4.4.1.3.2.2.2.cmml">H</mi><mrow id="S2.Ex24.m1.4.4.1.3.2.3" xref="S2.Ex24.m1.4.4.1.3.2.3.cmml"><mi id="S2.Ex24.m1.4.4.1.3.2.3.2" xref="S2.Ex24.m1.4.4.1.3.2.3.2.cmml">B</mi><mo id="S2.Ex24.m1.4.4.1.3.2.3.1" xref="S2.Ex24.m1.4.4.1.3.2.3.1.cmml">⁢</mo><mi id="S2.Ex24.m1.4.4.1.3.2.3.3" xref="S2.Ex24.m1.4.4.1.3.2.3.3.cmml">W</mi></mrow><mo 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id="S2.Ex24.m1.4.4.1.1.3.3.2.cmml" xref="S2.Ex24.m1.4.4.1.1.3.3.2">𝐵</ci><ci id="S2.Ex24.m1.4.4.1.1.3.3.3.cmml" xref="S2.Ex24.m1.4.4.1.1.3.3.3">𝑊</ci></apply></apply><list id="S2.Ex24.m1.4.4.1.1.1.2.cmml" xref="S2.Ex24.m1.4.4.1.1.1.1"><ci id="S2.Ex24.m1.3.3.cmml" xref="S2.Ex24.m1.3.3">𝒞</ci><apply id="S2.Ex24.m1.4.4.1.1.1.1.1.cmml" xref="S2.Ex24.m1.4.4.1.1.1.1.1"><times id="S2.Ex24.m1.4.4.1.1.1.1.1.1.cmml" xref="S2.Ex24.m1.4.4.1.1.1.1.1.1"></times><apply id="S2.Ex24.m1.4.4.1.1.1.1.1.2.cmml" xref="S2.Ex24.m1.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex24.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S2.Ex24.m1.4.4.1.1.1.1.1.2">superscript</csymbol><ci id="S2.Ex24.m1.4.4.1.1.1.1.1.2.2.cmml" xref="S2.Ex24.m1.4.4.1.1.1.1.1.2.2">𝜑</ci><times id="S2.Ex24.m1.4.4.1.1.1.1.1.2.3.cmml" xref="S2.Ex24.m1.4.4.1.1.1.1.1.2.3"></times></apply><ci id="S2.Ex24.m1.4.4.1.1.1.1.1.3.cmml" xref="S2.Ex24.m1.4.4.1.1.1.1.1.3">𝑀</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex24.m1.4c">\varphi^{*}:H^{*}_{BW}(\mathcal{D};M)\to H^{*}_{BW}(\mathcal{C};\varphi^{*}M)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex24.m1.4d">italic_φ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_D ; italic_M ) → italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_φ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p3.2">is an isomorphism for every natural system <math alttext="M:\mathfrak{F}\mathcal{D}\to R" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m1.1"><semantics id="S2.SS2.p3.2.m1.1a"><mrow id="S2.SS2.p3.2.m1.1.1" xref="S2.SS2.p3.2.m1.1.1.cmml"><mi id="S2.SS2.p3.2.m1.1.1.2" xref="S2.SS2.p3.2.m1.1.1.2.cmml">M</mi><mo id="S2.SS2.p3.2.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p3.2.m1.1.1.1.cmml">:</mo><mrow id="S2.SS2.p3.2.m1.1.1.3" xref="S2.SS2.p3.2.m1.1.1.3.cmml"><mrow id="S2.SS2.p3.2.m1.1.1.3.2" xref="S2.SS2.p3.2.m1.1.1.3.2.cmml"><mi id="S2.SS2.p3.2.m1.1.1.3.2.2" xref="S2.SS2.p3.2.m1.1.1.3.2.2.cmml">𝔉</mi><mo id="S2.SS2.p3.2.m1.1.1.3.2.1" xref="S2.SS2.p3.2.m1.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.2.m1.1.1.3.2.3" xref="S2.SS2.p3.2.m1.1.1.3.2.3.cmml">𝒟</mi></mrow><mo id="S2.SS2.p3.2.m1.1.1.3.1" stretchy="false" xref="S2.SS2.p3.2.m1.1.1.3.1.cmml">→</mo><mi id="S2.SS2.p3.2.m1.1.1.3.3" xref="S2.SS2.p3.2.m1.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m1.1b"><apply id="S2.SS2.p3.2.m1.1.1.cmml" xref="S2.SS2.p3.2.m1.1.1"><ci id="S2.SS2.p3.2.m1.1.1.1.cmml" xref="S2.SS2.p3.2.m1.1.1.1">:</ci><ci id="S2.SS2.p3.2.m1.1.1.2.cmml" xref="S2.SS2.p3.2.m1.1.1.2">𝑀</ci><apply id="S2.SS2.p3.2.m1.1.1.3.cmml" xref="S2.SS2.p3.2.m1.1.1.3"><ci id="S2.SS2.p3.2.m1.1.1.3.1.cmml" xref="S2.SS2.p3.2.m1.1.1.3.1">→</ci><apply id="S2.SS2.p3.2.m1.1.1.3.2.cmml" xref="S2.SS2.p3.2.m1.1.1.3.2"><times id="S2.SS2.p3.2.m1.1.1.3.2.1.cmml" xref="S2.SS2.p3.2.m1.1.1.3.2.1"></times><ci id="S2.SS2.p3.2.m1.1.1.3.2.2.cmml" xref="S2.SS2.p3.2.m1.1.1.3.2.2">𝔉</ci><ci id="S2.SS2.p3.2.m1.1.1.3.2.3.cmml" xref="S2.SS2.p3.2.m1.1.1.3.2.3">𝒟</ci></apply><ci id="S2.SS2.p3.2.m1.1.1.3.3.cmml" xref="S2.SS2.p3.2.m1.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m1.1c">M:\mathfrak{F}\mathcal{D}\to R</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m1.1d">italic_M : fraktur_F caligraphic_D → italic_R</annotation></semantics></math>-Mod (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib1" title="">1</a>, Theorem 1.11]</cite>).</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.1">Similar to the Quillen cohomology of small categories <math alttext="H^{*}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.2"><semantics id="S2.SS2.p4.1.m1.2a"><mrow id="S2.SS2.p4.1.m1.2.3" xref="S2.SS2.p4.1.m1.2.3.cmml"><msup id="S2.SS2.p4.1.m1.2.3.2" xref="S2.SS2.p4.1.m1.2.3.2.cmml"><mi id="S2.SS2.p4.1.m1.2.3.2.2" xref="S2.SS2.p4.1.m1.2.3.2.2.cmml">H</mi><mo id="S2.SS2.p4.1.m1.2.3.2.3" xref="S2.SS2.p4.1.m1.2.3.2.3.cmml">∗</mo></msup><mo id="S2.SS2.p4.1.m1.2.3.1" xref="S2.SS2.p4.1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p4.1.m1.2.3.3.2" xref="S2.SS2.p4.1.m1.2.3.3.1.cmml"><mo id="S2.SS2.p4.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p4.1.m1.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml">𝒞</mi><mo id="S2.SS2.p4.1.m1.2.3.3.2.2" xref="S2.SS2.p4.1.m1.2.3.3.1.cmml">;</mo><mi id="S2.SS2.p4.1.m1.2.2" xref="S2.SS2.p4.1.m1.2.2.cmml">M</mi><mo id="S2.SS2.p4.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS2.p4.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.2b"><apply id="S2.SS2.p4.1.m1.2.3.cmml" xref="S2.SS2.p4.1.m1.2.3"><times id="S2.SS2.p4.1.m1.2.3.1.cmml" xref="S2.SS2.p4.1.m1.2.3.1"></times><apply id="S2.SS2.p4.1.m1.2.3.2.cmml" xref="S2.SS2.p4.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p4.1.m1.2.3.2.1.cmml" xref="S2.SS2.p4.1.m1.2.3.2">superscript</csymbol><ci id="S2.SS2.p4.1.m1.2.3.2.2.cmml" xref="S2.SS2.p4.1.m1.2.3.2.2">𝐻</ci><times id="S2.SS2.p4.1.m1.2.3.2.3.cmml" xref="S2.SS2.p4.1.m1.2.3.2.3"></times></apply><list id="S2.SS2.p4.1.m1.2.3.3.1.cmml" xref="S2.SS2.p4.1.m1.2.3.3.2"><ci id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1">𝒞</ci><ci id="S2.SS2.p4.1.m1.2.2.cmml" xref="S2.SS2.p4.1.m1.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.2c">H^{*}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math>, Baues-Wirsching cohomology can also be interpreted as Ext-groups over a category of modules.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S2.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.1.1.1">Proposition 2.7</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem7.p1"> <p class="ltx_p" id="S2.Thmtheorem7.p1.6"><cite class="ltx_cite ltx_citemacro_cite"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem7.p1.6.7.1">[</span><a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib1" title="">1</a><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem7.p1.6.8.2">, Theorem 4.4]</span></cite><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem7.p1.6.6"> Let <math alttext="\underline{R}:\mathfrak{F}\mathcal{C}\to R" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.1.1.m1.1"><semantics id="S2.Thmtheorem7.p1.1.1.m1.1a"><mrow id="S2.Thmtheorem7.p1.1.1.m1.1.1" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml"><munder accentunder="true" id="S2.Thmtheorem7.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.2.cmml"><mi id="S2.Thmtheorem7.p1.1.1.m1.1.1.2.2" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.2.2.cmml">R</mi><mo id="S2.Thmtheorem7.p1.1.1.m1.1.1.2.1" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.2.1.cmml">¯</mo></munder><mo id="S2.Thmtheorem7.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.Thmtheorem7.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.cmml"><mrow id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.cmml"><mi id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.2" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.2.cmml">𝔉</mi><mo id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.1" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.3" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.3.cmml">𝒞</mi></mrow><mo id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.3" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.1.1.m1.1b"><apply id="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1"><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.1">:</ci><apply id="S2.Thmtheorem7.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.2"><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.2.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.2.1">¯</ci><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.2.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.2.2">𝑅</ci></apply><apply id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3"><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.1">→</ci><apply id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2"><times id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.1"></times><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.2">𝔉</ci><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.3">𝒞</ci></apply><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.1.1.m1.1c">\underline{R}:\mathfrak{F}\mathcal{C}\to R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.1.1.m1.1d">under¯ start_ARG italic_R end_ARG : fraktur_F caligraphic_C → italic_R</annotation></semantics></math>-Mod denote the constant functor for <math alttext="F\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.2.2.m2.1"><semantics id="S2.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem7.p1.2.2.m2.1.1" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">F</mi><mo id="S2.Thmtheorem7.p1.2.2.m2.1.1.1" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem7.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.2.2.m2.1b"><apply id="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1"><times id="S2.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.1"></times><ci id="S2.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.2">𝐹</ci><ci id="S2.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.2.2.m2.1c">F\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.2.2.m2.1d">italic_F caligraphic_C</annotation></semantics></math> over <math alttext="R" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.3.3.m3.1"><semantics id="S2.Thmtheorem7.p1.3.3.m3.1a"><mi id="S2.Thmtheorem7.p1.3.3.m3.1.1" xref="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.3.3.m3.1b"><ci id="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem7.p1.3.3.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.3.3.m3.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.3.3.m3.1d">italic_R</annotation></semantics></math>. For every natural system <math alttext="M" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.4.4.m4.1"><semantics id="S2.Thmtheorem7.p1.4.4.m4.1a"><mi id="S2.Thmtheorem7.p1.4.4.m4.1.1" xref="S2.Thmtheorem7.p1.4.4.m4.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.4.4.m4.1b"><ci id="S2.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.4.4.m4.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.4.4.m4.1d">italic_M</annotation></semantics></math> for <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.5.5.m5.1"><semantics id="S2.Thmtheorem7.p1.5.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem7.p1.5.5.m5.1.1" xref="S2.Thmtheorem7.p1.5.5.m5.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.5.5.m5.1b"><ci id="S2.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.5.5.m5.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.5.5.m5.1d">caligraphic_C</annotation></semantics></math>, and for <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.6.6.m6.1"><semantics id="S2.Thmtheorem7.p1.6.6.m6.1a"><mrow id="S2.Thmtheorem7.p1.6.6.m6.1.1" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.cmml"><mi id="S2.Thmtheorem7.p1.6.6.m6.1.1.2" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.2.cmml">n</mi><mo id="S2.Thmtheorem7.p1.6.6.m6.1.1.1" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.1.cmml">≥</mo><mn id="S2.Thmtheorem7.p1.6.6.m6.1.1.3" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.6.6.m6.1b"><apply id="S2.Thmtheorem7.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1"><geq id="S2.Thmtheorem7.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.1"></geq><ci id="S2.Thmtheorem7.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.2">𝑛</ci><cn id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.cmml" type="integer" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.6.6.m6.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.6.6.m6.1d">italic_n ≥ 0</annotation></semantics></math>, there is an isomorphism</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{n}_{BW}(\mathcal{C};M)\cong\mathrm{Ext}^{n}_{R\mathfrak{F}\mathcal{C}}(% \underline{R},M)" class="ltx_Math" display="block" id="S2.Ex25.m1.4"><semantics id="S2.Ex25.m1.4a"><mrow id="S2.Ex25.m1.4.5" xref="S2.Ex25.m1.4.5.cmml"><mrow id="S2.Ex25.m1.4.5.2" xref="S2.Ex25.m1.4.5.2.cmml"><msubsup id="S2.Ex25.m1.4.5.2.2" xref="S2.Ex25.m1.4.5.2.2.cmml"><mi id="S2.Ex25.m1.4.5.2.2.2.2" xref="S2.Ex25.m1.4.5.2.2.2.2.cmml">H</mi><mrow id="S2.Ex25.m1.4.5.2.2.3" xref="S2.Ex25.m1.4.5.2.2.3.cmml"><mi id="S2.Ex25.m1.4.5.2.2.3.2" xref="S2.Ex25.m1.4.5.2.2.3.2.cmml">B</mi><mo id="S2.Ex25.m1.4.5.2.2.3.1" xref="S2.Ex25.m1.4.5.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex25.m1.4.5.2.2.3.3" xref="S2.Ex25.m1.4.5.2.2.3.3.cmml">W</mi></mrow><mi id="S2.Ex25.m1.4.5.2.2.2.3" xref="S2.Ex25.m1.4.5.2.2.2.3.cmml">n</mi></msubsup><mo id="S2.Ex25.m1.4.5.2.1" xref="S2.Ex25.m1.4.5.2.1.cmml">⁢</mo><mrow id="S2.Ex25.m1.4.5.2.3.2" xref="S2.Ex25.m1.4.5.2.3.1.cmml"><mo id="S2.Ex25.m1.4.5.2.3.2.1" stretchy="false" xref="S2.Ex25.m1.4.5.2.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex25.m1.1.1" xref="S2.Ex25.m1.1.1.cmml">𝒞</mi><mo id="S2.Ex25.m1.4.5.2.3.2.2" xref="S2.Ex25.m1.4.5.2.3.1.cmml">;</mo><mi id="S2.Ex25.m1.2.2" xref="S2.Ex25.m1.2.2.cmml">M</mi><mo id="S2.Ex25.m1.4.5.2.3.2.3" stretchy="false" xref="S2.Ex25.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex25.m1.4.5.1" xref="S2.Ex25.m1.4.5.1.cmml">≅</mo><mrow id="S2.Ex25.m1.4.5.3" xref="S2.Ex25.m1.4.5.3.cmml"><msubsup id="S2.Ex25.m1.4.5.3.2" xref="S2.Ex25.m1.4.5.3.2.cmml"><mi id="S2.Ex25.m1.4.5.3.2.2.2" xref="S2.Ex25.m1.4.5.3.2.2.2.cmml">Ext</mi><mrow id="S2.Ex25.m1.4.5.3.2.3" xref="S2.Ex25.m1.4.5.3.2.3.cmml"><mi id="S2.Ex25.m1.4.5.3.2.3.2" xref="S2.Ex25.m1.4.5.3.2.3.2.cmml">R</mi><mo id="S2.Ex25.m1.4.5.3.2.3.1" xref="S2.Ex25.m1.4.5.3.2.3.1.cmml">⁢</mo><mi id="S2.Ex25.m1.4.5.3.2.3.3" xref="S2.Ex25.m1.4.5.3.2.3.3.cmml">𝔉</mi><mo id="S2.Ex25.m1.4.5.3.2.3.1a" xref="S2.Ex25.m1.4.5.3.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex25.m1.4.5.3.2.3.4" xref="S2.Ex25.m1.4.5.3.2.3.4.cmml">𝒞</mi></mrow><mi id="S2.Ex25.m1.4.5.3.2.2.3" xref="S2.Ex25.m1.4.5.3.2.2.3.cmml">n</mi></msubsup><mo id="S2.Ex25.m1.4.5.3.1" xref="S2.Ex25.m1.4.5.3.1.cmml">⁢</mo><mrow id="S2.Ex25.m1.4.5.3.3.2" xref="S2.Ex25.m1.4.5.3.3.1.cmml"><mo id="S2.Ex25.m1.4.5.3.3.2.1" stretchy="false" xref="S2.Ex25.m1.4.5.3.3.1.cmml">(</mo><munder accentunder="true" id="S2.Ex25.m1.3.3" xref="S2.Ex25.m1.3.3.cmml"><mi id="S2.Ex25.m1.3.3.2" xref="S2.Ex25.m1.3.3.2.cmml">R</mi><mo id="S2.Ex25.m1.3.3.1" xref="S2.Ex25.m1.3.3.1.cmml">¯</mo></munder><mo id="S2.Ex25.m1.4.5.3.3.2.2" xref="S2.Ex25.m1.4.5.3.3.1.cmml">,</mo><mi id="S2.Ex25.m1.4.4" xref="S2.Ex25.m1.4.4.cmml">M</mi><mo id="S2.Ex25.m1.4.5.3.3.2.3" stretchy="false" xref="S2.Ex25.m1.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex25.m1.4b"><apply id="S2.Ex25.m1.4.5.cmml" xref="S2.Ex25.m1.4.5"><approx id="S2.Ex25.m1.4.5.1.cmml" xref="S2.Ex25.m1.4.5.1"></approx><apply id="S2.Ex25.m1.4.5.2.cmml" xref="S2.Ex25.m1.4.5.2"><times id="S2.Ex25.m1.4.5.2.1.cmml" xref="S2.Ex25.m1.4.5.2.1"></times><apply id="S2.Ex25.m1.4.5.2.2.cmml" xref="S2.Ex25.m1.4.5.2.2"><csymbol cd="ambiguous" id="S2.Ex25.m1.4.5.2.2.1.cmml" xref="S2.Ex25.m1.4.5.2.2">subscript</csymbol><apply id="S2.Ex25.m1.4.5.2.2.2.cmml" xref="S2.Ex25.m1.4.5.2.2"><csymbol cd="ambiguous" id="S2.Ex25.m1.4.5.2.2.2.1.cmml" xref="S2.Ex25.m1.4.5.2.2">superscript</csymbol><ci id="S2.Ex25.m1.4.5.2.2.2.2.cmml" xref="S2.Ex25.m1.4.5.2.2.2.2">𝐻</ci><ci id="S2.Ex25.m1.4.5.2.2.2.3.cmml" xref="S2.Ex25.m1.4.5.2.2.2.3">𝑛</ci></apply><apply id="S2.Ex25.m1.4.5.2.2.3.cmml" xref="S2.Ex25.m1.4.5.2.2.3"><times id="S2.Ex25.m1.4.5.2.2.3.1.cmml" xref="S2.Ex25.m1.4.5.2.2.3.1"></times><ci id="S2.Ex25.m1.4.5.2.2.3.2.cmml" xref="S2.Ex25.m1.4.5.2.2.3.2">𝐵</ci><ci id="S2.Ex25.m1.4.5.2.2.3.3.cmml" xref="S2.Ex25.m1.4.5.2.2.3.3">𝑊</ci></apply></apply><list id="S2.Ex25.m1.4.5.2.3.1.cmml" xref="S2.Ex25.m1.4.5.2.3.2"><ci id="S2.Ex25.m1.1.1.cmml" xref="S2.Ex25.m1.1.1">𝒞</ci><ci id="S2.Ex25.m1.2.2.cmml" xref="S2.Ex25.m1.2.2">𝑀</ci></list></apply><apply id="S2.Ex25.m1.4.5.3.cmml" xref="S2.Ex25.m1.4.5.3"><times id="S2.Ex25.m1.4.5.3.1.cmml" xref="S2.Ex25.m1.4.5.3.1"></times><apply id="S2.Ex25.m1.4.5.3.2.cmml" xref="S2.Ex25.m1.4.5.3.2"><csymbol cd="ambiguous" id="S2.Ex25.m1.4.5.3.2.1.cmml" xref="S2.Ex25.m1.4.5.3.2">subscript</csymbol><apply id="S2.Ex25.m1.4.5.3.2.2.cmml" xref="S2.Ex25.m1.4.5.3.2"><csymbol cd="ambiguous" id="S2.Ex25.m1.4.5.3.2.2.1.cmml" xref="S2.Ex25.m1.4.5.3.2">superscript</csymbol><ci id="S2.Ex25.m1.4.5.3.2.2.2.cmml" xref="S2.Ex25.m1.4.5.3.2.2.2">Ext</ci><ci id="S2.Ex25.m1.4.5.3.2.2.3.cmml" xref="S2.Ex25.m1.4.5.3.2.2.3">𝑛</ci></apply><apply id="S2.Ex25.m1.4.5.3.2.3.cmml" xref="S2.Ex25.m1.4.5.3.2.3"><times id="S2.Ex25.m1.4.5.3.2.3.1.cmml" xref="S2.Ex25.m1.4.5.3.2.3.1"></times><ci id="S2.Ex25.m1.4.5.3.2.3.2.cmml" xref="S2.Ex25.m1.4.5.3.2.3.2">𝑅</ci><ci id="S2.Ex25.m1.4.5.3.2.3.3.cmml" xref="S2.Ex25.m1.4.5.3.2.3.3">𝔉</ci><ci id="S2.Ex25.m1.4.5.3.2.3.4.cmml" xref="S2.Ex25.m1.4.5.3.2.3.4">𝒞</ci></apply></apply><interval closure="open" id="S2.Ex25.m1.4.5.3.3.1.cmml" xref="S2.Ex25.m1.4.5.3.3.2"><apply id="S2.Ex25.m1.3.3.cmml" xref="S2.Ex25.m1.3.3"><ci id="S2.Ex25.m1.3.3.1.cmml" xref="S2.Ex25.m1.3.3.1">¯</ci><ci id="S2.Ex25.m1.3.3.2.cmml" xref="S2.Ex25.m1.3.3.2">𝑅</ci></apply><ci id="S2.Ex25.m1.4.4.cmml" xref="S2.Ex25.m1.4.4">𝑀</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex25.m1.4c">H^{n}_{BW}(\mathcal{C};M)\cong\mathrm{Ext}^{n}_{R\mathfrak{F}\mathcal{C}}(% \underline{R},M)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex25.m1.4d">italic_H start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M ) ≅ roman_Ext start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_R fraktur_F caligraphic_C end_POSTSUBSCRIPT ( under¯ start_ARG italic_R end_ARG , italic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem7.p1.7"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem7.p1.7.1">which is natural in <math alttext="M" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.7.1.m1.1"><semantics id="S2.Thmtheorem7.p1.7.1.m1.1a"><mi id="S2.Thmtheorem7.p1.7.1.m1.1.1" xref="S2.Thmtheorem7.p1.7.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.7.1.m1.1b"><ci id="S2.Thmtheorem7.p1.7.1.m1.1.1.cmml" xref="S2.Thmtheorem7.p1.7.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.7.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.7.1.m1.1d">italic_M</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.9">To prove Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S2.Thmtheorem7" title="Proposition 2.7. ‣ 2.2. Baues-Wirsching Cohomology ‣ 2. Modules and cohomology for small categories ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">2.7</span></a>, Baues and Wirsching construct a projective resolution <math alttext="P_{*}\to\underline{R}" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.1"><semantics id="S2.SS2.p5.1.m1.1a"><mrow id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml"><msub id="S2.SS2.p5.1.m1.1.1.2" xref="S2.SS2.p5.1.m1.1.1.2.cmml"><mi id="S2.SS2.p5.1.m1.1.1.2.2" xref="S2.SS2.p5.1.m1.1.1.2.2.cmml">P</mi><mo id="S2.SS2.p5.1.m1.1.1.2.3" xref="S2.SS2.p5.1.m1.1.1.2.3.cmml">∗</mo></msub><mo id="S2.SS2.p5.1.m1.1.1.1" stretchy="false" xref="S2.SS2.p5.1.m1.1.1.1.cmml">→</mo><munder accentunder="true" id="S2.SS2.p5.1.m1.1.1.3" xref="S2.SS2.p5.1.m1.1.1.3.cmml"><mi id="S2.SS2.p5.1.m1.1.1.3.2" xref="S2.SS2.p5.1.m1.1.1.3.2.cmml">R</mi><mo id="S2.SS2.p5.1.m1.1.1.3.1" xref="S2.SS2.p5.1.m1.1.1.3.1.cmml">¯</mo></munder></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.1b"><apply id="S2.SS2.p5.1.m1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1"><ci id="S2.SS2.p5.1.m1.1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1.1">→</ci><apply id="S2.SS2.p5.1.m1.1.1.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.1.1.2.1.cmml" xref="S2.SS2.p5.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS2.p5.1.m1.1.1.2.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2.2">𝑃</ci><times id="S2.SS2.p5.1.m1.1.1.2.3.cmml" xref="S2.SS2.p5.1.m1.1.1.2.3"></times></apply><apply id="S2.SS2.p5.1.m1.1.1.3.cmml" xref="S2.SS2.p5.1.m1.1.1.3"><ci id="S2.SS2.p5.1.m1.1.1.3.1.cmml" xref="S2.SS2.p5.1.m1.1.1.3.1">¯</ci><ci id="S2.SS2.p5.1.m1.1.1.3.2.cmml" xref="S2.SS2.p5.1.m1.1.1.3.2">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.1c">P_{*}\to\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.1d">italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → under¯ start_ARG italic_R end_ARG</annotation></semantics></math> of the constant functor <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.1"><semantics id="S2.SS2.p5.2.m2.1a"><munder accentunder="true" id="S2.SS2.p5.2.m2.1.1" xref="S2.SS2.p5.2.m2.1.1.cmml"><mi id="S2.SS2.p5.2.m2.1.1.2" xref="S2.SS2.p5.2.m2.1.1.2.cmml">R</mi><mo id="S2.SS2.p5.2.m2.1.1.1" xref="S2.SS2.p5.2.m2.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.2.m2.1b"><apply id="S2.SS2.p5.2.m2.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1"><ci id="S2.SS2.p5.2.m2.1.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1.1">¯</ci><ci id="S2.SS2.p5.2.m2.1.1.2.cmml" xref="S2.SS2.p5.2.m2.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.2.m2.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.2.m2.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as a <math alttext="R\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p5.3.m3.1"><semantics id="S2.SS2.p5.3.m3.1a"><mrow id="S2.SS2.p5.3.m3.1.1" xref="S2.SS2.p5.3.m3.1.1.cmml"><mi id="S2.SS2.p5.3.m3.1.1.2" xref="S2.SS2.p5.3.m3.1.1.2.cmml">R</mi><mo id="S2.SS2.p5.3.m3.1.1.1" xref="S2.SS2.p5.3.m3.1.1.1.cmml">⁢</mo><mi id="S2.SS2.p5.3.m3.1.1.3" xref="S2.SS2.p5.3.m3.1.1.3.cmml">𝔉</mi><mo id="S2.SS2.p5.3.m3.1.1.1a" xref="S2.SS2.p5.3.m3.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.3.m3.1.1.4" xref="S2.SS2.p5.3.m3.1.1.4.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.1b"><apply id="S2.SS2.p5.3.m3.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1"><times id="S2.SS2.p5.3.m3.1.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1.1"></times><ci id="S2.SS2.p5.3.m3.1.1.2.cmml" xref="S2.SS2.p5.3.m3.1.1.2">𝑅</ci><ci id="S2.SS2.p5.3.m3.1.1.3.cmml" xref="S2.SS2.p5.3.m3.1.1.3">𝔉</ci><ci id="S2.SS2.p5.3.m3.1.1.4.cmml" xref="S2.SS2.p5.3.m3.1.1.4">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.1c">R\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.1d">italic_R fraktur_F caligraphic_C</annotation></semantics></math>-module such that <math alttext="\mathrm{Hom}_{RF\mathcal{C}}(P_{*},M)" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.2"><semantics id="S2.SS2.p5.4.m4.2a"><mrow id="S2.SS2.p5.4.m4.2.2" xref="S2.SS2.p5.4.m4.2.2.cmml"><msub id="S2.SS2.p5.4.m4.2.2.3" xref="S2.SS2.p5.4.m4.2.2.3.cmml"><mi id="S2.SS2.p5.4.m4.2.2.3.2" xref="S2.SS2.p5.4.m4.2.2.3.2.cmml">Hom</mi><mrow id="S2.SS2.p5.4.m4.2.2.3.3" xref="S2.SS2.p5.4.m4.2.2.3.3.cmml"><mi id="S2.SS2.p5.4.m4.2.2.3.3.2" xref="S2.SS2.p5.4.m4.2.2.3.3.2.cmml">R</mi><mo id="S2.SS2.p5.4.m4.2.2.3.3.1" xref="S2.SS2.p5.4.m4.2.2.3.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.4.m4.2.2.3.3.3" xref="S2.SS2.p5.4.m4.2.2.3.3.3.cmml">F</mi><mo id="S2.SS2.p5.4.m4.2.2.3.3.1a" xref="S2.SS2.p5.4.m4.2.2.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.4.m4.2.2.3.3.4" xref="S2.SS2.p5.4.m4.2.2.3.3.4.cmml">𝒞</mi></mrow></msub><mo id="S2.SS2.p5.4.m4.2.2.2" xref="S2.SS2.p5.4.m4.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p5.4.m4.2.2.1.1" xref="S2.SS2.p5.4.m4.2.2.1.2.cmml"><mo id="S2.SS2.p5.4.m4.2.2.1.1.2" stretchy="false" xref="S2.SS2.p5.4.m4.2.2.1.2.cmml">(</mo><msub id="S2.SS2.p5.4.m4.2.2.1.1.1" xref="S2.SS2.p5.4.m4.2.2.1.1.1.cmml"><mi id="S2.SS2.p5.4.m4.2.2.1.1.1.2" xref="S2.SS2.p5.4.m4.2.2.1.1.1.2.cmml">P</mi><mo id="S2.SS2.p5.4.m4.2.2.1.1.1.3" xref="S2.SS2.p5.4.m4.2.2.1.1.1.3.cmml">∗</mo></msub><mo id="S2.SS2.p5.4.m4.2.2.1.1.3" xref="S2.SS2.p5.4.m4.2.2.1.2.cmml">,</mo><mi id="S2.SS2.p5.4.m4.1.1" xref="S2.SS2.p5.4.m4.1.1.cmml">M</mi><mo id="S2.SS2.p5.4.m4.2.2.1.1.4" stretchy="false" xref="S2.SS2.p5.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.4.m4.2b"><apply id="S2.SS2.p5.4.m4.2.2.cmml" xref="S2.SS2.p5.4.m4.2.2"><times id="S2.SS2.p5.4.m4.2.2.2.cmml" xref="S2.SS2.p5.4.m4.2.2.2"></times><apply id="S2.SS2.p5.4.m4.2.2.3.cmml" xref="S2.SS2.p5.4.m4.2.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.4.m4.2.2.3.1.cmml" xref="S2.SS2.p5.4.m4.2.2.3">subscript</csymbol><ci id="S2.SS2.p5.4.m4.2.2.3.2.cmml" xref="S2.SS2.p5.4.m4.2.2.3.2">Hom</ci><apply id="S2.SS2.p5.4.m4.2.2.3.3.cmml" xref="S2.SS2.p5.4.m4.2.2.3.3"><times id="S2.SS2.p5.4.m4.2.2.3.3.1.cmml" xref="S2.SS2.p5.4.m4.2.2.3.3.1"></times><ci id="S2.SS2.p5.4.m4.2.2.3.3.2.cmml" xref="S2.SS2.p5.4.m4.2.2.3.3.2">𝑅</ci><ci id="S2.SS2.p5.4.m4.2.2.3.3.3.cmml" xref="S2.SS2.p5.4.m4.2.2.3.3.3">𝐹</ci><ci id="S2.SS2.p5.4.m4.2.2.3.3.4.cmml" xref="S2.SS2.p5.4.m4.2.2.3.3.4">𝒞</ci></apply></apply><interval closure="open" id="S2.SS2.p5.4.m4.2.2.1.2.cmml" xref="S2.SS2.p5.4.m4.2.2.1.1"><apply id="S2.SS2.p5.4.m4.2.2.1.1.1.cmml" xref="S2.SS2.p5.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.4.m4.2.2.1.1.1.1.cmml" xref="S2.SS2.p5.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S2.SS2.p5.4.m4.2.2.1.1.1.2.cmml" xref="S2.SS2.p5.4.m4.2.2.1.1.1.2">𝑃</ci><times id="S2.SS2.p5.4.m4.2.2.1.1.1.3.cmml" xref="S2.SS2.p5.4.m4.2.2.1.1.1.3"></times></apply><ci id="S2.SS2.p5.4.m4.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1">𝑀</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.2c">\mathrm{Hom}_{RF\mathcal{C}}(P_{*},M)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.2d">roman_Hom start_POSTSUBSCRIPT italic_R italic_F caligraphic_C end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , italic_M )</annotation></semantics></math> is a cochain complex isomorphic to <math alttext="C^{*}_{BW}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m5.2"><semantics id="S2.SS2.p5.5.m5.2a"><mrow id="S2.SS2.p5.5.m5.2.3" xref="S2.SS2.p5.5.m5.2.3.cmml"><msubsup id="S2.SS2.p5.5.m5.2.3.2" xref="S2.SS2.p5.5.m5.2.3.2.cmml"><mi id="S2.SS2.p5.5.m5.2.3.2.2.2" xref="S2.SS2.p5.5.m5.2.3.2.2.2.cmml">C</mi><mrow id="S2.SS2.p5.5.m5.2.3.2.3" xref="S2.SS2.p5.5.m5.2.3.2.3.cmml"><mi id="S2.SS2.p5.5.m5.2.3.2.3.2" xref="S2.SS2.p5.5.m5.2.3.2.3.2.cmml">B</mi><mo id="S2.SS2.p5.5.m5.2.3.2.3.1" xref="S2.SS2.p5.5.m5.2.3.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.5.m5.2.3.2.3.3" xref="S2.SS2.p5.5.m5.2.3.2.3.3.cmml">W</mi></mrow><mo id="S2.SS2.p5.5.m5.2.3.2.2.3" xref="S2.SS2.p5.5.m5.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S2.SS2.p5.5.m5.2.3.1" xref="S2.SS2.p5.5.m5.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p5.5.m5.2.3.3.2" xref="S2.SS2.p5.5.m5.2.3.3.1.cmml"><mo id="S2.SS2.p5.5.m5.2.3.3.2.1" stretchy="false" xref="S2.SS2.p5.5.m5.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.5.m5.1.1" xref="S2.SS2.p5.5.m5.1.1.cmml">𝒞</mi><mo id="S2.SS2.p5.5.m5.2.3.3.2.2" xref="S2.SS2.p5.5.m5.2.3.3.1.cmml">;</mo><mi id="S2.SS2.p5.5.m5.2.2" xref="S2.SS2.p5.5.m5.2.2.cmml">M</mi><mo id="S2.SS2.p5.5.m5.2.3.3.2.3" stretchy="false" xref="S2.SS2.p5.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m5.2b"><apply id="S2.SS2.p5.5.m5.2.3.cmml" xref="S2.SS2.p5.5.m5.2.3"><times id="S2.SS2.p5.5.m5.2.3.1.cmml" xref="S2.SS2.p5.5.m5.2.3.1"></times><apply id="S2.SS2.p5.5.m5.2.3.2.cmml" xref="S2.SS2.p5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.2.3.2.1.cmml" xref="S2.SS2.p5.5.m5.2.3.2">subscript</csymbol><apply id="S2.SS2.p5.5.m5.2.3.2.2.cmml" xref="S2.SS2.p5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.2.3.2.2.1.cmml" xref="S2.SS2.p5.5.m5.2.3.2">superscript</csymbol><ci id="S2.SS2.p5.5.m5.2.3.2.2.2.cmml" xref="S2.SS2.p5.5.m5.2.3.2.2.2">𝐶</ci><times id="S2.SS2.p5.5.m5.2.3.2.2.3.cmml" xref="S2.SS2.p5.5.m5.2.3.2.2.3"></times></apply><apply id="S2.SS2.p5.5.m5.2.3.2.3.cmml" xref="S2.SS2.p5.5.m5.2.3.2.3"><times id="S2.SS2.p5.5.m5.2.3.2.3.1.cmml" xref="S2.SS2.p5.5.m5.2.3.2.3.1"></times><ci id="S2.SS2.p5.5.m5.2.3.2.3.2.cmml" xref="S2.SS2.p5.5.m5.2.3.2.3.2">𝐵</ci><ci id="S2.SS2.p5.5.m5.2.3.2.3.3.cmml" xref="S2.SS2.p5.5.m5.2.3.2.3.3">𝑊</ci></apply></apply><list id="S2.SS2.p5.5.m5.2.3.3.1.cmml" xref="S2.SS2.p5.5.m5.2.3.3.2"><ci id="S2.SS2.p5.5.m5.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1">𝒞</ci><ci id="S2.SS2.p5.5.m5.2.2.cmml" xref="S2.SS2.p5.5.m5.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m5.2c">C^{*}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m5.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math>. Note that for every object <math alttext="\alpha\in\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m6.1"><semantics id="S2.SS2.p5.6.m6.1a"><mrow id="S2.SS2.p5.6.m6.1.1" xref="S2.SS2.p5.6.m6.1.1.cmml"><mi id="S2.SS2.p5.6.m6.1.1.2" xref="S2.SS2.p5.6.m6.1.1.2.cmml">α</mi><mo id="S2.SS2.p5.6.m6.1.1.1" xref="S2.SS2.p5.6.m6.1.1.1.cmml">∈</mo><mrow id="S2.SS2.p5.6.m6.1.1.3" xref="S2.SS2.p5.6.m6.1.1.3.cmml"><mi id="S2.SS2.p5.6.m6.1.1.3.2" xref="S2.SS2.p5.6.m6.1.1.3.2.cmml">𝔉</mi><mo id="S2.SS2.p5.6.m6.1.1.3.1" xref="S2.SS2.p5.6.m6.1.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.6.m6.1.1.3.3" xref="S2.SS2.p5.6.m6.1.1.3.3.cmml">𝒞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m6.1b"><apply id="S2.SS2.p5.6.m6.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1"><in id="S2.SS2.p5.6.m6.1.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1.1"></in><ci id="S2.SS2.p5.6.m6.1.1.2.cmml" xref="S2.SS2.p5.6.m6.1.1.2">𝛼</ci><apply id="S2.SS2.p5.6.m6.1.1.3.cmml" xref="S2.SS2.p5.6.m6.1.1.3"><times id="S2.SS2.p5.6.m6.1.1.3.1.cmml" xref="S2.SS2.p5.6.m6.1.1.3.1"></times><ci id="S2.SS2.p5.6.m6.1.1.3.2.cmml" xref="S2.SS2.p5.6.m6.1.1.3.2">𝔉</ci><ci id="S2.SS2.p5.6.m6.1.1.3.3.cmml" xref="S2.SS2.p5.6.m6.1.1.3.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m6.1c">\alpha\in\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m6.1d">italic_α ∈ fraktur_F caligraphic_C</annotation></semantics></math>, the <math alttext="R\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p5.7.m7.1"><semantics id="S2.SS2.p5.7.m7.1a"><mrow id="S2.SS2.p5.7.m7.1.1" xref="S2.SS2.p5.7.m7.1.1.cmml"><mi id="S2.SS2.p5.7.m7.1.1.2" xref="S2.SS2.p5.7.m7.1.1.2.cmml">R</mi><mo id="S2.SS2.p5.7.m7.1.1.1" xref="S2.SS2.p5.7.m7.1.1.1.cmml">⁢</mo><mi id="S2.SS2.p5.7.m7.1.1.3" xref="S2.SS2.p5.7.m7.1.1.3.cmml">𝔉</mi><mo id="S2.SS2.p5.7.m7.1.1.1a" xref="S2.SS2.p5.7.m7.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.7.m7.1.1.4" xref="S2.SS2.p5.7.m7.1.1.4.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.7.m7.1b"><apply id="S2.SS2.p5.7.m7.1.1.cmml" xref="S2.SS2.p5.7.m7.1.1"><times id="S2.SS2.p5.7.m7.1.1.1.cmml" xref="S2.SS2.p5.7.m7.1.1.1"></times><ci id="S2.SS2.p5.7.m7.1.1.2.cmml" xref="S2.SS2.p5.7.m7.1.1.2">𝑅</ci><ci id="S2.SS2.p5.7.m7.1.1.3.cmml" xref="S2.SS2.p5.7.m7.1.1.3">𝔉</ci><ci id="S2.SS2.p5.7.m7.1.1.4.cmml" xref="S2.SS2.p5.7.m7.1.1.4">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.7.m7.1c">R\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.7.m7.1d">italic_R fraktur_F caligraphic_C</annotation></semantics></math>-module <math alttext="R\mathrm{Mor}_{\mathfrak{F}\mathcal{C}}(\alpha,?)" class="ltx_Math" display="inline" id="S2.SS2.p5.8.m8.2"><semantics id="S2.SS2.p5.8.m8.2a"><mrow id="S2.SS2.p5.8.m8.2.3" xref="S2.SS2.p5.8.m8.2.3.cmml"><mi id="S2.SS2.p5.8.m8.2.3.2" xref="S2.SS2.p5.8.m8.2.3.2.cmml">R</mi><mo id="S2.SS2.p5.8.m8.2.3.1" xref="S2.SS2.p5.8.m8.2.3.1.cmml">⁢</mo><msub id="S2.SS2.p5.8.m8.2.3.3" xref="S2.SS2.p5.8.m8.2.3.3.cmml"><mi id="S2.SS2.p5.8.m8.2.3.3.2" xref="S2.SS2.p5.8.m8.2.3.3.2.cmml">Mor</mi><mrow id="S2.SS2.p5.8.m8.2.3.3.3" xref="S2.SS2.p5.8.m8.2.3.3.3.cmml"><mi id="S2.SS2.p5.8.m8.2.3.3.3.2" xref="S2.SS2.p5.8.m8.2.3.3.3.2.cmml">𝔉</mi><mo id="S2.SS2.p5.8.m8.2.3.3.3.1" xref="S2.SS2.p5.8.m8.2.3.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.8.m8.2.3.3.3.3" xref="S2.SS2.p5.8.m8.2.3.3.3.3.cmml">𝒞</mi></mrow></msub><mo id="S2.SS2.p5.8.m8.2.3.1a" xref="S2.SS2.p5.8.m8.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p5.8.m8.2.3.4.2" xref="S2.SS2.p5.8.m8.2.3.4.1.cmml"><mo id="S2.SS2.p5.8.m8.2.3.4.2.1" stretchy="false" xref="S2.SS2.p5.8.m8.2.3.4.1.cmml">(</mo><mi id="S2.SS2.p5.8.m8.1.1" xref="S2.SS2.p5.8.m8.1.1.cmml">α</mi><mo id="S2.SS2.p5.8.m8.2.3.4.2.2" xref="S2.SS2.p5.8.m8.2.3.4.1.cmml">,</mo><mi id="S2.SS2.p5.8.m8.2.2" mathvariant="normal" xref="S2.SS2.p5.8.m8.2.2.cmml">?</mi><mo id="S2.SS2.p5.8.m8.2.3.4.2.3" stretchy="false" xref="S2.SS2.p5.8.m8.2.3.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.8.m8.2b"><apply id="S2.SS2.p5.8.m8.2.3.cmml" xref="S2.SS2.p5.8.m8.2.3"><times id="S2.SS2.p5.8.m8.2.3.1.cmml" xref="S2.SS2.p5.8.m8.2.3.1"></times><ci id="S2.SS2.p5.8.m8.2.3.2.cmml" xref="S2.SS2.p5.8.m8.2.3.2">𝑅</ci><apply id="S2.SS2.p5.8.m8.2.3.3.cmml" xref="S2.SS2.p5.8.m8.2.3.3"><csymbol cd="ambiguous" id="S2.SS2.p5.8.m8.2.3.3.1.cmml" xref="S2.SS2.p5.8.m8.2.3.3">subscript</csymbol><ci id="S2.SS2.p5.8.m8.2.3.3.2.cmml" xref="S2.SS2.p5.8.m8.2.3.3.2">Mor</ci><apply id="S2.SS2.p5.8.m8.2.3.3.3.cmml" xref="S2.SS2.p5.8.m8.2.3.3.3"><times id="S2.SS2.p5.8.m8.2.3.3.3.1.cmml" xref="S2.SS2.p5.8.m8.2.3.3.3.1"></times><ci id="S2.SS2.p5.8.m8.2.3.3.3.2.cmml" xref="S2.SS2.p5.8.m8.2.3.3.3.2">𝔉</ci><ci id="S2.SS2.p5.8.m8.2.3.3.3.3.cmml" xref="S2.SS2.p5.8.m8.2.3.3.3.3">𝒞</ci></apply></apply><interval closure="open" id="S2.SS2.p5.8.m8.2.3.4.1.cmml" xref="S2.SS2.p5.8.m8.2.3.4.2"><ci id="S2.SS2.p5.8.m8.1.1.cmml" xref="S2.SS2.p5.8.m8.1.1">𝛼</ci><ci id="S2.SS2.p5.8.m8.2.2.cmml" xref="S2.SS2.p5.8.m8.2.2">?</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.8.m8.2c">R\mathrm{Mor}_{\mathfrak{F}\mathcal{C}}(\alpha,?)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.8.m8.2d">italic_R roman_Mor start_POSTSUBSCRIPT fraktur_F caligraphic_C end_POSTSUBSCRIPT ( italic_α , ? )</annotation></semantics></math> is a projective <math alttext="R\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p5.9.m9.1"><semantics id="S2.SS2.p5.9.m9.1a"><mrow id="S2.SS2.p5.9.m9.1.1" xref="S2.SS2.p5.9.m9.1.1.cmml"><mi id="S2.SS2.p5.9.m9.1.1.2" xref="S2.SS2.p5.9.m9.1.1.2.cmml">R</mi><mo id="S2.SS2.p5.9.m9.1.1.1" xref="S2.SS2.p5.9.m9.1.1.1.cmml">⁢</mo><mi id="S2.SS2.p5.9.m9.1.1.3" xref="S2.SS2.p5.9.m9.1.1.3.cmml">𝔉</mi><mo id="S2.SS2.p5.9.m9.1.1.1a" xref="S2.SS2.p5.9.m9.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.9.m9.1.1.4" xref="S2.SS2.p5.9.m9.1.1.4.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.9.m9.1b"><apply id="S2.SS2.p5.9.m9.1.1.cmml" xref="S2.SS2.p5.9.m9.1.1"><times id="S2.SS2.p5.9.m9.1.1.1.cmml" xref="S2.SS2.p5.9.m9.1.1.1"></times><ci id="S2.SS2.p5.9.m9.1.1.2.cmml" xref="S2.SS2.p5.9.m9.1.1.2">𝑅</ci><ci id="S2.SS2.p5.9.m9.1.1.3.cmml" xref="S2.SS2.p5.9.m9.1.1.3">𝔉</ci><ci id="S2.SS2.p5.9.m9.1.1.4.cmml" xref="S2.SS2.p5.9.m9.1.1.4">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.9.m9.1c">R\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.9.m9.1d">italic_R fraktur_F caligraphic_C</annotation></semantics></math>-module. One can use these modules to construct the projective resolution (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib1" title="">1</a>, Theorem 4.4]</cite> for details).</p> </div> <div class="ltx_para" id="S2.SS2.p6"> <p class="ltx_p" id="S2.SS2.p6.11">The Baues-Wirsching cohomology generalizes the Quillen cohomology of a small category in the following sense: For a small category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p6.1.m1.1"><semantics id="S2.SS2.p6.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.1.m1.1.1" xref="S2.SS2.p6.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.1.m1.1b"><ci id="S2.SS2.p6.1.m1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.1.m1.1d">caligraphic_C</annotation></semantics></math>, there are functors <math alttext="S:\mathfrak{F}\mathcal{C}\to\mathcal{C}^{op}" class="ltx_Math" display="inline" id="S2.SS2.p6.2.m2.1"><semantics id="S2.SS2.p6.2.m2.1a"><mrow id="S2.SS2.p6.2.m2.1.1" xref="S2.SS2.p6.2.m2.1.1.cmml"><mi id="S2.SS2.p6.2.m2.1.1.2" xref="S2.SS2.p6.2.m2.1.1.2.cmml">S</mi><mo id="S2.SS2.p6.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p6.2.m2.1.1.1.cmml">:</mo><mrow id="S2.SS2.p6.2.m2.1.1.3" xref="S2.SS2.p6.2.m2.1.1.3.cmml"><mrow id="S2.SS2.p6.2.m2.1.1.3.2" xref="S2.SS2.p6.2.m2.1.1.3.2.cmml"><mi id="S2.SS2.p6.2.m2.1.1.3.2.2" xref="S2.SS2.p6.2.m2.1.1.3.2.2.cmml">𝔉</mi><mo id="S2.SS2.p6.2.m2.1.1.3.2.1" xref="S2.SS2.p6.2.m2.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.2.m2.1.1.3.2.3" xref="S2.SS2.p6.2.m2.1.1.3.2.3.cmml">𝒞</mi></mrow><mo id="S2.SS2.p6.2.m2.1.1.3.1" stretchy="false" xref="S2.SS2.p6.2.m2.1.1.3.1.cmml">→</mo><msup id="S2.SS2.p6.2.m2.1.1.3.3" xref="S2.SS2.p6.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.2.m2.1.1.3.3.2" xref="S2.SS2.p6.2.m2.1.1.3.3.2.cmml">𝒞</mi><mrow id="S2.SS2.p6.2.m2.1.1.3.3.3" xref="S2.SS2.p6.2.m2.1.1.3.3.3.cmml"><mi id="S2.SS2.p6.2.m2.1.1.3.3.3.2" xref="S2.SS2.p6.2.m2.1.1.3.3.3.2.cmml">o</mi><mo id="S2.SS2.p6.2.m2.1.1.3.3.3.1" xref="S2.SS2.p6.2.m2.1.1.3.3.3.1.cmml">⁢</mo><mi id="S2.SS2.p6.2.m2.1.1.3.3.3.3" xref="S2.SS2.p6.2.m2.1.1.3.3.3.3.cmml">p</mi></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.2.m2.1b"><apply id="S2.SS2.p6.2.m2.1.1.cmml" xref="S2.SS2.p6.2.m2.1.1"><ci id="S2.SS2.p6.2.m2.1.1.1.cmml" xref="S2.SS2.p6.2.m2.1.1.1">:</ci><ci id="S2.SS2.p6.2.m2.1.1.2.cmml" xref="S2.SS2.p6.2.m2.1.1.2">𝑆</ci><apply id="S2.SS2.p6.2.m2.1.1.3.cmml" xref="S2.SS2.p6.2.m2.1.1.3"><ci id="S2.SS2.p6.2.m2.1.1.3.1.cmml" xref="S2.SS2.p6.2.m2.1.1.3.1">→</ci><apply id="S2.SS2.p6.2.m2.1.1.3.2.cmml" xref="S2.SS2.p6.2.m2.1.1.3.2"><times id="S2.SS2.p6.2.m2.1.1.3.2.1.cmml" xref="S2.SS2.p6.2.m2.1.1.3.2.1"></times><ci id="S2.SS2.p6.2.m2.1.1.3.2.2.cmml" xref="S2.SS2.p6.2.m2.1.1.3.2.2">𝔉</ci><ci id="S2.SS2.p6.2.m2.1.1.3.2.3.cmml" xref="S2.SS2.p6.2.m2.1.1.3.2.3">𝒞</ci></apply><apply id="S2.SS2.p6.2.m2.1.1.3.3.cmml" xref="S2.SS2.p6.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p6.2.m2.1.1.3.3.1.cmml" xref="S2.SS2.p6.2.m2.1.1.3.3">superscript</csymbol><ci id="S2.SS2.p6.2.m2.1.1.3.3.2.cmml" xref="S2.SS2.p6.2.m2.1.1.3.3.2">𝒞</ci><apply id="S2.SS2.p6.2.m2.1.1.3.3.3.cmml" xref="S2.SS2.p6.2.m2.1.1.3.3.3"><times id="S2.SS2.p6.2.m2.1.1.3.3.3.1.cmml" xref="S2.SS2.p6.2.m2.1.1.3.3.3.1"></times><ci id="S2.SS2.p6.2.m2.1.1.3.3.3.2.cmml" xref="S2.SS2.p6.2.m2.1.1.3.3.3.2">𝑜</ci><ci id="S2.SS2.p6.2.m2.1.1.3.3.3.3.cmml" xref="S2.SS2.p6.2.m2.1.1.3.3.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.2.m2.1c">S:\mathfrak{F}\mathcal{C}\to\mathcal{C}^{op}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.2.m2.1d">italic_S : fraktur_F caligraphic_C → caligraphic_C start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="T:\mathfrak{F}\mathcal{C}\to\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p6.3.m3.1"><semantics id="S2.SS2.p6.3.m3.1a"><mrow id="S2.SS2.p6.3.m3.1.1" xref="S2.SS2.p6.3.m3.1.1.cmml"><mi id="S2.SS2.p6.3.m3.1.1.2" xref="S2.SS2.p6.3.m3.1.1.2.cmml">T</mi><mo id="S2.SS2.p6.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p6.3.m3.1.1.1.cmml">:</mo><mrow id="S2.SS2.p6.3.m3.1.1.3" xref="S2.SS2.p6.3.m3.1.1.3.cmml"><mrow id="S2.SS2.p6.3.m3.1.1.3.2" xref="S2.SS2.p6.3.m3.1.1.3.2.cmml"><mi id="S2.SS2.p6.3.m3.1.1.3.2.2" xref="S2.SS2.p6.3.m3.1.1.3.2.2.cmml">𝔉</mi><mo id="S2.SS2.p6.3.m3.1.1.3.2.1" xref="S2.SS2.p6.3.m3.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.3.m3.1.1.3.2.3" xref="S2.SS2.p6.3.m3.1.1.3.2.3.cmml">𝒞</mi></mrow><mo id="S2.SS2.p6.3.m3.1.1.3.1" stretchy="false" xref="S2.SS2.p6.3.m3.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.3.m3.1.1.3.3" xref="S2.SS2.p6.3.m3.1.1.3.3.cmml">𝒞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.3.m3.1b"><apply id="S2.SS2.p6.3.m3.1.1.cmml" xref="S2.SS2.p6.3.m3.1.1"><ci id="S2.SS2.p6.3.m3.1.1.1.cmml" xref="S2.SS2.p6.3.m3.1.1.1">:</ci><ci id="S2.SS2.p6.3.m3.1.1.2.cmml" xref="S2.SS2.p6.3.m3.1.1.2">𝑇</ci><apply id="S2.SS2.p6.3.m3.1.1.3.cmml" xref="S2.SS2.p6.3.m3.1.1.3"><ci id="S2.SS2.p6.3.m3.1.1.3.1.cmml" xref="S2.SS2.p6.3.m3.1.1.3.1">→</ci><apply id="S2.SS2.p6.3.m3.1.1.3.2.cmml" xref="S2.SS2.p6.3.m3.1.1.3.2"><times id="S2.SS2.p6.3.m3.1.1.3.2.1.cmml" xref="S2.SS2.p6.3.m3.1.1.3.2.1"></times><ci id="S2.SS2.p6.3.m3.1.1.3.2.2.cmml" xref="S2.SS2.p6.3.m3.1.1.3.2.2">𝔉</ci><ci id="S2.SS2.p6.3.m3.1.1.3.2.3.cmml" xref="S2.SS2.p6.3.m3.1.1.3.2.3">𝒞</ci></apply><ci id="S2.SS2.p6.3.m3.1.1.3.3.cmml" xref="S2.SS2.p6.3.m3.1.1.3.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.3.m3.1c">T:\mathfrak{F}\mathcal{C}\to\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.3.m3.1d">italic_T : fraktur_F caligraphic_C → caligraphic_C</annotation></semantics></math> defined by <math alttext="S(\alpha:x\to y)=x" class="ltx_math_unparsed" display="inline" id="S2.SS2.p6.4.m4.1"><semantics id="S2.SS2.p6.4.m4.1a"><mrow id="S2.SS2.p6.4.m4.1b"><mi id="S2.SS2.p6.4.m4.1.1">S</mi><mrow id="S2.SS2.p6.4.m4.1.2"><mo id="S2.SS2.p6.4.m4.1.2.1" stretchy="false">(</mo><mi id="S2.SS2.p6.4.m4.1.2.2">α</mi><mo id="S2.SS2.p6.4.m4.1.2.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.SS2.p6.4.m4.1.2.4">x</mi><mo id="S2.SS2.p6.4.m4.1.2.5" stretchy="false">→</mo><mi id="S2.SS2.p6.4.m4.1.2.6">y</mi><mo id="S2.SS2.p6.4.m4.1.2.7" stretchy="false">)</mo></mrow><mo id="S2.SS2.p6.4.m4.1.3">=</mo><mi id="S2.SS2.p6.4.m4.1.4">x</mi></mrow><annotation encoding="application/x-tex" id="S2.SS2.p6.4.m4.1c">S(\alpha:x\to y)=x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.4.m4.1d">italic_S ( italic_α : italic_x → italic_y ) = italic_x</annotation></semantics></math> and <math alttext="T(\alpha:x\to y)=y" class="ltx_math_unparsed" display="inline" id="S2.SS2.p6.5.m5.1"><semantics id="S2.SS2.p6.5.m5.1a"><mrow id="S2.SS2.p6.5.m5.1b"><mi id="S2.SS2.p6.5.m5.1.1">T</mi><mrow id="S2.SS2.p6.5.m5.1.2"><mo id="S2.SS2.p6.5.m5.1.2.1" stretchy="false">(</mo><mi id="S2.SS2.p6.5.m5.1.2.2">α</mi><mo id="S2.SS2.p6.5.m5.1.2.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.SS2.p6.5.m5.1.2.4">x</mi><mo id="S2.SS2.p6.5.m5.1.2.5" stretchy="false">→</mo><mi id="S2.SS2.p6.5.m5.1.2.6">y</mi><mo id="S2.SS2.p6.5.m5.1.2.7" stretchy="false">)</mo></mrow><mo id="S2.SS2.p6.5.m5.1.3">=</mo><mi id="S2.SS2.p6.5.m5.1.4">y</mi></mrow><annotation encoding="application/x-tex" id="S2.SS2.p6.5.m5.1c">T(\alpha:x\to y)=y</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.5.m5.1d">italic_T ( italic_α : italic_x → italic_y ) = italic_y</annotation></semantics></math> on objects. A morphism <math alttext="(u,v):\alpha\to\alpha^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.p6.6.m6.2"><semantics id="S2.SS2.p6.6.m6.2a"><mrow id="S2.SS2.p6.6.m6.2.3" xref="S2.SS2.p6.6.m6.2.3.cmml"><mrow id="S2.SS2.p6.6.m6.2.3.2.2" xref="S2.SS2.p6.6.m6.2.3.2.1.cmml"><mo id="S2.SS2.p6.6.m6.2.3.2.2.1" stretchy="false" xref="S2.SS2.p6.6.m6.2.3.2.1.cmml">(</mo><mi id="S2.SS2.p6.6.m6.1.1" xref="S2.SS2.p6.6.m6.1.1.cmml">u</mi><mo id="S2.SS2.p6.6.m6.2.3.2.2.2" xref="S2.SS2.p6.6.m6.2.3.2.1.cmml">,</mo><mi id="S2.SS2.p6.6.m6.2.2" xref="S2.SS2.p6.6.m6.2.2.cmml">v</mi><mo id="S2.SS2.p6.6.m6.2.3.2.2.3" rspace="0.278em" stretchy="false" xref="S2.SS2.p6.6.m6.2.3.2.1.cmml">)</mo></mrow><mo id="S2.SS2.p6.6.m6.2.3.1" rspace="0.278em" xref="S2.SS2.p6.6.m6.2.3.1.cmml">:</mo><mrow id="S2.SS2.p6.6.m6.2.3.3" xref="S2.SS2.p6.6.m6.2.3.3.cmml"><mi id="S2.SS2.p6.6.m6.2.3.3.2" xref="S2.SS2.p6.6.m6.2.3.3.2.cmml">α</mi><mo id="S2.SS2.p6.6.m6.2.3.3.1" stretchy="false" xref="S2.SS2.p6.6.m6.2.3.3.1.cmml">→</mo><msup id="S2.SS2.p6.6.m6.2.3.3.3" xref="S2.SS2.p6.6.m6.2.3.3.3.cmml"><mi id="S2.SS2.p6.6.m6.2.3.3.3.2" xref="S2.SS2.p6.6.m6.2.3.3.3.2.cmml">α</mi><mo id="S2.SS2.p6.6.m6.2.3.3.3.3" xref="S2.SS2.p6.6.m6.2.3.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.6.m6.2b"><apply id="S2.SS2.p6.6.m6.2.3.cmml" xref="S2.SS2.p6.6.m6.2.3"><ci id="S2.SS2.p6.6.m6.2.3.1.cmml" xref="S2.SS2.p6.6.m6.2.3.1">:</ci><interval closure="open" id="S2.SS2.p6.6.m6.2.3.2.1.cmml" xref="S2.SS2.p6.6.m6.2.3.2.2"><ci id="S2.SS2.p6.6.m6.1.1.cmml" xref="S2.SS2.p6.6.m6.1.1">𝑢</ci><ci id="S2.SS2.p6.6.m6.2.2.cmml" xref="S2.SS2.p6.6.m6.2.2">𝑣</ci></interval><apply id="S2.SS2.p6.6.m6.2.3.3.cmml" xref="S2.SS2.p6.6.m6.2.3.3"><ci id="S2.SS2.p6.6.m6.2.3.3.1.cmml" xref="S2.SS2.p6.6.m6.2.3.3.1">→</ci><ci id="S2.SS2.p6.6.m6.2.3.3.2.cmml" xref="S2.SS2.p6.6.m6.2.3.3.2">𝛼</ci><apply id="S2.SS2.p6.6.m6.2.3.3.3.cmml" xref="S2.SS2.p6.6.m6.2.3.3.3"><csymbol cd="ambiguous" id="S2.SS2.p6.6.m6.2.3.3.3.1.cmml" xref="S2.SS2.p6.6.m6.2.3.3.3">superscript</csymbol><ci id="S2.SS2.p6.6.m6.2.3.3.3.2.cmml" xref="S2.SS2.p6.6.m6.2.3.3.3.2">𝛼</ci><ci id="S2.SS2.p6.6.m6.2.3.3.3.3.cmml" xref="S2.SS2.p6.6.m6.2.3.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.6.m6.2c">(u,v):\alpha\to\alpha^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.6.m6.2d">( italic_u , italic_v ) : italic_α → italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS2.p6.7.m7.1"><semantics id="S2.SS2.p6.7.m7.1a"><mrow id="S2.SS2.p6.7.m7.1.1" xref="S2.SS2.p6.7.m7.1.1.cmml"><mi id="S2.SS2.p6.7.m7.1.1.2" xref="S2.SS2.p6.7.m7.1.1.2.cmml">𝔉</mi><mo id="S2.SS2.p6.7.m7.1.1.1" xref="S2.SS2.p6.7.m7.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.7.m7.1.1.3" xref="S2.SS2.p6.7.m7.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.7.m7.1b"><apply id="S2.SS2.p6.7.m7.1.1.cmml" xref="S2.SS2.p6.7.m7.1.1"><times id="S2.SS2.p6.7.m7.1.1.1.cmml" xref="S2.SS2.p6.7.m7.1.1.1"></times><ci id="S2.SS2.p6.7.m7.1.1.2.cmml" xref="S2.SS2.p6.7.m7.1.1.2">𝔉</ci><ci id="S2.SS2.p6.7.m7.1.1.3.cmml" xref="S2.SS2.p6.7.m7.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.7.m7.1c">\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.7.m7.1d">fraktur_F caligraphic_C</annotation></semantics></math> is taken to <math alttext="v:x^{\prime}\to x" class="ltx_Math" display="inline" id="S2.SS2.p6.8.m8.1"><semantics id="S2.SS2.p6.8.m8.1a"><mrow id="S2.SS2.p6.8.m8.1.1" xref="S2.SS2.p6.8.m8.1.1.cmml"><mi id="S2.SS2.p6.8.m8.1.1.2" xref="S2.SS2.p6.8.m8.1.1.2.cmml">v</mi><mo id="S2.SS2.p6.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p6.8.m8.1.1.1.cmml">:</mo><mrow id="S2.SS2.p6.8.m8.1.1.3" xref="S2.SS2.p6.8.m8.1.1.3.cmml"><msup id="S2.SS2.p6.8.m8.1.1.3.2" xref="S2.SS2.p6.8.m8.1.1.3.2.cmml"><mi id="S2.SS2.p6.8.m8.1.1.3.2.2" xref="S2.SS2.p6.8.m8.1.1.3.2.2.cmml">x</mi><mo id="S2.SS2.p6.8.m8.1.1.3.2.3" xref="S2.SS2.p6.8.m8.1.1.3.2.3.cmml">′</mo></msup><mo id="S2.SS2.p6.8.m8.1.1.3.1" stretchy="false" xref="S2.SS2.p6.8.m8.1.1.3.1.cmml">→</mo><mi id="S2.SS2.p6.8.m8.1.1.3.3" xref="S2.SS2.p6.8.m8.1.1.3.3.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.8.m8.1b"><apply id="S2.SS2.p6.8.m8.1.1.cmml" xref="S2.SS2.p6.8.m8.1.1"><ci id="S2.SS2.p6.8.m8.1.1.1.cmml" xref="S2.SS2.p6.8.m8.1.1.1">:</ci><ci id="S2.SS2.p6.8.m8.1.1.2.cmml" xref="S2.SS2.p6.8.m8.1.1.2">𝑣</ci><apply id="S2.SS2.p6.8.m8.1.1.3.cmml" xref="S2.SS2.p6.8.m8.1.1.3"><ci id="S2.SS2.p6.8.m8.1.1.3.1.cmml" xref="S2.SS2.p6.8.m8.1.1.3.1">→</ci><apply id="S2.SS2.p6.8.m8.1.1.3.2.cmml" xref="S2.SS2.p6.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p6.8.m8.1.1.3.2.1.cmml" xref="S2.SS2.p6.8.m8.1.1.3.2">superscript</csymbol><ci id="S2.SS2.p6.8.m8.1.1.3.2.2.cmml" xref="S2.SS2.p6.8.m8.1.1.3.2.2">𝑥</ci><ci id="S2.SS2.p6.8.m8.1.1.3.2.3.cmml" xref="S2.SS2.p6.8.m8.1.1.3.2.3">′</ci></apply><ci id="S2.SS2.p6.8.m8.1.1.3.3.cmml" xref="S2.SS2.p6.8.m8.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.8.m8.1c">v:x^{\prime}\to x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.8.m8.1d">italic_v : italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_x</annotation></semantics></math> and <math alttext="u:y\to y^{\prime}" class="ltx_Math" display="inline" id="S2.SS2.p6.9.m9.1"><semantics id="S2.SS2.p6.9.m9.1a"><mrow id="S2.SS2.p6.9.m9.1.1" xref="S2.SS2.p6.9.m9.1.1.cmml"><mi id="S2.SS2.p6.9.m9.1.1.2" xref="S2.SS2.p6.9.m9.1.1.2.cmml">u</mi><mo id="S2.SS2.p6.9.m9.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p6.9.m9.1.1.1.cmml">:</mo><mrow id="S2.SS2.p6.9.m9.1.1.3" xref="S2.SS2.p6.9.m9.1.1.3.cmml"><mi id="S2.SS2.p6.9.m9.1.1.3.2" xref="S2.SS2.p6.9.m9.1.1.3.2.cmml">y</mi><mo id="S2.SS2.p6.9.m9.1.1.3.1" stretchy="false" xref="S2.SS2.p6.9.m9.1.1.3.1.cmml">→</mo><msup id="S2.SS2.p6.9.m9.1.1.3.3" xref="S2.SS2.p6.9.m9.1.1.3.3.cmml"><mi id="S2.SS2.p6.9.m9.1.1.3.3.2" xref="S2.SS2.p6.9.m9.1.1.3.3.2.cmml">y</mi><mo id="S2.SS2.p6.9.m9.1.1.3.3.3" xref="S2.SS2.p6.9.m9.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.9.m9.1b"><apply id="S2.SS2.p6.9.m9.1.1.cmml" xref="S2.SS2.p6.9.m9.1.1"><ci id="S2.SS2.p6.9.m9.1.1.1.cmml" xref="S2.SS2.p6.9.m9.1.1.1">:</ci><ci id="S2.SS2.p6.9.m9.1.1.2.cmml" xref="S2.SS2.p6.9.m9.1.1.2">𝑢</ci><apply id="S2.SS2.p6.9.m9.1.1.3.cmml" xref="S2.SS2.p6.9.m9.1.1.3"><ci id="S2.SS2.p6.9.m9.1.1.3.1.cmml" xref="S2.SS2.p6.9.m9.1.1.3.1">→</ci><ci id="S2.SS2.p6.9.m9.1.1.3.2.cmml" xref="S2.SS2.p6.9.m9.1.1.3.2">𝑦</ci><apply id="S2.SS2.p6.9.m9.1.1.3.3.cmml" xref="S2.SS2.p6.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p6.9.m9.1.1.3.3.1.cmml" xref="S2.SS2.p6.9.m9.1.1.3.3">superscript</csymbol><ci id="S2.SS2.p6.9.m9.1.1.3.3.2.cmml" xref="S2.SS2.p6.9.m9.1.1.3.3.2">𝑦</ci><ci id="S2.SS2.p6.9.m9.1.1.3.3.3.cmml" xref="S2.SS2.p6.9.m9.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.9.m9.1c">u:y\to y^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.9.m9.1d">italic_u : italic_y → italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> under the functors <math alttext="S" class="ltx_Math" display="inline" id="S2.SS2.p6.10.m10.1"><semantics id="S2.SS2.p6.10.m10.1a"><mi id="S2.SS2.p6.10.m10.1.1" xref="S2.SS2.p6.10.m10.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.10.m10.1b"><ci id="S2.SS2.p6.10.m10.1.1.cmml" xref="S2.SS2.p6.10.m10.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.10.m10.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.10.m10.1d">italic_S</annotation></semantics></math> and <math alttext="T" class="ltx_Math" display="inline" id="S2.SS2.p6.11.m11.1"><semantics id="S2.SS2.p6.11.m11.1a"><mi id="S2.SS2.p6.11.m11.1.1" xref="S2.SS2.p6.11.m11.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.11.m11.1b"><ci id="S2.SS2.p6.11.m11.1.1.cmml" xref="S2.SS2.p6.11.m11.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.11.m11.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.11.m11.1d">italic_T</annotation></semantics></math>, respectively.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.1.1.1">Lemma 2.8</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.2.2"> </span>(Baues-Wirsching <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib1" title="">1</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem8.p1"> <p class="ltx_p" id="S2.Thmtheorem8.p1.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem8.p1.1.1">(i) If <math alttext="M" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.1.1.m1.1"><semantics id="S2.Thmtheorem8.p1.1.1.m1.1a"><mi id="S2.Thmtheorem8.p1.1.1.m1.1.1" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.1.1.m1.1b"><ci id="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.1.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.1.1.m1.1d">italic_M</annotation></semantics></math> is a natural system defined by the composition</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex26"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="M:\mathfrak{F}\mathcal{C}\smash{\,\mathop{\longrightarrow}\limits^{T}\,}% \mathcal{C}\smash{\,\mathop{\longrightarrow}\limits^{N}\,}R\text{-mod}," class="ltx_Math" display="block" id="S2.Ex26.m1.1"><semantics id="S2.Ex26.m1.1a"><mrow id="S2.Ex26.m1.1.1.1" xref="S2.Ex26.m1.1.1.1.1.cmml"><mrow id="S2.Ex26.m1.1.1.1.1" xref="S2.Ex26.m1.1.1.1.1.cmml"><mi id="S2.Ex26.m1.1.1.1.1.2" xref="S2.Ex26.m1.1.1.1.1.2.cmml">M</mi><mo id="S2.Ex26.m1.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex26.m1.1.1.1.1.1.cmml">:</mo><mrow id="S2.Ex26.m1.1.1.1.1.3" xref="S2.Ex26.m1.1.1.1.1.3.cmml"><mi id="S2.Ex26.m1.1.1.1.1.3.2" xref="S2.Ex26.m1.1.1.1.1.3.2.cmml">𝔉</mi><mo id="S2.Ex26.m1.1.1.1.1.3.1" xref="S2.Ex26.m1.1.1.1.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex26.m1.1.1.1.1.3.3" xref="S2.Ex26.m1.1.1.1.1.3.3.cmml">𝒞</mi><mo id="S2.Ex26.m1.1.1.1.1.3.1a" lspace="0.337em" xref="S2.Ex26.m1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex26.m1.1.1.1.1.3.4" xref="S2.Ex26.m1.1.1.1.1.3.4.cmml"><mover id="S2.Ex26.m1.1.1.1.1.3.4.1" xref="S2.Ex26.m1.1.1.1.1.3.4.1.cmml"><mo id="S2.Ex26.m1.1.1.1.1.3.4.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex26.m1.1.1.1.1.3.4.1.2.cmml">⟶</mo><mi id="S2.Ex26.m1.1.1.1.1.3.4.1.3" xref="S2.Ex26.m1.1.1.1.1.3.4.1.3.cmml">T</mi></mover><mrow id="S2.Ex26.m1.1.1.1.1.3.4.2" xref="S2.Ex26.m1.1.1.1.1.3.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex26.m1.1.1.1.1.3.4.2.2" xref="S2.Ex26.m1.1.1.1.1.3.4.2.2.cmml">𝒞</mi><mo id="S2.Ex26.m1.1.1.1.1.3.4.2.1" lspace="0.337em" xref="S2.Ex26.m1.1.1.1.1.3.4.2.1.cmml">⁢</mo><mrow id="S2.Ex26.m1.1.1.1.1.3.4.2.3" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.cmml"><mover id="S2.Ex26.m1.1.1.1.1.3.4.2.3.1" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.cmml"><mo id="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.2.cmml">⟶</mo><mi id="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.3" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.3.cmml">N</mi></mover><mrow id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.cmml"><mi id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.2" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.2.cmml">R</mi><mo id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.1" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.3" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.3a.cmml">-mod</mtext></mrow></mrow></mrow></mrow></mrow></mrow><mo id="S2.Ex26.m1.1.1.1.2" xref="S2.Ex26.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex26.m1.1b"><apply id="S2.Ex26.m1.1.1.1.1.cmml" xref="S2.Ex26.m1.1.1.1"><ci id="S2.Ex26.m1.1.1.1.1.1.cmml" xref="S2.Ex26.m1.1.1.1.1.1">:</ci><ci id="S2.Ex26.m1.1.1.1.1.2.cmml" xref="S2.Ex26.m1.1.1.1.1.2">𝑀</ci><apply id="S2.Ex26.m1.1.1.1.1.3.cmml" xref="S2.Ex26.m1.1.1.1.1.3"><times id="S2.Ex26.m1.1.1.1.1.3.1.cmml" xref="S2.Ex26.m1.1.1.1.1.3.1"></times><ci id="S2.Ex26.m1.1.1.1.1.3.2.cmml" xref="S2.Ex26.m1.1.1.1.1.3.2">𝔉</ci><ci id="S2.Ex26.m1.1.1.1.1.3.3.cmml" xref="S2.Ex26.m1.1.1.1.1.3.3">𝒞</ci><apply id="S2.Ex26.m1.1.1.1.1.3.4.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4"><apply id="S2.Ex26.m1.1.1.1.1.3.4.1.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.1"><csymbol cd="ambiguous" id="S2.Ex26.m1.1.1.1.1.3.4.1.1.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.1">superscript</csymbol><ci id="S2.Ex26.m1.1.1.1.1.3.4.1.2.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.1.2">⟶</ci><ci id="S2.Ex26.m1.1.1.1.1.3.4.1.3.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.1.3">𝑇</ci></apply><apply id="S2.Ex26.m1.1.1.1.1.3.4.2.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2"><times id="S2.Ex26.m1.1.1.1.1.3.4.2.1.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.1"></times><ci id="S2.Ex26.m1.1.1.1.1.3.4.2.2.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.2">𝒞</ci><apply id="S2.Ex26.m1.1.1.1.1.3.4.2.3.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3"><apply id="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.1"><csymbol cd="ambiguous" id="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.1.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.1">superscript</csymbol><ci id="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.2.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.2">⟶</ci><ci id="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.3.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.1.3">𝑁</ci></apply><apply id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2"><times id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.1.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.1"></times><ci id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.2.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.2">𝑅</ci><ci id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.3a.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.3"><mtext class="ltx_mathvariant_italic" id="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.3.cmml" xref="S2.Ex26.m1.1.1.1.1.3.4.2.3.2.3">-mod</mtext></ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex26.m1.1c">M:\mathfrak{F}\mathcal{C}\smash{\,\mathop{\longrightarrow}\limits^{T}\,}% \mathcal{C}\smash{\,\mathop{\longrightarrow}\limits^{N}\,}R\text{-mod},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex26.m1.1d">italic_M : fraktur_F caligraphic_C ⟶ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT caligraphic_C ⟶ start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_R -mod ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem8.p1.7"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem8.p1.7.6">then <math alttext="H^{*}_{BW}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.2.1.m1.2"><semantics id="S2.Thmtheorem8.p1.2.1.m1.2a"><mrow id="S2.Thmtheorem8.p1.2.1.m1.2.3" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.cmml"><msubsup id="S2.Thmtheorem8.p1.2.1.m1.2.3.2" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.cmml"><mi id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.2" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.2.cmml">H</mi><mrow id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.cmml"><mi id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.2" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.2.cmml">B</mi><mo id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.1" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.1.cmml">⁢</mo><mi id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.3" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.3.cmml">W</mi></mrow><mo id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.3" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S2.Thmtheorem8.p1.2.1.m1.2.3.1" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem8.p1.2.1.m1.2.3.3.2" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.3.1.cmml"><mo id="S2.Thmtheorem8.p1.2.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem8.p1.2.1.m1.1.1" xref="S2.Thmtheorem8.p1.2.1.m1.1.1.cmml">𝒞</mi><mo id="S2.Thmtheorem8.p1.2.1.m1.2.3.3.2.2" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.3.1.cmml">;</mo><mi id="S2.Thmtheorem8.p1.2.1.m1.2.2" xref="S2.Thmtheorem8.p1.2.1.m1.2.2.cmml">M</mi><mo id="S2.Thmtheorem8.p1.2.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.2.1.m1.2b"><apply id="S2.Thmtheorem8.p1.2.1.m1.2.3.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3"><times id="S2.Thmtheorem8.p1.2.1.m1.2.3.1.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.1"></times><apply id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.1.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2">subscript</csymbol><apply id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.1.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.2.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.2">𝐻</ci><times id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.3.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.2.3"></times></apply><apply id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3"><times id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.1.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.1"></times><ci id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.2.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.2">𝐵</ci><ci id="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.3.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.2.3.3">𝑊</ci></apply></apply><list id="S2.Thmtheorem8.p1.2.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.3.3.2"><ci id="S2.Thmtheorem8.p1.2.1.m1.1.1.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.1.1">𝒞</ci><ci id="S2.Thmtheorem8.p1.2.1.m1.2.2.cmml" xref="S2.Thmtheorem8.p1.2.1.m1.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.2.1.m1.2c">H^{*}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.2.1.m1.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> is isomorphic to the cohomology <math alttext="H^{*}(\mathcal{C};N)" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.3.2.m2.2"><semantics id="S2.Thmtheorem8.p1.3.2.m2.2a"><mrow id="S2.Thmtheorem8.p1.3.2.m2.2.3" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.cmml"><msup id="S2.Thmtheorem8.p1.3.2.m2.2.3.2" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.2.cmml"><mi id="S2.Thmtheorem8.p1.3.2.m2.2.3.2.2" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.2.2.cmml">H</mi><mo id="S2.Thmtheorem8.p1.3.2.m2.2.3.2.3" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.2.3.cmml">∗</mo></msup><mo id="S2.Thmtheorem8.p1.3.2.m2.2.3.1" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem8.p1.3.2.m2.2.3.3.2" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.3.1.cmml"><mo id="S2.Thmtheorem8.p1.3.2.m2.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem8.p1.3.2.m2.1.1" xref="S2.Thmtheorem8.p1.3.2.m2.1.1.cmml">𝒞</mi><mo id="S2.Thmtheorem8.p1.3.2.m2.2.3.3.2.2" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.3.1.cmml">;</mo><mi id="S2.Thmtheorem8.p1.3.2.m2.2.2" xref="S2.Thmtheorem8.p1.3.2.m2.2.2.cmml">N</mi><mo id="S2.Thmtheorem8.p1.3.2.m2.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.3.2.m2.2b"><apply id="S2.Thmtheorem8.p1.3.2.m2.2.3.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.2.3"><times id="S2.Thmtheorem8.p1.3.2.m2.2.3.1.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.1"></times><apply id="S2.Thmtheorem8.p1.3.2.m2.2.3.2.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.3.2.m2.2.3.2.1.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem8.p1.3.2.m2.2.3.2.2.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.2.2">𝐻</ci><times id="S2.Thmtheorem8.p1.3.2.m2.2.3.2.3.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.2.3"></times></apply><list id="S2.Thmtheorem8.p1.3.2.m2.2.3.3.1.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.2.3.3.2"><ci id="S2.Thmtheorem8.p1.3.2.m2.1.1.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.1.1">𝒞</ci><ci id="S2.Thmtheorem8.p1.3.2.m2.2.2.cmml" xref="S2.Thmtheorem8.p1.3.2.m2.2.2">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.3.2.m2.2c">H^{*}(\mathcal{C};N)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.3.2.m2.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_C ; italic_N )</annotation></semantics></math> of <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.4.3.m3.1"><semantics id="S2.Thmtheorem8.p1.4.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem8.p1.4.3.m3.1.1" xref="S2.Thmtheorem8.p1.4.3.m3.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.4.3.m3.1b"><ci id="S2.Thmtheorem8.p1.4.3.m3.1.1.cmml" xref="S2.Thmtheorem8.p1.4.3.m3.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.4.3.m3.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.4.3.m3.1d">caligraphic_C</annotation></semantics></math> with coefficients in the (covariant) <math alttext="R\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.5.4.m4.1"><semantics id="S2.Thmtheorem8.p1.5.4.m4.1a"><mrow id="S2.Thmtheorem8.p1.5.4.m4.1.1" xref="S2.Thmtheorem8.p1.5.4.m4.1.1.cmml"><mi id="S2.Thmtheorem8.p1.5.4.m4.1.1.2" xref="S2.Thmtheorem8.p1.5.4.m4.1.1.2.cmml">R</mi><mo id="S2.Thmtheorem8.p1.5.4.m4.1.1.1" xref="S2.Thmtheorem8.p1.5.4.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem8.p1.5.4.m4.1.1.3" xref="S2.Thmtheorem8.p1.5.4.m4.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.5.4.m4.1b"><apply id="S2.Thmtheorem8.p1.5.4.m4.1.1.cmml" xref="S2.Thmtheorem8.p1.5.4.m4.1.1"><times id="S2.Thmtheorem8.p1.5.4.m4.1.1.1.cmml" xref="S2.Thmtheorem8.p1.5.4.m4.1.1.1"></times><ci id="S2.Thmtheorem8.p1.5.4.m4.1.1.2.cmml" xref="S2.Thmtheorem8.p1.5.4.m4.1.1.2">𝑅</ci><ci id="S2.Thmtheorem8.p1.5.4.m4.1.1.3.cmml" xref="S2.Thmtheorem8.p1.5.4.m4.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.5.4.m4.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.5.4.m4.1d">italic_R caligraphic_C</annotation></semantics></math>-module <math alttext="N" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.6.5.m5.1"><semantics id="S2.Thmtheorem8.p1.6.5.m5.1a"><mi id="S2.Thmtheorem8.p1.6.5.m5.1.1" xref="S2.Thmtheorem8.p1.6.5.m5.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.6.5.m5.1b"><ci id="S2.Thmtheorem8.p1.6.5.m5.1.1.cmml" xref="S2.Thmtheorem8.p1.6.5.m5.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.6.5.m5.1c">N</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.6.5.m5.1d">italic_N</annotation></semantics></math>. <br class="ltx_break"/>(ii) If <math alttext="M" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.7.6.m6.1"><semantics id="S2.Thmtheorem8.p1.7.6.m6.1a"><mi id="S2.Thmtheorem8.p1.7.6.m6.1.1" xref="S2.Thmtheorem8.p1.7.6.m6.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.7.6.m6.1b"><ci id="S2.Thmtheorem8.p1.7.6.m6.1.1.cmml" xref="S2.Thmtheorem8.p1.7.6.m6.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.7.6.m6.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.7.6.m6.1d">italic_M</annotation></semantics></math> is a natural system defined by the composition</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex27"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="M:\mathfrak{F}\mathcal{C}\smash{\,\mathop{\longrightarrow}\limits^{S}\,}% \mathcal{C}^{op}\smash{\,\mathop{\longrightarrow}\limits^{N^{\prime}}\,}R\text% {-mod}," class="ltx_Math" display="block" id="S2.Ex27.m1.1"><semantics id="S2.Ex27.m1.1a"><mrow id="S2.Ex27.m1.1.1.1" xref="S2.Ex27.m1.1.1.1.1.cmml"><mrow id="S2.Ex27.m1.1.1.1.1" xref="S2.Ex27.m1.1.1.1.1.cmml"><mi id="S2.Ex27.m1.1.1.1.1.2" xref="S2.Ex27.m1.1.1.1.1.2.cmml">M</mi><mo id="S2.Ex27.m1.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex27.m1.1.1.1.1.1.cmml">:</mo><mrow id="S2.Ex27.m1.1.1.1.1.3" xref="S2.Ex27.m1.1.1.1.1.3.cmml"><mi id="S2.Ex27.m1.1.1.1.1.3.2" xref="S2.Ex27.m1.1.1.1.1.3.2.cmml">𝔉</mi><mo id="S2.Ex27.m1.1.1.1.1.3.1" xref="S2.Ex27.m1.1.1.1.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex27.m1.1.1.1.1.3.3" xref="S2.Ex27.m1.1.1.1.1.3.3.cmml">𝒞</mi><mo id="S2.Ex27.m1.1.1.1.1.3.1a" lspace="0.337em" xref="S2.Ex27.m1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex27.m1.1.1.1.1.3.4" xref="S2.Ex27.m1.1.1.1.1.3.4.cmml"><mover id="S2.Ex27.m1.1.1.1.1.3.4.1" xref="S2.Ex27.m1.1.1.1.1.3.4.1.cmml"><mo id="S2.Ex27.m1.1.1.1.1.3.4.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex27.m1.1.1.1.1.3.4.1.2.cmml">⟶</mo><mi id="S2.Ex27.m1.1.1.1.1.3.4.1.3" xref="S2.Ex27.m1.1.1.1.1.3.4.1.3.cmml">S</mi></mover><mrow id="S2.Ex27.m1.1.1.1.1.3.4.2" xref="S2.Ex27.m1.1.1.1.1.3.4.2.cmml"><msup id="S2.Ex27.m1.1.1.1.1.3.4.2.2" xref="S2.Ex27.m1.1.1.1.1.3.4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex27.m1.1.1.1.1.3.4.2.2.2" xref="S2.Ex27.m1.1.1.1.1.3.4.2.2.2.cmml">𝒞</mi><mrow id="S2.Ex27.m1.1.1.1.1.3.4.2.2.3" xref="S2.Ex27.m1.1.1.1.1.3.4.2.2.3.cmml"><mi id="S2.Ex27.m1.1.1.1.1.3.4.2.2.3.2" xref="S2.Ex27.m1.1.1.1.1.3.4.2.2.3.2.cmml">o</mi><mo id="S2.Ex27.m1.1.1.1.1.3.4.2.2.3.1" xref="S2.Ex27.m1.1.1.1.1.3.4.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex27.m1.1.1.1.1.3.4.2.2.3.3" xref="S2.Ex27.m1.1.1.1.1.3.4.2.2.3.3.cmml">p</mi></mrow></msup><mo id="S2.Ex27.m1.1.1.1.1.3.4.2.1" lspace="0.167em" xref="S2.Ex27.m1.1.1.1.1.3.4.2.1.cmml">⁢</mo><mrow id="S2.Ex27.m1.1.1.1.1.3.4.2.3" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.cmml"><mover id="S2.Ex27.m1.1.1.1.1.3.4.2.3.1" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.cmml"><mo id="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.2.cmml">⟶</mo><msup id="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.cmml"><mi id="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.2" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.2.cmml">N</mi><mo id="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.3" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.3.cmml">′</mo></msup></mover><mrow id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.cmml"><mi id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.2" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.2.cmml">R</mi><mo id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.1" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.3" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.3a.cmml">-mod</mtext></mrow></mrow></mrow></mrow></mrow></mrow><mo id="S2.Ex27.m1.1.1.1.2" xref="S2.Ex27.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex27.m1.1b"><apply id="S2.Ex27.m1.1.1.1.1.cmml" xref="S2.Ex27.m1.1.1.1"><ci id="S2.Ex27.m1.1.1.1.1.1.cmml" xref="S2.Ex27.m1.1.1.1.1.1">:</ci><ci id="S2.Ex27.m1.1.1.1.1.2.cmml" xref="S2.Ex27.m1.1.1.1.1.2">𝑀</ci><apply id="S2.Ex27.m1.1.1.1.1.3.cmml" xref="S2.Ex27.m1.1.1.1.1.3"><times 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id="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.1.cmml" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3">superscript</csymbol><ci id="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.2.cmml" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.2">𝑁</ci><ci id="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.3.cmml" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.1.3.3">′</ci></apply></apply><apply id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.cmml" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2"><times id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.1.cmml" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.1"></times><ci id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.2.cmml" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.2">𝑅</ci><ci id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.3a.cmml" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.3"><mtext class="ltx_mathvariant_italic" id="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.3.cmml" xref="S2.Ex27.m1.1.1.1.1.3.4.2.3.2.3">-mod</mtext></ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex27.m1.1c">M:\mathfrak{F}\mathcal{C}\smash{\,\mathop{\longrightarrow}\limits^{S}\,}% \mathcal{C}^{op}\smash{\,\mathop{\longrightarrow}\limits^{N^{\prime}}\,}R\text% {-mod},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex27.m1.1d">italic_M : fraktur_F caligraphic_C ⟶ start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT caligraphic_C start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT ⟶ start_POSTSUPERSCRIPT italic_N start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_R -mod ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem8.p1.12"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem8.p1.12.5">then <math alttext="H^{*}_{BW}(\mathcal{C};M)" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.8.1.m1.2"><semantics id="S2.Thmtheorem8.p1.8.1.m1.2a"><mrow id="S2.Thmtheorem8.p1.8.1.m1.2.3" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.cmml"><msubsup id="S2.Thmtheorem8.p1.8.1.m1.2.3.2" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.2.cmml"><mi 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id="S2.Thmtheorem8.p1.8.1.m1.2.3.3.2.2" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.3.1.cmml">;</mo><mi id="S2.Thmtheorem8.p1.8.1.m1.2.2" xref="S2.Thmtheorem8.p1.8.1.m1.2.2.cmml">M</mi><mo id="S2.Thmtheorem8.p1.8.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.8.1.m1.2b"><apply id="S2.Thmtheorem8.p1.8.1.m1.2.3.cmml" xref="S2.Thmtheorem8.p1.8.1.m1.2.3"><times id="S2.Thmtheorem8.p1.8.1.m1.2.3.1.cmml" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.1"></times><apply id="S2.Thmtheorem8.p1.8.1.m1.2.3.2.cmml" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.8.1.m1.2.3.2.1.cmml" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.2">subscript</csymbol><apply id="S2.Thmtheorem8.p1.8.1.m1.2.3.2.2.cmml" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.8.1.m1.2.3.2.2.1.cmml" xref="S2.Thmtheorem8.p1.8.1.m1.2.3.2">superscript</csymbol><ci 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id="S2.Thmtheorem8.p1.8.1.m1.2c">H^{*}_{BW}(\mathcal{C};M)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.8.1.m1.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M )</annotation></semantics></math> is isomorphic to the cohomology <math alttext="H^{*}(\mathcal{C};N^{\prime})" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.9.2.m2.2"><semantics id="S2.Thmtheorem8.p1.9.2.m2.2a"><mrow id="S2.Thmtheorem8.p1.9.2.m2.2.2" xref="S2.Thmtheorem8.p1.9.2.m2.2.2.cmml"><msup id="S2.Thmtheorem8.p1.9.2.m2.2.2.3" xref="S2.Thmtheorem8.p1.9.2.m2.2.2.3.cmml"><mi id="S2.Thmtheorem8.p1.9.2.m2.2.2.3.2" xref="S2.Thmtheorem8.p1.9.2.m2.2.2.3.2.cmml">H</mi><mo id="S2.Thmtheorem8.p1.9.2.m2.2.2.3.3" xref="S2.Thmtheorem8.p1.9.2.m2.2.2.3.3.cmml">∗</mo></msup><mo id="S2.Thmtheorem8.p1.9.2.m2.2.2.2" xref="S2.Thmtheorem8.p1.9.2.m2.2.2.2.cmml">⁢</mo><mrow id="S2.Thmtheorem8.p1.9.2.m2.2.2.1.1" 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xref="S2.Thmtheorem8.p1.9.2.m2.2.2.1.1.1.2">𝑁</ci><ci id="S2.Thmtheorem8.p1.9.2.m2.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem8.p1.9.2.m2.2.2.1.1.1.3">′</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.9.2.m2.2c">H^{*}(\mathcal{C};N^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.9.2.m2.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_C ; italic_N start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> of <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.10.3.m3.1"><semantics id="S2.Thmtheorem8.p1.10.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem8.p1.10.3.m3.1.1" xref="S2.Thmtheorem8.p1.10.3.m3.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.10.3.m3.1b"><ci id="S2.Thmtheorem8.p1.10.3.m3.1.1.cmml" xref="S2.Thmtheorem8.p1.10.3.m3.1.1">𝒞</ci></annotation-xml><annotation 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xref="S2.Thmtheorem8.p1.11.4.m4.1.1.1"></times><ci id="S2.Thmtheorem8.p1.11.4.m4.1.1.2.cmml" xref="S2.Thmtheorem8.p1.11.4.m4.1.1.2">𝑅</ci><ci id="S2.Thmtheorem8.p1.11.4.m4.1.1.3.cmml" xref="S2.Thmtheorem8.p1.11.4.m4.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.11.4.m4.1c">R\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.11.4.m4.1d">italic_R caligraphic_C</annotation></semantics></math>-module <math alttext="N^{\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.12.5.m5.1"><semantics id="S2.Thmtheorem8.p1.12.5.m5.1a"><msup id="S2.Thmtheorem8.p1.12.5.m5.1.1" xref="S2.Thmtheorem8.p1.12.5.m5.1.1.cmml"><mi id="S2.Thmtheorem8.p1.12.5.m5.1.1.2" xref="S2.Thmtheorem8.p1.12.5.m5.1.1.2.cmml">N</mi><mo id="S2.Thmtheorem8.p1.12.5.m5.1.1.3" xref="S2.Thmtheorem8.p1.12.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.12.5.m5.1b"><apply 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definitions for cohomology of simplicial sets with general coefficient systems and define Thomason cohomology for small categories. For more details on this material we refer the reader to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib5" title="">5</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib7" title="">7</a>]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib8" title="">8</a>]</cite>.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1. </span>Coefficient systems for simplicial sets</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.6">The <em class="ltx_emph ltx_font_italic" id="S3.SS1.p1.6.1">simplex category</em> <math alttext="\Delta" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mi id="S3.SS1.p1.1.m1.1.1" mathvariant="normal" xref="S3.SS1.p1.1.m1.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">\Delta</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">roman_Δ</annotation></semantics></math> is the category whose objects are ordered finite sets <math alttext="[n]=\{0,1,\dots,n\}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.5"><semantics id="S3.SS1.p1.2.m2.5a"><mrow id="S3.SS1.p1.2.m2.5.6" xref="S3.SS1.p1.2.m2.5.6.cmml"><mrow id="S3.SS1.p1.2.m2.5.6.2.2" xref="S3.SS1.p1.2.m2.5.6.2.1.cmml"><mo id="S3.SS1.p1.2.m2.5.6.2.2.1" stretchy="false" xref="S3.SS1.p1.2.m2.5.6.2.1.1.cmml">[</mo><mi id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">n</mi><mo id="S3.SS1.p1.2.m2.5.6.2.2.2" stretchy="false" xref="S3.SS1.p1.2.m2.5.6.2.1.1.cmml">]</mo></mrow><mo id="S3.SS1.p1.2.m2.5.6.1" xref="S3.SS1.p1.2.m2.5.6.1.cmml">=</mo><mrow id="S3.SS1.p1.2.m2.5.6.3.2" xref="S3.SS1.p1.2.m2.5.6.3.1.cmml"><mo id="S3.SS1.p1.2.m2.5.6.3.2.1" stretchy="false" xref="S3.SS1.p1.2.m2.5.6.3.1.cmml">{</mo><mn id="S3.SS1.p1.2.m2.2.2" xref="S3.SS1.p1.2.m2.2.2.cmml">0</mn><mo id="S3.SS1.p1.2.m2.5.6.3.2.2" xref="S3.SS1.p1.2.m2.5.6.3.1.cmml">,</mo><mn id="S3.SS1.p1.2.m2.3.3" xref="S3.SS1.p1.2.m2.3.3.cmml">1</mn><mo id="S3.SS1.p1.2.m2.5.6.3.2.3" xref="S3.SS1.p1.2.m2.5.6.3.1.cmml">,</mo><mi id="S3.SS1.p1.2.m2.4.4" mathvariant="normal" xref="S3.SS1.p1.2.m2.4.4.cmml">…</mi><mo id="S3.SS1.p1.2.m2.5.6.3.2.4" xref="S3.SS1.p1.2.m2.5.6.3.1.cmml">,</mo><mi id="S3.SS1.p1.2.m2.5.5" xref="S3.SS1.p1.2.m2.5.5.cmml">n</mi><mo id="S3.SS1.p1.2.m2.5.6.3.2.5" stretchy="false" xref="S3.SS1.p1.2.m2.5.6.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.5b"><apply id="S3.SS1.p1.2.m2.5.6.cmml" xref="S3.SS1.p1.2.m2.5.6"><eq id="S3.SS1.p1.2.m2.5.6.1.cmml" xref="S3.SS1.p1.2.m2.5.6.1"></eq><apply id="S3.SS1.p1.2.m2.5.6.2.1.cmml" xref="S3.SS1.p1.2.m2.5.6.2.2"><csymbol cd="latexml" id="S3.SS1.p1.2.m2.5.6.2.1.1.cmml" xref="S3.SS1.p1.2.m2.5.6.2.2.1">delimited-[]</csymbol><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝑛</ci></apply><set id="S3.SS1.p1.2.m2.5.6.3.1.cmml" xref="S3.SS1.p1.2.m2.5.6.3.2"><cn id="S3.SS1.p1.2.m2.2.2.cmml" type="integer" xref="S3.SS1.p1.2.m2.2.2">0</cn><cn id="S3.SS1.p1.2.m2.3.3.cmml" type="integer" xref="S3.SS1.p1.2.m2.3.3">1</cn><ci id="S3.SS1.p1.2.m2.4.4.cmml" xref="S3.SS1.p1.2.m2.4.4">…</ci><ci id="S3.SS1.p1.2.m2.5.5.cmml" xref="S3.SS1.p1.2.m2.5.5">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.5c">[n]=\{0,1,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.5d">[ italic_n ] = { 0 , 1 , … , italic_n }</annotation></semantics></math>, and whose morphisms are given by weakly order-preserving functions between them. 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xref="S3.SS1.p1.4.m4.1.1.5.cmml">≤</mo><mi id="S3.SS1.p1.4.m4.1.1.6" xref="S3.SS1.p1.4.m4.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><apply id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1"><and id="S3.SS1.p1.4.m4.1.1a.cmml" xref="S3.SS1.p1.4.m4.1.1"></and><apply id="S3.SS1.p1.4.m4.1.1b.cmml" xref="S3.SS1.p1.4.m4.1.1"><leq id="S3.SS1.p1.4.m4.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.3"></leq><cn id="S3.SS1.p1.4.m4.1.1.2.cmml" type="integer" xref="S3.SS1.p1.4.m4.1.1.2">0</cn><ci id="S3.SS1.p1.4.m4.1.1.4.cmml" xref="S3.SS1.p1.4.m4.1.1.4">𝑖</ci></apply><apply id="S3.SS1.p1.4.m4.1.1c.cmml" xref="S3.SS1.p1.4.m4.1.1"><leq id="S3.SS1.p1.4.m4.1.1.5.cmml" xref="S3.SS1.p1.4.m4.1.1.5"></leq><share href="https://arxiv.org/html/2503.14659v1#S3.SS1.p1.4.m4.1.1.4.cmml" id="S3.SS1.p1.4.m4.1.1d.cmml" xref="S3.SS1.p1.4.m4.1.1"></share><ci id="S3.SS1.p1.4.m4.1.1.6.cmml" xref="S3.SS1.p1.4.m4.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">0\leq i\leq n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">0 ≤ italic_i ≤ italic_n</annotation></semantics></math>, and codegeneracy maps <math alttext="s^{i}:[n+1]\to[n]" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.2"><semantics id="S3.SS1.p1.5.m5.2a"><mrow id="S3.SS1.p1.5.m5.2.2" xref="S3.SS1.p1.5.m5.2.2.cmml"><msup id="S3.SS1.p1.5.m5.2.2.3" xref="S3.SS1.p1.5.m5.2.2.3.cmml"><mi id="S3.SS1.p1.5.m5.2.2.3.2" xref="S3.SS1.p1.5.m5.2.2.3.2.cmml">s</mi><mi id="S3.SS1.p1.5.m5.2.2.3.3" xref="S3.SS1.p1.5.m5.2.2.3.3.cmml">i</mi></msup><mo id="S3.SS1.p1.5.m5.2.2.2" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p1.5.m5.2.2.2.cmml">:</mo><mrow id="S3.SS1.p1.5.m5.2.2.1" xref="S3.SS1.p1.5.m5.2.2.1.cmml"><mrow id="S3.SS1.p1.5.m5.2.2.1.1.1" xref="S3.SS1.p1.5.m5.2.2.1.1.2.cmml"><mo id="S3.SS1.p1.5.m5.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.p1.5.m5.2.2.1.1.2.1.cmml">[</mo><mrow id="S3.SS1.p1.5.m5.2.2.1.1.1.1" xref="S3.SS1.p1.5.m5.2.2.1.1.1.1.cmml"><mi id="S3.SS1.p1.5.m5.2.2.1.1.1.1.2" xref="S3.SS1.p1.5.m5.2.2.1.1.1.1.2.cmml">n</mi><mo id="S3.SS1.p1.5.m5.2.2.1.1.1.1.1" xref="S3.SS1.p1.5.m5.2.2.1.1.1.1.1.cmml">+</mo><mn id="S3.SS1.p1.5.m5.2.2.1.1.1.1.3" xref="S3.SS1.p1.5.m5.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.p1.5.m5.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.p1.5.m5.2.2.1.1.2.1.cmml">]</mo></mrow><mo id="S3.SS1.p1.5.m5.2.2.1.2" stretchy="false" xref="S3.SS1.p1.5.m5.2.2.1.2.cmml">→</mo><mrow id="S3.SS1.p1.5.m5.2.2.1.3.2" xref="S3.SS1.p1.5.m5.2.2.1.3.1.cmml"><mo id="S3.SS1.p1.5.m5.2.2.1.3.2.1" stretchy="false" xref="S3.SS1.p1.5.m5.2.2.1.3.1.1.cmml">[</mo><mi id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml">n</mi><mo id="S3.SS1.p1.5.m5.2.2.1.3.2.2" stretchy="false" xref="S3.SS1.p1.5.m5.2.2.1.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.2b"><apply id="S3.SS1.p1.5.m5.2.2.cmml" xref="S3.SS1.p1.5.m5.2.2"><ci id="S3.SS1.p1.5.m5.2.2.2.cmml" xref="S3.SS1.p1.5.m5.2.2.2">:</ci><apply id="S3.SS1.p1.5.m5.2.2.3.cmml" xref="S3.SS1.p1.5.m5.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.2.2.3.1.cmml" xref="S3.SS1.p1.5.m5.2.2.3">superscript</csymbol><ci id="S3.SS1.p1.5.m5.2.2.3.2.cmml" xref="S3.SS1.p1.5.m5.2.2.3.2">𝑠</ci><ci id="S3.SS1.p1.5.m5.2.2.3.3.cmml" xref="S3.SS1.p1.5.m5.2.2.3.3">𝑖</ci></apply><apply id="S3.SS1.p1.5.m5.2.2.1.cmml" xref="S3.SS1.p1.5.m5.2.2.1"><ci id="S3.SS1.p1.5.m5.2.2.1.2.cmml" xref="S3.SS1.p1.5.m5.2.2.1.2">→</ci><apply id="S3.SS1.p1.5.m5.2.2.1.1.2.cmml" xref="S3.SS1.p1.5.m5.2.2.1.1.1"><csymbol cd="latexml" id="S3.SS1.p1.5.m5.2.2.1.1.2.1.cmml" xref="S3.SS1.p1.5.m5.2.2.1.1.1.2">delimited-[]</csymbol><apply id="S3.SS1.p1.5.m5.2.2.1.1.1.1.cmml" xref="S3.SS1.p1.5.m5.2.2.1.1.1.1"><plus id="S3.SS1.p1.5.m5.2.2.1.1.1.1.1.cmml" xref="S3.SS1.p1.5.m5.2.2.1.1.1.1.1"></plus><ci id="S3.SS1.p1.5.m5.2.2.1.1.1.1.2.cmml" xref="S3.SS1.p1.5.m5.2.2.1.1.1.1.2">𝑛</ci><cn id="S3.SS1.p1.5.m5.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.5.m5.2.2.1.1.1.1.3">1</cn></apply></apply><apply id="S3.SS1.p1.5.m5.2.2.1.3.1.cmml" xref="S3.SS1.p1.5.m5.2.2.1.3.2"><csymbol cd="latexml" id="S3.SS1.p1.5.m5.2.2.1.3.1.1.cmml" xref="S3.SS1.p1.5.m5.2.2.1.3.2.1">delimited-[]</csymbol><ci id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.2c">s^{i}:[n+1]\to[n]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.2d">italic_s start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT : [ italic_n + 1 ] → [ italic_n ]</annotation></semantics></math>, <math alttext="0\leq i\leq n" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.1"><semantics id="S3.SS1.p1.6.m6.1a"><mrow id="S3.SS1.p1.6.m6.1.1" xref="S3.SS1.p1.6.m6.1.1.cmml"><mn id="S3.SS1.p1.6.m6.1.1.2" xref="S3.SS1.p1.6.m6.1.1.2.cmml">0</mn><mo id="S3.SS1.p1.6.m6.1.1.3" xref="S3.SS1.p1.6.m6.1.1.3.cmml">≤</mo><mi id="S3.SS1.p1.6.m6.1.1.4" xref="S3.SS1.p1.6.m6.1.1.4.cmml">i</mi><mo id="S3.SS1.p1.6.m6.1.1.5" xref="S3.SS1.p1.6.m6.1.1.5.cmml">≤</mo><mi id="S3.SS1.p1.6.m6.1.1.6" xref="S3.SS1.p1.6.m6.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.1b"><apply id="S3.SS1.p1.6.m6.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1"><and id="S3.SS1.p1.6.m6.1.1a.cmml" xref="S3.SS1.p1.6.m6.1.1"></and><apply id="S3.SS1.p1.6.m6.1.1b.cmml" xref="S3.SS1.p1.6.m6.1.1"><leq id="S3.SS1.p1.6.m6.1.1.3.cmml" xref="S3.SS1.p1.6.m6.1.1.3"></leq><cn id="S3.SS1.p1.6.m6.1.1.2.cmml" type="integer" xref="S3.SS1.p1.6.m6.1.1.2">0</cn><ci id="S3.SS1.p1.6.m6.1.1.4.cmml" xref="S3.SS1.p1.6.m6.1.1.4">𝑖</ci></apply><apply id="S3.SS1.p1.6.m6.1.1c.cmml" xref="S3.SS1.p1.6.m6.1.1"><leq id="S3.SS1.p1.6.m6.1.1.5.cmml" xref="S3.SS1.p1.6.m6.1.1.5"></leq><share href="https://arxiv.org/html/2503.14659v1#S3.SS1.p1.6.m6.1.1.4.cmml" id="S3.SS1.p1.6.m6.1.1d.cmml" xref="S3.SS1.p1.6.m6.1.1"></share><ci id="S3.SS1.p1.6.m6.1.1.6.cmml" xref="S3.SS1.p1.6.m6.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.1c">0\leq i\leq n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m6.1d">0 ≤ italic_i ≤ italic_n</annotation></semantics></math>, which satisfy certain relations (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib9" title="">9</a>]</cite>).</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.11">A <em class="ltx_emph ltx_font_italic" id="S3.SS1.p2.11.1">simplicial object</em> in the category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">caligraphic_C</annotation></semantics></math> is a functor <math alttext="\Delta^{op}\to\mathcal{C}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><msup id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2.2" mathvariant="normal" xref="S3.SS1.p2.2.m2.1.1.2.2.cmml">Δ</mi><mrow id="S3.SS1.p2.2.m2.1.1.2.3" xref="S3.SS1.p2.2.m2.1.1.2.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2.3.2" xref="S3.SS1.p2.2.m2.1.1.2.3.2.cmml">o</mi><mo id="S3.SS1.p2.2.m2.1.1.2.3.1" xref="S3.SS1.p2.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS1.p2.2.m2.1.1.2.3.3" xref="S3.SS1.p2.2.m2.1.1.2.3.3.cmml">p</mi></mrow></msup><mo id="S3.SS1.p2.2.m2.1.1.1" stretchy="false" xref="S3.SS1.p2.2.m2.1.1.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><ci id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1">→</ci><apply id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.2.1.cmml" xref="S3.SS1.p2.2.m2.1.1.2">superscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.2.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2.2">Δ</ci><apply id="S3.SS1.p2.2.m2.1.1.2.3.cmml" xref="S3.SS1.p2.2.m2.1.1.2.3"><times id="S3.SS1.p2.2.m2.1.1.2.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.2.3.1"></times><ci id="S3.SS1.p2.2.m2.1.1.2.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2.3.2">𝑜</ci><ci id="S3.SS1.p2.2.m2.1.1.2.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.2.3.3">𝑝</ci></apply></apply><ci id="S3.SS1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">\Delta^{op}\to\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → caligraphic_C</annotation></semantics></math>. A simplicial object in sets is called a simplicial set. We write <math alttext="X_{n}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><msub id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mi id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml">X</mi><mi id="S3.SS1.p2.3.m3.1.1.3" xref="S3.SS1.p2.3.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2">𝑋</ci><ci id="S3.SS1.p2.3.m3.1.1.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">X_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="X([n])" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.2"><semantics id="S3.SS1.p2.4.m4.2a"><mrow id="S3.SS1.p2.4.m4.2.2" xref="S3.SS1.p2.4.m4.2.2.cmml"><mi id="S3.SS1.p2.4.m4.2.2.3" xref="S3.SS1.p2.4.m4.2.2.3.cmml">X</mi><mo id="S3.SS1.p2.4.m4.2.2.2" xref="S3.SS1.p2.4.m4.2.2.2.cmml">⁢</mo><mrow id="S3.SS1.p2.4.m4.2.2.1.1" xref="S3.SS1.p2.4.m4.2.2.cmml"><mo id="S3.SS1.p2.4.m4.2.2.1.1.2" stretchy="false" xref="S3.SS1.p2.4.m4.2.2.cmml">(</mo><mrow id="S3.SS1.p2.4.m4.2.2.1.1.1.2" xref="S3.SS1.p2.4.m4.2.2.1.1.1.1.cmml"><mo id="S3.SS1.p2.4.m4.2.2.1.1.1.2.1" stretchy="false" xref="S3.SS1.p2.4.m4.2.2.1.1.1.1.1.cmml">[</mo><mi id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml">n</mi><mo id="S3.SS1.p2.4.m4.2.2.1.1.1.2.2" stretchy="false" xref="S3.SS1.p2.4.m4.2.2.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS1.p2.4.m4.2.2.1.1.3" stretchy="false" xref="S3.SS1.p2.4.m4.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.2b"><apply id="S3.SS1.p2.4.m4.2.2.cmml" xref="S3.SS1.p2.4.m4.2.2"><times id="S3.SS1.p2.4.m4.2.2.2.cmml" xref="S3.SS1.p2.4.m4.2.2.2"></times><ci id="S3.SS1.p2.4.m4.2.2.3.cmml" xref="S3.SS1.p2.4.m4.2.2.3">𝑋</ci><apply id="S3.SS1.p2.4.m4.2.2.1.1.1.1.cmml" xref="S3.SS1.p2.4.m4.2.2.1.1.1.2"><csymbol cd="latexml" id="S3.SS1.p2.4.m4.2.2.1.1.1.1.1.cmml" xref="S3.SS1.p2.4.m4.2.2.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.2c">X([n])</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.2d">italic_X ( [ italic_n ] )</annotation></semantics></math>. For a morphism <math alttext="f:[m]\to[n]" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m5.2"><semantics id="S3.SS1.p2.5.m5.2a"><mrow id="S3.SS1.p2.5.m5.2.3" xref="S3.SS1.p2.5.m5.2.3.cmml"><mi id="S3.SS1.p2.5.m5.2.3.2" xref="S3.SS1.p2.5.m5.2.3.2.cmml">f</mi><mo id="S3.SS1.p2.5.m5.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p2.5.m5.2.3.1.cmml">:</mo><mrow id="S3.SS1.p2.5.m5.2.3.3" xref="S3.SS1.p2.5.m5.2.3.3.cmml"><mrow id="S3.SS1.p2.5.m5.2.3.3.2.2" xref="S3.SS1.p2.5.m5.2.3.3.2.1.cmml"><mo id="S3.SS1.p2.5.m5.2.3.3.2.2.1" stretchy="false" xref="S3.SS1.p2.5.m5.2.3.3.2.1.1.cmml">[</mo><mi id="S3.SS1.p2.5.m5.1.1" xref="S3.SS1.p2.5.m5.1.1.cmml">m</mi><mo id="S3.SS1.p2.5.m5.2.3.3.2.2.2" stretchy="false" xref="S3.SS1.p2.5.m5.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S3.SS1.p2.5.m5.2.3.3.1" stretchy="false" xref="S3.SS1.p2.5.m5.2.3.3.1.cmml">→</mo><mrow id="S3.SS1.p2.5.m5.2.3.3.3.2" xref="S3.SS1.p2.5.m5.2.3.3.3.1.cmml"><mo id="S3.SS1.p2.5.m5.2.3.3.3.2.1" stretchy="false" xref="S3.SS1.p2.5.m5.2.3.3.3.1.1.cmml">[</mo><mi id="S3.SS1.p2.5.m5.2.2" xref="S3.SS1.p2.5.m5.2.2.cmml">n</mi><mo id="S3.SS1.p2.5.m5.2.3.3.3.2.2" stretchy="false" xref="S3.SS1.p2.5.m5.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m5.2b"><apply id="S3.SS1.p2.5.m5.2.3.cmml" xref="S3.SS1.p2.5.m5.2.3"><ci id="S3.SS1.p2.5.m5.2.3.1.cmml" xref="S3.SS1.p2.5.m5.2.3.1">:</ci><ci id="S3.SS1.p2.5.m5.2.3.2.cmml" xref="S3.SS1.p2.5.m5.2.3.2">𝑓</ci><apply id="S3.SS1.p2.5.m5.2.3.3.cmml" xref="S3.SS1.p2.5.m5.2.3.3"><ci id="S3.SS1.p2.5.m5.2.3.3.1.cmml" xref="S3.SS1.p2.5.m5.2.3.3.1">→</ci><apply id="S3.SS1.p2.5.m5.2.3.3.2.1.cmml" xref="S3.SS1.p2.5.m5.2.3.3.2.2"><csymbol cd="latexml" id="S3.SS1.p2.5.m5.2.3.3.2.1.1.cmml" xref="S3.SS1.p2.5.m5.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S3.SS1.p2.5.m5.1.1.cmml" xref="S3.SS1.p2.5.m5.1.1">𝑚</ci></apply><apply id="S3.SS1.p2.5.m5.2.3.3.3.1.cmml" xref="S3.SS1.p2.5.m5.2.3.3.3.2"><csymbol cd="latexml" id="S3.SS1.p2.5.m5.2.3.3.3.1.1.cmml" xref="S3.SS1.p2.5.m5.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S3.SS1.p2.5.m5.2.2.cmml" xref="S3.SS1.p2.5.m5.2.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m5.2c">f:[m]\to[n]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m5.2d">italic_f : [ italic_m ] → [ italic_n ]</annotation></semantics></math>, we denote the induced map <math alttext="X(f):X[n]\to X[m" class="ltx_math_unparsed" display="inline" id="S3.SS1.p2.6.m6.2"><semantics id="S3.SS1.p2.6.m6.2a"><mrow id="S3.SS1.p2.6.m6.2b"><mi id="S3.SS1.p2.6.m6.2.3">X</mi><mrow id="S3.SS1.p2.6.m6.2.4"><mo id="S3.SS1.p2.6.m6.2.4.1" stretchy="false">(</mo><mi id="S3.SS1.p2.6.m6.1.1">f</mi><mo id="S3.SS1.p2.6.m6.2.4.2" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S3.SS1.p2.6.m6.2.5" rspace="0.278em">:</mo><mi id="S3.SS1.p2.6.m6.2.6">X</mi><mrow id="S3.SS1.p2.6.m6.2.7"><mo id="S3.SS1.p2.6.m6.2.7.1" stretchy="false">[</mo><mi id="S3.SS1.p2.6.m6.2.2">n</mi><mo id="S3.SS1.p2.6.m6.2.7.2" stretchy="false">]</mo></mrow><mo id="S3.SS1.p2.6.m6.2.8" stretchy="false">→</mo><mi id="S3.SS1.p2.6.m6.2.9">X</mi><mrow id="S3.SS1.p2.6.m6.2.10"><mo id="S3.SS1.p2.6.m6.2.10.1" stretchy="false">[</mo><mi id="S3.SS1.p2.6.m6.2.10.2">m</mi></mrow></mrow><annotation encoding="application/x-tex" id="S3.SS1.p2.6.m6.2c">X(f):X[n]\to X[m</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.6.m6.2d">italic_X ( italic_f ) : italic_X [ italic_n ] → italic_X [ italic_m</annotation></semantics></math>] by <math alttext="f^{*}:X_{n}\to X_{m}" class="ltx_Math" display="inline" id="S3.SS1.p2.7.m7.1"><semantics id="S3.SS1.p2.7.m7.1a"><mrow id="S3.SS1.p2.7.m7.1.1" xref="S3.SS1.p2.7.m7.1.1.cmml"><msup id="S3.SS1.p2.7.m7.1.1.2" xref="S3.SS1.p2.7.m7.1.1.2.cmml"><mi id="S3.SS1.p2.7.m7.1.1.2.2" xref="S3.SS1.p2.7.m7.1.1.2.2.cmml">f</mi><mo id="S3.SS1.p2.7.m7.1.1.2.3" xref="S3.SS1.p2.7.m7.1.1.2.3.cmml">∗</mo></msup><mo id="S3.SS1.p2.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p2.7.m7.1.1.1.cmml">:</mo><mrow id="S3.SS1.p2.7.m7.1.1.3" xref="S3.SS1.p2.7.m7.1.1.3.cmml"><msub id="S3.SS1.p2.7.m7.1.1.3.2" xref="S3.SS1.p2.7.m7.1.1.3.2.cmml"><mi id="S3.SS1.p2.7.m7.1.1.3.2.2" xref="S3.SS1.p2.7.m7.1.1.3.2.2.cmml">X</mi><mi id="S3.SS1.p2.7.m7.1.1.3.2.3" xref="S3.SS1.p2.7.m7.1.1.3.2.3.cmml">n</mi></msub><mo id="S3.SS1.p2.7.m7.1.1.3.1" stretchy="false" xref="S3.SS1.p2.7.m7.1.1.3.1.cmml">→</mo><msub id="S3.SS1.p2.7.m7.1.1.3.3" xref="S3.SS1.p2.7.m7.1.1.3.3.cmml"><mi id="S3.SS1.p2.7.m7.1.1.3.3.2" xref="S3.SS1.p2.7.m7.1.1.3.3.2.cmml">X</mi><mi id="S3.SS1.p2.7.m7.1.1.3.3.3" xref="S3.SS1.p2.7.m7.1.1.3.3.3.cmml">m</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.7.m7.1b"><apply id="S3.SS1.p2.7.m7.1.1.cmml" xref="S3.SS1.p2.7.m7.1.1"><ci id="S3.SS1.p2.7.m7.1.1.1.cmml" xref="S3.SS1.p2.7.m7.1.1.1">:</ci><apply id="S3.SS1.p2.7.m7.1.1.2.cmml" xref="S3.SS1.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.7.m7.1.1.2.1.cmml" xref="S3.SS1.p2.7.m7.1.1.2">superscript</csymbol><ci id="S3.SS1.p2.7.m7.1.1.2.2.cmml" xref="S3.SS1.p2.7.m7.1.1.2.2">𝑓</ci><times id="S3.SS1.p2.7.m7.1.1.2.3.cmml" xref="S3.SS1.p2.7.m7.1.1.2.3"></times></apply><apply id="S3.SS1.p2.7.m7.1.1.3.cmml" xref="S3.SS1.p2.7.m7.1.1.3"><ci id="S3.SS1.p2.7.m7.1.1.3.1.cmml" xref="S3.SS1.p2.7.m7.1.1.3.1">→</ci><apply id="S3.SS1.p2.7.m7.1.1.3.2.cmml" xref="S3.SS1.p2.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.7.m7.1.1.3.2.1.cmml" xref="S3.SS1.p2.7.m7.1.1.3.2">subscript</csymbol><ci id="S3.SS1.p2.7.m7.1.1.3.2.2.cmml" xref="S3.SS1.p2.7.m7.1.1.3.2.2">𝑋</ci><ci id="S3.SS1.p2.7.m7.1.1.3.2.3.cmml" xref="S3.SS1.p2.7.m7.1.1.3.2.3">𝑛</ci></apply><apply id="S3.SS1.p2.7.m7.1.1.3.3.cmml" xref="S3.SS1.p2.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p2.7.m7.1.1.3.3.1.cmml" xref="S3.SS1.p2.7.m7.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p2.7.m7.1.1.3.3.2.cmml" xref="S3.SS1.p2.7.m7.1.1.3.3.2">𝑋</ci><ci id="S3.SS1.p2.7.m7.1.1.3.3.3.cmml" xref="S3.SS1.p2.7.m7.1.1.3.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.7.m7.1c">f^{*}:X_{n}\to X_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.7.m7.1d">italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_X start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. The induced maps <math alttext="(d^{i})^{*}:X_{n}\to X_{n-1}" class="ltx_Math" display="inline" id="S3.SS1.p2.8.m8.1"><semantics id="S3.SS1.p2.8.m8.1a"><mrow id="S3.SS1.p2.8.m8.1.1" xref="S3.SS1.p2.8.m8.1.1.cmml"><msup id="S3.SS1.p2.8.m8.1.1.1" xref="S3.SS1.p2.8.m8.1.1.1.cmml"><mrow id="S3.SS1.p2.8.m8.1.1.1.1.1" xref="S3.SS1.p2.8.m8.1.1.1.1.1.1.cmml"><mo id="S3.SS1.p2.8.m8.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p2.8.m8.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.p2.8.m8.1.1.1.1.1.1" xref="S3.SS1.p2.8.m8.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p2.8.m8.1.1.1.1.1.1.2" xref="S3.SS1.p2.8.m8.1.1.1.1.1.1.2.cmml">d</mi><mi id="S3.SS1.p2.8.m8.1.1.1.1.1.1.3" xref="S3.SS1.p2.8.m8.1.1.1.1.1.1.3.cmml">i</mi></msup><mo id="S3.SS1.p2.8.m8.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p2.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.p2.8.m8.1.1.1.3" xref="S3.SS1.p2.8.m8.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.p2.8.m8.1.1.2" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p2.8.m8.1.1.2.cmml">:</mo><mrow id="S3.SS1.p2.8.m8.1.1.3" xref="S3.SS1.p2.8.m8.1.1.3.cmml"><msub id="S3.SS1.p2.8.m8.1.1.3.2" xref="S3.SS1.p2.8.m8.1.1.3.2.cmml"><mi id="S3.SS1.p2.8.m8.1.1.3.2.2" xref="S3.SS1.p2.8.m8.1.1.3.2.2.cmml">X</mi><mi id="S3.SS1.p2.8.m8.1.1.3.2.3" xref="S3.SS1.p2.8.m8.1.1.3.2.3.cmml">n</mi></msub><mo id="S3.SS1.p2.8.m8.1.1.3.1" stretchy="false" xref="S3.SS1.p2.8.m8.1.1.3.1.cmml">→</mo><msub id="S3.SS1.p2.8.m8.1.1.3.3" xref="S3.SS1.p2.8.m8.1.1.3.3.cmml"><mi id="S3.SS1.p2.8.m8.1.1.3.3.2" xref="S3.SS1.p2.8.m8.1.1.3.3.2.cmml">X</mi><mrow id="S3.SS1.p2.8.m8.1.1.3.3.3" xref="S3.SS1.p2.8.m8.1.1.3.3.3.cmml"><mi id="S3.SS1.p2.8.m8.1.1.3.3.3.2" xref="S3.SS1.p2.8.m8.1.1.3.3.3.2.cmml">n</mi><mo id="S3.SS1.p2.8.m8.1.1.3.3.3.1" xref="S3.SS1.p2.8.m8.1.1.3.3.3.1.cmml">−</mo><mn id="S3.SS1.p2.8.m8.1.1.3.3.3.3" xref="S3.SS1.p2.8.m8.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.8.m8.1b"><apply id="S3.SS1.p2.8.m8.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1"><ci id="S3.SS1.p2.8.m8.1.1.2.cmml" xref="S3.SS1.p2.8.m8.1.1.2">:</ci><apply id="S3.SS1.p2.8.m8.1.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.8.m8.1.1.1.2.cmml" xref="S3.SS1.p2.8.m8.1.1.1">superscript</csymbol><apply id="S3.SS1.p2.8.m8.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.8.m8.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.p2.8.m8.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p2.8.m8.1.1.1.1.1.1.2">𝑑</ci><ci id="S3.SS1.p2.8.m8.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p2.8.m8.1.1.1.1.1.1.3">𝑖</ci></apply><times id="S3.SS1.p2.8.m8.1.1.1.3.cmml" xref="S3.SS1.p2.8.m8.1.1.1.3"></times></apply><apply id="S3.SS1.p2.8.m8.1.1.3.cmml" xref="S3.SS1.p2.8.m8.1.1.3"><ci id="S3.SS1.p2.8.m8.1.1.3.1.cmml" xref="S3.SS1.p2.8.m8.1.1.3.1">→</ci><apply id="S3.SS1.p2.8.m8.1.1.3.2.cmml" xref="S3.SS1.p2.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.8.m8.1.1.3.2.1.cmml" 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encoding="application/x-llamapun" id="S3.SS1.p2.8.m8.1d">( italic_d start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_X start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="(s^{i})^{*}:X_{n}\to X_{n+1}" class="ltx_Math" display="inline" id="S3.SS1.p2.9.m9.1"><semantics id="S3.SS1.p2.9.m9.1a"><mrow id="S3.SS1.p2.9.m9.1.1" xref="S3.SS1.p2.9.m9.1.1.cmml"><msup id="S3.SS1.p2.9.m9.1.1.1" xref="S3.SS1.p2.9.m9.1.1.1.cmml"><mrow id="S3.SS1.p2.9.m9.1.1.1.1.1" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.cmml"><mo id="S3.SS1.p2.9.m9.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.p2.9.m9.1.1.1.1.1.1" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p2.9.m9.1.1.1.1.1.1.2" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.2.cmml">s</mi><mi id="S3.SS1.p2.9.m9.1.1.1.1.1.1.3" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.3.cmml">i</mi></msup><mo id="S3.SS1.p2.9.m9.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.p2.9.m9.1.1.1.3" xref="S3.SS1.p2.9.m9.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.p2.9.m9.1.1.2" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p2.9.m9.1.1.2.cmml">:</mo><mrow id="S3.SS1.p2.9.m9.1.1.3" xref="S3.SS1.p2.9.m9.1.1.3.cmml"><msub id="S3.SS1.p2.9.m9.1.1.3.2" xref="S3.SS1.p2.9.m9.1.1.3.2.cmml"><mi id="S3.SS1.p2.9.m9.1.1.3.2.2" xref="S3.SS1.p2.9.m9.1.1.3.2.2.cmml">X</mi><mi id="S3.SS1.p2.9.m9.1.1.3.2.3" xref="S3.SS1.p2.9.m9.1.1.3.2.3.cmml">n</mi></msub><mo id="S3.SS1.p2.9.m9.1.1.3.1" stretchy="false" xref="S3.SS1.p2.9.m9.1.1.3.1.cmml">→</mo><msub id="S3.SS1.p2.9.m9.1.1.3.3" xref="S3.SS1.p2.9.m9.1.1.3.3.cmml"><mi id="S3.SS1.p2.9.m9.1.1.3.3.2" xref="S3.SS1.p2.9.m9.1.1.3.3.2.cmml">X</mi><mrow id="S3.SS1.p2.9.m9.1.1.3.3.3" xref="S3.SS1.p2.9.m9.1.1.3.3.3.cmml"><mi id="S3.SS1.p2.9.m9.1.1.3.3.3.2" xref="S3.SS1.p2.9.m9.1.1.3.3.3.2.cmml">n</mi><mo id="S3.SS1.p2.9.m9.1.1.3.3.3.1" xref="S3.SS1.p2.9.m9.1.1.3.3.3.1.cmml">+</mo><mn id="S3.SS1.p2.9.m9.1.1.3.3.3.3" xref="S3.SS1.p2.9.m9.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.9.m9.1b"><apply id="S3.SS1.p2.9.m9.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1"><ci id="S3.SS1.p2.9.m9.1.1.2.cmml" xref="S3.SS1.p2.9.m9.1.1.2">:</ci><apply id="S3.SS1.p2.9.m9.1.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.9.m9.1.1.1.2.cmml" xref="S3.SS1.p2.9.m9.1.1.1">superscript</csymbol><apply id="S3.SS1.p2.9.m9.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.p2.9.m9.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.2">𝑠</ci><ci id="S3.SS1.p2.9.m9.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.3">𝑖</ci></apply><times id="S3.SS1.p2.9.m9.1.1.1.3.cmml" xref="S3.SS1.p2.9.m9.1.1.1.3"></times></apply><apply id="S3.SS1.p2.9.m9.1.1.3.cmml" xref="S3.SS1.p2.9.m9.1.1.3"><ci id="S3.SS1.p2.9.m9.1.1.3.1.cmml" xref="S3.SS1.p2.9.m9.1.1.3.1">→</ci><apply id="S3.SS1.p2.9.m9.1.1.3.2.cmml" xref="S3.SS1.p2.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.9.m9.1.1.3.2.1.cmml" xref="S3.SS1.p2.9.m9.1.1.3.2">subscript</csymbol><ci id="S3.SS1.p2.9.m9.1.1.3.2.2.cmml" xref="S3.SS1.p2.9.m9.1.1.3.2.2">𝑋</ci><ci id="S3.SS1.p2.9.m9.1.1.3.2.3.cmml" xref="S3.SS1.p2.9.m9.1.1.3.2.3">𝑛</ci></apply><apply id="S3.SS1.p2.9.m9.1.1.3.3.cmml" xref="S3.SS1.p2.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p2.9.m9.1.1.3.3.1.cmml" xref="S3.SS1.p2.9.m9.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p2.9.m9.1.1.3.3.2.cmml" xref="S3.SS1.p2.9.m9.1.1.3.3.2">𝑋</ci><apply id="S3.SS1.p2.9.m9.1.1.3.3.3.cmml" xref="S3.SS1.p2.9.m9.1.1.3.3.3"><plus id="S3.SS1.p2.9.m9.1.1.3.3.3.1.cmml" xref="S3.SS1.p2.9.m9.1.1.3.3.3.1"></plus><ci id="S3.SS1.p2.9.m9.1.1.3.3.3.2.cmml" xref="S3.SS1.p2.9.m9.1.1.3.3.3.2">𝑛</ci><cn id="S3.SS1.p2.9.m9.1.1.3.3.3.3.cmml" type="integer" xref="S3.SS1.p2.9.m9.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.9.m9.1c">(s^{i})^{*}:X_{n}\to X_{n+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.9.m9.1d">( italic_s start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_X start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT</annotation></semantics></math> are denoted <math alttext="d_{i}" class="ltx_Math" display="inline" id="S3.SS1.p2.10.m10.1"><semantics id="S3.SS1.p2.10.m10.1a"><msub id="S3.SS1.p2.10.m10.1.1" xref="S3.SS1.p2.10.m10.1.1.cmml"><mi id="S3.SS1.p2.10.m10.1.1.2" xref="S3.SS1.p2.10.m10.1.1.2.cmml">d</mi><mi id="S3.SS1.p2.10.m10.1.1.3" xref="S3.SS1.p2.10.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.10.m10.1b"><apply id="S3.SS1.p2.10.m10.1.1.cmml" xref="S3.SS1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.10.m10.1.1.1.cmml" xref="S3.SS1.p2.10.m10.1.1">subscript</csymbol><ci id="S3.SS1.p2.10.m10.1.1.2.cmml" xref="S3.SS1.p2.10.m10.1.1.2">𝑑</ci><ci id="S3.SS1.p2.10.m10.1.1.3.cmml" xref="S3.SS1.p2.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.10.m10.1c">d_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.10.m10.1d">italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="s_{i}" class="ltx_Math" display="inline" id="S3.SS1.p2.11.m11.1"><semantics id="S3.SS1.p2.11.m11.1a"><msub id="S3.SS1.p2.11.m11.1.1" xref="S3.SS1.p2.11.m11.1.1.cmml"><mi id="S3.SS1.p2.11.m11.1.1.2" xref="S3.SS1.p2.11.m11.1.1.2.cmml">s</mi><mi id="S3.SS1.p2.11.m11.1.1.3" xref="S3.SS1.p2.11.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.11.m11.1b"><apply id="S3.SS1.p2.11.m11.1.1.cmml" xref="S3.SS1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.11.m11.1.1.1.cmml" xref="S3.SS1.p2.11.m11.1.1">subscript</csymbol><ci id="S3.SS1.p2.11.m11.1.1.2.cmml" xref="S3.SS1.p2.11.m11.1.1.2">𝑠</ci><ci id="S3.SS1.p2.11.m11.1.1.3.cmml" xref="S3.SS1.p2.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.11.m11.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.11.m11.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, respectively, and called the <em class="ltx_emph ltx_font_italic" id="S3.SS1.p2.11.2">boundary and degeneracy maps</em>.</p> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.3">A natural transformation <math alttext="X\to Y" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mrow id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml"><mi id="S3.SS1.p3.1.m1.1.1.2" xref="S3.SS1.p3.1.m1.1.1.2.cmml">X</mi><mo id="S3.SS1.p3.1.m1.1.1.1" stretchy="false" xref="S3.SS1.p3.1.m1.1.1.1.cmml">→</mo><mi id="S3.SS1.p3.1.m1.1.1.3" xref="S3.SS1.p3.1.m1.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><apply id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1"><ci id="S3.SS1.p3.1.m1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1.1">→</ci><ci id="S3.SS1.p3.1.m1.1.1.2.cmml" xref="S3.SS1.p3.1.m1.1.1.2">𝑋</ci><ci id="S3.SS1.p3.1.m1.1.1.3.cmml" xref="S3.SS1.p3.1.m1.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">X\to Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">italic_X → italic_Y</annotation></semantics></math> between two simplicial sets <math alttext="X,Y" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.2"><semantics id="S3.SS1.p3.2.m2.2a"><mrow id="S3.SS1.p3.2.m2.2.3.2" xref="S3.SS1.p3.2.m2.2.3.1.cmml"><mi id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml">X</mi><mo id="S3.SS1.p3.2.m2.2.3.2.1" xref="S3.SS1.p3.2.m2.2.3.1.cmml">,</mo><mi id="S3.SS1.p3.2.m2.2.2" xref="S3.SS1.p3.2.m2.2.2.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.2b"><list id="S3.SS1.p3.2.m2.2.3.1.cmml" xref="S3.SS1.p3.2.m2.2.3.2"><ci id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1">𝑋</ci><ci id="S3.SS1.p3.2.m2.2.2.cmml" xref="S3.SS1.p3.2.m2.2.2">𝑌</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.2c">X,Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.2d">italic_X , italic_Y</annotation></semantics></math> is called a <em class="ltx_emph ltx_font_italic" id="S3.SS1.p3.3.1">simplicial map</em>. The category of simplicial sets is denoted <math alttext="sSet" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><mrow id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml"><mi id="S3.SS1.p3.3.m3.1.1.2" xref="S3.SS1.p3.3.m3.1.1.2.cmml">s</mi><mo id="S3.SS1.p3.3.m3.1.1.1" xref="S3.SS1.p3.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p3.3.m3.1.1.3" xref="S3.SS1.p3.3.m3.1.1.3.cmml">S</mi><mo id="S3.SS1.p3.3.m3.1.1.1a" xref="S3.SS1.p3.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p3.3.m3.1.1.4" xref="S3.SS1.p3.3.m3.1.1.4.cmml">e</mi><mo id="S3.SS1.p3.3.m3.1.1.1b" xref="S3.SS1.p3.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p3.3.m3.1.1.5" xref="S3.SS1.p3.3.m3.1.1.5.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><apply id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1"><times id="S3.SS1.p3.3.m3.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1.1"></times><ci id="S3.SS1.p3.3.m3.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2">𝑠</ci><ci id="S3.SS1.p3.3.m3.1.1.3.cmml" xref="S3.SS1.p3.3.m3.1.1.3">𝑆</ci><ci id="S3.SS1.p3.3.m3.1.1.4.cmml" xref="S3.SS1.p3.3.m3.1.1.4">𝑒</ci><ci id="S3.SS1.p3.3.m3.1.1.5.cmml" xref="S3.SS1.p3.3.m3.1.1.5">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">sSet</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">italic_s italic_S italic_e italic_t</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.1.1.1">Definition 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.15">Given a simplicial set <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.m1.1"><semantics id="S3.Thmtheorem1.p1.1.m1.1a"><mi id="S3.Thmtheorem1.p1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.m1.1b"><ci id="S3.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.m1.1d">italic_X</annotation></semantics></math>, the <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem1.p1.2.1">category of simplices of <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.1.m1.1"><semantics id="S3.Thmtheorem1.p1.2.1.m1.1a"><mi id="S3.Thmtheorem1.p1.2.1.m1.1.1" xref="S3.Thmtheorem1.p1.2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.2.1.m1.1b"><ci id="S3.Thmtheorem1.p1.2.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.1.m1.1d">italic_X</annotation></semantics></math></em>, or the <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem1.p1.3.2">category of elements of <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.2.m1.1"><semantics id="S3.Thmtheorem1.p1.3.2.m1.1a"><mi id="S3.Thmtheorem1.p1.3.2.m1.1.1" xref="S3.Thmtheorem1.p1.3.2.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.2.m1.1b"><ci id="S3.Thmtheorem1.p1.3.2.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.3.2.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.2.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.2.m1.1d">italic_X</annotation></semantics></math></em>, is the category <math alttext="\Delta(X)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.4.m2.1"><semantics id="S3.Thmtheorem1.p1.4.m2.1a"><mrow id="S3.Thmtheorem1.p1.4.m2.1.2" xref="S3.Thmtheorem1.p1.4.m2.1.2.cmml"><mi id="S3.Thmtheorem1.p1.4.m2.1.2.2" mathvariant="normal" xref="S3.Thmtheorem1.p1.4.m2.1.2.2.cmml">Δ</mi><mo id="S3.Thmtheorem1.p1.4.m2.1.2.1" xref="S3.Thmtheorem1.p1.4.m2.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.4.m2.1.2.3.2" xref="S3.Thmtheorem1.p1.4.m2.1.2.cmml"><mo id="S3.Thmtheorem1.p1.4.m2.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.4.m2.1.2.cmml">(</mo><mi id="S3.Thmtheorem1.p1.4.m2.1.1" xref="S3.Thmtheorem1.p1.4.m2.1.1.cmml">X</mi><mo id="S3.Thmtheorem1.p1.4.m2.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.4.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.4.m2.1b"><apply id="S3.Thmtheorem1.p1.4.m2.1.2.cmml" xref="S3.Thmtheorem1.p1.4.m2.1.2"><times id="S3.Thmtheorem1.p1.4.m2.1.2.1.cmml" xref="S3.Thmtheorem1.p1.4.m2.1.2.1"></times><ci id="S3.Thmtheorem1.p1.4.m2.1.2.2.cmml" xref="S3.Thmtheorem1.p1.4.m2.1.2.2">Δ</ci><ci id="S3.Thmtheorem1.p1.4.m2.1.1.cmml" xref="S3.Thmtheorem1.p1.4.m2.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.4.m2.1c">\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.4.m2.1d">roman_Δ ( italic_X )</annotation></semantics></math> whose objects are the pairs <math alttext="([n],\sigma)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.5.m3.3"><semantics id="S3.Thmtheorem1.p1.5.m3.3a"><mrow id="S3.Thmtheorem1.p1.5.m3.3.3.1" xref="S3.Thmtheorem1.p1.5.m3.3.3.2.cmml"><mo id="S3.Thmtheorem1.p1.5.m3.3.3.1.2" stretchy="false" xref="S3.Thmtheorem1.p1.5.m3.3.3.2.cmml">(</mo><mrow id="S3.Thmtheorem1.p1.5.m3.3.3.1.1.2" xref="S3.Thmtheorem1.p1.5.m3.3.3.1.1.1.cmml"><mo id="S3.Thmtheorem1.p1.5.m3.3.3.1.1.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.5.m3.3.3.1.1.1.1.cmml">[</mo><mi id="S3.Thmtheorem1.p1.5.m3.1.1" xref="S3.Thmtheorem1.p1.5.m3.1.1.cmml">n</mi><mo id="S3.Thmtheorem1.p1.5.m3.3.3.1.1.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.5.m3.3.3.1.1.1.1.cmml">]</mo></mrow><mo id="S3.Thmtheorem1.p1.5.m3.3.3.1.3" xref="S3.Thmtheorem1.p1.5.m3.3.3.2.cmml">,</mo><mi id="S3.Thmtheorem1.p1.5.m3.2.2" xref="S3.Thmtheorem1.p1.5.m3.2.2.cmml">σ</mi><mo id="S3.Thmtheorem1.p1.5.m3.3.3.1.4" stretchy="false" xref="S3.Thmtheorem1.p1.5.m3.3.3.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.5.m3.3b"><interval closure="open" id="S3.Thmtheorem1.p1.5.m3.3.3.2.cmml" xref="S3.Thmtheorem1.p1.5.m3.3.3.1"><apply id="S3.Thmtheorem1.p1.5.m3.3.3.1.1.1.cmml" xref="S3.Thmtheorem1.p1.5.m3.3.3.1.1.2"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.5.m3.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.5.m3.3.3.1.1.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem1.p1.5.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.5.m3.1.1">𝑛</ci></apply><ci id="S3.Thmtheorem1.p1.5.m3.2.2.cmml" xref="S3.Thmtheorem1.p1.5.m3.2.2">𝜎</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.5.m3.3c">([n],\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.5.m3.3d">( [ italic_n ] , italic_σ )</annotation></semantics></math> where <math alttext="\sigma\in X_{n}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.6.m4.1"><semantics id="S3.Thmtheorem1.p1.6.m4.1a"><mrow id="S3.Thmtheorem1.p1.6.m4.1.1" xref="S3.Thmtheorem1.p1.6.m4.1.1.cmml"><mi id="S3.Thmtheorem1.p1.6.m4.1.1.2" xref="S3.Thmtheorem1.p1.6.m4.1.1.2.cmml">σ</mi><mo id="S3.Thmtheorem1.p1.6.m4.1.1.1" xref="S3.Thmtheorem1.p1.6.m4.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem1.p1.6.m4.1.1.3" xref="S3.Thmtheorem1.p1.6.m4.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.6.m4.1.1.3.2" xref="S3.Thmtheorem1.p1.6.m4.1.1.3.2.cmml">X</mi><mi id="S3.Thmtheorem1.p1.6.m4.1.1.3.3" xref="S3.Thmtheorem1.p1.6.m4.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.6.m4.1b"><apply id="S3.Thmtheorem1.p1.6.m4.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m4.1.1"><in id="S3.Thmtheorem1.p1.6.m4.1.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m4.1.1.1"></in><ci id="S3.Thmtheorem1.p1.6.m4.1.1.2.cmml" xref="S3.Thmtheorem1.p1.6.m4.1.1.2">𝜎</ci><apply id="S3.Thmtheorem1.p1.6.m4.1.1.3.cmml" xref="S3.Thmtheorem1.p1.6.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.6.m4.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.6.m4.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.6.m4.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.6.m4.1.1.3.2">𝑋</ci><ci id="S3.Thmtheorem1.p1.6.m4.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.6.m4.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.6.m4.1c">\sigma\in X_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.6.m4.1d">italic_σ ∈ italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, and the morphisms <math alttext="f:([m],\tau)\to([n],\sigma)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.7.m5.6"><semantics id="S3.Thmtheorem1.p1.7.m5.6a"><mrow id="S3.Thmtheorem1.p1.7.m5.6.6" xref="S3.Thmtheorem1.p1.7.m5.6.6.cmml"><mi id="S3.Thmtheorem1.p1.7.m5.6.6.4" xref="S3.Thmtheorem1.p1.7.m5.6.6.4.cmml">f</mi><mo id="S3.Thmtheorem1.p1.7.m5.6.6.3" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem1.p1.7.m5.6.6.3.cmml">:</mo><mrow id="S3.Thmtheorem1.p1.7.m5.6.6.2" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.cmml"><mrow id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.2.cmml"><mo id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.2.cmml">(</mo><mrow id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.2" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.1.1.cmml">[</mo><mi id="S3.Thmtheorem1.p1.7.m5.1.1" xref="S3.Thmtheorem1.p1.7.m5.1.1.cmml">m</mi><mo id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.3" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.2.cmml">,</mo><mi id="S3.Thmtheorem1.p1.7.m5.2.2" xref="S3.Thmtheorem1.p1.7.m5.2.2.cmml">τ</mi><mo id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.4" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.2.cmml">)</mo></mrow><mo id="S3.Thmtheorem1.p1.7.m5.6.6.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.3.cmml">→</mo><mrow id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.2.cmml"><mo id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.2" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.2.cmml">(</mo><mrow id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.2" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.1.1.cmml">[</mo><mi id="S3.Thmtheorem1.p1.7.m5.3.3" xref="S3.Thmtheorem1.p1.7.m5.3.3.cmml">n</mi><mo id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.1.1.cmml">]</mo></mrow><mo id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.3" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.2.cmml">,</mo><mi id="S3.Thmtheorem1.p1.7.m5.4.4" xref="S3.Thmtheorem1.p1.7.m5.4.4.cmml">σ</mi><mo id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.4" stretchy="false" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.7.m5.6b"><apply id="S3.Thmtheorem1.p1.7.m5.6.6.cmml" xref="S3.Thmtheorem1.p1.7.m5.6.6"><ci id="S3.Thmtheorem1.p1.7.m5.6.6.3.cmml" xref="S3.Thmtheorem1.p1.7.m5.6.6.3">:</ci><ci id="S3.Thmtheorem1.p1.7.m5.6.6.4.cmml" xref="S3.Thmtheorem1.p1.7.m5.6.6.4">𝑓</ci><apply id="S3.Thmtheorem1.p1.7.m5.6.6.2.cmml" xref="S3.Thmtheorem1.p1.7.m5.6.6.2"><ci id="S3.Thmtheorem1.p1.7.m5.6.6.2.3.cmml" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.3">→</ci><interval closure="open" id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.2.cmml" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1"><apply id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.2"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.7.m5.5.5.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem1.p1.7.m5.1.1.cmml" xref="S3.Thmtheorem1.p1.7.m5.1.1">𝑚</ci></apply><ci id="S3.Thmtheorem1.p1.7.m5.2.2.cmml" xref="S3.Thmtheorem1.p1.7.m5.2.2">𝜏</ci></interval><interval closure="open" id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.2.cmml" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1"><apply id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.1.cmml" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.2"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.7.m5.6.6.2.2.1.1.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem1.p1.7.m5.3.3.cmml" xref="S3.Thmtheorem1.p1.7.m5.3.3">𝑛</ci></apply><ci id="S3.Thmtheorem1.p1.7.m5.4.4.cmml" xref="S3.Thmtheorem1.p1.7.m5.4.4">𝜎</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.7.m5.6c">f:([m],\tau)\to([n],\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.7.m5.6d">italic_f : ( [ italic_m ] , italic_τ ) → ( [ italic_n ] , italic_σ )</annotation></semantics></math> in <math alttext="\Delta(X)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.8.m6.1"><semantics id="S3.Thmtheorem1.p1.8.m6.1a"><mrow id="S3.Thmtheorem1.p1.8.m6.1.2" xref="S3.Thmtheorem1.p1.8.m6.1.2.cmml"><mi id="S3.Thmtheorem1.p1.8.m6.1.2.2" mathvariant="normal" xref="S3.Thmtheorem1.p1.8.m6.1.2.2.cmml">Δ</mi><mo id="S3.Thmtheorem1.p1.8.m6.1.2.1" xref="S3.Thmtheorem1.p1.8.m6.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.8.m6.1.2.3.2" xref="S3.Thmtheorem1.p1.8.m6.1.2.cmml"><mo id="S3.Thmtheorem1.p1.8.m6.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.8.m6.1.2.cmml">(</mo><mi id="S3.Thmtheorem1.p1.8.m6.1.1" xref="S3.Thmtheorem1.p1.8.m6.1.1.cmml">X</mi><mo id="S3.Thmtheorem1.p1.8.m6.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.8.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.8.m6.1b"><apply id="S3.Thmtheorem1.p1.8.m6.1.2.cmml" xref="S3.Thmtheorem1.p1.8.m6.1.2"><times id="S3.Thmtheorem1.p1.8.m6.1.2.1.cmml" xref="S3.Thmtheorem1.p1.8.m6.1.2.1"></times><ci id="S3.Thmtheorem1.p1.8.m6.1.2.2.cmml" xref="S3.Thmtheorem1.p1.8.m6.1.2.2">Δ</ci><ci id="S3.Thmtheorem1.p1.8.m6.1.1.cmml" xref="S3.Thmtheorem1.p1.8.m6.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.8.m6.1c">\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.8.m6.1d">roman_Δ ( italic_X )</annotation></semantics></math> are given by the morphisms <math alttext="f:[m]\to[n]" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.9.m7.2"><semantics id="S3.Thmtheorem1.p1.9.m7.2a"><mrow id="S3.Thmtheorem1.p1.9.m7.2.3" xref="S3.Thmtheorem1.p1.9.m7.2.3.cmml"><mi id="S3.Thmtheorem1.p1.9.m7.2.3.2" xref="S3.Thmtheorem1.p1.9.m7.2.3.2.cmml">f</mi><mo id="S3.Thmtheorem1.p1.9.m7.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem1.p1.9.m7.2.3.1.cmml">:</mo><mrow id="S3.Thmtheorem1.p1.9.m7.2.3.3" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.cmml"><mrow id="S3.Thmtheorem1.p1.9.m7.2.3.3.2.2" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.2.1.cmml"><mo id="S3.Thmtheorem1.p1.9.m7.2.3.3.2.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.2.1.1.cmml">[</mo><mi id="S3.Thmtheorem1.p1.9.m7.1.1" xref="S3.Thmtheorem1.p1.9.m7.1.1.cmml">m</mi><mo id="S3.Thmtheorem1.p1.9.m7.2.3.3.2.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S3.Thmtheorem1.p1.9.m7.2.3.3.1" stretchy="false" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.1.cmml">→</mo><mrow id="S3.Thmtheorem1.p1.9.m7.2.3.3.3.2" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.3.1.cmml"><mo id="S3.Thmtheorem1.p1.9.m7.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.3.1.1.cmml">[</mo><mi id="S3.Thmtheorem1.p1.9.m7.2.2" xref="S3.Thmtheorem1.p1.9.m7.2.2.cmml">n</mi><mo id="S3.Thmtheorem1.p1.9.m7.2.3.3.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.9.m7.2b"><apply id="S3.Thmtheorem1.p1.9.m7.2.3.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3"><ci id="S3.Thmtheorem1.p1.9.m7.2.3.1.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3.1">:</ci><ci id="S3.Thmtheorem1.p1.9.m7.2.3.2.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3.2">𝑓</ci><apply id="S3.Thmtheorem1.p1.9.m7.2.3.3.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3.3"><ci id="S3.Thmtheorem1.p1.9.m7.2.3.3.1.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.1">→</ci><apply id="S3.Thmtheorem1.p1.9.m7.2.3.3.2.1.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.2.2"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.9.m7.2.3.3.2.1.1.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem1.p1.9.m7.1.1.cmml" xref="S3.Thmtheorem1.p1.9.m7.1.1">𝑚</ci></apply><apply id="S3.Thmtheorem1.p1.9.m7.2.3.3.3.1.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.3.2"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.9.m7.2.3.3.3.1.1.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.3.3.3.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem1.p1.9.m7.2.2.cmml" xref="S3.Thmtheorem1.p1.9.m7.2.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.9.m7.2c">f:[m]\to[n]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.9.m7.2d">italic_f : [ italic_m ] → [ italic_n ]</annotation></semantics></math> in <math alttext="\Delta" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.10.m8.1"><semantics id="S3.Thmtheorem1.p1.10.m8.1a"><mi id="S3.Thmtheorem1.p1.10.m8.1.1" mathvariant="normal" xref="S3.Thmtheorem1.p1.10.m8.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.10.m8.1b"><ci id="S3.Thmtheorem1.p1.10.m8.1.1.cmml" xref="S3.Thmtheorem1.p1.10.m8.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.10.m8.1c">\Delta</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.10.m8.1d">roman_Δ</annotation></semantics></math> such that <math alttext="f^{*}(\sigma)=\tau" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.11.m9.1"><semantics id="S3.Thmtheorem1.p1.11.m9.1a"><mrow id="S3.Thmtheorem1.p1.11.m9.1.2" xref="S3.Thmtheorem1.p1.11.m9.1.2.cmml"><mrow id="S3.Thmtheorem1.p1.11.m9.1.2.2" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.cmml"><msup id="S3.Thmtheorem1.p1.11.m9.1.2.2.2" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.2.cmml"><mi id="S3.Thmtheorem1.p1.11.m9.1.2.2.2.2" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.2.2.cmml">f</mi><mo id="S3.Thmtheorem1.p1.11.m9.1.2.2.2.3" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem1.p1.11.m9.1.2.2.1" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.11.m9.1.2.2.3.2" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.cmml"><mo id="S3.Thmtheorem1.p1.11.m9.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem1.p1.11.m9.1.1" xref="S3.Thmtheorem1.p1.11.m9.1.1.cmml">σ</mi><mo id="S3.Thmtheorem1.p1.11.m9.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem1.p1.11.m9.1.2.1" xref="S3.Thmtheorem1.p1.11.m9.1.2.1.cmml">=</mo><mi id="S3.Thmtheorem1.p1.11.m9.1.2.3" xref="S3.Thmtheorem1.p1.11.m9.1.2.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.11.m9.1b"><apply id="S3.Thmtheorem1.p1.11.m9.1.2.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2"><eq id="S3.Thmtheorem1.p1.11.m9.1.2.1.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2.1"></eq><apply id="S3.Thmtheorem1.p1.11.m9.1.2.2.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2.2"><times id="S3.Thmtheorem1.p1.11.m9.1.2.2.1.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.1"></times><apply id="S3.Thmtheorem1.p1.11.m9.1.2.2.2.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.11.m9.1.2.2.2.1.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.2">superscript</csymbol><ci id="S3.Thmtheorem1.p1.11.m9.1.2.2.2.2.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.2.2">𝑓</ci><times id="S3.Thmtheorem1.p1.11.m9.1.2.2.2.3.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2.2.2.3"></times></apply><ci id="S3.Thmtheorem1.p1.11.m9.1.1.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.1">𝜎</ci></apply><ci id="S3.Thmtheorem1.p1.11.m9.1.2.3.cmml" xref="S3.Thmtheorem1.p1.11.m9.1.2.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.11.m9.1c">f^{*}(\sigma)=\tau</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.11.m9.1d">italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ ) = italic_τ</annotation></semantics></math>. A functor <math alttext="\mathcal{M}:\Delta(X)\to R" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.12.m10.1"><semantics id="S3.Thmtheorem1.p1.12.m10.1a"><mrow id="S3.Thmtheorem1.p1.12.m10.1.2" xref="S3.Thmtheorem1.p1.12.m10.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem1.p1.12.m10.1.2.2" xref="S3.Thmtheorem1.p1.12.m10.1.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem1.p1.12.m10.1.2.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem1.p1.12.m10.1.2.1.cmml">:</mo><mrow id="S3.Thmtheorem1.p1.12.m10.1.2.3" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.cmml"><mrow id="S3.Thmtheorem1.p1.12.m10.1.2.3.2" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2.cmml"><mi id="S3.Thmtheorem1.p1.12.m10.1.2.3.2.2" mathvariant="normal" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2.2.cmml">Δ</mi><mo id="S3.Thmtheorem1.p1.12.m10.1.2.3.2.1" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.12.m10.1.2.3.2.3.2" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2.cmml"><mo id="S3.Thmtheorem1.p1.12.m10.1.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2.cmml">(</mo><mi id="S3.Thmtheorem1.p1.12.m10.1.1" xref="S3.Thmtheorem1.p1.12.m10.1.1.cmml">X</mi><mo id="S3.Thmtheorem1.p1.12.m10.1.2.3.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem1.p1.12.m10.1.2.3.1" stretchy="false" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.1.cmml">→</mo><mi id="S3.Thmtheorem1.p1.12.m10.1.2.3.3" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.12.m10.1b"><apply id="S3.Thmtheorem1.p1.12.m10.1.2.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2"><ci id="S3.Thmtheorem1.p1.12.m10.1.2.1.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2.1">:</ci><ci id="S3.Thmtheorem1.p1.12.m10.1.2.2.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2.2">ℳ</ci><apply id="S3.Thmtheorem1.p1.12.m10.1.2.3.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2.3"><ci id="S3.Thmtheorem1.p1.12.m10.1.2.3.1.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.1">→</ci><apply id="S3.Thmtheorem1.p1.12.m10.1.2.3.2.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2"><times id="S3.Thmtheorem1.p1.12.m10.1.2.3.2.1.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2.1"></times><ci id="S3.Thmtheorem1.p1.12.m10.1.2.3.2.2.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.2.2">Δ</ci><ci id="S3.Thmtheorem1.p1.12.m10.1.1.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.1">𝑋</ci></apply><ci id="S3.Thmtheorem1.p1.12.m10.1.2.3.3.cmml" xref="S3.Thmtheorem1.p1.12.m10.1.2.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.12.m10.1c">\mathcal{M}:\Delta(X)\to R</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.12.m10.1d">caligraphic_M : roman_Δ ( italic_X ) → italic_R</annotation></semantics></math>-Mod is called a <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem1.p1.15.3">coefficient system</em> <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.13.m11.1"><semantics id="S3.Thmtheorem1.p1.13.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem1.p1.13.m11.1.1" xref="S3.Thmtheorem1.p1.13.m11.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.13.m11.1b"><ci id="S3.Thmtheorem1.p1.13.m11.1.1.cmml" xref="S3.Thmtheorem1.p1.13.m11.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.13.m11.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.13.m11.1d">caligraphic_M</annotation></semantics></math> for the simplicial set <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.14.m12.1"><semantics id="S3.Thmtheorem1.p1.14.m12.1a"><mi id="S3.Thmtheorem1.p1.14.m12.1.1" xref="S3.Thmtheorem1.p1.14.m12.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.14.m12.1b"><ci id="S3.Thmtheorem1.p1.14.m12.1.1.cmml" xref="S3.Thmtheorem1.p1.14.m12.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.14.m12.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.14.m12.1d">italic_X</annotation></semantics></math> over the commutative ring <math alttext="R" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.15.m13.1"><semantics id="S3.Thmtheorem1.p1.15.m13.1a"><mi id="S3.Thmtheorem1.p1.15.m13.1.1" xref="S3.Thmtheorem1.p1.15.m13.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.15.m13.1b"><ci id="S3.Thmtheorem1.p1.15.m13.1.1.cmml" xref="S3.Thmtheorem1.p1.15.m13.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.15.m13.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.15.m13.1d">italic_R</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.7">We write <math alttext="\mathcal{M}(\sigma)" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.1"><semantics id="S3.SS1.p4.1.m1.1a"><mrow id="S3.SS1.p4.1.m1.1.2" xref="S3.SS1.p4.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.1.m1.1.2.2" xref="S3.SS1.p4.1.m1.1.2.2.cmml">ℳ</mi><mo id="S3.SS1.p4.1.m1.1.2.1" xref="S3.SS1.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p4.1.m1.1.2.3.2" xref="S3.SS1.p4.1.m1.1.2.cmml"><mo id="S3.SS1.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S3.SS1.p4.1.m1.1.2.cmml">(</mo><mi id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml">σ</mi><mo id="S3.SS1.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S3.SS1.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.1b"><apply id="S3.SS1.p4.1.m1.1.2.cmml" xref="S3.SS1.p4.1.m1.1.2"><times id="S3.SS1.p4.1.m1.1.2.1.cmml" xref="S3.SS1.p4.1.m1.1.2.1"></times><ci id="S3.SS1.p4.1.m1.1.2.2.cmml" xref="S3.SS1.p4.1.m1.1.2.2">ℳ</ci><ci id="S3.SS1.p4.1.m1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.1c">\mathcal{M}(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.1d">caligraphic_M ( italic_σ )</annotation></semantics></math> for <math alttext="\mathcal{M}([n],\sigma)" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.3"><semantics id="S3.SS1.p4.2.m2.3a"><mrow id="S3.SS1.p4.2.m2.3.3" xref="S3.SS1.p4.2.m2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.2.m2.3.3.3" xref="S3.SS1.p4.2.m2.3.3.3.cmml">ℳ</mi><mo id="S3.SS1.p4.2.m2.3.3.2" xref="S3.SS1.p4.2.m2.3.3.2.cmml">⁢</mo><mrow id="S3.SS1.p4.2.m2.3.3.1.1" xref="S3.SS1.p4.2.m2.3.3.1.2.cmml"><mo id="S3.SS1.p4.2.m2.3.3.1.1.2" stretchy="false" xref="S3.SS1.p4.2.m2.3.3.1.2.cmml">(</mo><mrow id="S3.SS1.p4.2.m2.3.3.1.1.1.2" xref="S3.SS1.p4.2.m2.3.3.1.1.1.1.cmml"><mo id="S3.SS1.p4.2.m2.3.3.1.1.1.2.1" stretchy="false" xref="S3.SS1.p4.2.m2.3.3.1.1.1.1.1.cmml">[</mo><mi id="S3.SS1.p4.2.m2.1.1" xref="S3.SS1.p4.2.m2.1.1.cmml">n</mi><mo id="S3.SS1.p4.2.m2.3.3.1.1.1.2.2" stretchy="false" xref="S3.SS1.p4.2.m2.3.3.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS1.p4.2.m2.3.3.1.1.3" xref="S3.SS1.p4.2.m2.3.3.1.2.cmml">,</mo><mi id="S3.SS1.p4.2.m2.2.2" xref="S3.SS1.p4.2.m2.2.2.cmml">σ</mi><mo id="S3.SS1.p4.2.m2.3.3.1.1.4" stretchy="false" xref="S3.SS1.p4.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.3b"><apply id="S3.SS1.p4.2.m2.3.3.cmml" xref="S3.SS1.p4.2.m2.3.3"><times id="S3.SS1.p4.2.m2.3.3.2.cmml" xref="S3.SS1.p4.2.m2.3.3.2"></times><ci id="S3.SS1.p4.2.m2.3.3.3.cmml" xref="S3.SS1.p4.2.m2.3.3.3">ℳ</ci><interval closure="open" id="S3.SS1.p4.2.m2.3.3.1.2.cmml" xref="S3.SS1.p4.2.m2.3.3.1.1"><apply id="S3.SS1.p4.2.m2.3.3.1.1.1.1.cmml" xref="S3.SS1.p4.2.m2.3.3.1.1.1.2"><csymbol cd="latexml" id="S3.SS1.p4.2.m2.3.3.1.1.1.1.1.cmml" xref="S3.SS1.p4.2.m2.3.3.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS1.p4.2.m2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1">𝑛</ci></apply><ci id="S3.SS1.p4.2.m2.2.2.cmml" xref="S3.SS1.p4.2.m2.2.2">𝜎</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.3c">\mathcal{M}([n],\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.3d">caligraphic_M ( [ italic_n ] , italic_σ )</annotation></semantics></math>, and for each morphism <math alttext="f:([m],\tau)\to([n],\sigma)" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m3.6"><semantics id="S3.SS1.p4.3.m3.6a"><mrow id="S3.SS1.p4.3.m3.6.6" xref="S3.SS1.p4.3.m3.6.6.cmml"><mi id="S3.SS1.p4.3.m3.6.6.4" xref="S3.SS1.p4.3.m3.6.6.4.cmml">f</mi><mo id="S3.SS1.p4.3.m3.6.6.3" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p4.3.m3.6.6.3.cmml">:</mo><mrow id="S3.SS1.p4.3.m3.6.6.2" xref="S3.SS1.p4.3.m3.6.6.2.cmml"><mrow id="S3.SS1.p4.3.m3.5.5.1.1.1" xref="S3.SS1.p4.3.m3.5.5.1.1.2.cmml"><mo id="S3.SS1.p4.3.m3.5.5.1.1.1.2" stretchy="false" xref="S3.SS1.p4.3.m3.5.5.1.1.2.cmml">(</mo><mrow id="S3.SS1.p4.3.m3.5.5.1.1.1.1.2" xref="S3.SS1.p4.3.m3.5.5.1.1.1.1.1.cmml"><mo id="S3.SS1.p4.3.m3.5.5.1.1.1.1.2.1" stretchy="false" xref="S3.SS1.p4.3.m3.5.5.1.1.1.1.1.1.cmml">[</mo><mi id="S3.SS1.p4.3.m3.1.1" xref="S3.SS1.p4.3.m3.1.1.cmml">m</mi><mo id="S3.SS1.p4.3.m3.5.5.1.1.1.1.2.2" stretchy="false" xref="S3.SS1.p4.3.m3.5.5.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS1.p4.3.m3.5.5.1.1.1.3" xref="S3.SS1.p4.3.m3.5.5.1.1.2.cmml">,</mo><mi id="S3.SS1.p4.3.m3.2.2" xref="S3.SS1.p4.3.m3.2.2.cmml">τ</mi><mo id="S3.SS1.p4.3.m3.5.5.1.1.1.4" stretchy="false" xref="S3.SS1.p4.3.m3.5.5.1.1.2.cmml">)</mo></mrow><mo id="S3.SS1.p4.3.m3.6.6.2.3" stretchy="false" xref="S3.SS1.p4.3.m3.6.6.2.3.cmml">→</mo><mrow id="S3.SS1.p4.3.m3.6.6.2.2.1" xref="S3.SS1.p4.3.m3.6.6.2.2.2.cmml"><mo id="S3.SS1.p4.3.m3.6.6.2.2.1.2" stretchy="false" xref="S3.SS1.p4.3.m3.6.6.2.2.2.cmml">(</mo><mrow id="S3.SS1.p4.3.m3.6.6.2.2.1.1.2" xref="S3.SS1.p4.3.m3.6.6.2.2.1.1.1.cmml"><mo id="S3.SS1.p4.3.m3.6.6.2.2.1.1.2.1" stretchy="false" xref="S3.SS1.p4.3.m3.6.6.2.2.1.1.1.1.cmml">[</mo><mi id="S3.SS1.p4.3.m3.3.3" xref="S3.SS1.p4.3.m3.3.3.cmml">n</mi><mo id="S3.SS1.p4.3.m3.6.6.2.2.1.1.2.2" stretchy="false" xref="S3.SS1.p4.3.m3.6.6.2.2.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS1.p4.3.m3.6.6.2.2.1.3" xref="S3.SS1.p4.3.m3.6.6.2.2.2.cmml">,</mo><mi id="S3.SS1.p4.3.m3.4.4" xref="S3.SS1.p4.3.m3.4.4.cmml">σ</mi><mo id="S3.SS1.p4.3.m3.6.6.2.2.1.4" stretchy="false" xref="S3.SS1.p4.3.m3.6.6.2.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m3.6b"><apply id="S3.SS1.p4.3.m3.6.6.cmml" xref="S3.SS1.p4.3.m3.6.6"><ci id="S3.SS1.p4.3.m3.6.6.3.cmml" xref="S3.SS1.p4.3.m3.6.6.3">:</ci><ci id="S3.SS1.p4.3.m3.6.6.4.cmml" xref="S3.SS1.p4.3.m3.6.6.4">𝑓</ci><apply id="S3.SS1.p4.3.m3.6.6.2.cmml" xref="S3.SS1.p4.3.m3.6.6.2"><ci id="S3.SS1.p4.3.m3.6.6.2.3.cmml" xref="S3.SS1.p4.3.m3.6.6.2.3">→</ci><interval closure="open" id="S3.SS1.p4.3.m3.5.5.1.1.2.cmml" xref="S3.SS1.p4.3.m3.5.5.1.1.1"><apply id="S3.SS1.p4.3.m3.5.5.1.1.1.1.1.cmml" xref="S3.SS1.p4.3.m3.5.5.1.1.1.1.2"><csymbol cd="latexml" id="S3.SS1.p4.3.m3.5.5.1.1.1.1.1.1.cmml" xref="S3.SS1.p4.3.m3.5.5.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS1.p4.3.m3.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1">𝑚</ci></apply><ci id="S3.SS1.p4.3.m3.2.2.cmml" xref="S3.SS1.p4.3.m3.2.2">𝜏</ci></interval><interval closure="open" id="S3.SS1.p4.3.m3.6.6.2.2.2.cmml" xref="S3.SS1.p4.3.m3.6.6.2.2.1"><apply id="S3.SS1.p4.3.m3.6.6.2.2.1.1.1.cmml" xref="S3.SS1.p4.3.m3.6.6.2.2.1.1.2"><csymbol cd="latexml" id="S3.SS1.p4.3.m3.6.6.2.2.1.1.1.1.cmml" xref="S3.SS1.p4.3.m3.6.6.2.2.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS1.p4.3.m3.3.3.cmml" xref="S3.SS1.p4.3.m3.3.3">𝑛</ci></apply><ci id="S3.SS1.p4.3.m3.4.4.cmml" xref="S3.SS1.p4.3.m3.4.4">𝜎</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m3.6c">f:([m],\tau)\to([n],\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m3.6d">italic_f : ( [ italic_m ] , italic_τ ) → ( [ italic_n ] , italic_σ )</annotation></semantics></math>, the induced map <math alttext="\mathcal{M}(f):\mathcal{M}(\tau)\to\mathcal{M}(\sigma)" class="ltx_Math" display="inline" id="S3.SS1.p4.4.m4.3"><semantics id="S3.SS1.p4.4.m4.3a"><mrow id="S3.SS1.p4.4.m4.3.4" xref="S3.SS1.p4.4.m4.3.4.cmml"><mrow id="S3.SS1.p4.4.m4.3.4.2" xref="S3.SS1.p4.4.m4.3.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.4.m4.3.4.2.2" xref="S3.SS1.p4.4.m4.3.4.2.2.cmml">ℳ</mi><mo id="S3.SS1.p4.4.m4.3.4.2.1" xref="S3.SS1.p4.4.m4.3.4.2.1.cmml">⁢</mo><mrow id="S3.SS1.p4.4.m4.3.4.2.3.2" xref="S3.SS1.p4.4.m4.3.4.2.cmml"><mo id="S3.SS1.p4.4.m4.3.4.2.3.2.1" stretchy="false" xref="S3.SS1.p4.4.m4.3.4.2.cmml">(</mo><mi id="S3.SS1.p4.4.m4.1.1" xref="S3.SS1.p4.4.m4.1.1.cmml">f</mi><mo id="S3.SS1.p4.4.m4.3.4.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.SS1.p4.4.m4.3.4.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p4.4.m4.3.4.1" rspace="0.278em" xref="S3.SS1.p4.4.m4.3.4.1.cmml">:</mo><mrow id="S3.SS1.p4.4.m4.3.4.3" xref="S3.SS1.p4.4.m4.3.4.3.cmml"><mrow id="S3.SS1.p4.4.m4.3.4.3.2" xref="S3.SS1.p4.4.m4.3.4.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.4.m4.3.4.3.2.2" xref="S3.SS1.p4.4.m4.3.4.3.2.2.cmml">ℳ</mi><mo id="S3.SS1.p4.4.m4.3.4.3.2.1" xref="S3.SS1.p4.4.m4.3.4.3.2.1.cmml">⁢</mo><mrow id="S3.SS1.p4.4.m4.3.4.3.2.3.2" xref="S3.SS1.p4.4.m4.3.4.3.2.cmml"><mo id="S3.SS1.p4.4.m4.3.4.3.2.3.2.1" stretchy="false" xref="S3.SS1.p4.4.m4.3.4.3.2.cmml">(</mo><mi id="S3.SS1.p4.4.m4.2.2" xref="S3.SS1.p4.4.m4.2.2.cmml">τ</mi><mo id="S3.SS1.p4.4.m4.3.4.3.2.3.2.2" stretchy="false" xref="S3.SS1.p4.4.m4.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p4.4.m4.3.4.3.1" stretchy="false" xref="S3.SS1.p4.4.m4.3.4.3.1.cmml">→</mo><mrow id="S3.SS1.p4.4.m4.3.4.3.3" xref="S3.SS1.p4.4.m4.3.4.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.4.m4.3.4.3.3.2" xref="S3.SS1.p4.4.m4.3.4.3.3.2.cmml">ℳ</mi><mo id="S3.SS1.p4.4.m4.3.4.3.3.1" xref="S3.SS1.p4.4.m4.3.4.3.3.1.cmml">⁢</mo><mrow id="S3.SS1.p4.4.m4.3.4.3.3.3.2" xref="S3.SS1.p4.4.m4.3.4.3.3.cmml"><mo id="S3.SS1.p4.4.m4.3.4.3.3.3.2.1" stretchy="false" xref="S3.SS1.p4.4.m4.3.4.3.3.cmml">(</mo><mi id="S3.SS1.p4.4.m4.3.3" xref="S3.SS1.p4.4.m4.3.3.cmml">σ</mi><mo id="S3.SS1.p4.4.m4.3.4.3.3.3.2.2" stretchy="false" xref="S3.SS1.p4.4.m4.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.4.m4.3b"><apply id="S3.SS1.p4.4.m4.3.4.cmml" xref="S3.SS1.p4.4.m4.3.4"><ci id="S3.SS1.p4.4.m4.3.4.1.cmml" xref="S3.SS1.p4.4.m4.3.4.1">:</ci><apply id="S3.SS1.p4.4.m4.3.4.2.cmml" xref="S3.SS1.p4.4.m4.3.4.2"><times id="S3.SS1.p4.4.m4.3.4.2.1.cmml" xref="S3.SS1.p4.4.m4.3.4.2.1"></times><ci id="S3.SS1.p4.4.m4.3.4.2.2.cmml" xref="S3.SS1.p4.4.m4.3.4.2.2">ℳ</ci><ci id="S3.SS1.p4.4.m4.1.1.cmml" xref="S3.SS1.p4.4.m4.1.1">𝑓</ci></apply><apply id="S3.SS1.p4.4.m4.3.4.3.cmml" xref="S3.SS1.p4.4.m4.3.4.3"><ci id="S3.SS1.p4.4.m4.3.4.3.1.cmml" xref="S3.SS1.p4.4.m4.3.4.3.1">→</ci><apply id="S3.SS1.p4.4.m4.3.4.3.2.cmml" xref="S3.SS1.p4.4.m4.3.4.3.2"><times id="S3.SS1.p4.4.m4.3.4.3.2.1.cmml" xref="S3.SS1.p4.4.m4.3.4.3.2.1"></times><ci id="S3.SS1.p4.4.m4.3.4.3.2.2.cmml" xref="S3.SS1.p4.4.m4.3.4.3.2.2">ℳ</ci><ci id="S3.SS1.p4.4.m4.2.2.cmml" xref="S3.SS1.p4.4.m4.2.2">𝜏</ci></apply><apply id="S3.SS1.p4.4.m4.3.4.3.3.cmml" xref="S3.SS1.p4.4.m4.3.4.3.3"><times id="S3.SS1.p4.4.m4.3.4.3.3.1.cmml" xref="S3.SS1.p4.4.m4.3.4.3.3.1"></times><ci id="S3.SS1.p4.4.m4.3.4.3.3.2.cmml" xref="S3.SS1.p4.4.m4.3.4.3.3.2">ℳ</ci><ci id="S3.SS1.p4.4.m4.3.3.cmml" xref="S3.SS1.p4.4.m4.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.4.m4.3c">\mathcal{M}(f):\mathcal{M}(\tau)\to\mathcal{M}(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.4.m4.3d">caligraphic_M ( italic_f ) : caligraphic_M ( italic_τ ) → caligraphic_M ( italic_σ )</annotation></semantics></math> is denoted <math alttext="f_{*}" class="ltx_Math" display="inline" id="S3.SS1.p4.5.m5.1"><semantics id="S3.SS1.p4.5.m5.1a"><msub id="S3.SS1.p4.5.m5.1.1" xref="S3.SS1.p4.5.m5.1.1.cmml"><mi id="S3.SS1.p4.5.m5.1.1.2" xref="S3.SS1.p4.5.m5.1.1.2.cmml">f</mi><mo id="S3.SS1.p4.5.m5.1.1.3" xref="S3.SS1.p4.5.m5.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.5.m5.1b"><apply id="S3.SS1.p4.5.m5.1.1.cmml" xref="S3.SS1.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.p4.5.m5.1.1.1.cmml" xref="S3.SS1.p4.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.p4.5.m5.1.1.2.cmml" xref="S3.SS1.p4.5.m5.1.1.2">𝑓</ci><times id="S3.SS1.p4.5.m5.1.1.3.cmml" xref="S3.SS1.p4.5.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.5.m5.1c">f_{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.5.m5.1d">italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>. In particular, for each <math alttext="i" class="ltx_Math" display="inline" id="S3.SS1.p4.6.m6.1"><semantics id="S3.SS1.p4.6.m6.1a"><mi id="S3.SS1.p4.6.m6.1.1" xref="S3.SS1.p4.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.6.m6.1b"><ci id="S3.SS1.p4.6.m6.1.1.cmml" xref="S3.SS1.p4.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.6.m6.1d">italic_i</annotation></semantics></math>, the coboundary and codegeneracy maps induce <math alttext="R" class="ltx_Math" display="inline" id="S3.SS1.p4.7.m7.1"><semantics id="S3.SS1.p4.7.m7.1a"><mi id="S3.SS1.p4.7.m7.1.1" xref="S3.SS1.p4.7.m7.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.7.m7.1b"><ci id="S3.SS1.p4.7.m7.1.1.cmml" xref="S3.SS1.p4.7.m7.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.7.m7.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.7.m7.1d">italic_R</annotation></semantics></math>-module homomorphisms</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="d^{i}_{*}:\mathcal{M}(d_{i}\sigma)\to\mathcal{M}(\sigma)\quad\text{ and }\quad s% ^{i}_{*}:\mathcal{M}(s_{i}\sigma)\to\mathcal{M}(\sigma)." class="ltx_Math" display="block" id="S3.Ex28.m1.4"><semantics id="S3.Ex28.m1.4a"><mrow id="S3.Ex28.m1.4.4.1" xref="S3.Ex28.m1.4.4.1.1.cmml"><mrow id="S3.Ex28.m1.4.4.1.1" xref="S3.Ex28.m1.4.4.1.1.cmml"><msubsup id="S3.Ex28.m1.4.4.1.1.6" xref="S3.Ex28.m1.4.4.1.1.6.cmml"><mi id="S3.Ex28.m1.4.4.1.1.6.2.2" xref="S3.Ex28.m1.4.4.1.1.6.2.2.cmml">d</mi><mo id="S3.Ex28.m1.4.4.1.1.6.3" xref="S3.Ex28.m1.4.4.1.1.6.3.cmml">∗</mo><mi id="S3.Ex28.m1.4.4.1.1.6.2.3" xref="S3.Ex28.m1.4.4.1.1.6.2.3.cmml">i</mi></msubsup><mo id="S3.Ex28.m1.4.4.1.1.7" lspace="0.278em" rspace="0.278em" xref="S3.Ex28.m1.4.4.1.1.7.cmml">:</mo><mrow id="S3.Ex28.m1.4.4.1.1.3" xref="S3.Ex28.m1.4.4.1.1.3.cmml"><mrow id="S3.Ex28.m1.4.4.1.1.1.1" xref="S3.Ex28.m1.4.4.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex28.m1.4.4.1.1.1.1.3" xref="S3.Ex28.m1.4.4.1.1.1.1.3.cmml">ℳ</mi><mo id="S3.Ex28.m1.4.4.1.1.1.1.2" xref="S3.Ex28.m1.4.4.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex28.m1.4.4.1.1.1.1.1.1" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex28.m1.4.4.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex28.m1.4.4.1.1.1.1.1.1.1" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.cmml"><msub id="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.2" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.2.2" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.2.2.cmml">d</mi><mi id="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.2.3" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.1" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.3" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.3.cmml">σ</mi></mrow><mo id="S3.Ex28.m1.4.4.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex28.m1.4.4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex28.m1.4.4.1.1.3.4" stretchy="false" xref="S3.Ex28.m1.4.4.1.1.3.4.cmml">→</mo><mrow id="S3.Ex28.m1.4.4.1.1.3.3.2" xref="S3.Ex28.m1.4.4.1.1.3.3.3.cmml"><mrow id="S3.Ex28.m1.4.4.1.1.2.2.1.1" xref="S3.Ex28.m1.4.4.1.1.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex28.m1.4.4.1.1.2.2.1.1.2" xref="S3.Ex28.m1.4.4.1.1.2.2.1.1.2.cmml">ℳ</mi><mo id="S3.Ex28.m1.4.4.1.1.2.2.1.1.1" xref="S3.Ex28.m1.4.4.1.1.2.2.1.1.1.cmml">⁢</mo><mrow id="S3.Ex28.m1.4.4.1.1.2.2.1.1.3.2" xref="S3.Ex28.m1.4.4.1.1.2.2.1.1.cmml"><mo id="S3.Ex28.m1.4.4.1.1.2.2.1.1.3.2.1" stretchy="false" xref="S3.Ex28.m1.4.4.1.1.2.2.1.1.cmml">(</mo><mi id="S3.Ex28.m1.1.1" xref="S3.Ex28.m1.1.1.cmml">σ</mi><mo id="S3.Ex28.m1.4.4.1.1.2.2.1.1.3.2.2" stretchy="false" xref="S3.Ex28.m1.4.4.1.1.2.2.1.1.cmml">)</mo></mrow></mrow><mspace id="S3.Ex28.m1.4.4.1.1.3.3.2.3" width="1em" xref="S3.Ex28.m1.4.4.1.1.3.3.3.cmml"></mspace><mtext id="S3.Ex28.m1.2.2" xref="S3.Ex28.m1.2.2a.cmml"> and </mtext><mspace id="S3.Ex28.m1.4.4.1.1.3.3.2.4" width="1em" xref="S3.Ex28.m1.4.4.1.1.3.3.3.cmml"></mspace><msubsup id="S3.Ex28.m1.4.4.1.1.3.3.2.2" xref="S3.Ex28.m1.4.4.1.1.3.3.2.2.cmml"><mi id="S3.Ex28.m1.4.4.1.1.3.3.2.2.2.2" xref="S3.Ex28.m1.4.4.1.1.3.3.2.2.2.2.cmml">s</mi><mo id="S3.Ex28.m1.4.4.1.1.3.3.2.2.3" xref="S3.Ex28.m1.4.4.1.1.3.3.2.2.3.cmml">∗</mo><mi id="S3.Ex28.m1.4.4.1.1.3.3.2.2.2.3" xref="S3.Ex28.m1.4.4.1.1.3.3.2.2.2.3.cmml">i</mi></msubsup></mrow></mrow><mo id="S3.Ex28.m1.4.4.1.1.8" lspace="0.278em" rspace="0.278em" xref="S3.Ex28.m1.4.4.1.1.8.cmml">:</mo><mrow id="S3.Ex28.m1.4.4.1.1.4" xref="S3.Ex28.m1.4.4.1.1.4.cmml"><mrow id="S3.Ex28.m1.4.4.1.1.4.1" xref="S3.Ex28.m1.4.4.1.1.4.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex28.m1.4.4.1.1.4.1.3" xref="S3.Ex28.m1.4.4.1.1.4.1.3.cmml">ℳ</mi><mo id="S3.Ex28.m1.4.4.1.1.4.1.2" xref="S3.Ex28.m1.4.4.1.1.4.1.2.cmml">⁢</mo><mrow id="S3.Ex28.m1.4.4.1.1.4.1.1.1" xref="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.cmml"><mo id="S3.Ex28.m1.4.4.1.1.4.1.1.1.2" stretchy="false" xref="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.cmml">(</mo><mrow id="S3.Ex28.m1.4.4.1.1.4.1.1.1.1" xref="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.cmml"><msub id="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.2" xref="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.2.cmml"><mi id="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.2.2" xref="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.2.2.cmml">s</mi><mi id="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.2.3" xref="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.1" xref="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.3" xref="S3.Ex28.m1.4.4.1.1.4.1.1.1.1.3.cmml">σ</mi></mrow><mo 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^{i}_{*}:\mathcal{M}(s_{i}\sigma)\to\mathcal{M}(\sigma).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex28.m1.4d">italic_d start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : caligraphic_M ( italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ ) → caligraphic_M ( italic_σ ) and italic_s start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : caligraphic_M ( italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ ) → caligraphic_M ( italic_σ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p4.10">A coefficient system is called a <em class="ltx_emph ltx_font_italic" id="S3.SS1.p4.10.1">local coefficient system</em> if for every morphism <math alttext="f:([m],\tau)\to([n],\sigma)" class="ltx_Math" display="inline" id="S3.SS1.p4.8.m1.6"><semantics id="S3.SS1.p4.8.m1.6a"><mrow 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xref="S3.SS1.p4.9.m2.1.2.cmml"><mo id="S3.SS1.p4.9.m2.1.2.3.2.1" stretchy="false" xref="S3.SS1.p4.9.m2.1.2.cmml">(</mo><mi id="S3.SS1.p4.9.m2.1.1" xref="S3.SS1.p4.9.m2.1.1.cmml">X</mi><mo id="S3.SS1.p4.9.m2.1.2.3.2.2" stretchy="false" xref="S3.SS1.p4.9.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.9.m2.1b"><apply id="S3.SS1.p4.9.m2.1.2.cmml" xref="S3.SS1.p4.9.m2.1.2"><times id="S3.SS1.p4.9.m2.1.2.1.cmml" xref="S3.SS1.p4.9.m2.1.2.1"></times><ci id="S3.SS1.p4.9.m2.1.2.2.cmml" xref="S3.SS1.p4.9.m2.1.2.2">Δ</ci><ci id="S3.SS1.p4.9.m2.1.1.cmml" xref="S3.SS1.p4.9.m2.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.9.m2.1c">\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.9.m2.1d">roman_Δ ( italic_X )</annotation></semantics></math>, the induced homomorphism <math alttext="f_{*}:\mathcal{M}(\tau)\to\mathcal{M}(\sigma)" class="ltx_Math" display="inline" id="S3.SS1.p4.10.m3.2"><semantics id="S3.SS1.p4.10.m3.2a"><mrow id="S3.SS1.p4.10.m3.2.3" xref="S3.SS1.p4.10.m3.2.3.cmml"><msub id="S3.SS1.p4.10.m3.2.3.2" xref="S3.SS1.p4.10.m3.2.3.2.cmml"><mi id="S3.SS1.p4.10.m3.2.3.2.2" xref="S3.SS1.p4.10.m3.2.3.2.2.cmml">f</mi><mo id="S3.SS1.p4.10.m3.2.3.2.3" xref="S3.SS1.p4.10.m3.2.3.2.3.cmml">∗</mo></msub><mo id="S3.SS1.p4.10.m3.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p4.10.m3.2.3.1.cmml">:</mo><mrow id="S3.SS1.p4.10.m3.2.3.3" xref="S3.SS1.p4.10.m3.2.3.3.cmml"><mrow id="S3.SS1.p4.10.m3.2.3.3.2" xref="S3.SS1.p4.10.m3.2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.10.m3.2.3.3.2.2" xref="S3.SS1.p4.10.m3.2.3.3.2.2.cmml">ℳ</mi><mo id="S3.SS1.p4.10.m3.2.3.3.2.1" xref="S3.SS1.p4.10.m3.2.3.3.2.1.cmml">⁢</mo><mrow id="S3.SS1.p4.10.m3.2.3.3.2.3.2" xref="S3.SS1.p4.10.m3.2.3.3.2.cmml"><mo id="S3.SS1.p4.10.m3.2.3.3.2.3.2.1" stretchy="false" xref="S3.SS1.p4.10.m3.2.3.3.2.cmml">(</mo><mi id="S3.SS1.p4.10.m3.1.1" xref="S3.SS1.p4.10.m3.1.1.cmml">τ</mi><mo id="S3.SS1.p4.10.m3.2.3.3.2.3.2.2" stretchy="false" xref="S3.SS1.p4.10.m3.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p4.10.m3.2.3.3.1" stretchy="false" xref="S3.SS1.p4.10.m3.2.3.3.1.cmml">→</mo><mrow id="S3.SS1.p4.10.m3.2.3.3.3" xref="S3.SS1.p4.10.m3.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.10.m3.2.3.3.3.2" xref="S3.SS1.p4.10.m3.2.3.3.3.2.cmml">ℳ</mi><mo id="S3.SS1.p4.10.m3.2.3.3.3.1" xref="S3.SS1.p4.10.m3.2.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS1.p4.10.m3.2.3.3.3.3.2" xref="S3.SS1.p4.10.m3.2.3.3.3.cmml"><mo id="S3.SS1.p4.10.m3.2.3.3.3.3.2.1" stretchy="false" xref="S3.SS1.p4.10.m3.2.3.3.3.cmml">(</mo><mi id="S3.SS1.p4.10.m3.2.2" xref="S3.SS1.p4.10.m3.2.2.cmml">σ</mi><mo id="S3.SS1.p4.10.m3.2.3.3.3.3.2.2" stretchy="false" xref="S3.SS1.p4.10.m3.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.10.m3.2b"><apply id="S3.SS1.p4.10.m3.2.3.cmml" xref="S3.SS1.p4.10.m3.2.3"><ci id="S3.SS1.p4.10.m3.2.3.1.cmml" xref="S3.SS1.p4.10.m3.2.3.1">:</ci><apply id="S3.SS1.p4.10.m3.2.3.2.cmml" xref="S3.SS1.p4.10.m3.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.p4.10.m3.2.3.2.1.cmml" xref="S3.SS1.p4.10.m3.2.3.2">subscript</csymbol><ci id="S3.SS1.p4.10.m3.2.3.2.2.cmml" xref="S3.SS1.p4.10.m3.2.3.2.2">𝑓</ci><times id="S3.SS1.p4.10.m3.2.3.2.3.cmml" xref="S3.SS1.p4.10.m3.2.3.2.3"></times></apply><apply id="S3.SS1.p4.10.m3.2.3.3.cmml" xref="S3.SS1.p4.10.m3.2.3.3"><ci id="S3.SS1.p4.10.m3.2.3.3.1.cmml" xref="S3.SS1.p4.10.m3.2.3.3.1">→</ci><apply id="S3.SS1.p4.10.m3.2.3.3.2.cmml" xref="S3.SS1.p4.10.m3.2.3.3.2"><times id="S3.SS1.p4.10.m3.2.3.3.2.1.cmml" xref="S3.SS1.p4.10.m3.2.3.3.2.1"></times><ci id="S3.SS1.p4.10.m3.2.3.3.2.2.cmml" xref="S3.SS1.p4.10.m3.2.3.3.2.2">ℳ</ci><ci id="S3.SS1.p4.10.m3.1.1.cmml" xref="S3.SS1.p4.10.m3.1.1">𝜏</ci></apply><apply id="S3.SS1.p4.10.m3.2.3.3.3.cmml" xref="S3.SS1.p4.10.m3.2.3.3.3"><times id="S3.SS1.p4.10.m3.2.3.3.3.1.cmml" xref="S3.SS1.p4.10.m3.2.3.3.3.1"></times><ci id="S3.SS1.p4.10.m3.2.3.3.3.2.cmml" xref="S3.SS1.p4.10.m3.2.3.3.3.2">ℳ</ci><ci id="S3.SS1.p4.10.m3.2.2.cmml" xref="S3.SS1.p4.10.m3.2.2">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.10.m3.2c">f_{*}:\mathcal{M}(\tau)\to\mathcal{M}(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.10.m3.2d">italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : caligraphic_M ( italic_τ ) → caligraphic_M ( italic_σ )</annotation></semantics></math> is an isomorphism.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2. </span>Cohomology of simplicial sets with general coefficients</h3> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.1.1.1">Definition 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.5">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.m1.1a"><mi id="S3.Thmtheorem2.p1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.m1.1b"><ci id="S3.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.m1.1d">italic_X</annotation></semantics></math> be a simplicial set and <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem2.p1.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.m2.1b"><ci id="S3.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.m2.1d">caligraphic_M</annotation></semantics></math> a coefficient system for <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.m3.1"><semantics id="S3.Thmtheorem2.p1.3.m3.1a"><mi id="S3.Thmtheorem2.p1.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.m3.1b"><ci id="S3.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.m3.1d">italic_X</annotation></semantics></math>. Consider the cochain complex <math alttext="C^{*}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.m4.2"><semantics id="S3.Thmtheorem2.p1.4.m4.2a"><mrow id="S3.Thmtheorem2.p1.4.m4.2.3" xref="S3.Thmtheorem2.p1.4.m4.2.3.cmml"><msup id="S3.Thmtheorem2.p1.4.m4.2.3.2" xref="S3.Thmtheorem2.p1.4.m4.2.3.2.cmml"><mi id="S3.Thmtheorem2.p1.4.m4.2.3.2.2" xref="S3.Thmtheorem2.p1.4.m4.2.3.2.2.cmml">C</mi><mo id="S3.Thmtheorem2.p1.4.m4.2.3.2.3" xref="S3.Thmtheorem2.p1.4.m4.2.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem2.p1.4.m4.2.3.1" xref="S3.Thmtheorem2.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.4.m4.2.3.3.2" xref="S3.Thmtheorem2.p1.4.m4.2.3.3.1.cmml"><mo id="S3.Thmtheorem2.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.4.m4.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem2.p1.4.m4.1.1" xref="S3.Thmtheorem2.p1.4.m4.1.1.cmml">X</mi><mo id="S3.Thmtheorem2.p1.4.m4.2.3.3.2.2" xref="S3.Thmtheorem2.p1.4.m4.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem2.p1.4.m4.2.2" xref="S3.Thmtheorem2.p1.4.m4.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem2.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.4.m4.2b"><apply id="S3.Thmtheorem2.p1.4.m4.2.3.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.3"><times id="S3.Thmtheorem2.p1.4.m4.2.3.1.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.3.1"></times><apply id="S3.Thmtheorem2.p1.4.m4.2.3.2.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.4.m4.2.3.2.1.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.4.m4.2.3.2.2.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.3.2.2">𝐶</ci><times id="S3.Thmtheorem2.p1.4.m4.2.3.2.3.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.3.2.3"></times></apply><list id="S3.Thmtheorem2.p1.4.m4.2.3.3.1.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.3.3.2"><ci id="S3.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p1.4.m4.1.1">𝑋</ci><ci id="S3.Thmtheorem2.p1.4.m4.2.2.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.m4.2c">C^{*}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.m4.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math> such that for each <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.5.m5.1"><semantics id="S3.Thmtheorem2.p1.5.m5.1a"><mrow id="S3.Thmtheorem2.p1.5.m5.1.1" xref="S3.Thmtheorem2.p1.5.m5.1.1.cmml"><mi id="S3.Thmtheorem2.p1.5.m5.1.1.2" xref="S3.Thmtheorem2.p1.5.m5.1.1.2.cmml">n</mi><mo id="S3.Thmtheorem2.p1.5.m5.1.1.1" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.cmml">≥</mo><mn id="S3.Thmtheorem2.p1.5.m5.1.1.3" xref="S3.Thmtheorem2.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.5.m5.1b"><apply id="S3.Thmtheorem2.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p1.5.m5.1.1"><geq id="S3.Thmtheorem2.p1.5.m5.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.m5.1.1.1"></geq><ci id="S3.Thmtheorem2.p1.5.m5.1.1.2.cmml" xref="S3.Thmtheorem2.p1.5.m5.1.1.2">𝑛</ci><cn id="S3.Thmtheorem2.p1.5.m5.1.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.m5.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.m5.1d">italic_n ≥ 0</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex29"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C^{n}(X;\mathcal{M})=\{f:X_{n}\to\prod_{\sigma\in X_{n}}\mathcal{M}(\sigma)\,|% \,f(\sigma)\in\mathcal{M}(\sigma)\}" class="ltx_Math" 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id="S3.Ex29.m1.7.7.1.1.1.5" xref="S3.Ex29.m1.7.7.1.1.1.5.cmml">∈</mo><mrow id="S3.Ex29.m1.7.7.1.1.1.6" xref="S3.Ex29.m1.7.7.1.1.1.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex29.m1.7.7.1.1.1.6.2" xref="S3.Ex29.m1.7.7.1.1.1.6.2.cmml">ℳ</mi><mo id="S3.Ex29.m1.7.7.1.1.1.6.1" xref="S3.Ex29.m1.7.7.1.1.1.6.1.cmml">⁢</mo><mrow id="S3.Ex29.m1.7.7.1.1.1.6.3.2" xref="S3.Ex29.m1.7.7.1.1.1.6.cmml"><mo id="S3.Ex29.m1.7.7.1.1.1.6.3.2.1" stretchy="false" xref="S3.Ex29.m1.7.7.1.1.1.6.cmml">(</mo><mi id="S3.Ex29.m1.5.5" xref="S3.Ex29.m1.5.5.cmml">σ</mi><mo id="S3.Ex29.m1.7.7.1.1.1.6.3.2.2" stretchy="false" xref="S3.Ex29.m1.7.7.1.1.1.6.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex29.m1.7.7.1.1.4" stretchy="false" xref="S3.Ex29.m1.7.7.1.2.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex29.m1.7b"><apply id="S3.Ex29.m1.7.7.cmml" xref="S3.Ex29.m1.7.7"><eq id="S3.Ex29.m1.7.7.2.cmml" xref="S3.Ex29.m1.7.7.2"></eq><apply id="S3.Ex29.m1.7.7.3.cmml" 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xref="S3.Ex29.m1.7.7.1.1.1.4.2.1.2">product</csymbol><apply id="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.1.3"><in id="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.1.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.1"></in><ci id="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.2.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.2">𝜎</ci><apply id="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.3.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.3"><csymbol cd="ambiguous" id="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.3.1.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.3">subscript</csymbol><ci id="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.3.2.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.3.2">𝑋</ci><ci id="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.3.3.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.1.3.3.3">𝑛</ci></apply></apply></apply><apply id="S3.Ex29.m1.7.7.1.1.1.4.2.2.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.2"><times id="S3.Ex29.m1.7.7.1.1.1.4.2.2.1.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.2.1"></times><ci id="S3.Ex29.m1.7.7.1.1.1.4.2.2.2.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.2.2.2">ℳ</ci><ci id="S3.Ex29.m1.3.3.cmml" xref="S3.Ex29.m1.3.3">𝜎</ci></apply></apply><apply id="S3.Ex29.m1.7.7.1.1.1.4.3.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.3"><times id="S3.Ex29.m1.7.7.1.1.1.4.3.1.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.3.1"></times><ci id="S3.Ex29.m1.7.7.1.1.1.4.3.2.cmml" xref="S3.Ex29.m1.7.7.1.1.1.4.3.2">𝑓</ci><ci id="S3.Ex29.m1.4.4.cmml" xref="S3.Ex29.m1.4.4">𝜎</ci></apply></apply></apply><apply id="S3.Ex29.m1.7.7.1.1.1c.cmml" xref="S3.Ex29.m1.7.7.1.1.1"><in id="S3.Ex29.m1.7.7.1.1.1.5.cmml" xref="S3.Ex29.m1.7.7.1.1.1.5"></in><share href="https://arxiv.org/html/2503.14659v1#S3.Ex29.m1.7.7.1.1.1.4.cmml" id="S3.Ex29.m1.7.7.1.1.1d.cmml" xref="S3.Ex29.m1.7.7.1.1.1"></share><apply id="S3.Ex29.m1.7.7.1.1.1.6.cmml" xref="S3.Ex29.m1.7.7.1.1.1.6"><times id="S3.Ex29.m1.7.7.1.1.1.6.1.cmml" xref="S3.Ex29.m1.7.7.1.1.1.6.1"></times><ci id="S3.Ex29.m1.7.7.1.1.1.6.2.cmml" xref="S3.Ex29.m1.7.7.1.1.1.6.2">ℳ</ci><ci id="S3.Ex29.m1.5.5.cmml" xref="S3.Ex29.m1.5.5">𝜎</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex29.m1.7c">C^{n}(X;\mathcal{M})=\{f:X_{n}\to\prod_{\sigma\in X_{n}}\mathcal{M}(\sigma)\,|% \,f(\sigma)\in\mathcal{M}(\sigma)\}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex29.m1.7d">italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M ) = { italic_f : italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → ∏ start_POSTSUBSCRIPT italic_σ ∈ italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_M ( italic_σ ) | italic_f ( italic_σ ) ∈ caligraphic_M ( italic_σ ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem2.p1.6">where the coboundary map <math alttext="\delta^{n-1}:C^{n-1}(X;\mathcal{M})\to C^{n}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.6.m1.4"><semantics id="S3.Thmtheorem2.p1.6.m1.4a"><mrow id="S3.Thmtheorem2.p1.6.m1.4.5" xref="S3.Thmtheorem2.p1.6.m1.4.5.cmml"><msup id="S3.Thmtheorem2.p1.6.m1.4.5.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.cmml"><mi id="S3.Thmtheorem2.p1.6.m1.4.5.2.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.2.cmml">δ</mi><mrow id="S3.Thmtheorem2.p1.6.m1.4.5.2.3" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.3.cmml"><mi id="S3.Thmtheorem2.p1.6.m1.4.5.2.3.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.3.2.cmml">n</mi><mo id="S3.Thmtheorem2.p1.6.m1.4.5.2.3.1" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.3.1.cmml">−</mo><mn id="S3.Thmtheorem2.p1.6.m1.4.5.2.3.3" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.3.3.cmml">1</mn></mrow></msup><mo id="S3.Thmtheorem2.p1.6.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem2.p1.6.m1.4.5.1.cmml">:</mo><mrow id="S3.Thmtheorem2.p1.6.m1.4.5.3" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.cmml"><mrow id="S3.Thmtheorem2.p1.6.m1.4.5.3.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.2.cmml"><msup 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id="S3.Thmtheorem2.p1.6.m1.4.5.3.2.3.2.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.2.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem2.p1.6.m1.2.2" xref="S3.Thmtheorem2.p1.6.m1.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem2.p1.6.m1.4.5.3.2.3.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p1.6.m1.4.5.3.1" stretchy="false" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.1.cmml">→</mo><mrow id="S3.Thmtheorem2.p1.6.m1.4.5.3.3" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.cmml"><msup id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.cmml"><mi id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.2.cmml">C</mi><mi id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.3" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.3.cmml">n</mi></msup><mo id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.1" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.1.cmml"><mo id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem2.p1.6.m1.3.3" xref="S3.Thmtheorem2.p1.6.m1.3.3.cmml">X</mi><mo id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.2.2" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem2.p1.6.m1.4.4" xref="S3.Thmtheorem2.p1.6.m1.4.4.cmml">ℳ</mi><mo id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.6.m1.4b"><apply id="S3.Thmtheorem2.p1.6.m1.4.5.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5"><ci id="S3.Thmtheorem2.p1.6.m1.4.5.1.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.1">:</ci><apply id="S3.Thmtheorem2.p1.6.m1.4.5.2.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.m1.4.5.2.1.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.6.m1.4.5.2.2.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.2">𝛿</ci><apply id="S3.Thmtheorem2.p1.6.m1.4.5.2.3.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.3"><minus id="S3.Thmtheorem2.p1.6.m1.4.5.2.3.1.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.3.1"></minus><ci id="S3.Thmtheorem2.p1.6.m1.4.5.2.3.2.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.3.2">𝑛</ci><cn id="S3.Thmtheorem2.p1.6.m1.4.5.2.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.6.m1.4.5.2.3.3">1</cn></apply></apply><apply id="S3.Thmtheorem2.p1.6.m1.4.5.3.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3"><ci id="S3.Thmtheorem2.p1.6.m1.4.5.3.1.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.1">→</ci><apply id="S3.Thmtheorem2.p1.6.m1.4.5.3.2.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.2"><times id="S3.Thmtheorem2.p1.6.m1.4.5.3.2.1.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.2.1"></times><apply id="S3.Thmtheorem2.p1.6.m1.4.5.3.2.2.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.2.2"><csymbol 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xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3"><times id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.1.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.1"></times><apply id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.1.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.2.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.2">𝐶</ci><ci id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.3.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.2.3">𝑛</ci></apply><list id="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.1.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.5.3.3.3.2"><ci id="S3.Thmtheorem2.p1.6.m1.3.3.cmml" xref="S3.Thmtheorem2.p1.6.m1.3.3">𝑋</ci><ci id="S3.Thmtheorem2.p1.6.m1.4.4.cmml" xref="S3.Thmtheorem2.p1.6.m1.4.4">ℳ</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.6.m1.4c">\delta^{n-1}:C^{n-1}(X;\mathcal{M})\to C^{n}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.6.m1.4d">italic_δ start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT : italic_C start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M ) → italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math> is given by</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex30"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\delta^{n-1}(f)(\sigma)=\sum_{i=0}^{n}(-1)^{i}d^{i}_{*}\bigl{(}f(d_{i}\sigma)% \bigr{)}" class="ltx_Math" display="block" id="S3.Ex30.m1.4"><semantics id="S3.Ex30.m1.4a"><mrow id="S3.Ex30.m1.4.4" xref="S3.Ex30.m1.4.4.cmml"><mrow id="S3.Ex30.m1.4.4.4" xref="S3.Ex30.m1.4.4.4.cmml"><msup id="S3.Ex30.m1.4.4.4.2" xref="S3.Ex30.m1.4.4.4.2.cmml"><mi 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xref="S3.Thmtheorem2.p1.7.m1.2.3.3.2.2">𝐶</ci><apply id="S3.Thmtheorem2.p1.7.m1.2.3.3.2.3.cmml" xref="S3.Thmtheorem2.p1.7.m1.2.3.3.2.3"><minus id="S3.Thmtheorem2.p1.7.m1.2.3.3.2.3.1.cmml" xref="S3.Thmtheorem2.p1.7.m1.2.3.3.2.3.1"></minus><ci id="S3.Thmtheorem2.p1.7.m1.2.3.3.2.3.2.cmml" xref="S3.Thmtheorem2.p1.7.m1.2.3.3.2.3.2">𝑛</ci><cn id="S3.Thmtheorem2.p1.7.m1.2.3.3.2.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.7.m1.2.3.3.2.3.3">1</cn></apply></apply><list id="S3.Thmtheorem2.p1.7.m1.2.3.3.3.1.cmml" xref="S3.Thmtheorem2.p1.7.m1.2.3.3.3.2"><ci id="S3.Thmtheorem2.p1.7.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.7.m1.1.1">𝑋</ci><ci id="S3.Thmtheorem2.p1.7.m1.2.2.cmml" xref="S3.Thmtheorem2.p1.7.m1.2.2">ℳ</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.7.m1.2c">f\in C^{n-1}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.7.m1.2d">italic_f ∈ italic_C start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem2.p1.10.3">cohomology <math alttext="H^{*}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.8.1.m1.2"><semantics id="S3.Thmtheorem2.p1.8.1.m1.2a"><mrow id="S3.Thmtheorem2.p1.8.1.m1.2.3" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.cmml"><msup id="S3.Thmtheorem2.p1.8.1.m1.2.3.2" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.2.cmml"><mi id="S3.Thmtheorem2.p1.8.1.m1.2.3.2.2" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.2.2.cmml">H</mi><mo id="S3.Thmtheorem2.p1.8.1.m1.2.3.2.3" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem2.p1.8.1.m1.2.3.1" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.8.1.m1.2.3.3.2" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem2.p1.8.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem2.p1.8.1.m1.1.1" xref="S3.Thmtheorem2.p1.8.1.m1.1.1.cmml">X</mi><mo id="S3.Thmtheorem2.p1.8.1.m1.2.3.3.2.2" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem2.p1.8.1.m1.2.2" xref="S3.Thmtheorem2.p1.8.1.m1.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem2.p1.8.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.8.1.m1.2b"><apply id="S3.Thmtheorem2.p1.8.1.m1.2.3.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.2.3"><times id="S3.Thmtheorem2.p1.8.1.m1.2.3.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.1"></times><apply id="S3.Thmtheorem2.p1.8.1.m1.2.3.2.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.8.1.m1.2.3.2.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.8.1.m1.2.3.2.2.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.2.2">𝐻</ci><times id="S3.Thmtheorem2.p1.8.1.m1.2.3.2.3.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.2.3"></times></apply><list id="S3.Thmtheorem2.p1.8.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.2.3.3.2"><ci id="S3.Thmtheorem2.p1.8.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.1.1">𝑋</ci><ci id="S3.Thmtheorem2.p1.8.1.m1.2.2.cmml" xref="S3.Thmtheorem2.p1.8.1.m1.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.8.1.m1.2c">H^{*}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.8.1.m1.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math> of the simplicial set <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.9.2.m2.1"><semantics id="S3.Thmtheorem2.p1.9.2.m2.1a"><mi id="S3.Thmtheorem2.p1.9.2.m2.1.1" xref="S3.Thmtheorem2.p1.9.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.9.2.m2.1b"><ci id="S3.Thmtheorem2.p1.9.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.9.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.9.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.9.2.m2.1d">italic_X</annotation></semantics></math> with coefficients in <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.10.3.m3.1"><semantics id="S3.Thmtheorem2.p1.10.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem2.p1.10.3.m3.1.1" xref="S3.Thmtheorem2.p1.10.3.m3.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.10.3.m3.1b"><ci id="S3.Thmtheorem2.p1.10.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.10.3.m3.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.10.3.m3.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.10.3.m3.1d">caligraphic_M</annotation></semantics></math></em> is the cohomology of the cochain complex <math alttext="C^{*}(X;M)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.11.m2.2"><semantics id="S3.Thmtheorem2.p1.11.m2.2a"><mrow id="S3.Thmtheorem2.p1.11.m2.2.3" xref="S3.Thmtheorem2.p1.11.m2.2.3.cmml"><msup id="S3.Thmtheorem2.p1.11.m2.2.3.2" xref="S3.Thmtheorem2.p1.11.m2.2.3.2.cmml"><mi id="S3.Thmtheorem2.p1.11.m2.2.3.2.2" xref="S3.Thmtheorem2.p1.11.m2.2.3.2.2.cmml">C</mi><mo id="S3.Thmtheorem2.p1.11.m2.2.3.2.3" xref="S3.Thmtheorem2.p1.11.m2.2.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem2.p1.11.m2.2.3.1" xref="S3.Thmtheorem2.p1.11.m2.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.11.m2.2.3.3.2" xref="S3.Thmtheorem2.p1.11.m2.2.3.3.1.cmml"><mo id="S3.Thmtheorem2.p1.11.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.11.m2.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem2.p1.11.m2.1.1" xref="S3.Thmtheorem2.p1.11.m2.1.1.cmml">X</mi><mo id="S3.Thmtheorem2.p1.11.m2.2.3.3.2.2" xref="S3.Thmtheorem2.p1.11.m2.2.3.3.1.cmml">;</mo><mi id="S3.Thmtheorem2.p1.11.m2.2.2" xref="S3.Thmtheorem2.p1.11.m2.2.2.cmml">M</mi><mo id="S3.Thmtheorem2.p1.11.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.11.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.11.m2.2b"><apply id="S3.Thmtheorem2.p1.11.m2.2.3.cmml" xref="S3.Thmtheorem2.p1.11.m2.2.3"><times id="S3.Thmtheorem2.p1.11.m2.2.3.1.cmml" xref="S3.Thmtheorem2.p1.11.m2.2.3.1"></times><apply id="S3.Thmtheorem2.p1.11.m2.2.3.2.cmml" xref="S3.Thmtheorem2.p1.11.m2.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.11.m2.2.3.2.1.cmml" xref="S3.Thmtheorem2.p1.11.m2.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.11.m2.2.3.2.2.cmml" xref="S3.Thmtheorem2.p1.11.m2.2.3.2.2">𝐶</ci><times id="S3.Thmtheorem2.p1.11.m2.2.3.2.3.cmml" xref="S3.Thmtheorem2.p1.11.m2.2.3.2.3"></times></apply><list id="S3.Thmtheorem2.p1.11.m2.2.3.3.1.cmml" xref="S3.Thmtheorem2.p1.11.m2.2.3.3.2"><ci id="S3.Thmtheorem2.p1.11.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.11.m2.1.1">𝑋</ci><ci id="S3.Thmtheorem2.p1.11.m2.2.2.cmml" xref="S3.Thmtheorem2.p1.11.m2.2.2">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.11.m2.2c">C^{*}(X;M)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.11.m2.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; italic_M )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.6">An alternative way to define cohomology of a simplicial set <math alttext="X" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mi id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><ci id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_X</annotation></semantics></math> with coefficients in <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><ci id="S3.SS2.p1.2.m2.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">caligraphic_M</annotation></semantics></math> is to define it as the cohomology of the category <math alttext="\Delta(X)" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mrow id="S3.SS2.p1.3.m3.1.2" xref="S3.SS2.p1.3.m3.1.2.cmml"><mi id="S3.SS2.p1.3.m3.1.2.2" mathvariant="normal" xref="S3.SS2.p1.3.m3.1.2.2.cmml">Δ</mi><mo id="S3.SS2.p1.3.m3.1.2.1" xref="S3.SS2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p1.3.m3.1.2.3.2" xref="S3.SS2.p1.3.m3.1.2.cmml"><mo id="S3.SS2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S3.SS2.p1.3.m3.1.2.cmml">(</mo><mi id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml">X</mi><mo id="S3.SS2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S3.SS2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><apply id="S3.SS2.p1.3.m3.1.2.cmml" xref="S3.SS2.p1.3.m3.1.2"><times id="S3.SS2.p1.3.m3.1.2.1.cmml" xref="S3.SS2.p1.3.m3.1.2.1"></times><ci id="S3.SS2.p1.3.m3.1.2.2.cmml" xref="S3.SS2.p1.3.m3.1.2.2">Δ</ci><ci id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">roman_Δ ( italic_X )</annotation></semantics></math> with coefficients in <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.1"><semantics id="S3.SS2.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.4.m4.1.1" xref="S3.SS2.p1.4.m4.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m4.1b"><ci id="S3.SS2.p1.4.m4.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.1d">caligraphic_M</annotation></semantics></math> by considering <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m5.1"><semantics id="S3.SS2.p1.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.5.m5.1.1" xref="S3.SS2.p1.5.m5.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m5.1b"><ci id="S3.SS2.p1.5.m5.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m5.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.5.m5.1d">caligraphic_M</annotation></semantics></math> as a covariant <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S3.SS2.p1.6.m6.1"><semantics id="S3.SS2.p1.6.m6.1a"><mrow id="S3.SS2.p1.6.m6.1.2" xref="S3.SS2.p1.6.m6.1.2.cmml"><mi id="S3.SS2.p1.6.m6.1.2.2" xref="S3.SS2.p1.6.m6.1.2.2.cmml">R</mi><mo id="S3.SS2.p1.6.m6.1.2.1" xref="S3.SS2.p1.6.m6.1.2.1.cmml">⁢</mo><mi id="S3.SS2.p1.6.m6.1.2.3" mathvariant="normal" xref="S3.SS2.p1.6.m6.1.2.3.cmml">Δ</mi><mo id="S3.SS2.p1.6.m6.1.2.1a" xref="S3.SS2.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p1.6.m6.1.2.4.2" xref="S3.SS2.p1.6.m6.1.2.cmml"><mo id="S3.SS2.p1.6.m6.1.2.4.2.1" stretchy="false" xref="S3.SS2.p1.6.m6.1.2.cmml">(</mo><mi id="S3.SS2.p1.6.m6.1.1" xref="S3.SS2.p1.6.m6.1.1.cmml">X</mi><mo id="S3.SS2.p1.6.m6.1.2.4.2.2" stretchy="false" xref="S3.SS2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.6.m6.1b"><apply id="S3.SS2.p1.6.m6.1.2.cmml" xref="S3.SS2.p1.6.m6.1.2"><times id="S3.SS2.p1.6.m6.1.2.1.cmml" xref="S3.SS2.p1.6.m6.1.2.1"></times><ci id="S3.SS2.p1.6.m6.1.2.2.cmml" xref="S3.SS2.p1.6.m6.1.2.2">𝑅</ci><ci id="S3.SS2.p1.6.m6.1.2.3.cmml" xref="S3.SS2.p1.6.m6.1.2.3">Δ</ci><ci id="S3.SS2.p1.6.m6.1.1.cmml" xref="S3.SS2.p1.6.m6.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.6.m6.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.6.m6.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module. It is a well-known fact that these two cohomology definitions are isomorphic (see Gabriel–Zisman <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib5" title="">5</a>, Appendix II]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.1.1.1">Proposition 3.3</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.9.9">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.1.m1.1"><semantics id="S3.Thmtheorem3.p1.1.1.m1.1a"><mi id="S3.Thmtheorem3.p1.1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.1.1.m1.1b"><ci id="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a simplicial set and <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.2.2.m2.1"><semantics id="S3.Thmtheorem3.p1.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem3.p1.2.2.m2.1.1" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.2.2.m2.1b"><ci id="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.2.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.2.2.m2.1d">caligraphic_M</annotation></semantics></math> be a coefficient system for <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.3.3.m3.1"><semantics id="S3.Thmtheorem3.p1.3.3.m3.1a"><mi id="S3.Thmtheorem3.p1.3.3.m3.1.1" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.3.3.m3.1b"><ci id="S3.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.3.3.m3.1d">italic_X</annotation></semantics></math> over <math alttext="R" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.4.4.m4.1"><semantics id="S3.Thmtheorem3.p1.4.4.m4.1a"><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.4.4.m4.1b"><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.4.4.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.4.4.m4.1d">italic_R</annotation></semantics></math>. The cohomology groups <math alttext="H^{*}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.5.5.m5.2"><semantics id="S3.Thmtheorem3.p1.5.5.m5.2a"><mrow id="S3.Thmtheorem3.p1.5.5.m5.2.3" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.cmml"><msup id="S3.Thmtheorem3.p1.5.5.m5.2.3.2" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2.cmml"><mi id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.2" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2.2.cmml">H</mi><mo id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.3" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem3.p1.5.5.m5.2.3.1" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.5.5.m5.2.3.3.2" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.3.1.cmml"><mo id="S3.Thmtheorem3.p1.5.5.m5.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem3.p1.5.5.m5.1.1" xref="S3.Thmtheorem3.p1.5.5.m5.1.1.cmml">X</mi><mo id="S3.Thmtheorem3.p1.5.5.m5.2.3.3.2.2" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem3.p1.5.5.m5.2.2" xref="S3.Thmtheorem3.p1.5.5.m5.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem3.p1.5.5.m5.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.5.5.m5.2b"><apply id="S3.Thmtheorem3.p1.5.5.m5.2.3.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3"><times id="S3.Thmtheorem3.p1.5.5.m5.2.3.1.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.1"></times><apply id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.1.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.2.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2.2">𝐻</ci><times id="S3.Thmtheorem3.p1.5.5.m5.2.3.2.3.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.2.3"></times></apply><list id="S3.Thmtheorem3.p1.5.5.m5.2.3.3.1.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.3.3.2"><ci id="S3.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.1.1">𝑋</ci><ci id="S3.Thmtheorem3.p1.5.5.m5.2.2.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.5.5.m5.2c">H^{*}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.5.5.m5.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math>, defined as the cohomology of the cochain complex <math alttext="C^{*}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.6.6.m6.2"><semantics id="S3.Thmtheorem3.p1.6.6.m6.2a"><mrow id="S3.Thmtheorem3.p1.6.6.m6.2.3" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.cmml"><msup id="S3.Thmtheorem3.p1.6.6.m6.2.3.2" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.2.cmml"><mi id="S3.Thmtheorem3.p1.6.6.m6.2.3.2.2" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.2.2.cmml">C</mi><mo id="S3.Thmtheorem3.p1.6.6.m6.2.3.2.3" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem3.p1.6.6.m6.2.3.1" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.6.6.m6.2.3.3.2" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.3.1.cmml"><mo id="S3.Thmtheorem3.p1.6.6.m6.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem3.p1.6.6.m6.1.1" xref="S3.Thmtheorem3.p1.6.6.m6.1.1.cmml">X</mi><mo id="S3.Thmtheorem3.p1.6.6.m6.2.3.3.2.2" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem3.p1.6.6.m6.2.2" xref="S3.Thmtheorem3.p1.6.6.m6.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem3.p1.6.6.m6.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.6.6.m6.2b"><apply id="S3.Thmtheorem3.p1.6.6.m6.2.3.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.2.3"><times id="S3.Thmtheorem3.p1.6.6.m6.2.3.1.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.1"></times><apply id="S3.Thmtheorem3.p1.6.6.m6.2.3.2.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.6.6.m6.2.3.2.1.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem3.p1.6.6.m6.2.3.2.2.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.2.2">𝐶</ci><times id="S3.Thmtheorem3.p1.6.6.m6.2.3.2.3.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.2.3"></times></apply><list id="S3.Thmtheorem3.p1.6.6.m6.2.3.3.1.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.2.3.3.2"><ci id="S3.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.1.1">𝑋</ci><ci id="S3.Thmtheorem3.p1.6.6.m6.2.2.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.6.6.m6.2c">C^{*}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.6.6.m6.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math>, are isomorphic to the cohomology groups <math alttext="H^{*}(\Delta(X);\mathcal{M})" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.7.7.m7.3"><semantics id="S3.Thmtheorem3.p1.7.7.m7.3a"><mrow id="S3.Thmtheorem3.p1.7.7.m7.3.3" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.cmml"><msup id="S3.Thmtheorem3.p1.7.7.m7.3.3.3" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.3.cmml"><mi id="S3.Thmtheorem3.p1.7.7.m7.3.3.3.2" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.3.2.cmml">H</mi><mo id="S3.Thmtheorem3.p1.7.7.m7.3.3.3.3" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.3.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem3.p1.7.7.m7.3.3.2" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.2.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.2.cmml"><mo id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.2" stretchy="false" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.2.cmml">(</mo><mrow id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.cmml"><mi id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.2" mathvariant="normal" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.2.cmml">Δ</mi><mo id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.1" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.3.2" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.cmml"><mo id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.3.2.1" stretchy="false" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.cmml">(</mo><mi id="S3.Thmtheorem3.p1.7.7.m7.1.1" xref="S3.Thmtheorem3.p1.7.7.m7.1.1.cmml">X</mi><mo id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.3.2.2" stretchy="false" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.3" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.2.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem3.p1.7.7.m7.2.2" xref="S3.Thmtheorem3.p1.7.7.m7.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.4" stretchy="false" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.7.7.m7.3b"><apply id="S3.Thmtheorem3.p1.7.7.m7.3.3.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3"><times id="S3.Thmtheorem3.p1.7.7.m7.3.3.2.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.2"></times><apply id="S3.Thmtheorem3.p1.7.7.m7.3.3.3.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.7.7.m7.3.3.3.1.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.3">superscript</csymbol><ci id="S3.Thmtheorem3.p1.7.7.m7.3.3.3.2.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.3.2">𝐻</ci><times id="S3.Thmtheorem3.p1.7.7.m7.3.3.3.3.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.3.3"></times></apply><list id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.2.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1"><apply id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1"><times id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.1"></times><ci id="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.3.3.1.1.1.2">Δ</ci><ci id="S3.Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.1.1">𝑋</ci></apply><ci id="S3.Thmtheorem3.p1.7.7.m7.2.2.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.7.7.m7.3c">H^{*}(\Delta(X);\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.7.7.m7.3d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_Δ ( italic_X ) ; caligraphic_M )</annotation></semantics></math> of the category <math alttext="\Delta(X)" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.8.8.m8.1"><semantics id="S3.Thmtheorem3.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem3.p1.8.8.m8.1.2" xref="S3.Thmtheorem3.p1.8.8.m8.1.2.cmml"><mi id="S3.Thmtheorem3.p1.8.8.m8.1.2.2" mathvariant="normal" xref="S3.Thmtheorem3.p1.8.8.m8.1.2.2.cmml">Δ</mi><mo id="S3.Thmtheorem3.p1.8.8.m8.1.2.1" xref="S3.Thmtheorem3.p1.8.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.8.8.m8.1.2.3.2" xref="S3.Thmtheorem3.p1.8.8.m8.1.2.cmml"><mo id="S3.Thmtheorem3.p1.8.8.m8.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem3.p1.8.8.m8.1.2.cmml">(</mo><mi id="S3.Thmtheorem3.p1.8.8.m8.1.1" xref="S3.Thmtheorem3.p1.8.8.m8.1.1.cmml">X</mi><mo id="S3.Thmtheorem3.p1.8.8.m8.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem3.p1.8.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.8.8.m8.1b"><apply id="S3.Thmtheorem3.p1.8.8.m8.1.2.cmml" xref="S3.Thmtheorem3.p1.8.8.m8.1.2"><times id="S3.Thmtheorem3.p1.8.8.m8.1.2.1.cmml" xref="S3.Thmtheorem3.p1.8.8.m8.1.2.1"></times><ci id="S3.Thmtheorem3.p1.8.8.m8.1.2.2.cmml" xref="S3.Thmtheorem3.p1.8.8.m8.1.2.2">Δ</ci><ci id="S3.Thmtheorem3.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem3.p1.8.8.m8.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.8.8.m8.1c">\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.8.8.m8.1d">roman_Δ ( italic_X )</annotation></semantics></math> with coefficients in the covariant module <math alttext="\mathcal{M}:\Delta(X)\to R" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.9.9.m9.1"><semantics id="S3.Thmtheorem3.p1.9.9.m9.1a"><mrow id="S3.Thmtheorem3.p1.9.9.m9.1.2" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem3.p1.9.9.m9.1.2.2" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem3.p1.9.9.m9.1.2.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.1.cmml">:</mo><mrow id="S3.Thmtheorem3.p1.9.9.m9.1.2.3" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.cmml"><mrow id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.cmml"><mi id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.2" mathvariant="normal" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.2.cmml">Δ</mi><mo id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.1" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.3.2" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.cmml"><mo id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.cmml">(</mo><mi id="S3.Thmtheorem3.p1.9.9.m9.1.1" xref="S3.Thmtheorem3.p1.9.9.m9.1.1.cmml">X</mi><mo id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.3.2.2" stretchy="false" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.1" stretchy="false" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.1.cmml">→</mo><mi id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.3" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.9.9.m9.1b"><apply id="S3.Thmtheorem3.p1.9.9.m9.1.2.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2"><ci id="S3.Thmtheorem3.p1.9.9.m9.1.2.1.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.1">:</ci><ci id="S3.Thmtheorem3.p1.9.9.m9.1.2.2.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.2">ℳ</ci><apply id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3"><ci id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.1.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.1">→</ci><apply id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2"><times id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.1.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.1"></times><ci id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.2.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.2.2">Δ</ci><ci id="S3.Thmtheorem3.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.1">𝑋</ci></apply><ci id="S3.Thmtheorem3.p1.9.9.m9.1.2.3.3.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.2.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.9.9.m9.1c">\mathcal{M}:\Delta(X)\to R</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.9.9.m9.1d">caligraphic_M : roman_Δ ( italic_X ) → italic_R</annotation></semantics></math>-Mod.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS2.1.p1"> <p class="ltx_p" id="S3.SS2.1.p1.6">The proof is similar to earlier proofs of this type. We show that there is a projective resolution <math alttext="P_{*}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.1.m1.1"><semantics id="S3.SS2.1.p1.1.m1.1a"><msub id="S3.SS2.1.p1.1.m1.1.1" xref="S3.SS2.1.p1.1.m1.1.1.cmml"><mi id="S3.SS2.1.p1.1.m1.1.1.2" xref="S3.SS2.1.p1.1.m1.1.1.2.cmml">P</mi><mo id="S3.SS2.1.p1.1.m1.1.1.3" xref="S3.SS2.1.p1.1.m1.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.1.m1.1b"><apply id="S3.SS2.1.p1.1.m1.1.1.cmml" xref="S3.SS2.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.1.m1.1.1.1.cmml" xref="S3.SS2.1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.1.m1.1.1.2.cmml" xref="S3.SS2.1.p1.1.m1.1.1.2">𝑃</ci><times id="S3.SS2.1.p1.1.m1.1.1.3.cmml" xref="S3.SS2.1.p1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.1.m1.1c">P_{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.1.m1.1d">italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.2.m2.1"><semantics id="S3.SS2.1.p1.2.m2.1a"><munder accentunder="true" id="S3.SS2.1.p1.2.m2.1.1" xref="S3.SS2.1.p1.2.m2.1.1.cmml"><mi id="S3.SS2.1.p1.2.m2.1.1.2" xref="S3.SS2.1.p1.2.m2.1.1.2.cmml">R</mi><mo id="S3.SS2.1.p1.2.m2.1.1.1" xref="S3.SS2.1.p1.2.m2.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.2.m2.1b"><apply id="S3.SS2.1.p1.2.m2.1.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1"><ci id="S3.SS2.1.p1.2.m2.1.1.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1">¯</ci><ci id="S3.SS2.1.p1.2.m2.1.1.2.cmml" xref="S3.SS2.1.p1.2.m2.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.2.m2.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.2.m2.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S3.SS2.1.p1.3.m3.1"><semantics id="S3.SS2.1.p1.3.m3.1a"><mrow id="S3.SS2.1.p1.3.m3.1.2" xref="S3.SS2.1.p1.3.m3.1.2.cmml"><mi id="S3.SS2.1.p1.3.m3.1.2.2" xref="S3.SS2.1.p1.3.m3.1.2.2.cmml">R</mi><mo id="S3.SS2.1.p1.3.m3.1.2.1" xref="S3.SS2.1.p1.3.m3.1.2.1.cmml">⁢</mo><mi id="S3.SS2.1.p1.3.m3.1.2.3" mathvariant="normal" xref="S3.SS2.1.p1.3.m3.1.2.3.cmml">Δ</mi><mo id="S3.SS2.1.p1.3.m3.1.2.1a" xref="S3.SS2.1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.3.m3.1.2.4.2" xref="S3.SS2.1.p1.3.m3.1.2.cmml"><mo id="S3.SS2.1.p1.3.m3.1.2.4.2.1" stretchy="false" xref="S3.SS2.1.p1.3.m3.1.2.cmml">(</mo><mi id="S3.SS2.1.p1.3.m3.1.1" xref="S3.SS2.1.p1.3.m3.1.1.cmml">X</mi><mo id="S3.SS2.1.p1.3.m3.1.2.4.2.2" stretchy="false" xref="S3.SS2.1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.3.m3.1b"><apply id="S3.SS2.1.p1.3.m3.1.2.cmml" xref="S3.SS2.1.p1.3.m3.1.2"><times id="S3.SS2.1.p1.3.m3.1.2.1.cmml" xref="S3.SS2.1.p1.3.m3.1.2.1"></times><ci id="S3.SS2.1.p1.3.m3.1.2.2.cmml" xref="S3.SS2.1.p1.3.m3.1.2.2">𝑅</ci><ci id="S3.SS2.1.p1.3.m3.1.2.3.cmml" xref="S3.SS2.1.p1.3.m3.1.2.3">Δ</ci><ci id="S3.SS2.1.p1.3.m3.1.1.cmml" xref="S3.SS2.1.p1.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.3.m3.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.3.m3.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module such that <math alttext="\mathrm{Hom}_{R\Delta(X)}(P_{*},\mathcal{M})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.4.m4.3"><semantics id="S3.SS2.1.p1.4.m4.3a"><mrow id="S3.SS2.1.p1.4.m4.3.3" xref="S3.SS2.1.p1.4.m4.3.3.cmml"><msub id="S3.SS2.1.p1.4.m4.3.3.3" xref="S3.SS2.1.p1.4.m4.3.3.3.cmml"><mi id="S3.SS2.1.p1.4.m4.3.3.3.2" xref="S3.SS2.1.p1.4.m4.3.3.3.2.cmml">Hom</mi><mrow id="S3.SS2.1.p1.4.m4.1.1.1" xref="S3.SS2.1.p1.4.m4.1.1.1.cmml"><mi id="S3.SS2.1.p1.4.m4.1.1.1.3" xref="S3.SS2.1.p1.4.m4.1.1.1.3.cmml">R</mi><mo id="S3.SS2.1.p1.4.m4.1.1.1.2" xref="S3.SS2.1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mi id="S3.SS2.1.p1.4.m4.1.1.1.4" mathvariant="normal" xref="S3.SS2.1.p1.4.m4.1.1.1.4.cmml">Δ</mi><mo id="S3.SS2.1.p1.4.m4.1.1.1.2a" xref="S3.SS2.1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.1.p1.4.m4.1.1.1.5.2" xref="S3.SS2.1.p1.4.m4.1.1.1.cmml"><mo id="S3.SS2.1.p1.4.m4.1.1.1.5.2.1" stretchy="false" xref="S3.SS2.1.p1.4.m4.1.1.1.cmml">(</mo><mi id="S3.SS2.1.p1.4.m4.1.1.1.1" xref="S3.SS2.1.p1.4.m4.1.1.1.1.cmml">X</mi><mo id="S3.SS2.1.p1.4.m4.1.1.1.5.2.2" stretchy="false" xref="S3.SS2.1.p1.4.m4.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S3.SS2.1.p1.4.m4.3.3.2" xref="S3.SS2.1.p1.4.m4.3.3.2.cmml">⁢</mo><mrow id="S3.SS2.1.p1.4.m4.3.3.1.1" xref="S3.SS2.1.p1.4.m4.3.3.1.2.cmml"><mo id="S3.SS2.1.p1.4.m4.3.3.1.1.2" stretchy="false" xref="S3.SS2.1.p1.4.m4.3.3.1.2.cmml">(</mo><msub id="S3.SS2.1.p1.4.m4.3.3.1.1.1" xref="S3.SS2.1.p1.4.m4.3.3.1.1.1.cmml"><mi id="S3.SS2.1.p1.4.m4.3.3.1.1.1.2" xref="S3.SS2.1.p1.4.m4.3.3.1.1.1.2.cmml">P</mi><mo id="S3.SS2.1.p1.4.m4.3.3.1.1.1.3" xref="S3.SS2.1.p1.4.m4.3.3.1.1.1.3.cmml">∗</mo></msub><mo id="S3.SS2.1.p1.4.m4.3.3.1.1.3" xref="S3.SS2.1.p1.4.m4.3.3.1.2.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.4.m4.2.2" xref="S3.SS2.1.p1.4.m4.2.2.cmml">ℳ</mi><mo id="S3.SS2.1.p1.4.m4.3.3.1.1.4" stretchy="false" xref="S3.SS2.1.p1.4.m4.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.4.m4.3b"><apply id="S3.SS2.1.p1.4.m4.3.3.cmml" xref="S3.SS2.1.p1.4.m4.3.3"><times id="S3.SS2.1.p1.4.m4.3.3.2.cmml" xref="S3.SS2.1.p1.4.m4.3.3.2"></times><apply id="S3.SS2.1.p1.4.m4.3.3.3.cmml" xref="S3.SS2.1.p1.4.m4.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.1.p1.4.m4.3.3.3.1.cmml" xref="S3.SS2.1.p1.4.m4.3.3.3">subscript</csymbol><ci id="S3.SS2.1.p1.4.m4.3.3.3.2.cmml" xref="S3.SS2.1.p1.4.m4.3.3.3.2">Hom</ci><apply id="S3.SS2.1.p1.4.m4.1.1.1.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1"><times id="S3.SS2.1.p1.4.m4.1.1.1.2.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.2"></times><ci id="S3.SS2.1.p1.4.m4.1.1.1.3.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.3">𝑅</ci><ci id="S3.SS2.1.p1.4.m4.1.1.1.4.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.4">Δ</ci><ci id="S3.SS2.1.p1.4.m4.1.1.1.1.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.1">𝑋</ci></apply></apply><interval closure="open" id="S3.SS2.1.p1.4.m4.3.3.1.2.cmml" xref="S3.SS2.1.p1.4.m4.3.3.1.1"><apply id="S3.SS2.1.p1.4.m4.3.3.1.1.1.cmml" xref="S3.SS2.1.p1.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.4.m4.3.3.1.1.1.1.cmml" xref="S3.SS2.1.p1.4.m4.3.3.1.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.4.m4.3.3.1.1.1.2.cmml" xref="S3.SS2.1.p1.4.m4.3.3.1.1.1.2">𝑃</ci><times id="S3.SS2.1.p1.4.m4.3.3.1.1.1.3.cmml" xref="S3.SS2.1.p1.4.m4.3.3.1.1.1.3"></times></apply><ci id="S3.SS2.1.p1.4.m4.2.2.cmml" xref="S3.SS2.1.p1.4.m4.2.2">ℳ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.4.m4.3c">\mathrm{Hom}_{R\Delta(X)}(P_{*},\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.4.m4.3d">roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_X ) end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , caligraphic_M )</annotation></semantics></math> is isomorphic to the cochain complex <math alttext="C^{*}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.5.m5.2"><semantics id="S3.SS2.1.p1.5.m5.2a"><mrow id="S3.SS2.1.p1.5.m5.2.3" xref="S3.SS2.1.p1.5.m5.2.3.cmml"><msup id="S3.SS2.1.p1.5.m5.2.3.2" xref="S3.SS2.1.p1.5.m5.2.3.2.cmml"><mi id="S3.SS2.1.p1.5.m5.2.3.2.2" xref="S3.SS2.1.p1.5.m5.2.3.2.2.cmml">C</mi><mo id="S3.SS2.1.p1.5.m5.2.3.2.3" xref="S3.SS2.1.p1.5.m5.2.3.2.3.cmml">∗</mo></msup><mo id="S3.SS2.1.p1.5.m5.2.3.1" xref="S3.SS2.1.p1.5.m5.2.3.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.5.m5.2.3.3.2" xref="S3.SS2.1.p1.5.m5.2.3.3.1.cmml"><mo id="S3.SS2.1.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S3.SS2.1.p1.5.m5.2.3.3.1.cmml">(</mo><mi id="S3.SS2.1.p1.5.m5.1.1" xref="S3.SS2.1.p1.5.m5.1.1.cmml">X</mi><mo id="S3.SS2.1.p1.5.m5.2.3.3.2.2" xref="S3.SS2.1.p1.5.m5.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.5.m5.2.2" xref="S3.SS2.1.p1.5.m5.2.2.cmml">ℳ</mi><mo id="S3.SS2.1.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S3.SS2.1.p1.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.5.m5.2b"><apply id="S3.SS2.1.p1.5.m5.2.3.cmml" xref="S3.SS2.1.p1.5.m5.2.3"><times id="S3.SS2.1.p1.5.m5.2.3.1.cmml" xref="S3.SS2.1.p1.5.m5.2.3.1"></times><apply id="S3.SS2.1.p1.5.m5.2.3.2.cmml" xref="S3.SS2.1.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.5.m5.2.3.2.1.cmml" xref="S3.SS2.1.p1.5.m5.2.3.2">superscript</csymbol><ci id="S3.SS2.1.p1.5.m5.2.3.2.2.cmml" xref="S3.SS2.1.p1.5.m5.2.3.2.2">𝐶</ci><times id="S3.SS2.1.p1.5.m5.2.3.2.3.cmml" xref="S3.SS2.1.p1.5.m5.2.3.2.3"></times></apply><list id="S3.SS2.1.p1.5.m5.2.3.3.1.cmml" xref="S3.SS2.1.p1.5.m5.2.3.3.2"><ci id="S3.SS2.1.p1.5.m5.1.1.cmml" xref="S3.SS2.1.p1.5.m5.1.1">𝑋</ci><ci id="S3.SS2.1.p1.5.m5.2.2.cmml" xref="S3.SS2.1.p1.5.m5.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.5.m5.2c">C^{*}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.5.m5.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math>. For each <math alttext="q\geq 0" class="ltx_Math" display="inline" id="S3.SS2.1.p1.6.m6.1"><semantics id="S3.SS2.1.p1.6.m6.1a"><mrow id="S3.SS2.1.p1.6.m6.1.1" xref="S3.SS2.1.p1.6.m6.1.1.cmml"><mi id="S3.SS2.1.p1.6.m6.1.1.2" xref="S3.SS2.1.p1.6.m6.1.1.2.cmml">q</mi><mo id="S3.SS2.1.p1.6.m6.1.1.1" xref="S3.SS2.1.p1.6.m6.1.1.1.cmml">≥</mo><mn id="S3.SS2.1.p1.6.m6.1.1.3" xref="S3.SS2.1.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.6.m6.1b"><apply id="S3.SS2.1.p1.6.m6.1.1.cmml" xref="S3.SS2.1.p1.6.m6.1.1"><geq id="S3.SS2.1.p1.6.m6.1.1.1.cmml" xref="S3.SS2.1.p1.6.m6.1.1.1"></geq><ci id="S3.SS2.1.p1.6.m6.1.1.2.cmml" xref="S3.SS2.1.p1.6.m6.1.1.2">𝑞</ci><cn id="S3.SS2.1.p1.6.m6.1.1.3.cmml" type="integer" xref="S3.SS2.1.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.6.m6.1c">q\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.6.m6.1d">italic_q ≥ 0</annotation></semantics></math>, let</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex31"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P_{q}=\bigoplus_{\sigma\in X_{q}}R\mathrm{Mor}_{\Delta(X)}(\sigma,?)." class="ltx_Math" display="block" id="S3.Ex31.m1.4"><semantics id="S3.Ex31.m1.4a"><mrow id="S3.Ex31.m1.4.4.1" xref="S3.Ex31.m1.4.4.1.1.cmml"><mrow id="S3.Ex31.m1.4.4.1.1" xref="S3.Ex31.m1.4.4.1.1.cmml"><msub id="S3.Ex31.m1.4.4.1.1.2" xref="S3.Ex31.m1.4.4.1.1.2.cmml"><mi id="S3.Ex31.m1.4.4.1.1.2.2" xref="S3.Ex31.m1.4.4.1.1.2.2.cmml">P</mi><mi id="S3.Ex31.m1.4.4.1.1.2.3" xref="S3.Ex31.m1.4.4.1.1.2.3.cmml">q</mi></msub><mo id="S3.Ex31.m1.4.4.1.1.1" rspace="0.111em" xref="S3.Ex31.m1.4.4.1.1.1.cmml">=</mo><mrow id="S3.Ex31.m1.4.4.1.1.3" xref="S3.Ex31.m1.4.4.1.1.3.cmml"><munder id="S3.Ex31.m1.4.4.1.1.3.1" xref="S3.Ex31.m1.4.4.1.1.3.1.cmml"><mo id="S3.Ex31.m1.4.4.1.1.3.1.2" movablelimits="false" xref="S3.Ex31.m1.4.4.1.1.3.1.2.cmml">⨁</mo><mrow id="S3.Ex31.m1.4.4.1.1.3.1.3" xref="S3.Ex31.m1.4.4.1.1.3.1.3.cmml"><mi id="S3.Ex31.m1.4.4.1.1.3.1.3.2" xref="S3.Ex31.m1.4.4.1.1.3.1.3.2.cmml">σ</mi><mo id="S3.Ex31.m1.4.4.1.1.3.1.3.1" xref="S3.Ex31.m1.4.4.1.1.3.1.3.1.cmml">∈</mo><msub id="S3.Ex31.m1.4.4.1.1.3.1.3.3" xref="S3.Ex31.m1.4.4.1.1.3.1.3.3.cmml"><mi id="S3.Ex31.m1.4.4.1.1.3.1.3.3.2" xref="S3.Ex31.m1.4.4.1.1.3.1.3.3.2.cmml">X</mi><mi id="S3.Ex31.m1.4.4.1.1.3.1.3.3.3" xref="S3.Ex31.m1.4.4.1.1.3.1.3.3.3.cmml">q</mi></msub></mrow></munder><mrow id="S3.Ex31.m1.4.4.1.1.3.2" xref="S3.Ex31.m1.4.4.1.1.3.2.cmml"><mi id="S3.Ex31.m1.4.4.1.1.3.2.2" xref="S3.Ex31.m1.4.4.1.1.3.2.2.cmml">R</mi><mo id="S3.Ex31.m1.4.4.1.1.3.2.1" xref="S3.Ex31.m1.4.4.1.1.3.2.1.cmml">⁢</mo><msub id="S3.Ex31.m1.4.4.1.1.3.2.3" xref="S3.Ex31.m1.4.4.1.1.3.2.3.cmml"><mi id="S3.Ex31.m1.4.4.1.1.3.2.3.2" xref="S3.Ex31.m1.4.4.1.1.3.2.3.2.cmml">Mor</mi><mrow id="S3.Ex31.m1.1.1.1" xref="S3.Ex31.m1.1.1.1.cmml"><mi id="S3.Ex31.m1.1.1.1.3" mathvariant="normal" xref="S3.Ex31.m1.1.1.1.3.cmml">Δ</mi><mo id="S3.Ex31.m1.1.1.1.2" xref="S3.Ex31.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex31.m1.1.1.1.4.2" xref="S3.Ex31.m1.1.1.1.cmml"><mo id="S3.Ex31.m1.1.1.1.4.2.1" stretchy="false" xref="S3.Ex31.m1.1.1.1.cmml">(</mo><mi id="S3.Ex31.m1.1.1.1.1" xref="S3.Ex31.m1.1.1.1.1.cmml">X</mi><mo id="S3.Ex31.m1.1.1.1.4.2.2" stretchy="false" xref="S3.Ex31.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S3.Ex31.m1.4.4.1.1.3.2.1a" xref="S3.Ex31.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.Ex31.m1.4.4.1.1.3.2.4.2" xref="S3.Ex31.m1.4.4.1.1.3.2.4.1.cmml"><mo id="S3.Ex31.m1.4.4.1.1.3.2.4.2.1" stretchy="false" xref="S3.Ex31.m1.4.4.1.1.3.2.4.1.cmml">(</mo><mi id="S3.Ex31.m1.2.2" xref="S3.Ex31.m1.2.2.cmml">σ</mi><mo id="S3.Ex31.m1.4.4.1.1.3.2.4.2.2" xref="S3.Ex31.m1.4.4.1.1.3.2.4.1.cmml">,</mo><mi id="S3.Ex31.m1.3.3" mathvariant="normal" xref="S3.Ex31.m1.3.3.cmml">?</mi><mo 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id="S3.SS2.1.p1.8.m2.1a"><mrow id="S3.SS2.1.p1.8.m2.1.2" xref="S3.SS2.1.p1.8.m2.1.2.cmml"><msub id="S3.SS2.1.p1.8.m2.1.2.2" xref="S3.SS2.1.p1.8.m2.1.2.2.cmml"><mi id="S3.SS2.1.p1.8.m2.1.2.2.2" xref="S3.SS2.1.p1.8.m2.1.2.2.2.cmml">P</mi><mi id="S3.SS2.1.p1.8.m2.1.2.2.3" xref="S3.SS2.1.p1.8.m2.1.2.2.3.cmml">q</mi></msub><mo id="S3.SS2.1.p1.8.m2.1.2.1" xref="S3.SS2.1.p1.8.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.8.m2.1.2.3.2" xref="S3.SS2.1.p1.8.m2.1.2.cmml"><mo id="S3.SS2.1.p1.8.m2.1.2.3.2.1" stretchy="false" xref="S3.SS2.1.p1.8.m2.1.2.cmml">(</mo><mi id="S3.SS2.1.p1.8.m2.1.1" xref="S3.SS2.1.p1.8.m2.1.1.cmml">θ</mi><mo id="S3.SS2.1.p1.8.m2.1.2.3.2.2" stretchy="false" xref="S3.SS2.1.p1.8.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.8.m2.1b"><apply id="S3.SS2.1.p1.8.m2.1.2.cmml" xref="S3.SS2.1.p1.8.m2.1.2"><times id="S3.SS2.1.p1.8.m2.1.2.1.cmml" xref="S3.SS2.1.p1.8.m2.1.2.1"></times><apply id="S3.SS2.1.p1.8.m2.1.2.2.cmml" xref="S3.SS2.1.p1.8.m2.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.8.m2.1.2.2.1.cmml" xref="S3.SS2.1.p1.8.m2.1.2.2">subscript</csymbol><ci id="S3.SS2.1.p1.8.m2.1.2.2.2.cmml" xref="S3.SS2.1.p1.8.m2.1.2.2.2">𝑃</ci><ci id="S3.SS2.1.p1.8.m2.1.2.2.3.cmml" xref="S3.SS2.1.p1.8.m2.1.2.2.3">𝑞</ci></apply><ci id="S3.SS2.1.p1.8.m2.1.1.cmml" xref="S3.SS2.1.p1.8.m2.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.8.m2.1c">P_{q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.8.m2.1d">italic_P start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> is a free <math alttext="R" class="ltx_Math" display="inline" id="S3.SS2.1.p1.9.m3.1"><semantics id="S3.SS2.1.p1.9.m3.1a"><mi id="S3.SS2.1.p1.9.m3.1.1" xref="S3.SS2.1.p1.9.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.9.m3.1b"><ci id="S3.SS2.1.p1.9.m3.1.1.cmml" xref="S3.SS2.1.p1.9.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.9.m3.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.9.m3.1d">italic_R</annotation></semantics></math>-module with basis given by</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex32"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}_{q}(\theta)=\{(\sigma,f)\,|\,\sigma\in X_{q},f:[q]\to[n]\text{ % such that }f^{*}(\theta)=\sigma\}." class="ltx_Math" display="block" id="S3.Ex32.m1.8"><semantics id="S3.Ex32.m1.8a"><mrow id="S3.Ex32.m1.8.8.1" xref="S3.Ex32.m1.8.8.1.1.cmml"><mrow id="S3.Ex32.m1.8.8.1.1" xref="S3.Ex32.m1.8.8.1.1.cmml"><mrow id="S3.Ex32.m1.8.8.1.1.4" xref="S3.Ex32.m1.8.8.1.1.4.cmml"><msub id="S3.Ex32.m1.8.8.1.1.4.2" xref="S3.Ex32.m1.8.8.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" 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xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.cmml"><mrow id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.2" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.1.cmml"><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.2.1" stretchy="false" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.1.1.cmml">[</mo><mi id="S3.Ex32.m1.6.6" xref="S3.Ex32.m1.6.6.cmml">n</mi><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.2.2" stretchy="false" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.1.1.cmml">]</mo></mrow><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.1" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.1.cmml">⁢</mo><mtext id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.3" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.3a.cmml"> such that </mtext><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.1a" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.1.cmml">⁢</mo><msup id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.cmml"><mi id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.2" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.2.cmml">f</mi><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.3" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.3.cmml">∗</mo></msup><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.1b" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.1.cmml">⁢</mo><mrow id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.5.2" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.cmml"><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.5.2.1" stretchy="false" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.cmml">(</mo><mi id="S3.Ex32.m1.7.7" xref="S3.Ex32.m1.7.7.cmml">θ</mi><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.5.2.2" stretchy="false" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.cmml">)</mo></mrow></mrow><mo id="S3.Ex32.m1.8.8.1.1.2.2.2.3.5" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.5.cmml">=</mo><mi id="S3.Ex32.m1.8.8.1.1.2.2.2.3.6" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.6.cmml">σ</mi></mrow></mrow><mo id="S3.Ex32.m1.8.8.1.1.2.2.5" stretchy="false" xref="S3.Ex32.m1.8.8.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S3.Ex32.m1.8.8.1.2" lspace="0em" xref="S3.Ex32.m1.8.8.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex32.m1.8b"><apply id="S3.Ex32.m1.8.8.1.1.cmml" xref="S3.Ex32.m1.8.8.1"><eq id="S3.Ex32.m1.8.8.1.1.3.cmml" xref="S3.Ex32.m1.8.8.1.1.3"></eq><apply id="S3.Ex32.m1.8.8.1.1.4.cmml" xref="S3.Ex32.m1.8.8.1.1.4"><times id="S3.Ex32.m1.8.8.1.1.4.1.cmml" xref="S3.Ex32.m1.8.8.1.1.4.1"></times><apply id="S3.Ex32.m1.8.8.1.1.4.2.cmml" xref="S3.Ex32.m1.8.8.1.1.4.2"><csymbol cd="ambiguous" id="S3.Ex32.m1.8.8.1.1.4.2.1.cmml" xref="S3.Ex32.m1.8.8.1.1.4.2">subscript</csymbol><ci id="S3.Ex32.m1.8.8.1.1.4.2.2.cmml" xref="S3.Ex32.m1.8.8.1.1.4.2.2">ℬ</ci><ci id="S3.Ex32.m1.8.8.1.1.4.2.3.cmml" xref="S3.Ex32.m1.8.8.1.1.4.2.3">𝑞</ci></apply><ci id="S3.Ex32.m1.1.1.cmml" xref="S3.Ex32.m1.1.1">𝜃</ci></apply><apply id="S3.Ex32.m1.8.8.1.1.2.3.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2"><csymbol cd="latexml" id="S3.Ex32.m1.8.8.1.1.2.3.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.3">conditional-set</csymbol><interval closure="open" id="S3.Ex32.m1.8.8.1.1.1.1.1.1.cmml" xref="S3.Ex32.m1.8.8.1.1.1.1.1.2"><ci id="S3.Ex32.m1.2.2.cmml" xref="S3.Ex32.m1.2.2">𝜎</ci><ci id="S3.Ex32.m1.3.3.cmml" xref="S3.Ex32.m1.3.3">𝑓</ci></interval><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2"><ci id="S3.Ex32.m1.8.8.1.1.2.2.2.2.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.2">:</ci><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.1"><in id="S3.Ex32.m1.8.8.1.1.2.2.2.1.2.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.1.2"></in><ci id="S3.Ex32.m1.8.8.1.1.2.2.2.1.3.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.1.3">𝜎</ci><list id="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.2.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1"><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1.1.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1.1">subscript</csymbol><ci id="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1.1.2.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1.1.2">𝑋</ci><ci id="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1.1.3.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.1.1.1.1.3">𝑞</ci></apply><ci id="S3.Ex32.m1.4.4.cmml" xref="S3.Ex32.m1.4.4">𝑓</ci></list></apply><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.3.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3"><and id="S3.Ex32.m1.8.8.1.1.2.2.2.3a.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3"></and><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.3b.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3"><ci id="S3.Ex32.m1.8.8.1.1.2.2.2.3.3.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.3">→</ci><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.3.2.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.2.2"><csymbol cd="latexml" id="S3.Ex32.m1.8.8.1.1.2.2.2.3.2.1.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.2.2.1">delimited-[]</csymbol><ci id="S3.Ex32.m1.5.5.cmml" xref="S3.Ex32.m1.5.5">𝑞</ci></apply><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4"><times id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.1"></times><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.2"><csymbol cd="latexml" id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.1.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.2.2.1">delimited-[]</csymbol><ci id="S3.Ex32.m1.6.6.cmml" xref="S3.Ex32.m1.6.6">𝑛</ci></apply><ci id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.3a.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.3"><mtext id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.3.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.3"> such that </mtext></ci><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4"><csymbol cd="ambiguous" id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.1.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4">superscript</csymbol><ci id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.2.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.2">𝑓</ci><times id="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.3.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.4.4.3"></times></apply><ci id="S3.Ex32.m1.7.7.cmml" xref="S3.Ex32.m1.7.7">𝜃</ci></apply></apply><apply id="S3.Ex32.m1.8.8.1.1.2.2.2.3c.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3"><eq id="S3.Ex32.m1.8.8.1.1.2.2.2.3.5.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.5"></eq><share href="https://arxiv.org/html/2503.14659v1#S3.Ex32.m1.8.8.1.1.2.2.2.3.4.cmml" id="S3.Ex32.m1.8.8.1.1.2.2.2.3d.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3"></share><ci id="S3.Ex32.m1.8.8.1.1.2.2.2.3.6.cmml" xref="S3.Ex32.m1.8.8.1.1.2.2.2.3.6">𝜎</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex32.m1.8c">\mathcal{B}_{q}(\theta)=\{(\sigma,f)\,|\,\sigma\in X_{q},f:[q]\to[n]\text{ % such that }f^{*}(\theta)=\sigma\}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex32.m1.8d">caligraphic_B start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_θ ) = { ( italic_σ , italic_f ) | italic_σ ∈ italic_X start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , italic_f : [ italic_q ] → [ italic_n ] such that italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_θ ) = italic_σ } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.1.p1.10">The boundary map <math alttext="\partial_{q}:P_{q}(\theta)\to P_{q-1}(\theta)" class="ltx_Math" display="inline" id="S3.SS2.1.p1.10.m1.2"><semantics id="S3.SS2.1.p1.10.m1.2a"><mrow id="S3.SS2.1.p1.10.m1.2.3" xref="S3.SS2.1.p1.10.m1.2.3.cmml"><msub id="S3.SS2.1.p1.10.m1.2.3.2" xref="S3.SS2.1.p1.10.m1.2.3.2.cmml"><mo id="S3.SS2.1.p1.10.m1.2.3.2.2" xref="S3.SS2.1.p1.10.m1.2.3.2.2.cmml">∂</mo><mi id="S3.SS2.1.p1.10.m1.2.3.2.3" xref="S3.SS2.1.p1.10.m1.2.3.2.3.cmml">q</mi></msub><mo id="S3.SS2.1.p1.10.m1.2.3.1" rspace="0.278em" xref="S3.SS2.1.p1.10.m1.2.3.1.cmml">:</mo><mrow id="S3.SS2.1.p1.10.m1.2.3.3" xref="S3.SS2.1.p1.10.m1.2.3.3.cmml"><mrow id="S3.SS2.1.p1.10.m1.2.3.3.2" xref="S3.SS2.1.p1.10.m1.2.3.3.2.cmml"><msub id="S3.SS2.1.p1.10.m1.2.3.3.2.2" xref="S3.SS2.1.p1.10.m1.2.3.3.2.2.cmml"><mi id="S3.SS2.1.p1.10.m1.2.3.3.2.2.2" xref="S3.SS2.1.p1.10.m1.2.3.3.2.2.2.cmml">P</mi><mi id="S3.SS2.1.p1.10.m1.2.3.3.2.2.3" xref="S3.SS2.1.p1.10.m1.2.3.3.2.2.3.cmml">q</mi></msub><mo id="S3.SS2.1.p1.10.m1.2.3.3.2.1" xref="S3.SS2.1.p1.10.m1.2.3.3.2.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.10.m1.2.3.3.2.3.2" xref="S3.SS2.1.p1.10.m1.2.3.3.2.cmml"><mo id="S3.SS2.1.p1.10.m1.2.3.3.2.3.2.1" stretchy="false" xref="S3.SS2.1.p1.10.m1.2.3.3.2.cmml">(</mo><mi id="S3.SS2.1.p1.10.m1.1.1" xref="S3.SS2.1.p1.10.m1.1.1.cmml">θ</mi><mo id="S3.SS2.1.p1.10.m1.2.3.3.2.3.2.2" stretchy="false" xref="S3.SS2.1.p1.10.m1.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS2.1.p1.10.m1.2.3.3.1" stretchy="false" xref="S3.SS2.1.p1.10.m1.2.3.3.1.cmml">→</mo><mrow id="S3.SS2.1.p1.10.m1.2.3.3.3" xref="S3.SS2.1.p1.10.m1.2.3.3.3.cmml"><msub id="S3.SS2.1.p1.10.m1.2.3.3.3.2" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.cmml"><mi id="S3.SS2.1.p1.10.m1.2.3.3.3.2.2" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.2.cmml">P</mi><mrow id="S3.SS2.1.p1.10.m1.2.3.3.3.2.3" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.cmml"><mi id="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.2" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.2.cmml">q</mi><mo id="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.1" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.1.cmml">−</mo><mn id="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.3" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS2.1.p1.10.m1.2.3.3.3.1" xref="S3.SS2.1.p1.10.m1.2.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.10.m1.2.3.3.3.3.2" xref="S3.SS2.1.p1.10.m1.2.3.3.3.cmml"><mo id="S3.SS2.1.p1.10.m1.2.3.3.3.3.2.1" stretchy="false" xref="S3.SS2.1.p1.10.m1.2.3.3.3.cmml">(</mo><mi id="S3.SS2.1.p1.10.m1.2.2" xref="S3.SS2.1.p1.10.m1.2.2.cmml">θ</mi><mo id="S3.SS2.1.p1.10.m1.2.3.3.3.3.2.2" stretchy="false" xref="S3.SS2.1.p1.10.m1.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.10.m1.2b"><apply id="S3.SS2.1.p1.10.m1.2.3.cmml" xref="S3.SS2.1.p1.10.m1.2.3"><ci id="S3.SS2.1.p1.10.m1.2.3.1.cmml" xref="S3.SS2.1.p1.10.m1.2.3.1">:</ci><apply id="S3.SS2.1.p1.10.m1.2.3.2.cmml" xref="S3.SS2.1.p1.10.m1.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.10.m1.2.3.2.1.cmml" xref="S3.SS2.1.p1.10.m1.2.3.2">subscript</csymbol><partialdiff id="S3.SS2.1.p1.10.m1.2.3.2.2.cmml" xref="S3.SS2.1.p1.10.m1.2.3.2.2"></partialdiff><ci id="S3.SS2.1.p1.10.m1.2.3.2.3.cmml" xref="S3.SS2.1.p1.10.m1.2.3.2.3">𝑞</ci></apply><apply id="S3.SS2.1.p1.10.m1.2.3.3.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3"><ci id="S3.SS2.1.p1.10.m1.2.3.3.1.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.1">→</ci><apply id="S3.SS2.1.p1.10.m1.2.3.3.2.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.2"><times id="S3.SS2.1.p1.10.m1.2.3.3.2.1.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.2.1"></times><apply id="S3.SS2.1.p1.10.m1.2.3.3.2.2.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.10.m1.2.3.3.2.2.1.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.2.2">subscript</csymbol><ci id="S3.SS2.1.p1.10.m1.2.3.3.2.2.2.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.2.2.2">𝑃</ci><ci id="S3.SS2.1.p1.10.m1.2.3.3.2.2.3.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.2.2.3">𝑞</ci></apply><ci id="S3.SS2.1.p1.10.m1.1.1.cmml" xref="S3.SS2.1.p1.10.m1.1.1">𝜃</ci></apply><apply id="S3.SS2.1.p1.10.m1.2.3.3.3.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.3"><times id="S3.SS2.1.p1.10.m1.2.3.3.3.1.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.3.1"></times><apply id="S3.SS2.1.p1.10.m1.2.3.3.3.2.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.10.m1.2.3.3.3.2.1.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2">subscript</csymbol><ci id="S3.SS2.1.p1.10.m1.2.3.3.3.2.2.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.2">𝑃</ci><apply id="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.3"><minus id="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.1.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.1"></minus><ci id="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.2.cmml" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.2">𝑞</ci><cn id="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.3.cmml" type="integer" xref="S3.SS2.1.p1.10.m1.2.3.3.3.2.3.3">1</cn></apply></apply><ci id="S3.SS2.1.p1.10.m1.2.2.cmml" xref="S3.SS2.1.p1.10.m1.2.2">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.10.m1.2c">\partial_{q}:P_{q}(\theta)\to P_{q-1}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.10.m1.2d">∂ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT : italic_P start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_θ ) → italic_P start_POSTSUBSCRIPT italic_q - 1 end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex33"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\partial_{q}(\sigma,f)=\sum_{i=0}^{q}(-1)^{i}(d_{i}\sigma,f\circ d^{i})." class="ltx_Math" display="block" id="S3.Ex33.m1.3"><semantics id="S3.Ex33.m1.3a"><mrow id="S3.Ex33.m1.3.3.1" xref="S3.Ex33.m1.3.3.1.1.cmml"><mrow id="S3.Ex33.m1.3.3.1.1" xref="S3.Ex33.m1.3.3.1.1.cmml"><mrow id="S3.Ex33.m1.3.3.1.1.5" xref="S3.Ex33.m1.3.3.1.1.5.cmml"><msub id="S3.Ex33.m1.3.3.1.1.5.1" xref="S3.Ex33.m1.3.3.1.1.5.1.cmml"><mo id="S3.Ex33.m1.3.3.1.1.5.1.2" xref="S3.Ex33.m1.3.3.1.1.5.1.2.cmml">∂</mo><mi id="S3.Ex33.m1.3.3.1.1.5.1.3" xref="S3.Ex33.m1.3.3.1.1.5.1.3.cmml">q</mi></msub><mrow id="S3.Ex33.m1.3.3.1.1.5.2.2" xref="S3.Ex33.m1.3.3.1.1.5.2.1.cmml"><mo id="S3.Ex33.m1.3.3.1.1.5.2.2.1" lspace="0em" stretchy="false" xref="S3.Ex33.m1.3.3.1.1.5.2.1.cmml">(</mo><mi id="S3.Ex33.m1.1.1" xref="S3.Ex33.m1.1.1.cmml">σ</mi><mo id="S3.Ex33.m1.3.3.1.1.5.2.2.2" xref="S3.Ex33.m1.3.3.1.1.5.2.1.cmml">,</mo><mi id="S3.Ex33.m1.2.2" xref="S3.Ex33.m1.2.2.cmml">f</mi><mo id="S3.Ex33.m1.3.3.1.1.5.2.2.3" stretchy="false" xref="S3.Ex33.m1.3.3.1.1.5.2.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex33.m1.3.3.1.1.4" rspace="0.111em" xref="S3.Ex33.m1.3.3.1.1.4.cmml">=</mo><mrow id="S3.Ex33.m1.3.3.1.1.3" xref="S3.Ex33.m1.3.3.1.1.3.cmml"><munderover 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xref="S3.Ex33.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex33.m1.3.3.1.1.1.1.1.1.1.1a" xref="S3.Ex33.m1.3.3.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.Ex33.m1.3.3.1.1.1.1.1.1.1.1.2" xref="S3.Ex33.m1.3.3.1.1.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S3.Ex33.m1.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex33.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.Ex33.m1.3.3.1.1.1.1.1.3" xref="S3.Ex33.m1.3.3.1.1.1.1.1.3.cmml">i</mi></msup><mo id="S3.Ex33.m1.3.3.1.1.3.3.4" xref="S3.Ex33.m1.3.3.1.1.3.3.4.cmml">⁢</mo><mrow id="S3.Ex33.m1.3.3.1.1.3.3.3.2" xref="S3.Ex33.m1.3.3.1.1.3.3.3.3.cmml"><mo id="S3.Ex33.m1.3.3.1.1.3.3.3.2.3" stretchy="false" xref="S3.Ex33.m1.3.3.1.1.3.3.3.3.cmml">(</mo><mrow id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.cmml"><msub id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.cmml"><mi id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.2" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.2.cmml">d</mi><mi id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.3" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.3.cmml">i</mi></msub><mo id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.1" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.1.cmml">⁢</mo><mi id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.3" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.3.cmml">σ</mi></mrow><mo id="S3.Ex33.m1.3.3.1.1.3.3.3.2.4" xref="S3.Ex33.m1.3.3.1.1.3.3.3.3.cmml">,</mo><mrow id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.cmml"><mi id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.2" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.2.cmml">f</mi><mo id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.1.cmml">∘</mo><msup id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3.cmml"><mi id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3.2" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3.2.cmml">d</mi><mi id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3.3" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3.3.cmml">i</mi></msup></mrow><mo id="S3.Ex33.m1.3.3.1.1.3.3.3.2.5" stretchy="false" 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xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.1"></times><apply id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.cmml" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.1.cmml" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2">subscript</csymbol><ci id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.2.cmml" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.2">𝑑</ci><ci id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.3.cmml" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.2.3">𝑖</ci></apply><ci id="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.3.cmml" xref="S3.Ex33.m1.3.3.1.1.2.2.2.1.1.3">𝜎</ci></apply><apply id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.cmml" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2"><compose id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.1.cmml" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.1"></compose><ci id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.2.cmml" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.2">𝑓</ci><apply id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3.cmml" xref="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3"><csymbol cd="ambiguous" id="S3.Ex33.m1.3.3.1.1.3.3.3.2.2.3.1.cmml" 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</tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS2.2.p2"> <p class="ltx_p" id="S3.SS2.2.p2.7">For each <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S3.SS2.2.p2.1.m1.1"><semantics id="S3.SS2.2.p2.1.m1.1a"><mrow id="S3.SS2.2.p2.1.m1.1.1" xref="S3.SS2.2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.2.p2.1.m1.1.1.2" xref="S3.SS2.2.p2.1.m1.1.1.2.cmml">n</mi><mo id="S3.SS2.2.p2.1.m1.1.1.1" xref="S3.SS2.2.p2.1.m1.1.1.1.cmml">≥</mo><mn id="S3.SS2.2.p2.1.m1.1.1.3" xref="S3.SS2.2.p2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.1.m1.1b"><apply id="S3.SS2.2.p2.1.m1.1.1.cmml" xref="S3.SS2.2.p2.1.m1.1.1"><geq id="S3.SS2.2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.2.p2.1.m1.1.1.1"></geq><ci id="S3.SS2.2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.2.p2.1.m1.1.1.2">𝑛</ci><cn id="S3.SS2.2.p2.1.m1.1.1.3.cmml" type="integer" xref="S3.SS2.2.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.1.m1.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.1.m1.1d">italic_n ≥ 0</annotation></semantics></math>, let <math alttext="\Delta[n]" class="ltx_Math" display="inline" id="S3.SS2.2.p2.2.m2.1"><semantics id="S3.SS2.2.p2.2.m2.1a"><mrow id="S3.SS2.2.p2.2.m2.1.2" xref="S3.SS2.2.p2.2.m2.1.2.cmml"><mi id="S3.SS2.2.p2.2.m2.1.2.2" mathvariant="normal" xref="S3.SS2.2.p2.2.m2.1.2.2.cmml">Δ</mi><mo id="S3.SS2.2.p2.2.m2.1.2.1" xref="S3.SS2.2.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.2.p2.2.m2.1.2.3.2" xref="S3.SS2.2.p2.2.m2.1.2.3.1.cmml"><mo id="S3.SS2.2.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS2.2.p2.2.m2.1.2.3.1.1.cmml">[</mo><mi id="S3.SS2.2.p2.2.m2.1.1" xref="S3.SS2.2.p2.2.m2.1.1.cmml">n</mi><mo id="S3.SS2.2.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS2.2.p2.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.2.m2.1b"><apply id="S3.SS2.2.p2.2.m2.1.2.cmml" xref="S3.SS2.2.p2.2.m2.1.2"><times id="S3.SS2.2.p2.2.m2.1.2.1.cmml" xref="S3.SS2.2.p2.2.m2.1.2.1"></times><ci id="S3.SS2.2.p2.2.m2.1.2.2.cmml" xref="S3.SS2.2.p2.2.m2.1.2.2">Δ</ci><apply id="S3.SS2.2.p2.2.m2.1.2.3.1.cmml" xref="S3.SS2.2.p2.2.m2.1.2.3.2"><csymbol cd="latexml" id="S3.SS2.2.p2.2.m2.1.2.3.1.1.cmml" xref="S3.SS2.2.p2.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS2.2.p2.2.m2.1.1.cmml" xref="S3.SS2.2.p2.2.m2.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.2.m2.1c">\Delta[n]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.2.m2.1d">roman_Δ [ italic_n ]</annotation></semantics></math> denote the simplicial set whose <math alttext="k" class="ltx_Math" display="inline" id="S3.SS2.2.p2.3.m3.1"><semantics id="S3.SS2.2.p2.3.m3.1a"><mi id="S3.SS2.2.p2.3.m3.1.1" xref="S3.SS2.2.p2.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.3.m3.1b"><ci id="S3.SS2.2.p2.3.m3.1.1.cmml" 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xref="S3.SS2.2.p2.4.m4.6.6.3.4.3.cmml">Δ</mi></msub><mo id="S3.SS2.2.p2.4.m4.6.6.3.3" xref="S3.SS2.2.p2.4.m4.6.6.3.3.cmml">⁢</mo><mrow id="S3.SS2.2.p2.4.m4.6.6.3.2.2" xref="S3.SS2.2.p2.4.m4.6.6.3.2.3.cmml"><mo id="S3.SS2.2.p2.4.m4.6.6.3.2.2.3" stretchy="false" xref="S3.SS2.2.p2.4.m4.6.6.3.2.3.cmml">(</mo><mrow id="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.2" xref="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.1.cmml"><mo id="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.2.1" stretchy="false" xref="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.1.1.cmml">[</mo><mi id="S3.SS2.2.p2.4.m4.2.2" xref="S3.SS2.2.p2.4.m4.2.2.cmml">k</mi><mo id="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.2.2" stretchy="false" xref="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS2.2.p2.4.m4.6.6.3.2.2.4" xref="S3.SS2.2.p2.4.m4.6.6.3.2.3.cmml">,</mo><mrow id="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.2" xref="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.1.cmml"><mo id="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.2.1" stretchy="false" xref="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.1.1.cmml">[</mo><mi id="S3.SS2.2.p2.4.m4.3.3" xref="S3.SS2.2.p2.4.m4.3.3.cmml">n</mi><mo id="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.2.2" stretchy="false" xref="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.1.1.cmml">]</mo></mrow><mo id="S3.SS2.2.p2.4.m4.6.6.3.2.2.5" stretchy="false" xref="S3.SS2.2.p2.4.m4.6.6.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.4.m4.6b"><apply id="S3.SS2.2.p2.4.m4.6.6.cmml" xref="S3.SS2.2.p2.4.m4.6.6"><eq id="S3.SS2.2.p2.4.m4.6.6.4.cmml" xref="S3.SS2.2.p2.4.m4.6.6.4"></eq><apply id="S3.SS2.2.p2.4.m4.4.4.1.cmml" xref="S3.SS2.2.p2.4.m4.4.4.1"><csymbol cd="ambiguous" id="S3.SS2.2.p2.4.m4.4.4.1.2.cmml" xref="S3.SS2.2.p2.4.m4.4.4.1">subscript</csymbol><apply id="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.cmml" xref="S3.SS2.2.p2.4.m4.4.4.1.1.1"><times id="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.1"></times><ci id="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.2.cmml" xref="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.2">Δ</ci><apply id="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.3.1.cmml" xref="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.3.2"><csymbol cd="latexml" id="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.3.1.1.cmml" xref="S3.SS2.2.p2.4.m4.4.4.1.1.1.1.3.2.1">delimited-[]</csymbol><ci id="S3.SS2.2.p2.4.m4.1.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1">𝑛</ci></apply></apply><ci id="S3.SS2.2.p2.4.m4.4.4.1.3.cmml" xref="S3.SS2.2.p2.4.m4.4.4.1.3">𝑘</ci></apply><apply id="S3.SS2.2.p2.4.m4.6.6.3.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3"><times id="S3.SS2.2.p2.4.m4.6.6.3.3.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3.3"></times><apply id="S3.SS2.2.p2.4.m4.6.6.3.4.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3.4"><csymbol cd="ambiguous" id="S3.SS2.2.p2.4.m4.6.6.3.4.1.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3.4">subscript</csymbol><ci id="S3.SS2.2.p2.4.m4.6.6.3.4.2.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3.4.2">Mor</ci><ci id="S3.SS2.2.p2.4.m4.6.6.3.4.3.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3.4.3">Δ</ci></apply><interval closure="open" id="S3.SS2.2.p2.4.m4.6.6.3.2.3.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3.2.2"><apply id="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.1.cmml" xref="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.2"><csymbol cd="latexml" id="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.4.m4.5.5.2.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS2.2.p2.4.m4.2.2.cmml" xref="S3.SS2.2.p2.4.m4.2.2">𝑘</ci></apply><apply id="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.1.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.2"><csymbol cd="latexml" id="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.1.1.cmml" xref="S3.SS2.2.p2.4.m4.6.6.3.2.2.2.2.1">delimited-[]</csymbol><ci id="S3.SS2.2.p2.4.m4.3.3.cmml" xref="S3.SS2.2.p2.4.m4.3.3">𝑛</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.4.m4.6c">(\Delta[n])_{k}=\mathrm{Mor}_{\Delta}([k],[n])</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.4.m4.6d">( roman_Δ [ italic_n ] ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = roman_Mor start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT ( [ italic_k ] , [ italic_n ] )</annotation></semantics></math>. Note that for every <math alttext="\theta\in X_{n}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.5.m5.1"><semantics id="S3.SS2.2.p2.5.m5.1a"><mrow id="S3.SS2.2.p2.5.m5.1.1" xref="S3.SS2.2.p2.5.m5.1.1.cmml"><mi id="S3.SS2.2.p2.5.m5.1.1.2" xref="S3.SS2.2.p2.5.m5.1.1.2.cmml">θ</mi><mo id="S3.SS2.2.p2.5.m5.1.1.1" xref="S3.SS2.2.p2.5.m5.1.1.1.cmml">∈</mo><msub id="S3.SS2.2.p2.5.m5.1.1.3" xref="S3.SS2.2.p2.5.m5.1.1.3.cmml"><mi id="S3.SS2.2.p2.5.m5.1.1.3.2" xref="S3.SS2.2.p2.5.m5.1.1.3.2.cmml">X</mi><mi id="S3.SS2.2.p2.5.m5.1.1.3.3" xref="S3.SS2.2.p2.5.m5.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.5.m5.1b"><apply id="S3.SS2.2.p2.5.m5.1.1.cmml" xref="S3.SS2.2.p2.5.m5.1.1"><in id="S3.SS2.2.p2.5.m5.1.1.1.cmml" xref="S3.SS2.2.p2.5.m5.1.1.1"></in><ci id="S3.SS2.2.p2.5.m5.1.1.2.cmml" xref="S3.SS2.2.p2.5.m5.1.1.2">𝜃</ci><apply id="S3.SS2.2.p2.5.m5.1.1.3.cmml" xref="S3.SS2.2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.5.m5.1.1.3.1.cmml" xref="S3.SS2.2.p2.5.m5.1.1.3">subscript</csymbol><ci id="S3.SS2.2.p2.5.m5.1.1.3.2.cmml" xref="S3.SS2.2.p2.5.m5.1.1.3.2">𝑋</ci><ci id="S3.SS2.2.p2.5.m5.1.1.3.3.cmml" xref="S3.SS2.2.p2.5.m5.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.5.m5.1c">\theta\in X_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.5.m5.1d">italic_θ ∈ italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, the set <math alttext="\mathcal{B}_{q}(\theta)" class="ltx_Math" display="inline" id="S3.SS2.2.p2.6.m6.1"><semantics id="S3.SS2.2.p2.6.m6.1a"><mrow id="S3.SS2.2.p2.6.m6.1.2" xref="S3.SS2.2.p2.6.m6.1.2.cmml"><msub id="S3.SS2.2.p2.6.m6.1.2.2" xref="S3.SS2.2.p2.6.m6.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.6.m6.1.2.2.2" xref="S3.SS2.2.p2.6.m6.1.2.2.2.cmml">ℬ</mi><mi id="S3.SS2.2.p2.6.m6.1.2.2.3" xref="S3.SS2.2.p2.6.m6.1.2.2.3.cmml">q</mi></msub><mo id="S3.SS2.2.p2.6.m6.1.2.1" xref="S3.SS2.2.p2.6.m6.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.2.p2.6.m6.1.2.3.2" xref="S3.SS2.2.p2.6.m6.1.2.cmml"><mo id="S3.SS2.2.p2.6.m6.1.2.3.2.1" stretchy="false" xref="S3.SS2.2.p2.6.m6.1.2.cmml">(</mo><mi id="S3.SS2.2.p2.6.m6.1.1" xref="S3.SS2.2.p2.6.m6.1.1.cmml">θ</mi><mo id="S3.SS2.2.p2.6.m6.1.2.3.2.2" stretchy="false" xref="S3.SS2.2.p2.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.6.m6.1b"><apply id="S3.SS2.2.p2.6.m6.1.2.cmml" xref="S3.SS2.2.p2.6.m6.1.2"><times id="S3.SS2.2.p2.6.m6.1.2.1.cmml" xref="S3.SS2.2.p2.6.m6.1.2.1"></times><apply id="S3.SS2.2.p2.6.m6.1.2.2.cmml" xref="S3.SS2.2.p2.6.m6.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.6.m6.1.2.2.1.cmml" xref="S3.SS2.2.p2.6.m6.1.2.2">subscript</csymbol><ci id="S3.SS2.2.p2.6.m6.1.2.2.2.cmml" xref="S3.SS2.2.p2.6.m6.1.2.2.2">ℬ</ci><ci id="S3.SS2.2.p2.6.m6.1.2.2.3.cmml" xref="S3.SS2.2.p2.6.m6.1.2.2.3">𝑞</ci></apply><ci id="S3.SS2.2.p2.6.m6.1.1.cmml" xref="S3.SS2.2.p2.6.m6.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.6.m6.1c">\mathcal{B}_{q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.6.m6.1d">caligraphic_B start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> is isomorphic to <math alttext="\mathrm{Mor}_{\Delta}([q],[n])" class="ltx_Math" display="inline" id="S3.SS2.2.p2.7.m7.4"><semantics id="S3.SS2.2.p2.7.m7.4a"><mrow id="S3.SS2.2.p2.7.m7.4.4" xref="S3.SS2.2.p2.7.m7.4.4.cmml"><msub id="S3.SS2.2.p2.7.m7.4.4.4" xref="S3.SS2.2.p2.7.m7.4.4.4.cmml"><mi id="S3.SS2.2.p2.7.m7.4.4.4.2" xref="S3.SS2.2.p2.7.m7.4.4.4.2.cmml">Mor</mi><mi id="S3.SS2.2.p2.7.m7.4.4.4.3" mathvariant="normal" xref="S3.SS2.2.p2.7.m7.4.4.4.3.cmml">Δ</mi></msub><mo id="S3.SS2.2.p2.7.m7.4.4.3" xref="S3.SS2.2.p2.7.m7.4.4.3.cmml">⁢</mo><mrow id="S3.SS2.2.p2.7.m7.4.4.2.2" xref="S3.SS2.2.p2.7.m7.4.4.2.3.cmml"><mo id="S3.SS2.2.p2.7.m7.4.4.2.2.3" stretchy="false" xref="S3.SS2.2.p2.7.m7.4.4.2.3.cmml">(</mo><mrow id="S3.SS2.2.p2.7.m7.3.3.1.1.1.2" xref="S3.SS2.2.p2.7.m7.3.3.1.1.1.1.cmml"><mo id="S3.SS2.2.p2.7.m7.3.3.1.1.1.2.1" stretchy="false" xref="S3.SS2.2.p2.7.m7.3.3.1.1.1.1.1.cmml">[</mo><mi id="S3.SS2.2.p2.7.m7.1.1" xref="S3.SS2.2.p2.7.m7.1.1.cmml">q</mi><mo id="S3.SS2.2.p2.7.m7.3.3.1.1.1.2.2" stretchy="false" xref="S3.SS2.2.p2.7.m7.3.3.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS2.2.p2.7.m7.4.4.2.2.4" xref="S3.SS2.2.p2.7.m7.4.4.2.3.cmml">,</mo><mrow id="S3.SS2.2.p2.7.m7.4.4.2.2.2.2" xref="S3.SS2.2.p2.7.m7.4.4.2.2.2.1.cmml"><mo id="S3.SS2.2.p2.7.m7.4.4.2.2.2.2.1" stretchy="false" xref="S3.SS2.2.p2.7.m7.4.4.2.2.2.1.1.cmml">[</mo><mi id="S3.SS2.2.p2.7.m7.2.2" xref="S3.SS2.2.p2.7.m7.2.2.cmml">n</mi><mo id="S3.SS2.2.p2.7.m7.4.4.2.2.2.2.2" stretchy="false" xref="S3.SS2.2.p2.7.m7.4.4.2.2.2.1.1.cmml">]</mo></mrow><mo id="S3.SS2.2.p2.7.m7.4.4.2.2.5" stretchy="false" xref="S3.SS2.2.p2.7.m7.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.7.m7.4b"><apply id="S3.SS2.2.p2.7.m7.4.4.cmml" xref="S3.SS2.2.p2.7.m7.4.4"><times id="S3.SS2.2.p2.7.m7.4.4.3.cmml" xref="S3.SS2.2.p2.7.m7.4.4.3"></times><apply id="S3.SS2.2.p2.7.m7.4.4.4.cmml" xref="S3.SS2.2.p2.7.m7.4.4.4"><csymbol cd="ambiguous" id="S3.SS2.2.p2.7.m7.4.4.4.1.cmml" xref="S3.SS2.2.p2.7.m7.4.4.4">subscript</csymbol><ci id="S3.SS2.2.p2.7.m7.4.4.4.2.cmml" xref="S3.SS2.2.p2.7.m7.4.4.4.2">Mor</ci><ci id="S3.SS2.2.p2.7.m7.4.4.4.3.cmml" xref="S3.SS2.2.p2.7.m7.4.4.4.3">Δ</ci></apply><interval closure="open" id="S3.SS2.2.p2.7.m7.4.4.2.3.cmml" xref="S3.SS2.2.p2.7.m7.4.4.2.2"><apply id="S3.SS2.2.p2.7.m7.3.3.1.1.1.1.cmml" xref="S3.SS2.2.p2.7.m7.3.3.1.1.1.2"><csymbol cd="latexml" id="S3.SS2.2.p2.7.m7.3.3.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.7.m7.3.3.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS2.2.p2.7.m7.1.1.cmml" xref="S3.SS2.2.p2.7.m7.1.1">𝑞</ci></apply><apply id="S3.SS2.2.p2.7.m7.4.4.2.2.2.1.cmml" xref="S3.SS2.2.p2.7.m7.4.4.2.2.2.2"><csymbol cd="latexml" id="S3.SS2.2.p2.7.m7.4.4.2.2.2.1.1.cmml" xref="S3.SS2.2.p2.7.m7.4.4.2.2.2.2.1">delimited-[]</csymbol><ci id="S3.SS2.2.p2.7.m7.2.2.cmml" xref="S3.SS2.2.p2.7.m7.2.2">𝑛</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.7.m7.4c">\mathrm{Mor}_{\Delta}([q],[n])</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.7.m7.4d">roman_Mor start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT ( [ italic_q ] , [ italic_n ] )</annotation></semantics></math>. Hence the homology of the chain complex</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex34"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{P}_{*}:\cdots\to P_{q}(\theta)\smash{\,\mathop{\longrightarrow}% \limits^{\partial_{q}}\,}P_{q-1}(\theta)\to\cdots\to P_{0}(\theta)\to R\to 0" class="ltx_Math" display="block" id="S3.Ex34.m1.3"><semantics id="S3.Ex34.m1.3a"><mrow id="S3.Ex34.m1.3.4" xref="S3.Ex34.m1.3.4.cmml"><msub id="S3.Ex34.m1.3.4.2" xref="S3.Ex34.m1.3.4.2.cmml"><mover accent="true" id="S3.Ex34.m1.3.4.2.2" xref="S3.Ex34.m1.3.4.2.2.cmml"><mi id="S3.Ex34.m1.3.4.2.2.2" xref="S3.Ex34.m1.3.4.2.2.2.cmml">P</mi><mo id="S3.Ex34.m1.3.4.2.2.1" xref="S3.Ex34.m1.3.4.2.2.1.cmml">~</mo></mover><mo id="S3.Ex34.m1.3.4.2.3" xref="S3.Ex34.m1.3.4.2.3.cmml">∗</mo></msub><mo id="S3.Ex34.m1.3.4.1" lspace="0.278em" rspace="0.278em" xref="S3.Ex34.m1.3.4.1.cmml">:</mo><mrow id="S3.Ex34.m1.3.4.3" xref="S3.Ex34.m1.3.4.3.cmml"><mi id="S3.Ex34.m1.3.4.3.2" mathvariant="normal" xref="S3.Ex34.m1.3.4.3.2.cmml">⋯</mi><mo id="S3.Ex34.m1.3.4.3.3" stretchy="false" xref="S3.Ex34.m1.3.4.3.3.cmml">→</mo><mrow id="S3.Ex34.m1.3.4.3.4" xref="S3.Ex34.m1.3.4.3.4.cmml"><msub id="S3.Ex34.m1.3.4.3.4.2" xref="S3.Ex34.m1.3.4.3.4.2.cmml"><mi id="S3.Ex34.m1.3.4.3.4.2.2" xref="S3.Ex34.m1.3.4.3.4.2.2.cmml">P</mi><mi id="S3.Ex34.m1.3.4.3.4.2.3" xref="S3.Ex34.m1.3.4.3.4.2.3.cmml">q</mi></msub><mo id="S3.Ex34.m1.3.4.3.4.1" xref="S3.Ex34.m1.3.4.3.4.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.3.4.3.4.3.2" xref="S3.Ex34.m1.3.4.3.4.cmml"><mo id="S3.Ex34.m1.3.4.3.4.3.2.1" stretchy="false" xref="S3.Ex34.m1.3.4.3.4.cmml">(</mo><mi id="S3.Ex34.m1.1.1" xref="S3.Ex34.m1.1.1.cmml">θ</mi><mo id="S3.Ex34.m1.3.4.3.4.3.2.2" stretchy="false" xref="S3.Ex34.m1.3.4.3.4.cmml">)</mo></mrow><mo id="S3.Ex34.m1.3.4.3.4.1a" lspace="0.337em" xref="S3.Ex34.m1.3.4.3.4.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.3.4.3.4.4" xref="S3.Ex34.m1.3.4.3.4.4.cmml"><mover id="S3.Ex34.m1.3.4.3.4.4.1" xref="S3.Ex34.m1.3.4.3.4.4.1.cmml"><mo id="S3.Ex34.m1.3.4.3.4.4.1.2" movablelimits="false" rspace="0.167em" xref="S3.Ex34.m1.3.4.3.4.4.1.2.cmml">⟶</mo><msub id="S3.Ex34.m1.3.4.3.4.4.1.3" xref="S3.Ex34.m1.3.4.3.4.4.1.3.cmml"><mo id="S3.Ex34.m1.3.4.3.4.4.1.3.2" xref="S3.Ex34.m1.3.4.3.4.4.1.3.2.cmml">∂</mo><mi id="S3.Ex34.m1.3.4.3.4.4.1.3.3" xref="S3.Ex34.m1.3.4.3.4.4.1.3.3.cmml">q</mi></msub></mover><mrow id="S3.Ex34.m1.3.4.3.4.4.2" xref="S3.Ex34.m1.3.4.3.4.4.2.cmml"><msub id="S3.Ex34.m1.3.4.3.4.4.2.2" xref="S3.Ex34.m1.3.4.3.4.4.2.2.cmml"><mi id="S3.Ex34.m1.3.4.3.4.4.2.2.2" xref="S3.Ex34.m1.3.4.3.4.4.2.2.2.cmml">P</mi><mrow id="S3.Ex34.m1.3.4.3.4.4.2.2.3" xref="S3.Ex34.m1.3.4.3.4.4.2.2.3.cmml"><mi id="S3.Ex34.m1.3.4.3.4.4.2.2.3.2" xref="S3.Ex34.m1.3.4.3.4.4.2.2.3.2.cmml">q</mi><mo id="S3.Ex34.m1.3.4.3.4.4.2.2.3.1" xref="S3.Ex34.m1.3.4.3.4.4.2.2.3.1.cmml">−</mo><mn id="S3.Ex34.m1.3.4.3.4.4.2.2.3.3" xref="S3.Ex34.m1.3.4.3.4.4.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.Ex34.m1.3.4.3.4.4.2.1" xref="S3.Ex34.m1.3.4.3.4.4.2.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.3.4.3.4.4.2.3.2" xref="S3.Ex34.m1.3.4.3.4.4.2.cmml"><mo id="S3.Ex34.m1.3.4.3.4.4.2.3.2.1" stretchy="false" xref="S3.Ex34.m1.3.4.3.4.4.2.cmml">(</mo><mi id="S3.Ex34.m1.2.2" xref="S3.Ex34.m1.2.2.cmml">θ</mi><mo id="S3.Ex34.m1.3.4.3.4.4.2.3.2.2" stretchy="false" xref="S3.Ex34.m1.3.4.3.4.4.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.Ex34.m1.3.4.3.5" stretchy="false" xref="S3.Ex34.m1.3.4.3.5.cmml">→</mo><mi id="S3.Ex34.m1.3.4.3.6" mathvariant="normal" xref="S3.Ex34.m1.3.4.3.6.cmml">⋯</mi><mo id="S3.Ex34.m1.3.4.3.7" stretchy="false" xref="S3.Ex34.m1.3.4.3.7.cmml">→</mo><mrow id="S3.Ex34.m1.3.4.3.8" xref="S3.Ex34.m1.3.4.3.8.cmml"><msub id="S3.Ex34.m1.3.4.3.8.2" xref="S3.Ex34.m1.3.4.3.8.2.cmml"><mi id="S3.Ex34.m1.3.4.3.8.2.2" xref="S3.Ex34.m1.3.4.3.8.2.2.cmml">P</mi><mn id="S3.Ex34.m1.3.4.3.8.2.3" xref="S3.Ex34.m1.3.4.3.8.2.3.cmml">0</mn></msub><mo id="S3.Ex34.m1.3.4.3.8.1" xref="S3.Ex34.m1.3.4.3.8.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.3.4.3.8.3.2" xref="S3.Ex34.m1.3.4.3.8.cmml"><mo id="S3.Ex34.m1.3.4.3.8.3.2.1" stretchy="false" xref="S3.Ex34.m1.3.4.3.8.cmml">(</mo><mi id="S3.Ex34.m1.3.3" xref="S3.Ex34.m1.3.3.cmml">θ</mi><mo id="S3.Ex34.m1.3.4.3.8.3.2.2" stretchy="false" xref="S3.Ex34.m1.3.4.3.8.cmml">)</mo></mrow></mrow><mo id="S3.Ex34.m1.3.4.3.9" stretchy="false" xref="S3.Ex34.m1.3.4.3.9.cmml">→</mo><mi id="S3.Ex34.m1.3.4.3.10" xref="S3.Ex34.m1.3.4.3.10.cmml">R</mi><mo id="S3.Ex34.m1.3.4.3.11" stretchy="false" xref="S3.Ex34.m1.3.4.3.11.cmml">→</mo><mn id="S3.Ex34.m1.3.4.3.12" xref="S3.Ex34.m1.3.4.3.12.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex34.m1.3b"><apply id="S3.Ex34.m1.3.4.cmml" xref="S3.Ex34.m1.3.4"><ci id="S3.Ex34.m1.3.4.1.cmml" xref="S3.Ex34.m1.3.4.1">:</ci><apply 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id="S3.Ex34.m1.3.4.3.5.cmml" xref="S3.Ex34.m1.3.4.3.5">→</ci><share href="https://arxiv.org/html/2503.14659v1#S3.Ex34.m1.3.4.3.4.cmml" id="S3.Ex34.m1.3.4.3d.cmml" xref="S3.Ex34.m1.3.4.3"></share><ci id="S3.Ex34.m1.3.4.3.6.cmml" xref="S3.Ex34.m1.3.4.3.6">⋯</ci></apply><apply id="S3.Ex34.m1.3.4.3e.cmml" xref="S3.Ex34.m1.3.4.3"><ci id="S3.Ex34.m1.3.4.3.7.cmml" xref="S3.Ex34.m1.3.4.3.7">→</ci><share href="https://arxiv.org/html/2503.14659v1#S3.Ex34.m1.3.4.3.6.cmml" id="S3.Ex34.m1.3.4.3f.cmml" xref="S3.Ex34.m1.3.4.3"></share><apply id="S3.Ex34.m1.3.4.3.8.cmml" xref="S3.Ex34.m1.3.4.3.8"><times id="S3.Ex34.m1.3.4.3.8.1.cmml" xref="S3.Ex34.m1.3.4.3.8.1"></times><apply id="S3.Ex34.m1.3.4.3.8.2.cmml" xref="S3.Ex34.m1.3.4.3.8.2"><csymbol cd="ambiguous" id="S3.Ex34.m1.3.4.3.8.2.1.cmml" xref="S3.Ex34.m1.3.4.3.8.2">subscript</csymbol><ci id="S3.Ex34.m1.3.4.3.8.2.2.cmml" xref="S3.Ex34.m1.3.4.3.8.2.2">𝑃</ci><cn id="S3.Ex34.m1.3.4.3.8.2.3.cmml" type="integer" xref="S3.Ex34.m1.3.4.3.8.2.3">0</cn></apply><ci id="S3.Ex34.m1.3.3.cmml" xref="S3.Ex34.m1.3.3">𝜃</ci></apply></apply><apply id="S3.Ex34.m1.3.4.3g.cmml" xref="S3.Ex34.m1.3.4.3"><ci id="S3.Ex34.m1.3.4.3.9.cmml" xref="S3.Ex34.m1.3.4.3.9">→</ci><share href="https://arxiv.org/html/2503.14659v1#S3.Ex34.m1.3.4.3.8.cmml" id="S3.Ex34.m1.3.4.3h.cmml" xref="S3.Ex34.m1.3.4.3"></share><ci id="S3.Ex34.m1.3.4.3.10.cmml" xref="S3.Ex34.m1.3.4.3.10">𝑅</ci></apply><apply id="S3.Ex34.m1.3.4.3i.cmml" xref="S3.Ex34.m1.3.4.3"><ci id="S3.Ex34.m1.3.4.3.11.cmml" xref="S3.Ex34.m1.3.4.3.11">→</ci><share href="https://arxiv.org/html/2503.14659v1#S3.Ex34.m1.3.4.3.10.cmml" id="S3.Ex34.m1.3.4.3j.cmml" xref="S3.Ex34.m1.3.4.3"></share><cn id="S3.Ex34.m1.3.4.3.12.cmml" type="integer" xref="S3.Ex34.m1.3.4.3.12">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex34.m1.3c">\widetilde{P}_{*}:\cdots\to P_{q}(\theta)\smash{\,\mathop{\longrightarrow}% \limits^{\partial_{q}}\,}P_{q-1}(\theta)\to\cdots\to P_{0}(\theta)\to R\to 0</annotation><annotation encoding="application/x-llamapun" id="S3.Ex34.m1.3d">over~ start_ARG italic_P end_ARG start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : ⋯ → italic_P start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_θ ) ⟶ start_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_q - 1 end_POSTSUBSCRIPT ( italic_θ ) → ⋯ → italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_θ ) → italic_R → 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.2.p2.16">is isomorphic to the reduced homology of simplicial set <math alttext="\Delta[n]" class="ltx_Math" display="inline" id="S3.SS2.2.p2.8.m1.1"><semantics id="S3.SS2.2.p2.8.m1.1a"><mrow id="S3.SS2.2.p2.8.m1.1.2" xref="S3.SS2.2.p2.8.m1.1.2.cmml"><mi id="S3.SS2.2.p2.8.m1.1.2.2" mathvariant="normal" xref="S3.SS2.2.p2.8.m1.1.2.2.cmml">Δ</mi><mo id="S3.SS2.2.p2.8.m1.1.2.1" xref="S3.SS2.2.p2.8.m1.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.2.p2.8.m1.1.2.3.2" xref="S3.SS2.2.p2.8.m1.1.2.3.1.cmml"><mo id="S3.SS2.2.p2.8.m1.1.2.3.2.1" stretchy="false" xref="S3.SS2.2.p2.8.m1.1.2.3.1.1.cmml">[</mo><mi id="S3.SS2.2.p2.8.m1.1.1" xref="S3.SS2.2.p2.8.m1.1.1.cmml">n</mi><mo id="S3.SS2.2.p2.8.m1.1.2.3.2.2" stretchy="false" xref="S3.SS2.2.p2.8.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.8.m1.1b"><apply id="S3.SS2.2.p2.8.m1.1.2.cmml" xref="S3.SS2.2.p2.8.m1.1.2"><times id="S3.SS2.2.p2.8.m1.1.2.1.cmml" xref="S3.SS2.2.p2.8.m1.1.2.1"></times><ci id="S3.SS2.2.p2.8.m1.1.2.2.cmml" xref="S3.SS2.2.p2.8.m1.1.2.2">Δ</ci><apply id="S3.SS2.2.p2.8.m1.1.2.3.1.cmml" xref="S3.SS2.2.p2.8.m1.1.2.3.2"><csymbol cd="latexml" id="S3.SS2.2.p2.8.m1.1.2.3.1.1.cmml" xref="S3.SS2.2.p2.8.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS2.2.p2.8.m1.1.1.cmml" xref="S3.SS2.2.p2.8.m1.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.8.m1.1c">\Delta[n]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.8.m1.1d">roman_Δ [ italic_n ]</annotation></semantics></math>. Since <math alttext="|\Delta[n]|=\Delta^{n}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.9.m2.2"><semantics id="S3.SS2.2.p2.9.m2.2a"><mrow id="S3.SS2.2.p2.9.m2.2.2" xref="S3.SS2.2.p2.9.m2.2.2.cmml"><mrow id="S3.SS2.2.p2.9.m2.2.2.1.1" xref="S3.SS2.2.p2.9.m2.2.2.1.2.cmml"><mo id="S3.SS2.2.p2.9.m2.2.2.1.1.2" stretchy="false" xref="S3.SS2.2.p2.9.m2.2.2.1.2.1.cmml">|</mo><mrow id="S3.SS2.2.p2.9.m2.2.2.1.1.1" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.cmml"><mi id="S3.SS2.2.p2.9.m2.2.2.1.1.1.2" mathvariant="normal" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.2.cmml">Δ</mi><mo id="S3.SS2.2.p2.9.m2.2.2.1.1.1.1" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.2" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.1.cmml"><mo id="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.1.1.cmml">[</mo><mi id="S3.SS2.2.p2.9.m2.1.1" xref="S3.SS2.2.p2.9.m2.1.1.cmml">n</mi><mo id="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="S3.SS2.2.p2.9.m2.2.2.1.1.3" stretchy="false" xref="S3.SS2.2.p2.9.m2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.SS2.2.p2.9.m2.2.2.2" xref="S3.SS2.2.p2.9.m2.2.2.2.cmml">=</mo><msup id="S3.SS2.2.p2.9.m2.2.2.3" xref="S3.SS2.2.p2.9.m2.2.2.3.cmml"><mi id="S3.SS2.2.p2.9.m2.2.2.3.2" mathvariant="normal" xref="S3.SS2.2.p2.9.m2.2.2.3.2.cmml">Δ</mi><mi id="S3.SS2.2.p2.9.m2.2.2.3.3" xref="S3.SS2.2.p2.9.m2.2.2.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.9.m2.2b"><apply id="S3.SS2.2.p2.9.m2.2.2.cmml" xref="S3.SS2.2.p2.9.m2.2.2"><eq id="S3.SS2.2.p2.9.m2.2.2.2.cmml" xref="S3.SS2.2.p2.9.m2.2.2.2"></eq><apply id="S3.SS2.2.p2.9.m2.2.2.1.2.cmml" xref="S3.SS2.2.p2.9.m2.2.2.1.1"><abs id="S3.SS2.2.p2.9.m2.2.2.1.2.1.cmml" xref="S3.SS2.2.p2.9.m2.2.2.1.1.2"></abs><apply id="S3.SS2.2.p2.9.m2.2.2.1.1.1.cmml" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1"><times id="S3.SS2.2.p2.9.m2.2.2.1.1.1.1.cmml" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.1"></times><ci id="S3.SS2.2.p2.9.m2.2.2.1.1.1.2.cmml" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.2">Δ</ci><apply id="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.1.cmml" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.2"><csymbol cd="latexml" id="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.1.1.cmml" xref="S3.SS2.2.p2.9.m2.2.2.1.1.1.3.2.1">delimited-[]</csymbol><ci id="S3.SS2.2.p2.9.m2.1.1.cmml" xref="S3.SS2.2.p2.9.m2.1.1">𝑛</ci></apply></apply></apply><apply id="S3.SS2.2.p2.9.m2.2.2.3.cmml" xref="S3.SS2.2.p2.9.m2.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.9.m2.2.2.3.1.cmml" xref="S3.SS2.2.p2.9.m2.2.2.3">superscript</csymbol><ci id="S3.SS2.2.p2.9.m2.2.2.3.2.cmml" xref="S3.SS2.2.p2.9.m2.2.2.3.2">Δ</ci><ci id="S3.SS2.2.p2.9.m2.2.2.3.3.cmml" xref="S3.SS2.2.p2.9.m2.2.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.9.m2.2c">|\Delta[n]|=\Delta^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.9.m2.2d">| roman_Δ [ italic_n ] | = roman_Δ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is contractible, this reduced homology is trivial. Hence <math alttext="P_{*}\to\underline{R}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.10.m3.1"><semantics id="S3.SS2.2.p2.10.m3.1a"><mrow id="S3.SS2.2.p2.10.m3.1.1" xref="S3.SS2.2.p2.10.m3.1.1.cmml"><msub id="S3.SS2.2.p2.10.m3.1.1.2" xref="S3.SS2.2.p2.10.m3.1.1.2.cmml"><mi id="S3.SS2.2.p2.10.m3.1.1.2.2" xref="S3.SS2.2.p2.10.m3.1.1.2.2.cmml">P</mi><mo id="S3.SS2.2.p2.10.m3.1.1.2.3" xref="S3.SS2.2.p2.10.m3.1.1.2.3.cmml">∗</mo></msub><mo id="S3.SS2.2.p2.10.m3.1.1.1" stretchy="false" xref="S3.SS2.2.p2.10.m3.1.1.1.cmml">→</mo><munder accentunder="true" id="S3.SS2.2.p2.10.m3.1.1.3" xref="S3.SS2.2.p2.10.m3.1.1.3.cmml"><mi id="S3.SS2.2.p2.10.m3.1.1.3.2" xref="S3.SS2.2.p2.10.m3.1.1.3.2.cmml">R</mi><mo id="S3.SS2.2.p2.10.m3.1.1.3.1" xref="S3.SS2.2.p2.10.m3.1.1.3.1.cmml">¯</mo></munder></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.10.m3.1b"><apply id="S3.SS2.2.p2.10.m3.1.1.cmml" xref="S3.SS2.2.p2.10.m3.1.1"><ci id="S3.SS2.2.p2.10.m3.1.1.1.cmml" xref="S3.SS2.2.p2.10.m3.1.1.1">→</ci><apply id="S3.SS2.2.p2.10.m3.1.1.2.cmml" xref="S3.SS2.2.p2.10.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.10.m3.1.1.2.1.cmml" xref="S3.SS2.2.p2.10.m3.1.1.2">subscript</csymbol><ci id="S3.SS2.2.p2.10.m3.1.1.2.2.cmml" xref="S3.SS2.2.p2.10.m3.1.1.2.2">𝑃</ci><times id="S3.SS2.2.p2.10.m3.1.1.2.3.cmml" xref="S3.SS2.2.p2.10.m3.1.1.2.3"></times></apply><apply id="S3.SS2.2.p2.10.m3.1.1.3.cmml" xref="S3.SS2.2.p2.10.m3.1.1.3"><ci id="S3.SS2.2.p2.10.m3.1.1.3.1.cmml" xref="S3.SS2.2.p2.10.m3.1.1.3.1">¯</ci><ci id="S3.SS2.2.p2.10.m3.1.1.3.2.cmml" xref="S3.SS2.2.p2.10.m3.1.1.3.2">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.10.m3.1c">P_{*}\to\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.10.m3.1d">italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → under¯ start_ARG italic_R end_ARG</annotation></semantics></math> is a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.11.m4.1"><semantics id="S3.SS2.2.p2.11.m4.1a"><munder accentunder="true" id="S3.SS2.2.p2.11.m4.1.1" xref="S3.SS2.2.p2.11.m4.1.1.cmml"><mi id="S3.SS2.2.p2.11.m4.1.1.2" xref="S3.SS2.2.p2.11.m4.1.1.2.cmml">R</mi><mo id="S3.SS2.2.p2.11.m4.1.1.1" xref="S3.SS2.2.p2.11.m4.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.11.m4.1b"><apply id="S3.SS2.2.p2.11.m4.1.1.cmml" xref="S3.SS2.2.p2.11.m4.1.1"><ci id="S3.SS2.2.p2.11.m4.1.1.1.cmml" xref="S3.SS2.2.p2.11.m4.1.1.1">¯</ci><ci id="S3.SS2.2.p2.11.m4.1.1.2.cmml" xref="S3.SS2.2.p2.11.m4.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.11.m4.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.11.m4.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S3.SS2.2.p2.12.m5.1"><semantics id="S3.SS2.2.p2.12.m5.1a"><mrow id="S3.SS2.2.p2.12.m5.1.2" xref="S3.SS2.2.p2.12.m5.1.2.cmml"><mi id="S3.SS2.2.p2.12.m5.1.2.2" xref="S3.SS2.2.p2.12.m5.1.2.2.cmml">R</mi><mo id="S3.SS2.2.p2.12.m5.1.2.1" xref="S3.SS2.2.p2.12.m5.1.2.1.cmml">⁢</mo><mi id="S3.SS2.2.p2.12.m5.1.2.3" mathvariant="normal" xref="S3.SS2.2.p2.12.m5.1.2.3.cmml">Δ</mi><mo id="S3.SS2.2.p2.12.m5.1.2.1a" xref="S3.SS2.2.p2.12.m5.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.2.p2.12.m5.1.2.4.2" xref="S3.SS2.2.p2.12.m5.1.2.cmml"><mo id="S3.SS2.2.p2.12.m5.1.2.4.2.1" stretchy="false" xref="S3.SS2.2.p2.12.m5.1.2.cmml">(</mo><mi id="S3.SS2.2.p2.12.m5.1.1" xref="S3.SS2.2.p2.12.m5.1.1.cmml">X</mi><mo id="S3.SS2.2.p2.12.m5.1.2.4.2.2" stretchy="false" xref="S3.SS2.2.p2.12.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.12.m5.1b"><apply id="S3.SS2.2.p2.12.m5.1.2.cmml" xref="S3.SS2.2.p2.12.m5.1.2"><times id="S3.SS2.2.p2.12.m5.1.2.1.cmml" xref="S3.SS2.2.p2.12.m5.1.2.1"></times><ci id="S3.SS2.2.p2.12.m5.1.2.2.cmml" xref="S3.SS2.2.p2.12.m5.1.2.2">𝑅</ci><ci id="S3.SS2.2.p2.12.m5.1.2.3.cmml" xref="S3.SS2.2.p2.12.m5.1.2.3">Δ</ci><ci id="S3.SS2.2.p2.12.m5.1.1.cmml" xref="S3.SS2.2.p2.12.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.12.m5.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.12.m5.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module. It is easy to see that for every <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S3.SS2.2.p2.13.m6.1"><semantics id="S3.SS2.2.p2.13.m6.1a"><mrow id="S3.SS2.2.p2.13.m6.1.2" xref="S3.SS2.2.p2.13.m6.1.2.cmml"><mi id="S3.SS2.2.p2.13.m6.1.2.2" xref="S3.SS2.2.p2.13.m6.1.2.2.cmml">R</mi><mo id="S3.SS2.2.p2.13.m6.1.2.1" xref="S3.SS2.2.p2.13.m6.1.2.1.cmml">⁢</mo><mi id="S3.SS2.2.p2.13.m6.1.2.3" mathvariant="normal" xref="S3.SS2.2.p2.13.m6.1.2.3.cmml">Δ</mi><mo id="S3.SS2.2.p2.13.m6.1.2.1a" xref="S3.SS2.2.p2.13.m6.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.2.p2.13.m6.1.2.4.2" xref="S3.SS2.2.p2.13.m6.1.2.cmml"><mo id="S3.SS2.2.p2.13.m6.1.2.4.2.1" stretchy="false" xref="S3.SS2.2.p2.13.m6.1.2.cmml">(</mo><mi id="S3.SS2.2.p2.13.m6.1.1" xref="S3.SS2.2.p2.13.m6.1.1.cmml">X</mi><mo id="S3.SS2.2.p2.13.m6.1.2.4.2.2" stretchy="false" xref="S3.SS2.2.p2.13.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.13.m6.1b"><apply id="S3.SS2.2.p2.13.m6.1.2.cmml" xref="S3.SS2.2.p2.13.m6.1.2"><times id="S3.SS2.2.p2.13.m6.1.2.1.cmml" xref="S3.SS2.2.p2.13.m6.1.2.1"></times><ci id="S3.SS2.2.p2.13.m6.1.2.2.cmml" xref="S3.SS2.2.p2.13.m6.1.2.2">𝑅</ci><ci id="S3.SS2.2.p2.13.m6.1.2.3.cmml" xref="S3.SS2.2.p2.13.m6.1.2.3">Δ</ci><ci id="S3.SS2.2.p2.13.m6.1.1.cmml" xref="S3.SS2.2.p2.13.m6.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.13.m6.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.13.m6.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.14.m7.1"><semantics id="S3.SS2.2.p2.14.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.14.m7.1.1" xref="S3.SS2.2.p2.14.m7.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.14.m7.1b"><ci id="S3.SS2.2.p2.14.m7.1.1.cmml" xref="S3.SS2.2.p2.14.m7.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.14.m7.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.14.m7.1d">caligraphic_M</annotation></semantics></math>, the cochain complex <math alttext="\mathrm{Hom}_{R\Delta(X)}(P_{*},\mathcal{M})" class="ltx_Math" display="inline" id="S3.SS2.2.p2.15.m8.3"><semantics id="S3.SS2.2.p2.15.m8.3a"><mrow id="S3.SS2.2.p2.15.m8.3.3" xref="S3.SS2.2.p2.15.m8.3.3.cmml"><msub id="S3.SS2.2.p2.15.m8.3.3.3" xref="S3.SS2.2.p2.15.m8.3.3.3.cmml"><mi id="S3.SS2.2.p2.15.m8.3.3.3.2" xref="S3.SS2.2.p2.15.m8.3.3.3.2.cmml">Hom</mi><mrow id="S3.SS2.2.p2.15.m8.1.1.1" xref="S3.SS2.2.p2.15.m8.1.1.1.cmml"><mi id="S3.SS2.2.p2.15.m8.1.1.1.3" xref="S3.SS2.2.p2.15.m8.1.1.1.3.cmml">R</mi><mo id="S3.SS2.2.p2.15.m8.1.1.1.2" xref="S3.SS2.2.p2.15.m8.1.1.1.2.cmml">⁢</mo><mi id="S3.SS2.2.p2.15.m8.1.1.1.4" mathvariant="normal" xref="S3.SS2.2.p2.15.m8.1.1.1.4.cmml">Δ</mi><mo id="S3.SS2.2.p2.15.m8.1.1.1.2a" xref="S3.SS2.2.p2.15.m8.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.2.p2.15.m8.1.1.1.5.2" xref="S3.SS2.2.p2.15.m8.1.1.1.cmml"><mo id="S3.SS2.2.p2.15.m8.1.1.1.5.2.1" stretchy="false" xref="S3.SS2.2.p2.15.m8.1.1.1.cmml">(</mo><mi id="S3.SS2.2.p2.15.m8.1.1.1.1" xref="S3.SS2.2.p2.15.m8.1.1.1.1.cmml">X</mi><mo id="S3.SS2.2.p2.15.m8.1.1.1.5.2.2" stretchy="false" xref="S3.SS2.2.p2.15.m8.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S3.SS2.2.p2.15.m8.3.3.2" xref="S3.SS2.2.p2.15.m8.3.3.2.cmml">⁢</mo><mrow id="S3.SS2.2.p2.15.m8.3.3.1.1" xref="S3.SS2.2.p2.15.m8.3.3.1.2.cmml"><mo id="S3.SS2.2.p2.15.m8.3.3.1.1.2" stretchy="false" xref="S3.SS2.2.p2.15.m8.3.3.1.2.cmml">(</mo><msub id="S3.SS2.2.p2.15.m8.3.3.1.1.1" xref="S3.SS2.2.p2.15.m8.3.3.1.1.1.cmml"><mi id="S3.SS2.2.p2.15.m8.3.3.1.1.1.2" xref="S3.SS2.2.p2.15.m8.3.3.1.1.1.2.cmml">P</mi><mo id="S3.SS2.2.p2.15.m8.3.3.1.1.1.3" xref="S3.SS2.2.p2.15.m8.3.3.1.1.1.3.cmml">∗</mo></msub><mo id="S3.SS2.2.p2.15.m8.3.3.1.1.3" xref="S3.SS2.2.p2.15.m8.3.3.1.2.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.15.m8.2.2" xref="S3.SS2.2.p2.15.m8.2.2.cmml">ℳ</mi><mo id="S3.SS2.2.p2.15.m8.3.3.1.1.4" stretchy="false" xref="S3.SS2.2.p2.15.m8.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.15.m8.3b"><apply id="S3.SS2.2.p2.15.m8.3.3.cmml" xref="S3.SS2.2.p2.15.m8.3.3"><times id="S3.SS2.2.p2.15.m8.3.3.2.cmml" xref="S3.SS2.2.p2.15.m8.3.3.2"></times><apply id="S3.SS2.2.p2.15.m8.3.3.3.cmml" xref="S3.SS2.2.p2.15.m8.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.15.m8.3.3.3.1.cmml" xref="S3.SS2.2.p2.15.m8.3.3.3">subscript</csymbol><ci id="S3.SS2.2.p2.15.m8.3.3.3.2.cmml" xref="S3.SS2.2.p2.15.m8.3.3.3.2">Hom</ci><apply id="S3.SS2.2.p2.15.m8.1.1.1.cmml" xref="S3.SS2.2.p2.15.m8.1.1.1"><times id="S3.SS2.2.p2.15.m8.1.1.1.2.cmml" xref="S3.SS2.2.p2.15.m8.1.1.1.2"></times><ci id="S3.SS2.2.p2.15.m8.1.1.1.3.cmml" xref="S3.SS2.2.p2.15.m8.1.1.1.3">𝑅</ci><ci id="S3.SS2.2.p2.15.m8.1.1.1.4.cmml" xref="S3.SS2.2.p2.15.m8.1.1.1.4">Δ</ci><ci id="S3.SS2.2.p2.15.m8.1.1.1.1.cmml" xref="S3.SS2.2.p2.15.m8.1.1.1.1">𝑋</ci></apply></apply><interval closure="open" id="S3.SS2.2.p2.15.m8.3.3.1.2.cmml" xref="S3.SS2.2.p2.15.m8.3.3.1.1"><apply id="S3.SS2.2.p2.15.m8.3.3.1.1.1.cmml" xref="S3.SS2.2.p2.15.m8.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.2.p2.15.m8.3.3.1.1.1.1.cmml" xref="S3.SS2.2.p2.15.m8.3.3.1.1.1">subscript</csymbol><ci id="S3.SS2.2.p2.15.m8.3.3.1.1.1.2.cmml" xref="S3.SS2.2.p2.15.m8.3.3.1.1.1.2">𝑃</ci><times id="S3.SS2.2.p2.15.m8.3.3.1.1.1.3.cmml" xref="S3.SS2.2.p2.15.m8.3.3.1.1.1.3"></times></apply><ci id="S3.SS2.2.p2.15.m8.2.2.cmml" xref="S3.SS2.2.p2.15.m8.2.2">ℳ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.15.m8.3c">\mathrm{Hom}_{R\Delta(X)}(P_{*},\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.15.m8.3d">roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_X ) end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , caligraphic_M )</annotation></semantics></math> is isomorphic to <math alttext="C^{*}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S3.SS2.2.p2.16.m9.2"><semantics id="S3.SS2.2.p2.16.m9.2a"><mrow id="S3.SS2.2.p2.16.m9.2.3" xref="S3.SS2.2.p2.16.m9.2.3.cmml"><msup id="S3.SS2.2.p2.16.m9.2.3.2" xref="S3.SS2.2.p2.16.m9.2.3.2.cmml"><mi id="S3.SS2.2.p2.16.m9.2.3.2.2" xref="S3.SS2.2.p2.16.m9.2.3.2.2.cmml">C</mi><mo id="S3.SS2.2.p2.16.m9.2.3.2.3" xref="S3.SS2.2.p2.16.m9.2.3.2.3.cmml">∗</mo></msup><mo id="S3.SS2.2.p2.16.m9.2.3.1" xref="S3.SS2.2.p2.16.m9.2.3.1.cmml">⁢</mo><mrow id="S3.SS2.2.p2.16.m9.2.3.3.2" xref="S3.SS2.2.p2.16.m9.2.3.3.1.cmml"><mo id="S3.SS2.2.p2.16.m9.2.3.3.2.1" stretchy="false" xref="S3.SS2.2.p2.16.m9.2.3.3.1.cmml">(</mo><mi id="S3.SS2.2.p2.16.m9.1.1" xref="S3.SS2.2.p2.16.m9.1.1.cmml">X</mi><mo id="S3.SS2.2.p2.16.m9.2.3.3.2.2" xref="S3.SS2.2.p2.16.m9.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.16.m9.2.2" xref="S3.SS2.2.p2.16.m9.2.2.cmml">ℳ</mi><mo id="S3.SS2.2.p2.16.m9.2.3.3.2.3" stretchy="false" xref="S3.SS2.2.p2.16.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.16.m9.2b"><apply id="S3.SS2.2.p2.16.m9.2.3.cmml" xref="S3.SS2.2.p2.16.m9.2.3"><times id="S3.SS2.2.p2.16.m9.2.3.1.cmml" xref="S3.SS2.2.p2.16.m9.2.3.1"></times><apply id="S3.SS2.2.p2.16.m9.2.3.2.cmml" xref="S3.SS2.2.p2.16.m9.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.16.m9.2.3.2.1.cmml" xref="S3.SS2.2.p2.16.m9.2.3.2">superscript</csymbol><ci id="S3.SS2.2.p2.16.m9.2.3.2.2.cmml" xref="S3.SS2.2.p2.16.m9.2.3.2.2">𝐶</ci><times id="S3.SS2.2.p2.16.m9.2.3.2.3.cmml" xref="S3.SS2.2.p2.16.m9.2.3.2.3"></times></apply><list id="S3.SS2.2.p2.16.m9.2.3.3.1.cmml" xref="S3.SS2.2.p2.16.m9.2.3.3.2"><ci id="S3.SS2.2.p2.16.m9.1.1.cmml" xref="S3.SS2.2.p2.16.m9.1.1">𝑋</ci><ci id="S3.SS2.2.p2.16.m9.2.2.cmml" xref="S3.SS2.2.p2.16.m9.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.16.m9.2c">C^{*}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.16.m9.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3. </span>Induced Maps on Cohomology</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.5">Let <math alttext="\lambda:X\to Y" class="ltx_Math" display="inline" id="S3.SS3.p1.1.m1.1"><semantics id="S3.SS3.p1.1.m1.1a"><mrow id="S3.SS3.p1.1.m1.1.1" xref="S3.SS3.p1.1.m1.1.1.cmml"><mi id="S3.SS3.p1.1.m1.1.1.2" xref="S3.SS3.p1.1.m1.1.1.2.cmml">λ</mi><mo id="S3.SS3.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.SS3.p1.1.m1.1.1.3" xref="S3.SS3.p1.1.m1.1.1.3.cmml"><mi id="S3.SS3.p1.1.m1.1.1.3.2" xref="S3.SS3.p1.1.m1.1.1.3.2.cmml">X</mi><mo id="S3.SS3.p1.1.m1.1.1.3.1" stretchy="false" xref="S3.SS3.p1.1.m1.1.1.3.1.cmml">→</mo><mi id="S3.SS3.p1.1.m1.1.1.3.3" xref="S3.SS3.p1.1.m1.1.1.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.1.m1.1b"><apply id="S3.SS3.p1.1.m1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1"><ci id="S3.SS3.p1.1.m1.1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1.1">:</ci><ci id="S3.SS3.p1.1.m1.1.1.2.cmml" xref="S3.SS3.p1.1.m1.1.1.2">𝜆</ci><apply id="S3.SS3.p1.1.m1.1.1.3.cmml" xref="S3.SS3.p1.1.m1.1.1.3"><ci id="S3.SS3.p1.1.m1.1.1.3.1.cmml" xref="S3.SS3.p1.1.m1.1.1.3.1">→</ci><ci id="S3.SS3.p1.1.m1.1.1.3.2.cmml" xref="S3.SS3.p1.1.m1.1.1.3.2">𝑋</ci><ci id="S3.SS3.p1.1.m1.1.1.3.3.cmml" xref="S3.SS3.p1.1.m1.1.1.3.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.1.m1.1c">\lambda:X\to Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.1.m1.1d">italic_λ : italic_X → italic_Y</annotation></semantics></math> be a simplicial map between two simplicial sets. The map <math alttext="\lambda" class="ltx_Math" display="inline" id="S3.SS3.p1.2.m2.1"><semantics id="S3.SS3.p1.2.m2.1a"><mi id="S3.SS3.p1.2.m2.1.1" xref="S3.SS3.p1.2.m2.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.2.m2.1b"><ci id="S3.SS3.p1.2.m2.1.1.cmml" xref="S3.SS3.p1.2.m2.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.2.m2.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.2.m2.1d">italic_λ</annotation></semantics></math> induces a functor <math alttext="\Delta(\lambda):\Delta(X)\to\Delta(Y)" class="ltx_Math" display="inline" id="S3.SS3.p1.3.m3.3"><semantics id="S3.SS3.p1.3.m3.3a"><mrow id="S3.SS3.p1.3.m3.3.4" xref="S3.SS3.p1.3.m3.3.4.cmml"><mrow id="S3.SS3.p1.3.m3.3.4.2" xref="S3.SS3.p1.3.m3.3.4.2.cmml"><mi id="S3.SS3.p1.3.m3.3.4.2.2" mathvariant="normal" xref="S3.SS3.p1.3.m3.3.4.2.2.cmml">Δ</mi><mo id="S3.SS3.p1.3.m3.3.4.2.1" xref="S3.SS3.p1.3.m3.3.4.2.1.cmml">⁢</mo><mrow id="S3.SS3.p1.3.m3.3.4.2.3.2" xref="S3.SS3.p1.3.m3.3.4.2.cmml"><mo id="S3.SS3.p1.3.m3.3.4.2.3.2.1" stretchy="false" xref="S3.SS3.p1.3.m3.3.4.2.cmml">(</mo><mi id="S3.SS3.p1.3.m3.1.1" xref="S3.SS3.p1.3.m3.1.1.cmml">λ</mi><mo id="S3.SS3.p1.3.m3.3.4.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.SS3.p1.3.m3.3.4.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p1.3.m3.3.4.1" rspace="0.278em" xref="S3.SS3.p1.3.m3.3.4.1.cmml">:</mo><mrow id="S3.SS3.p1.3.m3.3.4.3" xref="S3.SS3.p1.3.m3.3.4.3.cmml"><mrow id="S3.SS3.p1.3.m3.3.4.3.2" xref="S3.SS3.p1.3.m3.3.4.3.2.cmml"><mi id="S3.SS3.p1.3.m3.3.4.3.2.2" mathvariant="normal" xref="S3.SS3.p1.3.m3.3.4.3.2.2.cmml">Δ</mi><mo id="S3.SS3.p1.3.m3.3.4.3.2.1" xref="S3.SS3.p1.3.m3.3.4.3.2.1.cmml">⁢</mo><mrow id="S3.SS3.p1.3.m3.3.4.3.2.3.2" xref="S3.SS3.p1.3.m3.3.4.3.2.cmml"><mo id="S3.SS3.p1.3.m3.3.4.3.2.3.2.1" stretchy="false" xref="S3.SS3.p1.3.m3.3.4.3.2.cmml">(</mo><mi id="S3.SS3.p1.3.m3.2.2" xref="S3.SS3.p1.3.m3.2.2.cmml">X</mi><mo id="S3.SS3.p1.3.m3.3.4.3.2.3.2.2" stretchy="false" xref="S3.SS3.p1.3.m3.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p1.3.m3.3.4.3.1" stretchy="false" xref="S3.SS3.p1.3.m3.3.4.3.1.cmml">→</mo><mrow id="S3.SS3.p1.3.m3.3.4.3.3" xref="S3.SS3.p1.3.m3.3.4.3.3.cmml"><mi id="S3.SS3.p1.3.m3.3.4.3.3.2" mathvariant="normal" xref="S3.SS3.p1.3.m3.3.4.3.3.2.cmml">Δ</mi><mo id="S3.SS3.p1.3.m3.3.4.3.3.1" xref="S3.SS3.p1.3.m3.3.4.3.3.1.cmml">⁢</mo><mrow id="S3.SS3.p1.3.m3.3.4.3.3.3.2" xref="S3.SS3.p1.3.m3.3.4.3.3.cmml"><mo id="S3.SS3.p1.3.m3.3.4.3.3.3.2.1" stretchy="false" xref="S3.SS3.p1.3.m3.3.4.3.3.cmml">(</mo><mi id="S3.SS3.p1.3.m3.3.3" xref="S3.SS3.p1.3.m3.3.3.cmml">Y</mi><mo id="S3.SS3.p1.3.m3.3.4.3.3.3.2.2" stretchy="false" xref="S3.SS3.p1.3.m3.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.3.m3.3b"><apply id="S3.SS3.p1.3.m3.3.4.cmml" xref="S3.SS3.p1.3.m3.3.4"><ci id="S3.SS3.p1.3.m3.3.4.1.cmml" xref="S3.SS3.p1.3.m3.3.4.1">:</ci><apply id="S3.SS3.p1.3.m3.3.4.2.cmml" xref="S3.SS3.p1.3.m3.3.4.2"><times id="S3.SS3.p1.3.m3.3.4.2.1.cmml" xref="S3.SS3.p1.3.m3.3.4.2.1"></times><ci id="S3.SS3.p1.3.m3.3.4.2.2.cmml" xref="S3.SS3.p1.3.m3.3.4.2.2">Δ</ci><ci id="S3.SS3.p1.3.m3.1.1.cmml" xref="S3.SS3.p1.3.m3.1.1">𝜆</ci></apply><apply id="S3.SS3.p1.3.m3.3.4.3.cmml" xref="S3.SS3.p1.3.m3.3.4.3"><ci id="S3.SS3.p1.3.m3.3.4.3.1.cmml" xref="S3.SS3.p1.3.m3.3.4.3.1">→</ci><apply id="S3.SS3.p1.3.m3.3.4.3.2.cmml" xref="S3.SS3.p1.3.m3.3.4.3.2"><times id="S3.SS3.p1.3.m3.3.4.3.2.1.cmml" xref="S3.SS3.p1.3.m3.3.4.3.2.1"></times><ci id="S3.SS3.p1.3.m3.3.4.3.2.2.cmml" xref="S3.SS3.p1.3.m3.3.4.3.2.2">Δ</ci><ci id="S3.SS3.p1.3.m3.2.2.cmml" xref="S3.SS3.p1.3.m3.2.2">𝑋</ci></apply><apply id="S3.SS3.p1.3.m3.3.4.3.3.cmml" xref="S3.SS3.p1.3.m3.3.4.3.3"><times id="S3.SS3.p1.3.m3.3.4.3.3.1.cmml" xref="S3.SS3.p1.3.m3.3.4.3.3.1"></times><ci id="S3.SS3.p1.3.m3.3.4.3.3.2.cmml" xref="S3.SS3.p1.3.m3.3.4.3.3.2">Δ</ci><ci id="S3.SS3.p1.3.m3.3.3.cmml" xref="S3.SS3.p1.3.m3.3.3">𝑌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.3.m3.3c">\Delta(\lambda):\Delta(X)\to\Delta(Y)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.3.m3.3d">roman_Δ ( italic_λ ) : roman_Δ ( italic_X ) → roman_Δ ( italic_Y )</annotation></semantics></math>. For every coefficient system <math alttext="\mathcal{M}:\Delta(Y)\to R" class="ltx_Math" display="inline" id="S3.SS3.p1.4.m4.1"><semantics id="S3.SS3.p1.4.m4.1a"><mrow id="S3.SS3.p1.4.m4.1.2" xref="S3.SS3.p1.4.m4.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p1.4.m4.1.2.2" xref="S3.SS3.p1.4.m4.1.2.2.cmml">ℳ</mi><mo id="S3.SS3.p1.4.m4.1.2.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.p1.4.m4.1.2.1.cmml">:</mo><mrow id="S3.SS3.p1.4.m4.1.2.3" xref="S3.SS3.p1.4.m4.1.2.3.cmml"><mrow id="S3.SS3.p1.4.m4.1.2.3.2" xref="S3.SS3.p1.4.m4.1.2.3.2.cmml"><mi id="S3.SS3.p1.4.m4.1.2.3.2.2" mathvariant="normal" xref="S3.SS3.p1.4.m4.1.2.3.2.2.cmml">Δ</mi><mo id="S3.SS3.p1.4.m4.1.2.3.2.1" xref="S3.SS3.p1.4.m4.1.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS3.p1.4.m4.1.2.3.2.3.2" xref="S3.SS3.p1.4.m4.1.2.3.2.cmml"><mo id="S3.SS3.p1.4.m4.1.2.3.2.3.2.1" stretchy="false" xref="S3.SS3.p1.4.m4.1.2.3.2.cmml">(</mo><mi id="S3.SS3.p1.4.m4.1.1" xref="S3.SS3.p1.4.m4.1.1.cmml">Y</mi><mo id="S3.SS3.p1.4.m4.1.2.3.2.3.2.2" stretchy="false" xref="S3.SS3.p1.4.m4.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p1.4.m4.1.2.3.1" stretchy="false" xref="S3.SS3.p1.4.m4.1.2.3.1.cmml">→</mo><mi id="S3.SS3.p1.4.m4.1.2.3.3" xref="S3.SS3.p1.4.m4.1.2.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.4.m4.1b"><apply id="S3.SS3.p1.4.m4.1.2.cmml" xref="S3.SS3.p1.4.m4.1.2"><ci id="S3.SS3.p1.4.m4.1.2.1.cmml" xref="S3.SS3.p1.4.m4.1.2.1">:</ci><ci id="S3.SS3.p1.4.m4.1.2.2.cmml" xref="S3.SS3.p1.4.m4.1.2.2">ℳ</ci><apply id="S3.SS3.p1.4.m4.1.2.3.cmml" xref="S3.SS3.p1.4.m4.1.2.3"><ci id="S3.SS3.p1.4.m4.1.2.3.1.cmml" xref="S3.SS3.p1.4.m4.1.2.3.1">→</ci><apply id="S3.SS3.p1.4.m4.1.2.3.2.cmml" xref="S3.SS3.p1.4.m4.1.2.3.2"><times id="S3.SS3.p1.4.m4.1.2.3.2.1.cmml" xref="S3.SS3.p1.4.m4.1.2.3.2.1"></times><ci id="S3.SS3.p1.4.m4.1.2.3.2.2.cmml" xref="S3.SS3.p1.4.m4.1.2.3.2.2">Δ</ci><ci id="S3.SS3.p1.4.m4.1.1.cmml" xref="S3.SS3.p1.4.m4.1.1">𝑌</ci></apply><ci id="S3.SS3.p1.4.m4.1.2.3.3.cmml" xref="S3.SS3.p1.4.m4.1.2.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.4.m4.1c">\mathcal{M}:\Delta(Y)\to R</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.4.m4.1d">caligraphic_M : roman_Δ ( italic_Y ) → italic_R</annotation></semantics></math>-Mod for <math alttext="Y" class="ltx_Math" display="inline" id="S3.SS3.p1.5.m5.1"><semantics id="S3.SS3.p1.5.m5.1a"><mi id="S3.SS3.p1.5.m5.1.1" xref="S3.SS3.p1.5.m5.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.5.m5.1b"><ci id="S3.SS3.p1.5.m5.1.1.cmml" xref="S3.SS3.p1.5.m5.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.5.m5.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.5.m5.1d">italic_Y</annotation></semantics></math>, the composition</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex35"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lambda^{*}\mathcal{M}:\Delta(X)\xrightarrow{\Delta(\lambda)}\Delta(Y)% \xrightarrow{\mathcal{M}}R\text{-Mod}" class="ltx_Math" display="block" id="S3.Ex35.m1.3"><semantics id="S3.Ex35.m1.3a"><mrow id="S3.Ex35.m1.3.4" xref="S3.Ex35.m1.3.4.cmml"><mrow id="S3.Ex35.m1.3.4.2" xref="S3.Ex35.m1.3.4.2.cmml"><msup id="S3.Ex35.m1.3.4.2.2" xref="S3.Ex35.m1.3.4.2.2.cmml"><mi id="S3.Ex35.m1.3.4.2.2.2" xref="S3.Ex35.m1.3.4.2.2.2.cmml">λ</mi><mo id="S3.Ex35.m1.3.4.2.2.3" xref="S3.Ex35.m1.3.4.2.2.3.cmml">∗</mo></msup><mo id="S3.Ex35.m1.3.4.2.1" xref="S3.Ex35.m1.3.4.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex35.m1.3.4.2.3" xref="S3.Ex35.m1.3.4.2.3.cmml">ℳ</mi></mrow><mo id="S3.Ex35.m1.3.4.1" lspace="0.278em" rspace="0.278em" xref="S3.Ex35.m1.3.4.1.cmml">:</mo><mrow id="S3.Ex35.m1.3.4.3" xref="S3.Ex35.m1.3.4.3.cmml"><mrow id="S3.Ex35.m1.3.4.3.2" xref="S3.Ex35.m1.3.4.3.2.cmml"><mi id="S3.Ex35.m1.3.4.3.2.2" mathvariant="normal" xref="S3.Ex35.m1.3.4.3.2.2.cmml">Δ</mi><mo id="S3.Ex35.m1.3.4.3.2.1" xref="S3.Ex35.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S3.Ex35.m1.3.4.3.2.3.2" xref="S3.Ex35.m1.3.4.3.2.cmml"><mo id="S3.Ex35.m1.3.4.3.2.3.2.1" stretchy="false" xref="S3.Ex35.m1.3.4.3.2.cmml">(</mo><mi id="S3.Ex35.m1.2.2" xref="S3.Ex35.m1.2.2.cmml">X</mi><mo id="S3.Ex35.m1.3.4.3.2.3.2.2" stretchy="false" xref="S3.Ex35.m1.3.4.3.2.cmml">)</mo></mrow></mrow><mover accent="true" id="S3.Ex35.m1.1.1" xref="S3.Ex35.m1.1.1.cmml"><mo id="S3.Ex35.m1.1.1.2" stretchy="false" xref="S3.Ex35.m1.1.1.2.cmml">→</mo><mrow id="S3.Ex35.m1.1.1.1" xref="S3.Ex35.m1.1.1.1.cmml"><mi id="S3.Ex35.m1.1.1.1.3" mathvariant="normal" xref="S3.Ex35.m1.1.1.1.3.cmml">Δ</mi><mo id="S3.Ex35.m1.1.1.1.2" xref="S3.Ex35.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex35.m1.1.1.1.4.2" xref="S3.Ex35.m1.1.1.1.cmml"><mo id="S3.Ex35.m1.1.1.1.4.2.1" stretchy="false" xref="S3.Ex35.m1.1.1.1.cmml">(</mo><mi id="S3.Ex35.m1.1.1.1.1" xref="S3.Ex35.m1.1.1.1.1.cmml">λ</mi><mo id="S3.Ex35.m1.1.1.1.4.2.2" stretchy="false" xref="S3.Ex35.m1.1.1.1.cmml">)</mo></mrow></mrow></mover><mrow id="S3.Ex35.m1.3.4.3.3" xref="S3.Ex35.m1.3.4.3.3.cmml"><mi id="S3.Ex35.m1.3.4.3.3.2" mathvariant="normal" xref="S3.Ex35.m1.3.4.3.3.2.cmml">Δ</mi><mo id="S3.Ex35.m1.3.4.3.3.1" xref="S3.Ex35.m1.3.4.3.3.1.cmml">⁢</mo><mrow id="S3.Ex35.m1.3.4.3.3.3.2" xref="S3.Ex35.m1.3.4.3.3.cmml"><mo id="S3.Ex35.m1.3.4.3.3.3.2.1" stretchy="false" xref="S3.Ex35.m1.3.4.3.3.cmml">(</mo><mi id="S3.Ex35.m1.3.3" xref="S3.Ex35.m1.3.3.cmml">Y</mi><mo id="S3.Ex35.m1.3.4.3.3.3.2.2" stretchy="false" xref="S3.Ex35.m1.3.4.3.3.cmml">)</mo></mrow></mrow><mover accent="true" id="S3.Ex35.m1.3.4.3.4" xref="S3.Ex35.m1.3.4.3.4.cmml"><mo id="S3.Ex35.m1.3.4.3.4.2" stretchy="false" xref="S3.Ex35.m1.3.4.3.4.2.cmml">→</mo><mo class="ltx_font_mathcaligraphic" id="S3.Ex35.m1.3.4.3.4.1" xref="S3.Ex35.m1.3.4.3.4.1.cmml">ℳ</mo></mover><mrow id="S3.Ex35.m1.3.4.3.5" xref="S3.Ex35.m1.3.4.3.5.cmml"><mi id="S3.Ex35.m1.3.4.3.5.2" xref="S3.Ex35.m1.3.4.3.5.2.cmml">R</mi><mo id="S3.Ex35.m1.3.4.3.5.1" xref="S3.Ex35.m1.3.4.3.5.1.cmml">⁢</mo><mtext id="S3.Ex35.m1.3.4.3.5.3" xref="S3.Ex35.m1.3.4.3.5.3a.cmml">-Mod</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex35.m1.3b"><apply id="S3.Ex35.m1.3.4.cmml" xref="S3.Ex35.m1.3.4"><ci id="S3.Ex35.m1.3.4.1.cmml" xref="S3.Ex35.m1.3.4.1">:</ci><apply id="S3.Ex35.m1.3.4.2.cmml" xref="S3.Ex35.m1.3.4.2"><times id="S3.Ex35.m1.3.4.2.1.cmml" xref="S3.Ex35.m1.3.4.2.1"></times><apply id="S3.Ex35.m1.3.4.2.2.cmml" xref="S3.Ex35.m1.3.4.2.2"><csymbol cd="ambiguous" id="S3.Ex35.m1.3.4.2.2.1.cmml" xref="S3.Ex35.m1.3.4.2.2">superscript</csymbol><ci id="S3.Ex35.m1.3.4.2.2.2.cmml" xref="S3.Ex35.m1.3.4.2.2.2">𝜆</ci><times id="S3.Ex35.m1.3.4.2.2.3.cmml" xref="S3.Ex35.m1.3.4.2.2.3"></times></apply><ci id="S3.Ex35.m1.3.4.2.3.cmml" xref="S3.Ex35.m1.3.4.2.3">ℳ</ci></apply><apply id="S3.Ex35.m1.3.4.3.cmml" xref="S3.Ex35.m1.3.4.3"><and id="S3.Ex35.m1.3.4.3a.cmml" xref="S3.Ex35.m1.3.4.3"></and><apply id="S3.Ex35.m1.3.4.3b.cmml" xref="S3.Ex35.m1.3.4.3"><apply id="S3.Ex35.m1.1.1.cmml" xref="S3.Ex35.m1.1.1"><apply id="S3.Ex35.m1.1.1.1.cmml" xref="S3.Ex35.m1.1.1.1"><times id="S3.Ex35.m1.1.1.1.2.cmml" xref="S3.Ex35.m1.1.1.1.2"></times><ci id="S3.Ex35.m1.1.1.1.3.cmml" xref="S3.Ex35.m1.1.1.1.3">Δ</ci><ci id="S3.Ex35.m1.1.1.1.1.cmml" xref="S3.Ex35.m1.1.1.1.1">𝜆</ci></apply><ci id="S3.Ex35.m1.1.1.2.cmml" xref="S3.Ex35.m1.1.1.2">→</ci></apply><apply id="S3.Ex35.m1.3.4.3.2.cmml" xref="S3.Ex35.m1.3.4.3.2"><times id="S3.Ex35.m1.3.4.3.2.1.cmml" xref="S3.Ex35.m1.3.4.3.2.1"></times><ci id="S3.Ex35.m1.3.4.3.2.2.cmml" xref="S3.Ex35.m1.3.4.3.2.2">Δ</ci><ci id="S3.Ex35.m1.2.2.cmml" xref="S3.Ex35.m1.2.2">𝑋</ci></apply><apply id="S3.Ex35.m1.3.4.3.3.cmml" xref="S3.Ex35.m1.3.4.3.3"><times id="S3.Ex35.m1.3.4.3.3.1.cmml" xref="S3.Ex35.m1.3.4.3.3.1"></times><ci id="S3.Ex35.m1.3.4.3.3.2.cmml" xref="S3.Ex35.m1.3.4.3.3.2">Δ</ci><ci id="S3.Ex35.m1.3.3.cmml" xref="S3.Ex35.m1.3.3">𝑌</ci></apply></apply><apply id="S3.Ex35.m1.3.4.3c.cmml" xref="S3.Ex35.m1.3.4.3"><apply id="S3.Ex35.m1.3.4.3.4.cmml" xref="S3.Ex35.m1.3.4.3.4"><ci id="S3.Ex35.m1.3.4.3.4.1.cmml" xref="S3.Ex35.m1.3.4.3.4.1">ℳ</ci><ci id="S3.Ex35.m1.3.4.3.4.2.cmml" xref="S3.Ex35.m1.3.4.3.4.2">→</ci></apply><share href="https://arxiv.org/html/2503.14659v1#S3.Ex35.m1.3.4.3.3.cmml" id="S3.Ex35.m1.3.4.3d.cmml" xref="S3.Ex35.m1.3.4.3"></share><apply id="S3.Ex35.m1.3.4.3.5.cmml" xref="S3.Ex35.m1.3.4.3.5"><times id="S3.Ex35.m1.3.4.3.5.1.cmml" xref="S3.Ex35.m1.3.4.3.5.1"></times><ci id="S3.Ex35.m1.3.4.3.5.2.cmml" xref="S3.Ex35.m1.3.4.3.5.2">𝑅</ci><ci id="S3.Ex35.m1.3.4.3.5.3a.cmml" xref="S3.Ex35.m1.3.4.3.5.3"><mtext id="S3.Ex35.m1.3.4.3.5.3.cmml" xref="S3.Ex35.m1.3.4.3.5.3">-Mod</mtext></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex35.m1.3c">\lambda^{*}\mathcal{M}:\Delta(X)\xrightarrow{\Delta(\lambda)}\Delta(Y)% \xrightarrow{\mathcal{M}}R\text{-Mod}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex35.m1.3d">italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M : roman_Δ ( italic_X ) start_ARROW start_OVERACCENT roman_Δ ( italic_λ ) end_OVERACCENT → end_ARROW roman_Δ ( italic_Y ) start_ARROW overcaligraphic_M → end_ARROW italic_R -Mod</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p1.6">is a coefficient system for <math alttext="X" class="ltx_Math" display="inline" id="S3.SS3.p1.6.m1.1"><semantics id="S3.SS3.p1.6.m1.1a"><mi id="S3.SS3.p1.6.m1.1.1" xref="S3.SS3.p1.6.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.6.m1.1b"><ci id="S3.SS3.p1.6.m1.1.1.cmml" xref="S3.SS3.p1.6.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.6.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.6.m1.1d">italic_X</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.1.1.1">Lemma 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.2.2">The simplicial map <math alttext="\lambda:X\to Y" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.1"><semantics id="S3.Thmtheorem4.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem4.p1.1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">λ</mi><mo id="S3.Thmtheorem4.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmtheorem4.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.2" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml">X</mi><mo id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.1.m1.1b"><apply id="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1"><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.1">:</ci><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.2">𝜆</ci><apply id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3"><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.1">→</ci><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.2">𝑋</ci><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.3.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.1.m1.1c">\lambda:X\to Y</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.1.m1.1d">italic_λ : italic_X → italic_Y</annotation></semantics></math> induces an <math alttext="R" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.2.m2.1a"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.1b"><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.2.m2.1d">italic_R</annotation></semantics></math>-module homomorphism</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex36"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lambda^{*}:H^{*}(Y;\mathcal{M})\to H^{*}(X;\lambda^{*}\mathcal{M})" class="ltx_Math" display="block" id="S3.Ex36.m1.4"><semantics id="S3.Ex36.m1.4a"><mrow id="S3.Ex36.m1.4.4" xref="S3.Ex36.m1.4.4.cmml"><msup id="S3.Ex36.m1.4.4.3" xref="S3.Ex36.m1.4.4.3.cmml"><mi id="S3.Ex36.m1.4.4.3.2" xref="S3.Ex36.m1.4.4.3.2.cmml">λ</mi><mo id="S3.Ex36.m1.4.4.3.3" xref="S3.Ex36.m1.4.4.3.3.cmml">∗</mo></msup><mo id="S3.Ex36.m1.4.4.2" lspace="0.278em" rspace="0.278em" xref="S3.Ex36.m1.4.4.2.cmml">:</mo><mrow id="S3.Ex36.m1.4.4.1" xref="S3.Ex36.m1.4.4.1.cmml"><mrow id="S3.Ex36.m1.4.4.1.3" xref="S3.Ex36.m1.4.4.1.3.cmml"><msup id="S3.Ex36.m1.4.4.1.3.2" xref="S3.Ex36.m1.4.4.1.3.2.cmml"><mi id="S3.Ex36.m1.4.4.1.3.2.2" xref="S3.Ex36.m1.4.4.1.3.2.2.cmml">H</mi><mo id="S3.Ex36.m1.4.4.1.3.2.3" xref="S3.Ex36.m1.4.4.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Ex36.m1.4.4.1.3.1" xref="S3.Ex36.m1.4.4.1.3.1.cmml">⁢</mo><mrow id="S3.Ex36.m1.4.4.1.3.3.2" xref="S3.Ex36.m1.4.4.1.3.3.1.cmml"><mo id="S3.Ex36.m1.4.4.1.3.3.2.1" stretchy="false" xref="S3.Ex36.m1.4.4.1.3.3.1.cmml">(</mo><mi id="S3.Ex36.m1.1.1" xref="S3.Ex36.m1.1.1.cmml">Y</mi><mo id="S3.Ex36.m1.4.4.1.3.3.2.2" xref="S3.Ex36.m1.4.4.1.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex36.m1.2.2" xref="S3.Ex36.m1.2.2.cmml">ℳ</mi><mo id="S3.Ex36.m1.4.4.1.3.3.2.3" stretchy="false" xref="S3.Ex36.m1.4.4.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex36.m1.4.4.1.2" stretchy="false" xref="S3.Ex36.m1.4.4.1.2.cmml">→</mo><mrow id="S3.Ex36.m1.4.4.1.1" xref="S3.Ex36.m1.4.4.1.1.cmml"><msup id="S3.Ex36.m1.4.4.1.1.3" xref="S3.Ex36.m1.4.4.1.1.3.cmml"><mi id="S3.Ex36.m1.4.4.1.1.3.2" xref="S3.Ex36.m1.4.4.1.1.3.2.cmml">H</mi><mo id="S3.Ex36.m1.4.4.1.1.3.3" xref="S3.Ex36.m1.4.4.1.1.3.3.cmml">∗</mo></msup><mo id="S3.Ex36.m1.4.4.1.1.2" xref="S3.Ex36.m1.4.4.1.1.2.cmml">⁢</mo><mrow id="S3.Ex36.m1.4.4.1.1.1.1" xref="S3.Ex36.m1.4.4.1.1.1.2.cmml"><mo id="S3.Ex36.m1.4.4.1.1.1.1.2" stretchy="false" xref="S3.Ex36.m1.4.4.1.1.1.2.cmml">(</mo><mi id="S3.Ex36.m1.3.3" xref="S3.Ex36.m1.3.3.cmml">X</mi><mo id="S3.Ex36.m1.4.4.1.1.1.1.3" xref="S3.Ex36.m1.4.4.1.1.1.2.cmml">;</mo><mrow id="S3.Ex36.m1.4.4.1.1.1.1.1" xref="S3.Ex36.m1.4.4.1.1.1.1.1.cmml"><msup id="S3.Ex36.m1.4.4.1.1.1.1.1.2" xref="S3.Ex36.m1.4.4.1.1.1.1.1.2.cmml"><mi id="S3.Ex36.m1.4.4.1.1.1.1.1.2.2" xref="S3.Ex36.m1.4.4.1.1.1.1.1.2.2.cmml">λ</mi><mo id="S3.Ex36.m1.4.4.1.1.1.1.1.2.3" xref="S3.Ex36.m1.4.4.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S3.Ex36.m1.4.4.1.1.1.1.1.1" xref="S3.Ex36.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex36.m1.4.4.1.1.1.1.1.3" xref="S3.Ex36.m1.4.4.1.1.1.1.1.3.cmml">ℳ</mi></mrow><mo id="S3.Ex36.m1.4.4.1.1.1.1.4" stretchy="false" xref="S3.Ex36.m1.4.4.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex36.m1.4b"><apply id="S3.Ex36.m1.4.4.cmml" xref="S3.Ex36.m1.4.4"><ci id="S3.Ex36.m1.4.4.2.cmml" xref="S3.Ex36.m1.4.4.2">:</ci><apply id="S3.Ex36.m1.4.4.3.cmml" xref="S3.Ex36.m1.4.4.3"><csymbol cd="ambiguous" id="S3.Ex36.m1.4.4.3.1.cmml" xref="S3.Ex36.m1.4.4.3">superscript</csymbol><ci id="S3.Ex36.m1.4.4.3.2.cmml" xref="S3.Ex36.m1.4.4.3.2">𝜆</ci><times id="S3.Ex36.m1.4.4.3.3.cmml" xref="S3.Ex36.m1.4.4.3.3"></times></apply><apply id="S3.Ex36.m1.4.4.1.cmml" xref="S3.Ex36.m1.4.4.1"><ci id="S3.Ex36.m1.4.4.1.2.cmml" xref="S3.Ex36.m1.4.4.1.2">→</ci><apply id="S3.Ex36.m1.4.4.1.3.cmml" xref="S3.Ex36.m1.4.4.1.3"><times id="S3.Ex36.m1.4.4.1.3.1.cmml" xref="S3.Ex36.m1.4.4.1.3.1"></times><apply id="S3.Ex36.m1.4.4.1.3.2.cmml" xref="S3.Ex36.m1.4.4.1.3.2"><csymbol cd="ambiguous" id="S3.Ex36.m1.4.4.1.3.2.1.cmml" xref="S3.Ex36.m1.4.4.1.3.2">superscript</csymbol><ci id="S3.Ex36.m1.4.4.1.3.2.2.cmml" xref="S3.Ex36.m1.4.4.1.3.2.2">𝐻</ci><times id="S3.Ex36.m1.4.4.1.3.2.3.cmml" xref="S3.Ex36.m1.4.4.1.3.2.3"></times></apply><list id="S3.Ex36.m1.4.4.1.3.3.1.cmml" xref="S3.Ex36.m1.4.4.1.3.3.2"><ci id="S3.Ex36.m1.1.1.cmml" xref="S3.Ex36.m1.1.1">𝑌</ci><ci id="S3.Ex36.m1.2.2.cmml" xref="S3.Ex36.m1.2.2">ℳ</ci></list></apply><apply id="S3.Ex36.m1.4.4.1.1.cmml" xref="S3.Ex36.m1.4.4.1.1"><times id="S3.Ex36.m1.4.4.1.1.2.cmml" xref="S3.Ex36.m1.4.4.1.1.2"></times><apply id="S3.Ex36.m1.4.4.1.1.3.cmml" xref="S3.Ex36.m1.4.4.1.1.3"><csymbol cd="ambiguous" id="S3.Ex36.m1.4.4.1.1.3.1.cmml" xref="S3.Ex36.m1.4.4.1.1.3">superscript</csymbol><ci id="S3.Ex36.m1.4.4.1.1.3.2.cmml" xref="S3.Ex36.m1.4.4.1.1.3.2">𝐻</ci><times id="S3.Ex36.m1.4.4.1.1.3.3.cmml" xref="S3.Ex36.m1.4.4.1.1.3.3"></times></apply><list id="S3.Ex36.m1.4.4.1.1.1.2.cmml" xref="S3.Ex36.m1.4.4.1.1.1.1"><ci id="S3.Ex36.m1.3.3.cmml" xref="S3.Ex36.m1.3.3">𝑋</ci><apply id="S3.Ex36.m1.4.4.1.1.1.1.1.cmml" xref="S3.Ex36.m1.4.4.1.1.1.1.1"><times id="S3.Ex36.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.Ex36.m1.4.4.1.1.1.1.1.1"></times><apply id="S3.Ex36.m1.4.4.1.1.1.1.1.2.cmml" xref="S3.Ex36.m1.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex36.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S3.Ex36.m1.4.4.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Ex36.m1.4.4.1.1.1.1.1.2.2.cmml" xref="S3.Ex36.m1.4.4.1.1.1.1.1.2.2">𝜆</ci><times id="S3.Ex36.m1.4.4.1.1.1.1.1.2.3.cmml" xref="S3.Ex36.m1.4.4.1.1.1.1.1.2.3"></times></apply><ci id="S3.Ex36.m1.4.4.1.1.1.1.1.3.cmml" xref="S3.Ex36.m1.4.4.1.1.1.1.1.3">ℳ</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex36.m1.4c">\lambda^{*}:H^{*}(Y;\mathcal{M})\to H^{*}(X;\lambda^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex36.m1.4d">italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_Y ; caligraphic_M ) → italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem4.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.4.2">for every coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.3.1.m1.1"><semantics id="S3.Thmtheorem4.p1.3.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem4.p1.3.1.m1.1.1" xref="S3.Thmtheorem4.p1.3.1.m1.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.3.1.m1.1b"><ci id="S3.Thmtheorem4.p1.3.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.1.m1.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.1.m1.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.1.m1.1d">caligraphic_M</annotation></semantics></math> for <math alttext="Y" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.2.m2.1"><semantics id="S3.Thmtheorem4.p1.4.2.m2.1a"><mi id="S3.Thmtheorem4.p1.4.2.m2.1.1" xref="S3.Thmtheorem4.p1.4.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.2.m2.1b"><ci id="S3.Thmtheorem4.p1.4.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.4.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.2.m2.1d">italic_Y</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.1.p1"> <p class="ltx_p" id="S3.SS3.1.p1.1">The simplicial map <math alttext="\lambda" class="ltx_Math" display="inline" id="S3.SS3.1.p1.1.m1.1"><semantics id="S3.SS3.1.p1.1.m1.1a"><mi id="S3.SS3.1.p1.1.m1.1.1" xref="S3.SS3.1.p1.1.m1.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.1.m1.1b"><ci id="S3.SS3.1.p1.1.m1.1.1.cmml" xref="S3.SS3.1.p1.1.m1.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.1.m1.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.1.m1.1d">italic_λ</annotation></semantics></math> induces a chain map</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex37"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lambda^{*}:C^{n}(Y,\mathcal{M})\to C^{n}(X;\lambda^{*}\mathcal{M})" class="ltx_Math" display="block" id="S3.Ex37.m1.4"><semantics id="S3.Ex37.m1.4a"><mrow id="S3.Ex37.m1.4.4" xref="S3.Ex37.m1.4.4.cmml"><msup id="S3.Ex37.m1.4.4.3" xref="S3.Ex37.m1.4.4.3.cmml"><mi id="S3.Ex37.m1.4.4.3.2" xref="S3.Ex37.m1.4.4.3.2.cmml">λ</mi><mo id="S3.Ex37.m1.4.4.3.3" xref="S3.Ex37.m1.4.4.3.3.cmml">∗</mo></msup><mo id="S3.Ex37.m1.4.4.2" lspace="0.278em" rspace="0.278em" xref="S3.Ex37.m1.4.4.2.cmml">:</mo><mrow id="S3.Ex37.m1.4.4.1" xref="S3.Ex37.m1.4.4.1.cmml"><mrow id="S3.Ex37.m1.4.4.1.3" xref="S3.Ex37.m1.4.4.1.3.cmml"><msup id="S3.Ex37.m1.4.4.1.3.2" xref="S3.Ex37.m1.4.4.1.3.2.cmml"><mi id="S3.Ex37.m1.4.4.1.3.2.2" xref="S3.Ex37.m1.4.4.1.3.2.2.cmml">C</mi><mi id="S3.Ex37.m1.4.4.1.3.2.3" xref="S3.Ex37.m1.4.4.1.3.2.3.cmml">n</mi></msup><mo id="S3.Ex37.m1.4.4.1.3.1" xref="S3.Ex37.m1.4.4.1.3.1.cmml">⁢</mo><mrow id="S3.Ex37.m1.4.4.1.3.3.2" xref="S3.Ex37.m1.4.4.1.3.3.1.cmml"><mo id="S3.Ex37.m1.4.4.1.3.3.2.1" stretchy="false" xref="S3.Ex37.m1.4.4.1.3.3.1.cmml">(</mo><mi id="S3.Ex37.m1.1.1" xref="S3.Ex37.m1.1.1.cmml">Y</mi><mo id="S3.Ex37.m1.4.4.1.3.3.2.2" xref="S3.Ex37.m1.4.4.1.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex37.m1.2.2" xref="S3.Ex37.m1.2.2.cmml">ℳ</mi><mo id="S3.Ex37.m1.4.4.1.3.3.2.3" stretchy="false" xref="S3.Ex37.m1.4.4.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex37.m1.4.4.1.2" stretchy="false" xref="S3.Ex37.m1.4.4.1.2.cmml">→</mo><mrow id="S3.Ex37.m1.4.4.1.1" xref="S3.Ex37.m1.4.4.1.1.cmml"><msup id="S3.Ex37.m1.4.4.1.1.3" xref="S3.Ex37.m1.4.4.1.1.3.cmml"><mi id="S3.Ex37.m1.4.4.1.1.3.2" xref="S3.Ex37.m1.4.4.1.1.3.2.cmml">C</mi><mi id="S3.Ex37.m1.4.4.1.1.3.3" xref="S3.Ex37.m1.4.4.1.1.3.3.cmml">n</mi></msup><mo id="S3.Ex37.m1.4.4.1.1.2" xref="S3.Ex37.m1.4.4.1.1.2.cmml">⁢</mo><mrow id="S3.Ex37.m1.4.4.1.1.1.1" xref="S3.Ex37.m1.4.4.1.1.1.2.cmml"><mo id="S3.Ex37.m1.4.4.1.1.1.1.2" stretchy="false" xref="S3.Ex37.m1.4.4.1.1.1.2.cmml">(</mo><mi id="S3.Ex37.m1.3.3" xref="S3.Ex37.m1.3.3.cmml">X</mi><mo id="S3.Ex37.m1.4.4.1.1.1.1.3" xref="S3.Ex37.m1.4.4.1.1.1.2.cmml">;</mo><mrow id="S3.Ex37.m1.4.4.1.1.1.1.1" xref="S3.Ex37.m1.4.4.1.1.1.1.1.cmml"><msup id="S3.Ex37.m1.4.4.1.1.1.1.1.2" xref="S3.Ex37.m1.4.4.1.1.1.1.1.2.cmml"><mi id="S3.Ex37.m1.4.4.1.1.1.1.1.2.2" xref="S3.Ex37.m1.4.4.1.1.1.1.1.2.2.cmml">λ</mi><mo id="S3.Ex37.m1.4.4.1.1.1.1.1.2.3" xref="S3.Ex37.m1.4.4.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S3.Ex37.m1.4.4.1.1.1.1.1.1" xref="S3.Ex37.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex37.m1.4.4.1.1.1.1.1.3" xref="S3.Ex37.m1.4.4.1.1.1.1.1.3.cmml">ℳ</mi></mrow><mo id="S3.Ex37.m1.4.4.1.1.1.1.4" stretchy="false" xref="S3.Ex37.m1.4.4.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex37.m1.4b"><apply id="S3.Ex37.m1.4.4.cmml" xref="S3.Ex37.m1.4.4"><ci id="S3.Ex37.m1.4.4.2.cmml" xref="S3.Ex37.m1.4.4.2">:</ci><apply id="S3.Ex37.m1.4.4.3.cmml" xref="S3.Ex37.m1.4.4.3"><csymbol cd="ambiguous" id="S3.Ex37.m1.4.4.3.1.cmml" xref="S3.Ex37.m1.4.4.3">superscript</csymbol><ci id="S3.Ex37.m1.4.4.3.2.cmml" xref="S3.Ex37.m1.4.4.3.2">𝜆</ci><times id="S3.Ex37.m1.4.4.3.3.cmml" xref="S3.Ex37.m1.4.4.3.3"></times></apply><apply id="S3.Ex37.m1.4.4.1.cmml" xref="S3.Ex37.m1.4.4.1"><ci id="S3.Ex37.m1.4.4.1.2.cmml" xref="S3.Ex37.m1.4.4.1.2">→</ci><apply id="S3.Ex37.m1.4.4.1.3.cmml" xref="S3.Ex37.m1.4.4.1.3"><times id="S3.Ex37.m1.4.4.1.3.1.cmml" xref="S3.Ex37.m1.4.4.1.3.1"></times><apply id="S3.Ex37.m1.4.4.1.3.2.cmml" xref="S3.Ex37.m1.4.4.1.3.2"><csymbol cd="ambiguous" id="S3.Ex37.m1.4.4.1.3.2.1.cmml" xref="S3.Ex37.m1.4.4.1.3.2">superscript</csymbol><ci id="S3.Ex37.m1.4.4.1.3.2.2.cmml" xref="S3.Ex37.m1.4.4.1.3.2.2">𝐶</ci><ci id="S3.Ex37.m1.4.4.1.3.2.3.cmml" xref="S3.Ex37.m1.4.4.1.3.2.3">𝑛</ci></apply><interval closure="open" id="S3.Ex37.m1.4.4.1.3.3.1.cmml" xref="S3.Ex37.m1.4.4.1.3.3.2"><ci id="S3.Ex37.m1.1.1.cmml" xref="S3.Ex37.m1.1.1">𝑌</ci><ci id="S3.Ex37.m1.2.2.cmml" xref="S3.Ex37.m1.2.2">ℳ</ci></interval></apply><apply id="S3.Ex37.m1.4.4.1.1.cmml" xref="S3.Ex37.m1.4.4.1.1"><times id="S3.Ex37.m1.4.4.1.1.2.cmml" xref="S3.Ex37.m1.4.4.1.1.2"></times><apply id="S3.Ex37.m1.4.4.1.1.3.cmml" xref="S3.Ex37.m1.4.4.1.1.3"><csymbol cd="ambiguous" id="S3.Ex37.m1.4.4.1.1.3.1.cmml" xref="S3.Ex37.m1.4.4.1.1.3">superscript</csymbol><ci id="S3.Ex37.m1.4.4.1.1.3.2.cmml" xref="S3.Ex37.m1.4.4.1.1.3.2">𝐶</ci><ci id="S3.Ex37.m1.4.4.1.1.3.3.cmml" xref="S3.Ex37.m1.4.4.1.1.3.3">𝑛</ci></apply><list id="S3.Ex37.m1.4.4.1.1.1.2.cmml" xref="S3.Ex37.m1.4.4.1.1.1.1"><ci id="S3.Ex37.m1.3.3.cmml" xref="S3.Ex37.m1.3.3">𝑋</ci><apply id="S3.Ex37.m1.4.4.1.1.1.1.1.cmml" xref="S3.Ex37.m1.4.4.1.1.1.1.1"><times id="S3.Ex37.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.Ex37.m1.4.4.1.1.1.1.1.1"></times><apply id="S3.Ex37.m1.4.4.1.1.1.1.1.2.cmml" xref="S3.Ex37.m1.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex37.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S3.Ex37.m1.4.4.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Ex37.m1.4.4.1.1.1.1.1.2.2.cmml" xref="S3.Ex37.m1.4.4.1.1.1.1.1.2.2">𝜆</ci><times id="S3.Ex37.m1.4.4.1.1.1.1.1.2.3.cmml" xref="S3.Ex37.m1.4.4.1.1.1.1.1.2.3"></times></apply><ci id="S3.Ex37.m1.4.4.1.1.1.1.1.3.cmml" xref="S3.Ex37.m1.4.4.1.1.1.1.1.3">ℳ</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex37.m1.4c">\lambda^{*}:C^{n}(Y,\mathcal{M})\to C^{n}(X;\lambda^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex37.m1.4d">italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_Y , caligraphic_M ) → italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_X ; italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.1.p1.4">defined by <math alttext="\lambda^{*}(f)(\sigma)=f(\lambda(\sigma))" class="ltx_Math" display="inline" id="S3.SS3.1.p1.2.m1.4"><semantics id="S3.SS3.1.p1.2.m1.4a"><mrow id="S3.SS3.1.p1.2.m1.4.4" xref="S3.SS3.1.p1.2.m1.4.4.cmml"><mrow id="S3.SS3.1.p1.2.m1.4.4.3" xref="S3.SS3.1.p1.2.m1.4.4.3.cmml"><msup id="S3.SS3.1.p1.2.m1.4.4.3.2" xref="S3.SS3.1.p1.2.m1.4.4.3.2.cmml"><mi id="S3.SS3.1.p1.2.m1.4.4.3.2.2" xref="S3.SS3.1.p1.2.m1.4.4.3.2.2.cmml">λ</mi><mo id="S3.SS3.1.p1.2.m1.4.4.3.2.3" xref="S3.SS3.1.p1.2.m1.4.4.3.2.3.cmml">∗</mo></msup><mo id="S3.SS3.1.p1.2.m1.4.4.3.1" xref="S3.SS3.1.p1.2.m1.4.4.3.1.cmml">⁢</mo><mrow id="S3.SS3.1.p1.2.m1.4.4.3.3.2" xref="S3.SS3.1.p1.2.m1.4.4.3.cmml"><mo id="S3.SS3.1.p1.2.m1.4.4.3.3.2.1" stretchy="false" xref="S3.SS3.1.p1.2.m1.4.4.3.cmml">(</mo><mi id="S3.SS3.1.p1.2.m1.1.1" xref="S3.SS3.1.p1.2.m1.1.1.cmml">f</mi><mo id="S3.SS3.1.p1.2.m1.4.4.3.3.2.2" stretchy="false" xref="S3.SS3.1.p1.2.m1.4.4.3.cmml">)</mo></mrow><mo id="S3.SS3.1.p1.2.m1.4.4.3.1a" xref="S3.SS3.1.p1.2.m1.4.4.3.1.cmml">⁢</mo><mrow id="S3.SS3.1.p1.2.m1.4.4.3.4.2" xref="S3.SS3.1.p1.2.m1.4.4.3.cmml"><mo id="S3.SS3.1.p1.2.m1.4.4.3.4.2.1" stretchy="false" xref="S3.SS3.1.p1.2.m1.4.4.3.cmml">(</mo><mi id="S3.SS3.1.p1.2.m1.2.2" xref="S3.SS3.1.p1.2.m1.2.2.cmml">σ</mi><mo id="S3.SS3.1.p1.2.m1.4.4.3.4.2.2" stretchy="false" xref="S3.SS3.1.p1.2.m1.4.4.3.cmml">)</mo></mrow></mrow><mo id="S3.SS3.1.p1.2.m1.4.4.2" xref="S3.SS3.1.p1.2.m1.4.4.2.cmml">=</mo><mrow id="S3.SS3.1.p1.2.m1.4.4.1" xref="S3.SS3.1.p1.2.m1.4.4.1.cmml"><mi id="S3.SS3.1.p1.2.m1.4.4.1.3" xref="S3.SS3.1.p1.2.m1.4.4.1.3.cmml">f</mi><mo id="S3.SS3.1.p1.2.m1.4.4.1.2" xref="S3.SS3.1.p1.2.m1.4.4.1.2.cmml">⁢</mo><mrow id="S3.SS3.1.p1.2.m1.4.4.1.1.1" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.cmml"><mo id="S3.SS3.1.p1.2.m1.4.4.1.1.1.2" stretchy="false" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.cmml"><mi id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.2" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.2.cmml">λ</mi><mo id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.1" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.3.2" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.cmml"><mo id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.cmml">(</mo><mi id="S3.SS3.1.p1.2.m1.3.3" xref="S3.SS3.1.p1.2.m1.3.3.cmml">σ</mi><mo id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.1.p1.2.m1.4.4.1.1.1.3" stretchy="false" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.2.m1.4b"><apply id="S3.SS3.1.p1.2.m1.4.4.cmml" xref="S3.SS3.1.p1.2.m1.4.4"><eq id="S3.SS3.1.p1.2.m1.4.4.2.cmml" xref="S3.SS3.1.p1.2.m1.4.4.2"></eq><apply id="S3.SS3.1.p1.2.m1.4.4.3.cmml" xref="S3.SS3.1.p1.2.m1.4.4.3"><times id="S3.SS3.1.p1.2.m1.4.4.3.1.cmml" xref="S3.SS3.1.p1.2.m1.4.4.3.1"></times><apply id="S3.SS3.1.p1.2.m1.4.4.3.2.cmml" xref="S3.SS3.1.p1.2.m1.4.4.3.2"><csymbol cd="ambiguous" id="S3.SS3.1.p1.2.m1.4.4.3.2.1.cmml" xref="S3.SS3.1.p1.2.m1.4.4.3.2">superscript</csymbol><ci id="S3.SS3.1.p1.2.m1.4.4.3.2.2.cmml" xref="S3.SS3.1.p1.2.m1.4.4.3.2.2">𝜆</ci><times id="S3.SS3.1.p1.2.m1.4.4.3.2.3.cmml" xref="S3.SS3.1.p1.2.m1.4.4.3.2.3"></times></apply><ci id="S3.SS3.1.p1.2.m1.1.1.cmml" xref="S3.SS3.1.p1.2.m1.1.1">𝑓</ci><ci id="S3.SS3.1.p1.2.m1.2.2.cmml" xref="S3.SS3.1.p1.2.m1.2.2">𝜎</ci></apply><apply id="S3.SS3.1.p1.2.m1.4.4.1.cmml" xref="S3.SS3.1.p1.2.m1.4.4.1"><times id="S3.SS3.1.p1.2.m1.4.4.1.2.cmml" xref="S3.SS3.1.p1.2.m1.4.4.1.2"></times><ci id="S3.SS3.1.p1.2.m1.4.4.1.3.cmml" xref="S3.SS3.1.p1.2.m1.4.4.1.3">𝑓</ci><apply id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.cmml" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1"><times id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.1.cmml" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.1"></times><ci id="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.2.cmml" xref="S3.SS3.1.p1.2.m1.4.4.1.1.1.1.2">𝜆</ci><ci id="S3.SS3.1.p1.2.m1.3.3.cmml" xref="S3.SS3.1.p1.2.m1.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.2.m1.4c">\lambda^{*}(f)(\sigma)=f(\lambda(\sigma))</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.2.m1.4d">italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_f ) ( italic_σ ) = italic_f ( italic_λ ( italic_σ ) )</annotation></semantics></math> for every simplex <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS3.1.p1.3.m2.1"><semantics id="S3.SS3.1.p1.3.m2.1a"><mi id="S3.SS3.1.p1.3.m2.1.1" xref="S3.SS3.1.p1.3.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.3.m2.1b"><ci id="S3.SS3.1.p1.3.m2.1.1.cmml" xref="S3.SS3.1.p1.3.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.3.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.3.m2.1d">italic_σ</annotation></semantics></math> in <math alttext="X" class="ltx_Math" display="inline" id="S3.SS3.1.p1.4.m3.1"><semantics id="S3.SS3.1.p1.4.m3.1a"><mi id="S3.SS3.1.p1.4.m3.1.1" xref="S3.SS3.1.p1.4.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.4.m3.1b"><ci id="S3.SS3.1.p1.4.m3.1.1.cmml" xref="S3.SS3.1.p1.4.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.4.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.4.m3.1d">italic_X</annotation></semantics></math>. Note that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex38"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lambda^{*}(f)(\sigma)=f(\lambda(\sigma))\in\mathcal{M}(\lambda(\sigma))=(% \lambda^{*}\mathcal{M})(\sigma)." class="ltx_Math" display="block" id="S3.Ex38.m1.6"><semantics id="S3.Ex38.m1.6a"><mrow id="S3.Ex38.m1.6.6.1" xref="S3.Ex38.m1.6.6.1.1.cmml"><mrow id="S3.Ex38.m1.6.6.1.1" xref="S3.Ex38.m1.6.6.1.1.cmml"><mrow id="S3.Ex38.m1.6.6.1.1.5" xref="S3.Ex38.m1.6.6.1.1.5.cmml"><msup id="S3.Ex38.m1.6.6.1.1.5.2" xref="S3.Ex38.m1.6.6.1.1.5.2.cmml"><mi id="S3.Ex38.m1.6.6.1.1.5.2.2" xref="S3.Ex38.m1.6.6.1.1.5.2.2.cmml">λ</mi><mo id="S3.Ex38.m1.6.6.1.1.5.2.3" xref="S3.Ex38.m1.6.6.1.1.5.2.3.cmml">∗</mo></msup><mo id="S3.Ex38.m1.6.6.1.1.5.1" xref="S3.Ex38.m1.6.6.1.1.5.1.cmml">⁢</mo><mrow id="S3.Ex38.m1.6.6.1.1.5.3.2" xref="S3.Ex38.m1.6.6.1.1.5.cmml"><mo id="S3.Ex38.m1.6.6.1.1.5.3.2.1" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.5.cmml">(</mo><mi id="S3.Ex38.m1.1.1" xref="S3.Ex38.m1.1.1.cmml">f</mi><mo id="S3.Ex38.m1.6.6.1.1.5.3.2.2" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.5.cmml">)</mo></mrow><mo id="S3.Ex38.m1.6.6.1.1.5.1a" xref="S3.Ex38.m1.6.6.1.1.5.1.cmml">⁢</mo><mrow id="S3.Ex38.m1.6.6.1.1.5.4.2" xref="S3.Ex38.m1.6.6.1.1.5.cmml"><mo id="S3.Ex38.m1.6.6.1.1.5.4.2.1" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.5.cmml">(</mo><mi id="S3.Ex38.m1.2.2" xref="S3.Ex38.m1.2.2.cmml">σ</mi><mo id="S3.Ex38.m1.6.6.1.1.5.4.2.2" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.5.cmml">)</mo></mrow></mrow><mo id="S3.Ex38.m1.6.6.1.1.6" xref="S3.Ex38.m1.6.6.1.1.6.cmml">=</mo><mrow id="S3.Ex38.m1.6.6.1.1.1" xref="S3.Ex38.m1.6.6.1.1.1.cmml"><mi id="S3.Ex38.m1.6.6.1.1.1.3" xref="S3.Ex38.m1.6.6.1.1.1.3.cmml">f</mi><mo id="S3.Ex38.m1.6.6.1.1.1.2" xref="S3.Ex38.m1.6.6.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex38.m1.6.6.1.1.1.1.1" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.cmml"><mo id="S3.Ex38.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex38.m1.6.6.1.1.1.1.1.1" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.cmml"><mi id="S3.Ex38.m1.6.6.1.1.1.1.1.1.2" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.2.cmml">λ</mi><mo id="S3.Ex38.m1.6.6.1.1.1.1.1.1.1" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Ex38.m1.6.6.1.1.1.1.1.1.3.2" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.cmml"><mo id="S3.Ex38.m1.6.6.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.cmml">(</mo><mi id="S3.Ex38.m1.3.3" xref="S3.Ex38.m1.3.3.cmml">σ</mi><mo id="S3.Ex38.m1.6.6.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex38.m1.6.6.1.1.1.1.1.3" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex38.m1.6.6.1.1.7" xref="S3.Ex38.m1.6.6.1.1.7.cmml">∈</mo><mrow id="S3.Ex38.m1.6.6.1.1.2" xref="S3.Ex38.m1.6.6.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex38.m1.6.6.1.1.2.3" xref="S3.Ex38.m1.6.6.1.1.2.3.cmml">ℳ</mi><mo id="S3.Ex38.m1.6.6.1.1.2.2" xref="S3.Ex38.m1.6.6.1.1.2.2.cmml">⁢</mo><mrow id="S3.Ex38.m1.6.6.1.1.2.1.1" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.cmml"><mo id="S3.Ex38.m1.6.6.1.1.2.1.1.2" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.cmml">(</mo><mrow id="S3.Ex38.m1.6.6.1.1.2.1.1.1" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.cmml"><mi id="S3.Ex38.m1.6.6.1.1.2.1.1.1.2" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.2.cmml">λ</mi><mo id="S3.Ex38.m1.6.6.1.1.2.1.1.1.1" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.Ex38.m1.6.6.1.1.2.1.1.1.3.2" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.cmml"><mo id="S3.Ex38.m1.6.6.1.1.2.1.1.1.3.2.1" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.cmml">(</mo><mi id="S3.Ex38.m1.4.4" xref="S3.Ex38.m1.4.4.cmml">σ</mi><mo id="S3.Ex38.m1.6.6.1.1.2.1.1.1.3.2.2" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex38.m1.6.6.1.1.2.1.1.3" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex38.m1.6.6.1.1.8" xref="S3.Ex38.m1.6.6.1.1.8.cmml">=</mo><mrow id="S3.Ex38.m1.6.6.1.1.3" xref="S3.Ex38.m1.6.6.1.1.3.cmml"><mrow id="S3.Ex38.m1.6.6.1.1.3.1.1" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.cmml"><mo id="S3.Ex38.m1.6.6.1.1.3.1.1.2" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.cmml">(</mo><mrow id="S3.Ex38.m1.6.6.1.1.3.1.1.1" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.cmml"><msup id="S3.Ex38.m1.6.6.1.1.3.1.1.1.2" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.cmml"><mi id="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.2" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.2.cmml">λ</mi><mo id="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.3" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.3.cmml">∗</mo></msup><mo id="S3.Ex38.m1.6.6.1.1.3.1.1.1.1" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex38.m1.6.6.1.1.3.1.1.1.3" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.3.cmml">ℳ</mi></mrow><mo id="S3.Ex38.m1.6.6.1.1.3.1.1.3" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex38.m1.6.6.1.1.3.2" xref="S3.Ex38.m1.6.6.1.1.3.2.cmml">⁢</mo><mrow id="S3.Ex38.m1.6.6.1.1.3.3.2" xref="S3.Ex38.m1.6.6.1.1.3.cmml"><mo id="S3.Ex38.m1.6.6.1.1.3.3.2.1" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.3.cmml">(</mo><mi id="S3.Ex38.m1.5.5" xref="S3.Ex38.m1.5.5.cmml">σ</mi><mo id="S3.Ex38.m1.6.6.1.1.3.3.2.2" stretchy="false" xref="S3.Ex38.m1.6.6.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex38.m1.6.6.1.2" lspace="0em" xref="S3.Ex38.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex38.m1.6b"><apply id="S3.Ex38.m1.6.6.1.1.cmml" xref="S3.Ex38.m1.6.6.1"><and id="S3.Ex38.m1.6.6.1.1a.cmml" xref="S3.Ex38.m1.6.6.1"></and><apply id="S3.Ex38.m1.6.6.1.1b.cmml" xref="S3.Ex38.m1.6.6.1"><eq id="S3.Ex38.m1.6.6.1.1.6.cmml" xref="S3.Ex38.m1.6.6.1.1.6"></eq><apply id="S3.Ex38.m1.6.6.1.1.5.cmml" xref="S3.Ex38.m1.6.6.1.1.5"><times id="S3.Ex38.m1.6.6.1.1.5.1.cmml" 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xref="S3.Ex38.m1.6.6.1.1.8"></eq><share href="https://arxiv.org/html/2503.14659v1#S3.Ex38.m1.6.6.1.1.2.cmml" id="S3.Ex38.m1.6.6.1.1f.cmml" xref="S3.Ex38.m1.6.6.1"></share><apply id="S3.Ex38.m1.6.6.1.1.3.cmml" xref="S3.Ex38.m1.6.6.1.1.3"><times id="S3.Ex38.m1.6.6.1.1.3.2.cmml" xref="S3.Ex38.m1.6.6.1.1.3.2"></times><apply id="S3.Ex38.m1.6.6.1.1.3.1.1.1.cmml" xref="S3.Ex38.m1.6.6.1.1.3.1.1"><times id="S3.Ex38.m1.6.6.1.1.3.1.1.1.1.cmml" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.1"></times><apply id="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.cmml" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.1.cmml" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.2">superscript</csymbol><ci id="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.2.cmml" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.2">𝜆</ci><times id="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.3.cmml" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.2.3"></times></apply><ci id="S3.Ex38.m1.6.6.1.1.3.1.1.1.3.cmml" xref="S3.Ex38.m1.6.6.1.1.3.1.1.1.3">ℳ</ci></apply><ci id="S3.Ex38.m1.5.5.cmml" xref="S3.Ex38.m1.5.5">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex38.m1.6c">\lambda^{*}(f)(\sigma)=f(\lambda(\sigma))\in\mathcal{M}(\lambda(\sigma))=(% \lambda^{*}\mathcal{M})(\sigma).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex38.m1.6d">italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_f ) ( italic_σ ) = italic_f ( italic_λ ( italic_σ ) ) ∈ caligraphic_M ( italic_λ ( italic_σ ) ) = ( italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M ) ( italic_σ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.1.p1.5">The chain map <math alttext="\lambda^{*}" class="ltx_Math" display="inline" id="S3.SS3.1.p1.5.m1.1"><semantics id="S3.SS3.1.p1.5.m1.1a"><msup id="S3.SS3.1.p1.5.m1.1.1" xref="S3.SS3.1.p1.5.m1.1.1.cmml"><mi id="S3.SS3.1.p1.5.m1.1.1.2" xref="S3.SS3.1.p1.5.m1.1.1.2.cmml">λ</mi><mo id="S3.SS3.1.p1.5.m1.1.1.3" xref="S3.SS3.1.p1.5.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.5.m1.1b"><apply id="S3.SS3.1.p1.5.m1.1.1.cmml" xref="S3.SS3.1.p1.5.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.1.p1.5.m1.1.1.1.cmml" xref="S3.SS3.1.p1.5.m1.1.1">superscript</csymbol><ci id="S3.SS3.1.p1.5.m1.1.1.2.cmml" xref="S3.SS3.1.p1.5.m1.1.1.2">𝜆</ci><times id="S3.SS3.1.p1.5.m1.1.1.3.cmml" xref="S3.SS3.1.p1.5.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.5.m1.1c">\lambda^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.5.m1.1d">italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> induces the desired homomorphism between the corresponding cohomology modules. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.1.1.1">Definition 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem5.p1"> <p class="ltx_p" id="S3.Thmtheorem5.p1.9">The <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem5.p1.9.1">category of simplicial sets with coefficients</em> is defined as the category whose objects are the pairs <math alttext="(X,\mathcal{M})" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.m1.2"><semantics id="S3.Thmtheorem5.p1.1.m1.2a"><mrow id="S3.Thmtheorem5.p1.1.m1.2.3.2" xref="S3.Thmtheorem5.p1.1.m1.2.3.1.cmml"><mo id="S3.Thmtheorem5.p1.1.m1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.1.m1.2.3.1.cmml">(</mo><mi id="S3.Thmtheorem5.p1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.m1.1.1.cmml">X</mi><mo id="S3.Thmtheorem5.p1.1.m1.2.3.2.2" xref="S3.Thmtheorem5.p1.1.m1.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.1.m1.2.2" xref="S3.Thmtheorem5.p1.1.m1.2.2.cmml">ℳ</mi><mo id="S3.Thmtheorem5.p1.1.m1.2.3.2.3" stretchy="false" xref="S3.Thmtheorem5.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.m1.2b"><interval closure="open" id="S3.Thmtheorem5.p1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem5.p1.1.m1.2.3.2"><ci id="S3.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem5.p1.1.m1.1.1">𝑋</ci><ci id="S3.Thmtheorem5.p1.1.m1.2.2.cmml" xref="S3.Thmtheorem5.p1.1.m1.2.2">ℳ</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.m1.2c">(X,\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.m1.2d">( italic_X , caligraphic_M )</annotation></semantics></math> where <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.m2.1a"><mi id="S3.Thmtheorem5.p1.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.m2.1b"><ci id="S3.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.m2.1d">italic_X</annotation></semantics></math> is a simplicial set and <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.m3.1"><semantics id="S3.Thmtheorem5.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.m3.1b"><ci id="S3.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.m3.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.m3.1d">caligraphic_M</annotation></semantics></math> is a coefficient system on <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.m4.1"><semantics id="S3.Thmtheorem5.p1.4.m4.1a"><mi id="S3.Thmtheorem5.p1.4.m4.1.1" xref="S3.Thmtheorem5.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.m4.1b"><ci id="S3.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.m4.1d">italic_X</annotation></semantics></math>. A morphism of spaces of coefficients <math alttext="(X,\mathcal{M}_{X})\to(Y,\mathcal{M}_{Y})" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.5.m5.4"><semantics id="S3.Thmtheorem5.p1.5.m5.4a"><mrow id="S3.Thmtheorem5.p1.5.m5.4.4" xref="S3.Thmtheorem5.p1.5.m5.4.4.cmml"><mrow id="S3.Thmtheorem5.p1.5.m5.3.3.1.1" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.2.cmml"><mo id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.2.cmml">(</mo><mi id="S3.Thmtheorem5.p1.5.m5.1.1" xref="S3.Thmtheorem5.p1.5.m5.1.1.cmml">X</mi><mo id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.3" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.2.cmml">,</mo><msub id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.2" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.2.cmml">ℳ</mi><mi id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.3" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.3.cmml">X</mi></msub><mo id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.4" stretchy="false" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.2.cmml">)</mo></mrow><mo id="S3.Thmtheorem5.p1.5.m5.4.4.3" stretchy="false" xref="S3.Thmtheorem5.p1.5.m5.4.4.3.cmml">→</mo><mrow id="S3.Thmtheorem5.p1.5.m5.4.4.2.1" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.2.cmml"><mo id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.2" stretchy="false" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.2.cmml">(</mo><mi id="S3.Thmtheorem5.p1.5.m5.2.2" xref="S3.Thmtheorem5.p1.5.m5.2.2.cmml">Y</mi><mo id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.3" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.2.cmml">,</mo><msub id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.2" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.2.cmml">ℳ</mi><mi id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.3" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.3.cmml">Y</mi></msub><mo id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.4" stretchy="false" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.5.m5.4b"><apply id="S3.Thmtheorem5.p1.5.m5.4.4.cmml" xref="S3.Thmtheorem5.p1.5.m5.4.4"><ci id="S3.Thmtheorem5.p1.5.m5.4.4.3.cmml" xref="S3.Thmtheorem5.p1.5.m5.4.4.3">→</ci><interval closure="open" id="S3.Thmtheorem5.p1.5.m5.3.3.1.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.1"><ci id="S3.Thmtheorem5.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1">𝑋</ci><apply id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.2">ℳ</ci><ci id="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.3.cmml" xref="S3.Thmtheorem5.p1.5.m5.3.3.1.1.1.3">𝑋</ci></apply></interval><interval closure="open" id="S3.Thmtheorem5.p1.5.m5.4.4.2.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.1"><ci id="S3.Thmtheorem5.p1.5.m5.2.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.2.2">𝑌</ci><apply id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.2">ℳ</ci><ci id="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.3.cmml" xref="S3.Thmtheorem5.p1.5.m5.4.4.2.1.1.3">𝑌</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.5.m5.4c">(X,\mathcal{M}_{X})\to(Y,\mathcal{M}_{Y})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.5.m5.4d">( italic_X , caligraphic_M start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) → ( italic_Y , caligraphic_M start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT )</annotation></semantics></math> is a pair of maps <math alttext="(\lambda,\theta)" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.6.m6.2"><semantics id="S3.Thmtheorem5.p1.6.m6.2a"><mrow id="S3.Thmtheorem5.p1.6.m6.2.3.2" xref="S3.Thmtheorem5.p1.6.m6.2.3.1.cmml"><mo id="S3.Thmtheorem5.p1.6.m6.2.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.6.m6.2.3.1.cmml">(</mo><mi id="S3.Thmtheorem5.p1.6.m6.1.1" xref="S3.Thmtheorem5.p1.6.m6.1.1.cmml">λ</mi><mo id="S3.Thmtheorem5.p1.6.m6.2.3.2.2" xref="S3.Thmtheorem5.p1.6.m6.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.6.m6.2.2" xref="S3.Thmtheorem5.p1.6.m6.2.2.cmml">θ</mi><mo id="S3.Thmtheorem5.p1.6.m6.2.3.2.3" stretchy="false" xref="S3.Thmtheorem5.p1.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.6.m6.2b"><interval closure="open" id="S3.Thmtheorem5.p1.6.m6.2.3.1.cmml" xref="S3.Thmtheorem5.p1.6.m6.2.3.2"><ci id="S3.Thmtheorem5.p1.6.m6.1.1.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.1">𝜆</ci><ci id="S3.Thmtheorem5.p1.6.m6.2.2.cmml" xref="S3.Thmtheorem5.p1.6.m6.2.2">𝜃</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.6.m6.2c">(\lambda,\theta)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.6.m6.2d">( italic_λ , italic_θ )</annotation></semantics></math> where <math alttext="\lambda:X\to Y" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.7.m7.1"><semantics id="S3.Thmtheorem5.p1.7.m7.1a"><mrow id="S3.Thmtheorem5.p1.7.m7.1.1" xref="S3.Thmtheorem5.p1.7.m7.1.1.cmml"><mi id="S3.Thmtheorem5.p1.7.m7.1.1.2" xref="S3.Thmtheorem5.p1.7.m7.1.1.2.cmml">λ</mi><mo id="S3.Thmtheorem5.p1.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem5.p1.7.m7.1.1.1.cmml">:</mo><mrow id="S3.Thmtheorem5.p1.7.m7.1.1.3" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.cmml"><mi id="S3.Thmtheorem5.p1.7.m7.1.1.3.2" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.2.cmml">X</mi><mo id="S3.Thmtheorem5.p1.7.m7.1.1.3.1" stretchy="false" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.1.cmml">→</mo><mi id="S3.Thmtheorem5.p1.7.m7.1.1.3.3" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.7.m7.1b"><apply id="S3.Thmtheorem5.p1.7.m7.1.1.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1"><ci id="S3.Thmtheorem5.p1.7.m7.1.1.1.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.1">:</ci><ci id="S3.Thmtheorem5.p1.7.m7.1.1.2.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.2">𝜆</ci><apply id="S3.Thmtheorem5.p1.7.m7.1.1.3.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.3"><ci id="S3.Thmtheorem5.p1.7.m7.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.1">→</ci><ci id="S3.Thmtheorem5.p1.7.m7.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.2">𝑋</ci><ci id="S3.Thmtheorem5.p1.7.m7.1.1.3.3.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.7.m7.1c">\lambda:X\to Y</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.7.m7.1d">italic_λ : italic_X → italic_Y</annotation></semantics></math> is a simplicial map and <math alttext="\theta:\lambda^{*}\mathcal{M}_{Y}\to\mathcal{M}_{X}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.8.m8.1"><semantics id="S3.Thmtheorem5.p1.8.m8.1a"><mrow id="S3.Thmtheorem5.p1.8.m8.1.1" xref="S3.Thmtheorem5.p1.8.m8.1.1.cmml"><mi id="S3.Thmtheorem5.p1.8.m8.1.1.2" xref="S3.Thmtheorem5.p1.8.m8.1.1.2.cmml">θ</mi><mo id="S3.Thmtheorem5.p1.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem5.p1.8.m8.1.1.1.cmml">:</mo><mrow id="S3.Thmtheorem5.p1.8.m8.1.1.3" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.cmml"><mrow id="S3.Thmtheorem5.p1.8.m8.1.1.3.2" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.cmml"><msup id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.cmml"><mi id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.2" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.2.cmml">λ</mi><mo id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.3" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.1" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.1.cmml">⁢</mo><msub id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.2" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.2.cmml">ℳ</mi><mi id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.3" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.3.cmml">Y</mi></msub></mrow><mo id="S3.Thmtheorem5.p1.8.m8.1.1.3.1" stretchy="false" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.1.cmml">→</mo><msub id="S3.Thmtheorem5.p1.8.m8.1.1.3.3" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.8.m8.1.1.3.3.2" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.3.2.cmml">ℳ</mi><mi id="S3.Thmtheorem5.p1.8.m8.1.1.3.3.3" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.3.3.cmml">X</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.8.m8.1b"><apply id="S3.Thmtheorem5.p1.8.m8.1.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1"><ci id="S3.Thmtheorem5.p1.8.m8.1.1.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.1">:</ci><ci id="S3.Thmtheorem5.p1.8.m8.1.1.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.2">𝜃</ci><apply id="S3.Thmtheorem5.p1.8.m8.1.1.3.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3"><ci id="S3.Thmtheorem5.p1.8.m8.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.1">→</ci><apply id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2"><times id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.1"></times><apply id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2">superscript</csymbol><ci id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.2">𝜆</ci><times id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.3.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.2.3"></times></apply><apply id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3">subscript</csymbol><ci id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.2">ℳ</ci><ci id="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.3.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.2.3.3">𝑌</ci></apply></apply><apply id="S3.Thmtheorem5.p1.8.m8.1.1.3.3.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.8.m8.1.1.3.3.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.3">subscript</csymbol><ci id="S3.Thmtheorem5.p1.8.m8.1.1.3.3.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.3.2">ℳ</ci><ci id="S3.Thmtheorem5.p1.8.m8.1.1.3.3.3.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1.3.3.3">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.8.m8.1c">\theta:\lambda^{*}\mathcal{M}_{Y}\to\mathcal{M}_{X}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.8.m8.1d">italic_θ : italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT → caligraphic_M start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> is a morphism of <math alttext="\Delta(X)-" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.9.m9.1"><semantics id="S3.Thmtheorem5.p1.9.m9.1a"><mrow id="S3.Thmtheorem5.p1.9.m9.1.2" xref="S3.Thmtheorem5.p1.9.m9.1.2.cmml"><mrow id="S3.Thmtheorem5.p1.9.m9.1.2.2" xref="S3.Thmtheorem5.p1.9.m9.1.2.2.cmml"><mi id="S3.Thmtheorem5.p1.9.m9.1.2.2.2" mathvariant="normal" xref="S3.Thmtheorem5.p1.9.m9.1.2.2.2.cmml">Δ</mi><mo id="S3.Thmtheorem5.p1.9.m9.1.2.2.1" xref="S3.Thmtheorem5.p1.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem5.p1.9.m9.1.2.2.3.2" xref="S3.Thmtheorem5.p1.9.m9.1.2.2.cmml"><mo id="S3.Thmtheorem5.p1.9.m9.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.9.m9.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem5.p1.9.m9.1.1" xref="S3.Thmtheorem5.p1.9.m9.1.1.cmml">X</mi><mo id="S3.Thmtheorem5.p1.9.m9.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem5.p1.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem5.p1.9.m9.1.2.3" xref="S3.Thmtheorem5.p1.9.m9.1.2.3.cmml">−</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.9.m9.1b"><apply id="S3.Thmtheorem5.p1.9.m9.1.2.cmml" xref="S3.Thmtheorem5.p1.9.m9.1.2"><csymbol cd="latexml" id="S3.Thmtheorem5.p1.9.m9.1.2.1.cmml" xref="S3.Thmtheorem5.p1.9.m9.1.2">limit-from</csymbol><apply id="S3.Thmtheorem5.p1.9.m9.1.2.2.cmml" xref="S3.Thmtheorem5.p1.9.m9.1.2.2"><times id="S3.Thmtheorem5.p1.9.m9.1.2.2.1.cmml" xref="S3.Thmtheorem5.p1.9.m9.1.2.2.1"></times><ci id="S3.Thmtheorem5.p1.9.m9.1.2.2.2.cmml" xref="S3.Thmtheorem5.p1.9.m9.1.2.2.2">Δ</ci><ci id="S3.Thmtheorem5.p1.9.m9.1.1.cmml" xref="S3.Thmtheorem5.p1.9.m9.1.1">𝑋</ci></apply><minus id="S3.Thmtheorem5.p1.9.m9.1.2.3.cmml" xref="S3.Thmtheorem5.p1.9.m9.1.2.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.9.m9.1c">\Delta(X)-</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.9.m9.1d">roman_Δ ( italic_X ) -</annotation></semantics></math>modules.</p> </div> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1">A morphism of pairs <math alttext="(\lambda,\theta):(X,\mathcal{M}_{X})\to(Y,\mathcal{M}_{Y})" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.6"><semantics id="S3.SS3.p2.1.m1.6a"><mrow id="S3.SS3.p2.1.m1.6.6" xref="S3.SS3.p2.1.m1.6.6.cmml"><mrow id="S3.SS3.p2.1.m1.6.6.4.2" xref="S3.SS3.p2.1.m1.6.6.4.1.cmml"><mo id="S3.SS3.p2.1.m1.6.6.4.2.1" stretchy="false" xref="S3.SS3.p2.1.m1.6.6.4.1.cmml">(</mo><mi id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml">λ</mi><mo id="S3.SS3.p2.1.m1.6.6.4.2.2" xref="S3.SS3.p2.1.m1.6.6.4.1.cmml">,</mo><mi id="S3.SS3.p2.1.m1.2.2" xref="S3.SS3.p2.1.m1.2.2.cmml">θ</mi><mo id="S3.SS3.p2.1.m1.6.6.4.2.3" rspace="0.278em" stretchy="false" xref="S3.SS3.p2.1.m1.6.6.4.1.cmml">)</mo></mrow><mo id="S3.SS3.p2.1.m1.6.6.3" rspace="0.278em" xref="S3.SS3.p2.1.m1.6.6.3.cmml">:</mo><mrow id="S3.SS3.p2.1.m1.6.6.2" xref="S3.SS3.p2.1.m1.6.6.2.cmml"><mrow id="S3.SS3.p2.1.m1.5.5.1.1.1" xref="S3.SS3.p2.1.m1.5.5.1.1.2.cmml"><mo id="S3.SS3.p2.1.m1.5.5.1.1.1.2" stretchy="false" xref="S3.SS3.p2.1.m1.5.5.1.1.2.cmml">(</mo><mi id="S3.SS3.p2.1.m1.3.3" xref="S3.SS3.p2.1.m1.3.3.cmml">X</mi><mo id="S3.SS3.p2.1.m1.5.5.1.1.1.3" xref="S3.SS3.p2.1.m1.5.5.1.1.2.cmml">,</mo><msub id="S3.SS3.p2.1.m1.5.5.1.1.1.1" xref="S3.SS3.p2.1.m1.5.5.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.1.m1.5.5.1.1.1.1.2" xref="S3.SS3.p2.1.m1.5.5.1.1.1.1.2.cmml">ℳ</mi><mi id="S3.SS3.p2.1.m1.5.5.1.1.1.1.3" xref="S3.SS3.p2.1.m1.5.5.1.1.1.1.3.cmml">X</mi></msub><mo id="S3.SS3.p2.1.m1.5.5.1.1.1.4" stretchy="false" xref="S3.SS3.p2.1.m1.5.5.1.1.2.cmml">)</mo></mrow><mo id="S3.SS3.p2.1.m1.6.6.2.3" stretchy="false" xref="S3.SS3.p2.1.m1.6.6.2.3.cmml">→</mo><mrow id="S3.SS3.p2.1.m1.6.6.2.2.1" xref="S3.SS3.p2.1.m1.6.6.2.2.2.cmml"><mo id="S3.SS3.p2.1.m1.6.6.2.2.1.2" stretchy="false" xref="S3.SS3.p2.1.m1.6.6.2.2.2.cmml">(</mo><mi id="S3.SS3.p2.1.m1.4.4" xref="S3.SS3.p2.1.m1.4.4.cmml">Y</mi><mo id="S3.SS3.p2.1.m1.6.6.2.2.1.3" xref="S3.SS3.p2.1.m1.6.6.2.2.2.cmml">,</mo><msub id="S3.SS3.p2.1.m1.6.6.2.2.1.1" xref="S3.SS3.p2.1.m1.6.6.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.1.m1.6.6.2.2.1.1.2" xref="S3.SS3.p2.1.m1.6.6.2.2.1.1.2.cmml">ℳ</mi><mi id="S3.SS3.p2.1.m1.6.6.2.2.1.1.3" xref="S3.SS3.p2.1.m1.6.6.2.2.1.1.3.cmml">Y</mi></msub><mo id="S3.SS3.p2.1.m1.6.6.2.2.1.4" stretchy="false" xref="S3.SS3.p2.1.m1.6.6.2.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.6b"><apply id="S3.SS3.p2.1.m1.6.6.cmml" xref="S3.SS3.p2.1.m1.6.6"><ci id="S3.SS3.p2.1.m1.6.6.3.cmml" xref="S3.SS3.p2.1.m1.6.6.3">:</ci><interval closure="open" id="S3.SS3.p2.1.m1.6.6.4.1.cmml" xref="S3.SS3.p2.1.m1.6.6.4.2"><ci id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">𝜆</ci><ci id="S3.SS3.p2.1.m1.2.2.cmml" xref="S3.SS3.p2.1.m1.2.2">𝜃</ci></interval><apply id="S3.SS3.p2.1.m1.6.6.2.cmml" xref="S3.SS3.p2.1.m1.6.6.2"><ci id="S3.SS3.p2.1.m1.6.6.2.3.cmml" xref="S3.SS3.p2.1.m1.6.6.2.3">→</ci><interval closure="open" id="S3.SS3.p2.1.m1.5.5.1.1.2.cmml" xref="S3.SS3.p2.1.m1.5.5.1.1.1"><ci id="S3.SS3.p2.1.m1.3.3.cmml" xref="S3.SS3.p2.1.m1.3.3">𝑋</ci><apply id="S3.SS3.p2.1.m1.5.5.1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.5.5.1.1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.5.5.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p2.1.m1.5.5.1.1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.5.5.1.1.1.1.2">ℳ</ci><ci id="S3.SS3.p2.1.m1.5.5.1.1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.5.5.1.1.1.1.3">𝑋</ci></apply></interval><interval closure="open" id="S3.SS3.p2.1.m1.6.6.2.2.2.cmml" xref="S3.SS3.p2.1.m1.6.6.2.2.1"><ci id="S3.SS3.p2.1.m1.4.4.cmml" xref="S3.SS3.p2.1.m1.4.4">𝑌</ci><apply id="S3.SS3.p2.1.m1.6.6.2.2.1.1.cmml" xref="S3.SS3.p2.1.m1.6.6.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.6.6.2.2.1.1.1.cmml" xref="S3.SS3.p2.1.m1.6.6.2.2.1.1">subscript</csymbol><ci id="S3.SS3.p2.1.m1.6.6.2.2.1.1.2.cmml" xref="S3.SS3.p2.1.m1.6.6.2.2.1.1.2">ℳ</ci><ci id="S3.SS3.p2.1.m1.6.6.2.2.1.1.3.cmml" xref="S3.SS3.p2.1.m1.6.6.2.2.1.1.3">𝑌</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.6c">(\lambda,\theta):(X,\mathcal{M}_{X})\to(Y,\mathcal{M}_{Y})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.6d">( italic_λ , italic_θ ) : ( italic_X , caligraphic_M start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) → ( italic_Y , caligraphic_M start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT )</annotation></semantics></math> induces a homomorphism of cohomology groups</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex39"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(\lambda,\theta)^{*}:H^{*}(Y,\mathcal{M}_{Y})\smash{\,\mathop{\longrightarrow}% \limits^{\lambda^{*}}\,}H^{*}(X;\lambda^{*}\mathcal{M}_{Y})\smash{\,\mathop{% \longrightarrow}\limits^{\theta_{*}}\,}H^{*}(X;\mathcal{M}_{X})." class="ltx_Math" display="block" id="S3.Ex39.m1.6"><semantics id="S3.Ex39.m1.6a"><mrow id="S3.Ex39.m1.6.6.1" xref="S3.Ex39.m1.6.6.1.1.cmml"><mrow id="S3.Ex39.m1.6.6.1.1" xref="S3.Ex39.m1.6.6.1.1.cmml"><msup id="S3.Ex39.m1.6.6.1.1.5" xref="S3.Ex39.m1.6.6.1.1.5.cmml"><mrow id="S3.Ex39.m1.6.6.1.1.5.2.2" xref="S3.Ex39.m1.6.6.1.1.5.2.1.cmml"><mo id="S3.Ex39.m1.6.6.1.1.5.2.2.1" stretchy="false" xref="S3.Ex39.m1.6.6.1.1.5.2.1.cmml">(</mo><mi id="S3.Ex39.m1.1.1" xref="S3.Ex39.m1.1.1.cmml">λ</mi><mo id="S3.Ex39.m1.6.6.1.1.5.2.2.2" xref="S3.Ex39.m1.6.6.1.1.5.2.1.cmml">,</mo><mi id="S3.Ex39.m1.2.2" xref="S3.Ex39.m1.2.2.cmml">θ</mi><mo id="S3.Ex39.m1.6.6.1.1.5.2.2.3" stretchy="false" xref="S3.Ex39.m1.6.6.1.1.5.2.1.cmml">)</mo></mrow><mo id="S3.Ex39.m1.6.6.1.1.5.3" xref="S3.Ex39.m1.6.6.1.1.5.3.cmml">∗</mo></msup><mo id="S3.Ex39.m1.6.6.1.1.4" lspace="0.278em" rspace="0.278em" xref="S3.Ex39.m1.6.6.1.1.4.cmml">:</mo><mrow id="S3.Ex39.m1.6.6.1.1.3" xref="S3.Ex39.m1.6.6.1.1.3.cmml"><msup id="S3.Ex39.m1.6.6.1.1.3.5" xref="S3.Ex39.m1.6.6.1.1.3.5.cmml"><mi id="S3.Ex39.m1.6.6.1.1.3.5.2" xref="S3.Ex39.m1.6.6.1.1.3.5.2.cmml">H</mi><mo id="S3.Ex39.m1.6.6.1.1.3.5.3" xref="S3.Ex39.m1.6.6.1.1.3.5.3.cmml">∗</mo></msup><mo id="S3.Ex39.m1.6.6.1.1.3.4" xref="S3.Ex39.m1.6.6.1.1.3.4.cmml">⁢</mo><mrow id="S3.Ex39.m1.6.6.1.1.1.1.1" xref="S3.Ex39.m1.6.6.1.1.1.1.2.cmml"><mo id="S3.Ex39.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S3.Ex39.m1.6.6.1.1.1.1.2.cmml">(</mo><mi id="S3.Ex39.m1.3.3" xref="S3.Ex39.m1.3.3.cmml">Y</mi><mo id="S3.Ex39.m1.6.6.1.1.1.1.1.3" xref="S3.Ex39.m1.6.6.1.1.1.1.2.cmml">,</mo><msub id="S3.Ex39.m1.6.6.1.1.1.1.1.1" xref="S3.Ex39.m1.6.6.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex39.m1.6.6.1.1.1.1.1.1.2" xref="S3.Ex39.m1.6.6.1.1.1.1.1.1.2.cmml">ℳ</mi><mi id="S3.Ex39.m1.6.6.1.1.1.1.1.1.3" xref="S3.Ex39.m1.6.6.1.1.1.1.1.1.3.cmml">Y</mi></msub><mo id="S3.Ex39.m1.6.6.1.1.1.1.1.4" stretchy="false" xref="S3.Ex39.m1.6.6.1.1.1.1.2.cmml">)</mo></mrow><mo id="S3.Ex39.m1.6.6.1.1.3.4a" lspace="0.337em" xref="S3.Ex39.m1.6.6.1.1.3.4.cmml">⁢</mo><mrow id="S3.Ex39.m1.6.6.1.1.3.3" xref="S3.Ex39.m1.6.6.1.1.3.3.cmml"><mover id="S3.Ex39.m1.6.6.1.1.3.3.3" xref="S3.Ex39.m1.6.6.1.1.3.3.3.cmml"><mo id="S3.Ex39.m1.6.6.1.1.3.3.3.2" movablelimits="false" rspace="0.167em" xref="S3.Ex39.m1.6.6.1.1.3.3.3.2.cmml">⟶</mo><msup id="S3.Ex39.m1.6.6.1.1.3.3.3.3" 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xref="S3.Ex39.m1.6.6.1.1.3.3.2.2.1.1.1.1">subscript</csymbol><ci id="S3.Ex39.m1.6.6.1.1.3.3.2.2.1.1.1.1.2.cmml" xref="S3.Ex39.m1.6.6.1.1.3.3.2.2.1.1.1.1.2">ℳ</ci><ci id="S3.Ex39.m1.6.6.1.1.3.3.2.2.1.1.1.1.3.cmml" xref="S3.Ex39.m1.6.6.1.1.3.3.2.2.1.1.1.1.3">𝑋</ci></apply></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex39.m1.6c">(\lambda,\theta)^{*}:H^{*}(Y,\mathcal{M}_{Y})\smash{\,\mathop{\longrightarrow}% \limits^{\lambda^{*}}\,}H^{*}(X;\lambda^{*}\mathcal{M}_{Y})\smash{\,\mathop{% \longrightarrow}\limits^{\theta_{*}}\,}H^{*}(X;\mathcal{M}_{X}).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex39.m1.6d">( italic_λ , italic_θ ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_Y , caligraphic_M start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) ⟶ start_POSTSUPERSCRIPT italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; italic_λ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) ⟶ start_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.4. </span>The Thomason cohomology of a category</h3> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.1.1.1">Definition 3.6</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem6.p1"> <p class="ltx_p" id="S3.Thmtheorem6.p1.7">Let <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.m1.1"><semantics id="S3.Thmtheorem6.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.m1.1b"><ci id="S3.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.m1.1d">caligraphic_C</annotation></semantics></math> be a small category. A coefficient system for the simplicial set <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.m2.1"><semantics id="S3.Thmtheorem6.p1.2.m2.1a"><mrow id="S3.Thmtheorem6.p1.2.m2.1.1" xref="S3.Thmtheorem6.p1.2.m2.1.1.cmml"><mi id="S3.Thmtheorem6.p1.2.m2.1.1.2" xref="S3.Thmtheorem6.p1.2.m2.1.1.2.cmml">N</mi><mo id="S3.Thmtheorem6.p1.2.m2.1.1.1" xref="S3.Thmtheorem6.p1.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.2.m2.1.1.3" xref="S3.Thmtheorem6.p1.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.m2.1b"><apply id="S3.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1"><times id="S3.Thmtheorem6.p1.2.m2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1.1"></times><ci id="S3.Thmtheorem6.p1.2.m2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1.2">𝑁</ci><ci id="S3.Thmtheorem6.p1.2.m2.1.1.3.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.m2.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.m2.1d">italic_N caligraphic_C</annotation></semantics></math> is called a <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem6.p1.3.1">Thomason natural system for <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.3.1.m1.1"><semantics id="S3.Thmtheorem6.p1.3.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.3.1.m1.1.1" xref="S3.Thmtheorem6.p1.3.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.3.1.m1.1b"><ci id="S3.Thmtheorem6.p1.3.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.1.m1.1d">caligraphic_C</annotation></semantics></math></em>. Given a Thomason system <math alttext="\mathcal{M}:\Delta(N\mathcal{C})\to R" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.m3.1"><semantics id="S3.Thmtheorem6.p1.4.m3.1a"><mrow id="S3.Thmtheorem6.p1.4.m3.1.1" xref="S3.Thmtheorem6.p1.4.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.4.m3.1.1.3" xref="S3.Thmtheorem6.p1.4.m3.1.1.3.cmml">ℳ</mi><mo id="S3.Thmtheorem6.p1.4.m3.1.1.2" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem6.p1.4.m3.1.1.2.cmml">:</mo><mrow id="S3.Thmtheorem6.p1.4.m3.1.1.1" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.cmml"><mrow id="S3.Thmtheorem6.p1.4.m3.1.1.1.1" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.cmml"><mi id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.3" mathvariant="normal" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.3.cmml">Δ</mi><mo id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.2" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.2" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.1" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.3" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem6.p1.4.m3.1.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.2.cmml">→</mo><mi id="S3.Thmtheorem6.p1.4.m3.1.1.1.3" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.m3.1b"><apply id="S3.Thmtheorem6.p1.4.m3.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1"><ci id="S3.Thmtheorem6.p1.4.m3.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.2">:</ci><ci id="S3.Thmtheorem6.p1.4.m3.1.1.3.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.3">ℳ</ci><apply id="S3.Thmtheorem6.p1.4.m3.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1"><ci id="S3.Thmtheorem6.p1.4.m3.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.2">→</ci><apply id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1"><times id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.2"></times><ci id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.3">Δ</ci><apply id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1"><times id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.1"></times><ci id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.2">𝑁</ci><ci id="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.1.1.1.1.3">𝒞</ci></apply></apply><ci id="S3.Thmtheorem6.p1.4.m3.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.1.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.m3.1c">\mathcal{M}:\Delta(N\mathcal{C})\to R</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.m3.1d">caligraphic_M : roman_Δ ( italic_N caligraphic_C ) → italic_R</annotation></semantics></math>-mod, the cohomology of the simplicial set <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.5.m4.1"><semantics id="S3.Thmtheorem6.p1.5.m4.1a"><mrow id="S3.Thmtheorem6.p1.5.m4.1.1" xref="S3.Thmtheorem6.p1.5.m4.1.1.cmml"><mi id="S3.Thmtheorem6.p1.5.m4.1.1.2" xref="S3.Thmtheorem6.p1.5.m4.1.1.2.cmml">N</mi><mo id="S3.Thmtheorem6.p1.5.m4.1.1.1" xref="S3.Thmtheorem6.p1.5.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.5.m4.1.1.3" xref="S3.Thmtheorem6.p1.5.m4.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.5.m4.1b"><apply id="S3.Thmtheorem6.p1.5.m4.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m4.1.1"><times id="S3.Thmtheorem6.p1.5.m4.1.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m4.1.1.1"></times><ci id="S3.Thmtheorem6.p1.5.m4.1.1.2.cmml" xref="S3.Thmtheorem6.p1.5.m4.1.1.2">𝑁</ci><ci id="S3.Thmtheorem6.p1.5.m4.1.1.3.cmml" xref="S3.Thmtheorem6.p1.5.m4.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.5.m4.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.5.m4.1d">italic_N caligraphic_C</annotation></semantics></math> with coefficients in <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.6.m5.1"><semantics id="S3.Thmtheorem6.p1.6.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.6.m5.1.1" xref="S3.Thmtheorem6.p1.6.m5.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.6.m5.1b"><ci id="S3.Thmtheorem6.p1.6.m5.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m5.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.6.m5.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.6.m5.1d">caligraphic_M</annotation></semantics></math> is called the <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem6.p1.7.2">Thomason cohomology</em> of <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.7.m6.1"><semantics id="S3.Thmtheorem6.p1.7.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.7.m6.1.1" xref="S3.Thmtheorem6.p1.7.m6.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.7.m6.1b"><ci id="S3.Thmtheorem6.p1.7.m6.1.1.cmml" xref="S3.Thmtheorem6.p1.7.m6.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.7.m6.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.7.m6.1d">caligraphic_C</annotation></semantics></math> and is denoted</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex40"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{*}_{Th}(\mathcal{C};\mathcal{M}):=H^{*}(N\mathcal{C};\mathcal{M})." class="ltx_Math" display="block" id="S3.Ex40.m1.4"><semantics id="S3.Ex40.m1.4a"><mrow id="S3.Ex40.m1.4.4.1" xref="S3.Ex40.m1.4.4.1.1.cmml"><mrow id="S3.Ex40.m1.4.4.1.1" xref="S3.Ex40.m1.4.4.1.1.cmml"><mrow id="S3.Ex40.m1.4.4.1.1.3" xref="S3.Ex40.m1.4.4.1.1.3.cmml"><msubsup id="S3.Ex40.m1.4.4.1.1.3.2" xref="S3.Ex40.m1.4.4.1.1.3.2.cmml"><mi id="S3.Ex40.m1.4.4.1.1.3.2.2.2" xref="S3.Ex40.m1.4.4.1.1.3.2.2.2.cmml">H</mi><mrow id="S3.Ex40.m1.4.4.1.1.3.2.3" xref="S3.Ex40.m1.4.4.1.1.3.2.3.cmml"><mi id="S3.Ex40.m1.4.4.1.1.3.2.3.2" xref="S3.Ex40.m1.4.4.1.1.3.2.3.2.cmml">T</mi><mo id="S3.Ex40.m1.4.4.1.1.3.2.3.1" xref="S3.Ex40.m1.4.4.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.Ex40.m1.4.4.1.1.3.2.3.3" xref="S3.Ex40.m1.4.4.1.1.3.2.3.3.cmml">h</mi></mrow><mo id="S3.Ex40.m1.4.4.1.1.3.2.2.3" xref="S3.Ex40.m1.4.4.1.1.3.2.2.3.cmml">∗</mo></msubsup><mo id="S3.Ex40.m1.4.4.1.1.3.1" xref="S3.Ex40.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex40.m1.4.4.1.1.3.3.2" xref="S3.Ex40.m1.4.4.1.1.3.3.1.cmml"><mo id="S3.Ex40.m1.4.4.1.1.3.3.2.1" stretchy="false" xref="S3.Ex40.m1.4.4.1.1.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex40.m1.1.1" xref="S3.Ex40.m1.1.1.cmml">𝒞</mi><mo id="S3.Ex40.m1.4.4.1.1.3.3.2.2" xref="S3.Ex40.m1.4.4.1.1.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex40.m1.2.2" xref="S3.Ex40.m1.2.2.cmml">ℳ</mi><mo id="S3.Ex40.m1.4.4.1.1.3.3.2.3" rspace="0.278em" stretchy="false" xref="S3.Ex40.m1.4.4.1.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex40.m1.4.4.1.1.2" rspace="0.278em" xref="S3.Ex40.m1.4.4.1.1.2.cmml">:=</mo><mrow id="S3.Ex40.m1.4.4.1.1.1" xref="S3.Ex40.m1.4.4.1.1.1.cmml"><msup id="S3.Ex40.m1.4.4.1.1.1.3" xref="S3.Ex40.m1.4.4.1.1.1.3.cmml"><mi id="S3.Ex40.m1.4.4.1.1.1.3.2" xref="S3.Ex40.m1.4.4.1.1.1.3.2.cmml">H</mi><mo id="S3.Ex40.m1.4.4.1.1.1.3.3" xref="S3.Ex40.m1.4.4.1.1.1.3.3.cmml">∗</mo></msup><mo id="S3.Ex40.m1.4.4.1.1.1.2" xref="S3.Ex40.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex40.m1.4.4.1.1.1.1.1" xref="S3.Ex40.m1.4.4.1.1.1.1.2.cmml"><mo id="S3.Ex40.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.Ex40.m1.4.4.1.1.1.1.2.cmml">(</mo><mrow id="S3.Ex40.m1.4.4.1.1.1.1.1.1" xref="S3.Ex40.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S3.Ex40.m1.4.4.1.1.1.1.1.1.2" xref="S3.Ex40.m1.4.4.1.1.1.1.1.1.2.cmml">N</mi><mo id="S3.Ex40.m1.4.4.1.1.1.1.1.1.1" xref="S3.Ex40.m1.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" 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encoding="application/x-tex" id="S3.Ex40.m1.4c">H^{*}_{Th}(\mathcal{C};\mathcal{M}):=H^{*}(N\mathcal{C};\mathcal{M}).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex40.m1.4d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; caligraphic_M ) := italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_N caligraphic_C ; caligraphic_M ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.1">The Thomason cohomology of a category was first introduced by Galvez-Carrillo, Neumann, and Tonks in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib7" title="">7</a>]</cite>. Thomason cohomology generalizes the Baues-Wirsching cohomology of a category in the following sense:</p> </div> <div class="ltx_para" id="S3.SS4.p2"> <p class="ltx_p" id="S3.SS4.p2.10">Let <math alttext="\chi:\Delta(N\mathcal{C})\to\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S3.SS4.p2.1.m1.1"><semantics id="S3.SS4.p2.1.m1.1a"><mrow id="S3.SS4.p2.1.m1.1.1" xref="S3.SS4.p2.1.m1.1.1.cmml"><mi id="S3.SS4.p2.1.m1.1.1.3" xref="S3.SS4.p2.1.m1.1.1.3.cmml">χ</mi><mo id="S3.SS4.p2.1.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S3.SS4.p2.1.m1.1.1.2.cmml">:</mo><mrow id="S3.SS4.p2.1.m1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.cmml"><mrow id="S3.SS4.p2.1.m1.1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.1.cmml"><mi id="S3.SS4.p2.1.m1.1.1.1.1.3" mathvariant="normal" xref="S3.SS4.p2.1.m1.1.1.1.1.3.cmml">Δ</mi><mo id="S3.SS4.p2.1.m1.1.1.1.1.2" xref="S3.SS4.p2.1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.p2.1.m1.1.1.1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.p2.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.2" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.3" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S3.SS4.p2.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.p2.1.m1.1.1.1.2" stretchy="false" xref="S3.SS4.p2.1.m1.1.1.1.2.cmml">→</mo><mrow id="S3.SS4.p2.1.m1.1.1.1.3" xref="S3.SS4.p2.1.m1.1.1.1.3.cmml"><mi id="S3.SS4.p2.1.m1.1.1.1.3.2" xref="S3.SS4.p2.1.m1.1.1.1.3.2.cmml">𝔉</mi><mo id="S3.SS4.p2.1.m1.1.1.1.3.1" xref="S3.SS4.p2.1.m1.1.1.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.1.m1.1.1.1.3.3" xref="S3.SS4.p2.1.m1.1.1.1.3.3.cmml">𝒞</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.1.m1.1b"><apply id="S3.SS4.p2.1.m1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1"><ci id="S3.SS4.p2.1.m1.1.1.2.cmml" xref="S3.SS4.p2.1.m1.1.1.2">:</ci><ci id="S3.SS4.p2.1.m1.1.1.3.cmml" xref="S3.SS4.p2.1.m1.1.1.3">𝜒</ci><apply id="S3.SS4.p2.1.m1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1"><ci id="S3.SS4.p2.1.m1.1.1.1.2.cmml" xref="S3.SS4.p2.1.m1.1.1.1.2">→</ci><apply id="S3.SS4.p2.1.m1.1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1"><times id="S3.SS4.p2.1.m1.1.1.1.1.2.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.2"></times><ci id="S3.SS4.p2.1.m1.1.1.1.1.3.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.3">Δ</ci><apply id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1"><times id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.1"></times><ci id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.2">𝑁</ci><ci id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.3">𝒞</ci></apply></apply><apply id="S3.SS4.p2.1.m1.1.1.1.3.cmml" xref="S3.SS4.p2.1.m1.1.1.1.3"><times id="S3.SS4.p2.1.m1.1.1.1.3.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.3.1"></times><ci id="S3.SS4.p2.1.m1.1.1.1.3.2.cmml" xref="S3.SS4.p2.1.m1.1.1.1.3.2">𝔉</ci><ci id="S3.SS4.p2.1.m1.1.1.1.3.3.cmml" xref="S3.SS4.p2.1.m1.1.1.1.3.3">𝒞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.1.m1.1c">\chi:\Delta(N\mathcal{C})\to\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.1.m1.1d">italic_χ : roman_Δ ( italic_N caligraphic_C ) → fraktur_F caligraphic_C</annotation></semantics></math> be the functor that sends <math alttext="\sigma=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})" class="ltx_Math" display="inline" id="S3.SS4.p2.2.m2.1"><semantics 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id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.2.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.2">⟶</ci><apply id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.3.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.3.1.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.3">subscript</csymbol><ci id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.3.2.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.3.2">𝛼</ci><ci id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.3.3.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.1.3.3">𝑛</ci></apply></apply><apply id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.2.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.2.1.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.2">subscript</csymbol><ci id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.2.2.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.2.2">𝑐</ci><ci id="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.2.3.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1.1.3.2.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.2.m2.1c">\sigma=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.2.m2.1d">italic_σ = ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> in <math alttext="N\mathcal{C}_{n}" class="ltx_Math" display="inline" id="S3.SS4.p2.3.m3.1"><semantics id="S3.SS4.p2.3.m3.1a"><mrow id="S3.SS4.p2.3.m3.1.1" xref="S3.SS4.p2.3.m3.1.1.cmml"><mi id="S3.SS4.p2.3.m3.1.1.2" xref="S3.SS4.p2.3.m3.1.1.2.cmml">N</mi><mo id="S3.SS4.p2.3.m3.1.1.1" xref="S3.SS4.p2.3.m3.1.1.1.cmml">⁢</mo><msub id="S3.SS4.p2.3.m3.1.1.3" xref="S3.SS4.p2.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.3.m3.1.1.3.2" xref="S3.SS4.p2.3.m3.1.1.3.2.cmml">𝒞</mi><mi id="S3.SS4.p2.3.m3.1.1.3.3" xref="S3.SS4.p2.3.m3.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.3.m3.1b"><apply id="S3.SS4.p2.3.m3.1.1.cmml" xref="S3.SS4.p2.3.m3.1.1"><times id="S3.SS4.p2.3.m3.1.1.1.cmml" xref="S3.SS4.p2.3.m3.1.1.1"></times><ci id="S3.SS4.p2.3.m3.1.1.2.cmml" xref="S3.SS4.p2.3.m3.1.1.2">𝑁</ci><apply id="S3.SS4.p2.3.m3.1.1.3.cmml" xref="S3.SS4.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.3.m3.1.1.3.1.cmml" xref="S3.SS4.p2.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS4.p2.3.m3.1.1.3.2.cmml" xref="S3.SS4.p2.3.m3.1.1.3.2">𝒞</ci><ci id="S3.SS4.p2.3.m3.1.1.3.3.cmml" xref="S3.SS4.p2.3.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.3.m3.1c">N\mathcal{C}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.3.m3.1d">italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\sigma(0,n):=\alpha_{n}\cdots\alpha_{1}:c_{0}\to c_{n}" class="ltx_Math" display="inline" id="S3.SS4.p2.4.m4.2"><semantics id="S3.SS4.p2.4.m4.2a"><mrow id="S3.SS4.p2.4.m4.2.3" xref="S3.SS4.p2.4.m4.2.3.cmml"><mrow id="S3.SS4.p2.4.m4.2.3.2" xref="S3.SS4.p2.4.m4.2.3.2.cmml"><mrow id="S3.SS4.p2.4.m4.2.3.2.2" xref="S3.SS4.p2.4.m4.2.3.2.2.cmml"><mi id="S3.SS4.p2.4.m4.2.3.2.2.2" xref="S3.SS4.p2.4.m4.2.3.2.2.2.cmml">σ</mi><mo id="S3.SS4.p2.4.m4.2.3.2.2.1" xref="S3.SS4.p2.4.m4.2.3.2.2.1.cmml">⁢</mo><mrow id="S3.SS4.p2.4.m4.2.3.2.2.3.2" xref="S3.SS4.p2.4.m4.2.3.2.2.3.1.cmml"><mo id="S3.SS4.p2.4.m4.2.3.2.2.3.2.1" stretchy="false" xref="S3.SS4.p2.4.m4.2.3.2.2.3.1.cmml">(</mo><mn id="S3.SS4.p2.4.m4.1.1" xref="S3.SS4.p2.4.m4.1.1.cmml">0</mn><mo id="S3.SS4.p2.4.m4.2.3.2.2.3.2.2" xref="S3.SS4.p2.4.m4.2.3.2.2.3.1.cmml">,</mo><mi id="S3.SS4.p2.4.m4.2.2" xref="S3.SS4.p2.4.m4.2.2.cmml">n</mi><mo id="S3.SS4.p2.4.m4.2.3.2.2.3.2.3" rspace="0.278em" stretchy="false" xref="S3.SS4.p2.4.m4.2.3.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.p2.4.m4.2.3.2.1" rspace="0.278em" xref="S3.SS4.p2.4.m4.2.3.2.1.cmml">:=</mo><mrow id="S3.SS4.p2.4.m4.2.3.2.3" xref="S3.SS4.p2.4.m4.2.3.2.3.cmml"><msub id="S3.SS4.p2.4.m4.2.3.2.3.2" xref="S3.SS4.p2.4.m4.2.3.2.3.2.cmml"><mi id="S3.SS4.p2.4.m4.2.3.2.3.2.2" xref="S3.SS4.p2.4.m4.2.3.2.3.2.2.cmml">α</mi><mi id="S3.SS4.p2.4.m4.2.3.2.3.2.3" xref="S3.SS4.p2.4.m4.2.3.2.3.2.3.cmml">n</mi></msub><mo id="S3.SS4.p2.4.m4.2.3.2.3.1" xref="S3.SS4.p2.4.m4.2.3.2.3.1.cmml">⁢</mo><mi id="S3.SS4.p2.4.m4.2.3.2.3.3" mathvariant="normal" xref="S3.SS4.p2.4.m4.2.3.2.3.3.cmml">⋯</mi><mo id="S3.SS4.p2.4.m4.2.3.2.3.1a" xref="S3.SS4.p2.4.m4.2.3.2.3.1.cmml">⁢</mo><msub id="S3.SS4.p2.4.m4.2.3.2.3.4" xref="S3.SS4.p2.4.m4.2.3.2.3.4.cmml"><mi id="S3.SS4.p2.4.m4.2.3.2.3.4.2" xref="S3.SS4.p2.4.m4.2.3.2.3.4.2.cmml">α</mi><mn id="S3.SS4.p2.4.m4.2.3.2.3.4.3" xref="S3.SS4.p2.4.m4.2.3.2.3.4.3.cmml">1</mn></msub></mrow></mrow><mo id="S3.SS4.p2.4.m4.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.SS4.p2.4.m4.2.3.1.cmml">:</mo><mrow id="S3.SS4.p2.4.m4.2.3.3" xref="S3.SS4.p2.4.m4.2.3.3.cmml"><msub id="S3.SS4.p2.4.m4.2.3.3.2" xref="S3.SS4.p2.4.m4.2.3.3.2.cmml"><mi id="S3.SS4.p2.4.m4.2.3.3.2.2" xref="S3.SS4.p2.4.m4.2.3.3.2.2.cmml">c</mi><mn id="S3.SS4.p2.4.m4.2.3.3.2.3" xref="S3.SS4.p2.4.m4.2.3.3.2.3.cmml">0</mn></msub><mo id="S3.SS4.p2.4.m4.2.3.3.1" stretchy="false" xref="S3.SS4.p2.4.m4.2.3.3.1.cmml">→</mo><msub id="S3.SS4.p2.4.m4.2.3.3.3" xref="S3.SS4.p2.4.m4.2.3.3.3.cmml"><mi id="S3.SS4.p2.4.m4.2.3.3.3.2" xref="S3.SS4.p2.4.m4.2.3.3.3.2.cmml">c</mi><mi id="S3.SS4.p2.4.m4.2.3.3.3.3" xref="S3.SS4.p2.4.m4.2.3.3.3.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.4.m4.2b"><apply id="S3.SS4.p2.4.m4.2.3.cmml" xref="S3.SS4.p2.4.m4.2.3"><ci id="S3.SS4.p2.4.m4.2.3.1.cmml" xref="S3.SS4.p2.4.m4.2.3.1">:</ci><apply id="S3.SS4.p2.4.m4.2.3.2.cmml" xref="S3.SS4.p2.4.m4.2.3.2"><csymbol cd="latexml" id="S3.SS4.p2.4.m4.2.3.2.1.cmml" xref="S3.SS4.p2.4.m4.2.3.2.1">assign</csymbol><apply id="S3.SS4.p2.4.m4.2.3.2.2.cmml" xref="S3.SS4.p2.4.m4.2.3.2.2"><times id="S3.SS4.p2.4.m4.2.3.2.2.1.cmml" xref="S3.SS4.p2.4.m4.2.3.2.2.1"></times><ci id="S3.SS4.p2.4.m4.2.3.2.2.2.cmml" xref="S3.SS4.p2.4.m4.2.3.2.2.2">𝜎</ci><interval closure="open" id="S3.SS4.p2.4.m4.2.3.2.2.3.1.cmml" xref="S3.SS4.p2.4.m4.2.3.2.2.3.2"><cn id="S3.SS4.p2.4.m4.1.1.cmml" type="integer" xref="S3.SS4.p2.4.m4.1.1">0</cn><ci id="S3.SS4.p2.4.m4.2.2.cmml" xref="S3.SS4.p2.4.m4.2.2">𝑛</ci></interval></apply><apply id="S3.SS4.p2.4.m4.2.3.2.3.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3"><times id="S3.SS4.p2.4.m4.2.3.2.3.1.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.1"></times><apply id="S3.SS4.p2.4.m4.2.3.2.3.2.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.2"><csymbol cd="ambiguous" id="S3.SS4.p2.4.m4.2.3.2.3.2.1.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.2">subscript</csymbol><ci id="S3.SS4.p2.4.m4.2.3.2.3.2.2.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.2.2">𝛼</ci><ci id="S3.SS4.p2.4.m4.2.3.2.3.2.3.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.2.3">𝑛</ci></apply><ci id="S3.SS4.p2.4.m4.2.3.2.3.3.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.3">⋯</ci><apply id="S3.SS4.p2.4.m4.2.3.2.3.4.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.4"><csymbol cd="ambiguous" id="S3.SS4.p2.4.m4.2.3.2.3.4.1.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.4">subscript</csymbol><ci id="S3.SS4.p2.4.m4.2.3.2.3.4.2.cmml" xref="S3.SS4.p2.4.m4.2.3.2.3.4.2">𝛼</ci><cn id="S3.SS4.p2.4.m4.2.3.2.3.4.3.cmml" type="integer" xref="S3.SS4.p2.4.m4.2.3.2.3.4.3">1</cn></apply></apply></apply><apply id="S3.SS4.p2.4.m4.2.3.3.cmml" xref="S3.SS4.p2.4.m4.2.3.3"><ci id="S3.SS4.p2.4.m4.2.3.3.1.cmml" xref="S3.SS4.p2.4.m4.2.3.3.1">→</ci><apply id="S3.SS4.p2.4.m4.2.3.3.2.cmml" xref="S3.SS4.p2.4.m4.2.3.3.2"><csymbol cd="ambiguous" id="S3.SS4.p2.4.m4.2.3.3.2.1.cmml" xref="S3.SS4.p2.4.m4.2.3.3.2">subscript</csymbol><ci id="S3.SS4.p2.4.m4.2.3.3.2.2.cmml" xref="S3.SS4.p2.4.m4.2.3.3.2.2">𝑐</ci><cn id="S3.SS4.p2.4.m4.2.3.3.2.3.cmml" type="integer" xref="S3.SS4.p2.4.m4.2.3.3.2.3">0</cn></apply><apply id="S3.SS4.p2.4.m4.2.3.3.3.cmml" xref="S3.SS4.p2.4.m4.2.3.3.3"><csymbol cd="ambiguous" id="S3.SS4.p2.4.m4.2.3.3.3.1.cmml" xref="S3.SS4.p2.4.m4.2.3.3.3">subscript</csymbol><ci id="S3.SS4.p2.4.m4.2.3.3.3.2.cmml" xref="S3.SS4.p2.4.m4.2.3.3.3.2">𝑐</ci><ci id="S3.SS4.p2.4.m4.2.3.3.3.3.cmml" xref="S3.SS4.p2.4.m4.2.3.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.4.m4.2c">\sigma(0,n):=\alpha_{n}\cdots\alpha_{1}:c_{0}\to c_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.4.m4.2d">italic_σ ( 0 , italic_n ) := italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathfrak{F}\mathcal{C}" class="ltx_Math" display="inline" id="S3.SS4.p2.5.m5.1"><semantics id="S3.SS4.p2.5.m5.1a"><mrow id="S3.SS4.p2.5.m5.1.1" xref="S3.SS4.p2.5.m5.1.1.cmml"><mi id="S3.SS4.p2.5.m5.1.1.2" xref="S3.SS4.p2.5.m5.1.1.2.cmml">𝔉</mi><mo id="S3.SS4.p2.5.m5.1.1.1" xref="S3.SS4.p2.5.m5.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.5.m5.1.1.3" xref="S3.SS4.p2.5.m5.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.5.m5.1b"><apply id="S3.SS4.p2.5.m5.1.1.cmml" xref="S3.SS4.p2.5.m5.1.1"><times id="S3.SS4.p2.5.m5.1.1.1.cmml" xref="S3.SS4.p2.5.m5.1.1.1"></times><ci id="S3.SS4.p2.5.m5.1.1.2.cmml" xref="S3.SS4.p2.5.m5.1.1.2">𝔉</ci><ci id="S3.SS4.p2.5.m5.1.1.3.cmml" xref="S3.SS4.p2.5.m5.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.5.m5.1c">\mathfrak{F}\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.5.m5.1d">fraktur_F caligraphic_C</annotation></semantics></math>. For each morphism <math alttext="f:([m],\tau)\to([n],\sigma)" class="ltx_Math" display="inline" id="S3.SS4.p2.6.m6.6"><semantics id="S3.SS4.p2.6.m6.6a"><mrow id="S3.SS4.p2.6.m6.6.6" xref="S3.SS4.p2.6.m6.6.6.cmml"><mi id="S3.SS4.p2.6.m6.6.6.4" xref="S3.SS4.p2.6.m6.6.6.4.cmml">f</mi><mo id="S3.SS4.p2.6.m6.6.6.3" lspace="0.278em" rspace="0.278em" xref="S3.SS4.p2.6.m6.6.6.3.cmml">:</mo><mrow id="S3.SS4.p2.6.m6.6.6.2" xref="S3.SS4.p2.6.m6.6.6.2.cmml"><mrow id="S3.SS4.p2.6.m6.5.5.1.1.1" xref="S3.SS4.p2.6.m6.5.5.1.1.2.cmml"><mo id="S3.SS4.p2.6.m6.5.5.1.1.1.2" stretchy="false" xref="S3.SS4.p2.6.m6.5.5.1.1.2.cmml">(</mo><mrow id="S3.SS4.p2.6.m6.5.5.1.1.1.1.2" xref="S3.SS4.p2.6.m6.5.5.1.1.1.1.1.cmml"><mo id="S3.SS4.p2.6.m6.5.5.1.1.1.1.2.1" stretchy="false" xref="S3.SS4.p2.6.m6.5.5.1.1.1.1.1.1.cmml">[</mo><mi id="S3.SS4.p2.6.m6.1.1" xref="S3.SS4.p2.6.m6.1.1.cmml">m</mi><mo id="S3.SS4.p2.6.m6.5.5.1.1.1.1.2.2" stretchy="false" xref="S3.SS4.p2.6.m6.5.5.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS4.p2.6.m6.5.5.1.1.1.3" xref="S3.SS4.p2.6.m6.5.5.1.1.2.cmml">,</mo><mi id="S3.SS4.p2.6.m6.2.2" xref="S3.SS4.p2.6.m6.2.2.cmml">τ</mi><mo id="S3.SS4.p2.6.m6.5.5.1.1.1.4" stretchy="false" xref="S3.SS4.p2.6.m6.5.5.1.1.2.cmml">)</mo></mrow><mo id="S3.SS4.p2.6.m6.6.6.2.3" stretchy="false" xref="S3.SS4.p2.6.m6.6.6.2.3.cmml">→</mo><mrow id="S3.SS4.p2.6.m6.6.6.2.2.1" xref="S3.SS4.p2.6.m6.6.6.2.2.2.cmml"><mo id="S3.SS4.p2.6.m6.6.6.2.2.1.2" stretchy="false" xref="S3.SS4.p2.6.m6.6.6.2.2.2.cmml">(</mo><mrow id="S3.SS4.p2.6.m6.6.6.2.2.1.1.2" xref="S3.SS4.p2.6.m6.6.6.2.2.1.1.1.cmml"><mo id="S3.SS4.p2.6.m6.6.6.2.2.1.1.2.1" stretchy="false" xref="S3.SS4.p2.6.m6.6.6.2.2.1.1.1.1.cmml">[</mo><mi id="S3.SS4.p2.6.m6.3.3" xref="S3.SS4.p2.6.m6.3.3.cmml">n</mi><mo id="S3.SS4.p2.6.m6.6.6.2.2.1.1.2.2" stretchy="false" xref="S3.SS4.p2.6.m6.6.6.2.2.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS4.p2.6.m6.6.6.2.2.1.3" xref="S3.SS4.p2.6.m6.6.6.2.2.2.cmml">,</mo><mi id="S3.SS4.p2.6.m6.4.4" xref="S3.SS4.p2.6.m6.4.4.cmml">σ</mi><mo id="S3.SS4.p2.6.m6.6.6.2.2.1.4" stretchy="false" xref="S3.SS4.p2.6.m6.6.6.2.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.6.m6.6b"><apply id="S3.SS4.p2.6.m6.6.6.cmml" xref="S3.SS4.p2.6.m6.6.6"><ci id="S3.SS4.p2.6.m6.6.6.3.cmml" xref="S3.SS4.p2.6.m6.6.6.3">:</ci><ci id="S3.SS4.p2.6.m6.6.6.4.cmml" xref="S3.SS4.p2.6.m6.6.6.4">𝑓</ci><apply id="S3.SS4.p2.6.m6.6.6.2.cmml" xref="S3.SS4.p2.6.m6.6.6.2"><ci id="S3.SS4.p2.6.m6.6.6.2.3.cmml" xref="S3.SS4.p2.6.m6.6.6.2.3">→</ci><interval closure="open" id="S3.SS4.p2.6.m6.5.5.1.1.2.cmml" xref="S3.SS4.p2.6.m6.5.5.1.1.1"><apply id="S3.SS4.p2.6.m6.5.5.1.1.1.1.1.cmml" xref="S3.SS4.p2.6.m6.5.5.1.1.1.1.2"><csymbol cd="latexml" id="S3.SS4.p2.6.m6.5.5.1.1.1.1.1.1.cmml" xref="S3.SS4.p2.6.m6.5.5.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS4.p2.6.m6.1.1.cmml" xref="S3.SS4.p2.6.m6.1.1">𝑚</ci></apply><ci id="S3.SS4.p2.6.m6.2.2.cmml" xref="S3.SS4.p2.6.m6.2.2">𝜏</ci></interval><interval closure="open" id="S3.SS4.p2.6.m6.6.6.2.2.2.cmml" xref="S3.SS4.p2.6.m6.6.6.2.2.1"><apply id="S3.SS4.p2.6.m6.6.6.2.2.1.1.1.cmml" xref="S3.SS4.p2.6.m6.6.6.2.2.1.1.2"><csymbol cd="latexml" id="S3.SS4.p2.6.m6.6.6.2.2.1.1.1.1.cmml" xref="S3.SS4.p2.6.m6.6.6.2.2.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS4.p2.6.m6.3.3.cmml" xref="S3.SS4.p2.6.m6.3.3">𝑛</ci></apply><ci id="S3.SS4.p2.6.m6.4.4.cmml" xref="S3.SS4.p2.6.m6.4.4">𝜎</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.6.m6.6c">f:([m],\tau)\to([n],\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.6.m6.6d">italic_f : ( [ italic_m ] , italic_τ ) → ( [ italic_n ] , italic_σ )</annotation></semantics></math> in <math alttext="\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S3.SS4.p2.7.m7.1"><semantics id="S3.SS4.p2.7.m7.1a"><mrow id="S3.SS4.p2.7.m7.1.1" xref="S3.SS4.p2.7.m7.1.1.cmml"><mi id="S3.SS4.p2.7.m7.1.1.3" mathvariant="normal" xref="S3.SS4.p2.7.m7.1.1.3.cmml">Δ</mi><mo id="S3.SS4.p2.7.m7.1.1.2" xref="S3.SS4.p2.7.m7.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.p2.7.m7.1.1.1.1" xref="S3.SS4.p2.7.m7.1.1.1.1.1.cmml"><mo id="S3.SS4.p2.7.m7.1.1.1.1.2" stretchy="false" xref="S3.SS4.p2.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p2.7.m7.1.1.1.1.1" xref="S3.SS4.p2.7.m7.1.1.1.1.1.cmml"><mi id="S3.SS4.p2.7.m7.1.1.1.1.1.2" xref="S3.SS4.p2.7.m7.1.1.1.1.1.2.cmml">N</mi><mo id="S3.SS4.p2.7.m7.1.1.1.1.1.1" xref="S3.SS4.p2.7.m7.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.7.m7.1.1.1.1.1.3" xref="S3.SS4.p2.7.m7.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S3.SS4.p2.7.m7.1.1.1.1.3" stretchy="false" xref="S3.SS4.p2.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.7.m7.1b"><apply id="S3.SS4.p2.7.m7.1.1.cmml" xref="S3.SS4.p2.7.m7.1.1"><times id="S3.SS4.p2.7.m7.1.1.2.cmml" xref="S3.SS4.p2.7.m7.1.1.2"></times><ci id="S3.SS4.p2.7.m7.1.1.3.cmml" xref="S3.SS4.p2.7.m7.1.1.3">Δ</ci><apply id="S3.SS4.p2.7.m7.1.1.1.1.1.cmml" xref="S3.SS4.p2.7.m7.1.1.1.1"><times id="S3.SS4.p2.7.m7.1.1.1.1.1.1.cmml" xref="S3.SS4.p2.7.m7.1.1.1.1.1.1"></times><ci id="S3.SS4.p2.7.m7.1.1.1.1.1.2.cmml" xref="S3.SS4.p2.7.m7.1.1.1.1.1.2">𝑁</ci><ci id="S3.SS4.p2.7.m7.1.1.1.1.1.3.cmml" xref="S3.SS4.p2.7.m7.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.7.m7.1c">\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.7.m7.1d">roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>, where <math alttext="\tau=(c_{0}^{\prime}\smash{\,\mathop{\longrightarrow}\limits^{\beta_{1}}\,}% \cdots\smash{\,\mathop{\longrightarrow}\limits^{\beta_{m}}\,}c_{m}^{\prime})" class="ltx_Math" display="inline" id="S3.SS4.p2.8.m8.1"><semantics id="S3.SS4.p2.8.m8.1a"><mrow id="S3.SS4.p2.8.m8.1.1" xref="S3.SS4.p2.8.m8.1.1.cmml"><mi id="S3.SS4.p2.8.m8.1.1.3" xref="S3.SS4.p2.8.m8.1.1.3.cmml">τ</mi><mo id="S3.SS4.p2.8.m8.1.1.2" xref="S3.SS4.p2.8.m8.1.1.2.cmml">=</mo><mrow id="S3.SS4.p2.8.m8.1.1.1.1" xref="S3.SS4.p2.8.m8.1.1.1.1.1.cmml"><mo id="S3.SS4.p2.8.m8.1.1.1.1.2" stretchy="false" xref="S3.SS4.p2.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p2.8.m8.1.1.1.1.1" xref="S3.SS4.p2.8.m8.1.1.1.1.1.cmml"><msubsup id="S3.SS4.p2.8.m8.1.1.1.1.1.2" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2.cmml"><mi id="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.2" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.2.cmml">c</mi><mn id="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.3.cmml">0</mn><mo id="S3.SS4.p2.8.m8.1.1.1.1.1.2.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.SS4.p2.8.m8.1.1.1.1.1.1" lspace="0.167em" xref="S3.SS4.p2.8.m8.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.p2.8.m8.1.1.1.1.1.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.cmml"><mover id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.cmml"><mo id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.2.cmml">⟶</mo><msub id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.cmml"><mi id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.2" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.2.cmml">β</mi><mn id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.3.cmml">1</mn></msub></mover><mrow id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.cmml"><mi id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.2" mathvariant="normal" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.2.cmml">⋯</mi><mo id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.1" lspace="0.337em" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.cmml"><mover id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.cmml"><mo id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.cmml"><mi id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.2" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.2.cmml">β</mi><mi id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.3.cmml">m</mi></msub></mover><msubsup id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.cmml"><mi id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.2" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.2.cmml">c</mi><mi id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.3.cmml">m</mi><mo id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.3" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.3.cmml">′</mo></msubsup></mrow></mrow></mrow></mrow><mo id="S3.SS4.p2.8.m8.1.1.1.1.3" stretchy="false" xref="S3.SS4.p2.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.8.m8.1b"><apply id="S3.SS4.p2.8.m8.1.1.cmml" xref="S3.SS4.p2.8.m8.1.1"><eq id="S3.SS4.p2.8.m8.1.1.2.cmml" xref="S3.SS4.p2.8.m8.1.1.2"></eq><ci id="S3.SS4.p2.8.m8.1.1.3.cmml" xref="S3.SS4.p2.8.m8.1.1.3">𝜏</ci><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1"><times id="S3.SS4.p2.8.m8.1.1.1.1.1.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.1"></times><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.1.1.1.1.1.2.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2">superscript</csymbol><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.2">𝑐</ci><cn id="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2.2.3">0</cn></apply><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.2.3.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.2.3">′</ci></apply><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3"><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1">superscript</csymbol><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.2">⟶</ci><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3">subscript</csymbol><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.2">𝛽</ci><cn id="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.3.cmml" type="integer" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.1.3.3">1</cn></apply></apply><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2"><times id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.1"></times><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.2">⋯</ci><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3"><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1">superscript</csymbol><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.2">⟶</ci><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3">subscript</csymbol><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.2">𝛽</ci><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.3.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.1.3.3">𝑚</ci></apply></apply><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2">superscript</csymbol><apply id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2">subscript</csymbol><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.2.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.2">𝑐</ci><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.3.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.2.3">𝑚</ci></apply><ci id="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.3.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1.1.3.2.3.2.3">′</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.8.m8.1c">\tau=(c_{0}^{\prime}\smash{\,\mathop{\longrightarrow}\limits^{\beta_{1}}\,}% \cdots\smash{\,\mathop{\longrightarrow}\limits^{\beta_{m}}\,}c_{m}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.8.m8.1d">italic_τ = ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟶ start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\sigma=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})" class="ltx_Math" display="inline" id="S3.SS4.p2.9.m9.1"><semantics id="S3.SS4.p2.9.m9.1a"><mrow id="S3.SS4.p2.9.m9.1.1" xref="S3.SS4.p2.9.m9.1.1.cmml"><mi id="S3.SS4.p2.9.m9.1.1.3" xref="S3.SS4.p2.9.m9.1.1.3.cmml">σ</mi><mo id="S3.SS4.p2.9.m9.1.1.2" xref="S3.SS4.p2.9.m9.1.1.2.cmml">=</mo><mrow id="S3.SS4.p2.9.m9.1.1.1.1" xref="S3.SS4.p2.9.m9.1.1.1.1.1.cmml"><mo id="S3.SS4.p2.9.m9.1.1.1.1.2" stretchy="false" xref="S3.SS4.p2.9.m9.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p2.9.m9.1.1.1.1.1" xref="S3.SS4.p2.9.m9.1.1.1.1.1.cmml"><msub id="S3.SS4.p2.9.m9.1.1.1.1.1.2" xref="S3.SS4.p2.9.m9.1.1.1.1.1.2.cmml"><mi id="S3.SS4.p2.9.m9.1.1.1.1.1.2.2" xref="S3.SS4.p2.9.m9.1.1.1.1.1.2.2.cmml">c</mi><mn id="S3.SS4.p2.9.m9.1.1.1.1.1.2.3" xref="S3.SS4.p2.9.m9.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS4.p2.9.m9.1.1.1.1.1.1" lspace="0.167em" xref="S3.SS4.p2.9.m9.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.p2.9.m9.1.1.1.1.1.3" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.cmml"><mover id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.cmml"><mo id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.2.cmml">⟶</mo><msub id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.cmml"><mi id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.2" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.2.cmml">α</mi><mn id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.3" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.3.cmml">1</mn></msub></mover><mrow id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.cmml"><mi id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.2" mathvariant="normal" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.2.cmml">⋯</mi><mo id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.1" lspace="0.337em" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.cmml"><mover id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.cmml"><mo id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.cmml"><mi id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.2" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.2.cmml">α</mi><mi id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.3" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.3.cmml">n</mi></msub></mover><msub id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.cmml"><mi id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.2" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.2.cmml">c</mi><mi id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.3" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.3.cmml">n</mi></msub></mrow></mrow></mrow></mrow><mo id="S3.SS4.p2.9.m9.1.1.1.1.3" stretchy="false" xref="S3.SS4.p2.9.m9.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.9.m9.1b"><apply id="S3.SS4.p2.9.m9.1.1.cmml" xref="S3.SS4.p2.9.m9.1.1"><eq id="S3.SS4.p2.9.m9.1.1.2.cmml" xref="S3.SS4.p2.9.m9.1.1.2"></eq><ci id="S3.SS4.p2.9.m9.1.1.3.cmml" xref="S3.SS4.p2.9.m9.1.1.3">𝜎</ci><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1"><times id="S3.SS4.p2.9.m9.1.1.1.1.1.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.1"></times><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p2.9.m9.1.1.1.1.1.2.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.2.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.2.2">𝑐</ci><cn id="S3.SS4.p2.9.m9.1.1.1.1.1.2.3.cmml" type="integer" xref="S3.SS4.p2.9.m9.1.1.1.1.1.2.3">0</cn></apply><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.3.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3"><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1">superscript</csymbol><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.2">⟶</ci><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3">subscript</csymbol><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.2">𝛼</ci><cn id="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.3.cmml" type="integer" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.1.3.3">1</cn></apply></apply><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2"><times id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.1"></times><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.2">⋯</ci><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3"><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1">superscript</csymbol><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.2">⟶</ci><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3">subscript</csymbol><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.2">𝛼</ci><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.3.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.1.3.3">𝑛</ci></apply></apply><apply id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.1.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2">subscript</csymbol><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.2.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.2">𝑐</ci><ci id="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.3.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1.1.3.2.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.9.m9.1c">\sigma=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{n}}\,}c_{n})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.9.m9.1d">italic_σ = ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>, we define <math alttext="\chi(f)" class="ltx_Math" display="inline" id="S3.SS4.p2.10.m10.1"><semantics id="S3.SS4.p2.10.m10.1a"><mrow id="S3.SS4.p2.10.m10.1.2" xref="S3.SS4.p2.10.m10.1.2.cmml"><mi id="S3.SS4.p2.10.m10.1.2.2" xref="S3.SS4.p2.10.m10.1.2.2.cmml">χ</mi><mo id="S3.SS4.p2.10.m10.1.2.1" xref="S3.SS4.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S3.SS4.p2.10.m10.1.2.3.2" xref="S3.SS4.p2.10.m10.1.2.cmml"><mo id="S3.SS4.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S3.SS4.p2.10.m10.1.2.cmml">(</mo><mi id="S3.SS4.p2.10.m10.1.1" xref="S3.SS4.p2.10.m10.1.1.cmml">f</mi><mo id="S3.SS4.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S3.SS4.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.10.m10.1b"><apply id="S3.SS4.p2.10.m10.1.2.cmml" xref="S3.SS4.p2.10.m10.1.2"><times id="S3.SS4.p2.10.m10.1.2.1.cmml" xref="S3.SS4.p2.10.m10.1.2.1"></times><ci id="S3.SS4.p2.10.m10.1.2.2.cmml" xref="S3.SS4.p2.10.m10.1.2.2">𝜒</ci><ci id="S3.SS4.p2.10.m10.1.1.cmml" xref="S3.SS4.p2.10.m10.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.10.m10.1c">\chi(f)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.10.m10.1d">italic_χ ( italic_f )</annotation></semantics></math> to be the morphism</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex41"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(\alpha_{n}\cdots\alpha_{f(m)+1},\alpha_{f(0)}\cdots\alpha_{1}):\beta_{m}% \cdots\beta_{1}\to\alpha_{n}\cdots\alpha_{1}." class="ltx_Math" display="block" id="S3.Ex41.m1.3"><semantics id="S3.Ex41.m1.3a"><mrow id="S3.Ex41.m1.3.3.1" xref="S3.Ex41.m1.3.3.1.1.cmml"><mrow id="S3.Ex41.m1.3.3.1.1" xref="S3.Ex41.m1.3.3.1.1.cmml"><mrow id="S3.Ex41.m1.3.3.1.1.2.2" xref="S3.Ex41.m1.3.3.1.1.2.3.cmml"><mo id="S3.Ex41.m1.3.3.1.1.2.2.3" stretchy="false" xref="S3.Ex41.m1.3.3.1.1.2.3.cmml">(</mo><mrow id="S3.Ex41.m1.3.3.1.1.1.1.1" xref="S3.Ex41.m1.3.3.1.1.1.1.1.cmml"><msub id="S3.Ex41.m1.3.3.1.1.1.1.1.2" xref="S3.Ex41.m1.3.3.1.1.1.1.1.2.cmml"><mi id="S3.Ex41.m1.3.3.1.1.1.1.1.2.2" xref="S3.Ex41.m1.3.3.1.1.1.1.1.2.2.cmml">α</mi><mi id="S3.Ex41.m1.3.3.1.1.1.1.1.2.3" xref="S3.Ex41.m1.3.3.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S3.Ex41.m1.3.3.1.1.1.1.1.1" xref="S3.Ex41.m1.3.3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Ex41.m1.3.3.1.1.1.1.1.3" mathvariant="normal" xref="S3.Ex41.m1.3.3.1.1.1.1.1.3.cmml">⋯</mi><mo id="S3.Ex41.m1.3.3.1.1.1.1.1.1a" xref="S3.Ex41.m1.3.3.1.1.1.1.1.1.cmml">⁢</mo><msub id="S3.Ex41.m1.3.3.1.1.1.1.1.4" xref="S3.Ex41.m1.3.3.1.1.1.1.1.4.cmml"><mi id="S3.Ex41.m1.3.3.1.1.1.1.1.4.2" xref="S3.Ex41.m1.3.3.1.1.1.1.1.4.2.cmml">α</mi><mrow id="S3.Ex41.m1.1.1.1" xref="S3.Ex41.m1.1.1.1.cmml"><mrow id="S3.Ex41.m1.1.1.1.3" xref="S3.Ex41.m1.1.1.1.3.cmml"><mi id="S3.Ex41.m1.1.1.1.3.2" xref="S3.Ex41.m1.1.1.1.3.2.cmml">f</mi><mo id="S3.Ex41.m1.1.1.1.3.1" xref="S3.Ex41.m1.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex41.m1.1.1.1.3.3.2" xref="S3.Ex41.m1.1.1.1.3.cmml"><mo id="S3.Ex41.m1.1.1.1.3.3.2.1" stretchy="false" xref="S3.Ex41.m1.1.1.1.3.cmml">(</mo><mi id="S3.Ex41.m1.1.1.1.1" xref="S3.Ex41.m1.1.1.1.1.cmml">m</mi><mo id="S3.Ex41.m1.1.1.1.3.3.2.2" stretchy="false" xref="S3.Ex41.m1.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex41.m1.1.1.1.2" xref="S3.Ex41.m1.1.1.1.2.cmml">+</mo><mn id="S3.Ex41.m1.1.1.1.4" xref="S3.Ex41.m1.1.1.1.4.cmml">1</mn></mrow></msub></mrow><mo id="S3.Ex41.m1.3.3.1.1.2.2.4" xref="S3.Ex41.m1.3.3.1.1.2.3.cmml">,</mo><mrow id="S3.Ex41.m1.3.3.1.1.2.2.2" xref="S3.Ex41.m1.3.3.1.1.2.2.2.cmml"><msub id="S3.Ex41.m1.3.3.1.1.2.2.2.2" xref="S3.Ex41.m1.3.3.1.1.2.2.2.2.cmml"><mi id="S3.Ex41.m1.3.3.1.1.2.2.2.2.2" xref="S3.Ex41.m1.3.3.1.1.2.2.2.2.2.cmml">α</mi><mrow id="S3.Ex41.m1.2.2.1" xref="S3.Ex41.m1.2.2.1.cmml"><mi id="S3.Ex41.m1.2.2.1.3" xref="S3.Ex41.m1.2.2.1.3.cmml">f</mi><mo id="S3.Ex41.m1.2.2.1.2" xref="S3.Ex41.m1.2.2.1.2.cmml">⁢</mo><mrow id="S3.Ex41.m1.2.2.1.4.2" xref="S3.Ex41.m1.2.2.1.cmml"><mo id="S3.Ex41.m1.2.2.1.4.2.1" stretchy="false" xref="S3.Ex41.m1.2.2.1.cmml">(</mo><mn id="S3.Ex41.m1.2.2.1.1" xref="S3.Ex41.m1.2.2.1.1.cmml">0</mn><mo id="S3.Ex41.m1.2.2.1.4.2.2" stretchy="false" xref="S3.Ex41.m1.2.2.1.cmml">)</mo></mrow></mrow></msub><mo id="S3.Ex41.m1.3.3.1.1.2.2.2.1" 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italic_n end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT italic_f ( italic_m ) + 1 end_POSTSUBSCRIPT , italic_α start_POSTSUBSCRIPT italic_f ( 0 ) end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) : italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⋯ italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_α start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.1.1.1">Lemma 3.7</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.2.2"> </span>(<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib7" title="">7</a>, Thm 2.1]</cite>)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem7.p1"> <p class="ltx_p" id="S3.Thmtheorem7.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem7.p1.2.2">If <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.1.m1.1"><semantics id="S3.Thmtheorem7.p1.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem7.p1.1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.1.m1.1b"><ci id="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.1.m1.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.1.m1.1d">caligraphic_M</annotation></semantics></math> is a coefficient system for <math alttext="N\mathcal{C}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.2.m2.1"><semantics id="S3.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem7.p1.2.2.m2.1.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">N</mi><mo id="S3.Thmtheorem7.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem7.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">𝒞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.2.m2.1b"><apply id="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1"><times id="S3.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.1"></times><ci id="S3.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.2">𝑁</ci><ci id="S3.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.2.m2.1c">N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.2.m2.1d">italic_N caligraphic_C</annotation></semantics></math> defined by the composition</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex42"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{M}:\Delta(N\mathcal{C})\smash{\,\mathop{\longrightarrow}\limits^{\chi% }\,}\mathfrak{F}\mathcal{C}\smash{\,\mathop{\longrightarrow}\limits^{M}\,}R% \text{-Mod}" class="ltx_Math" display="block" id="S3.Ex42.m1.1"><semantics id="S3.Ex42.m1.1a"><mrow id="S3.Ex42.m1.1.1" xref="S3.Ex42.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex42.m1.1.1.3" xref="S3.Ex42.m1.1.1.3.cmml">ℳ</mi><mo id="S3.Ex42.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S3.Ex42.m1.1.1.2.cmml">:</mo><mrow id="S3.Ex42.m1.1.1.1" xref="S3.Ex42.m1.1.1.1.cmml"><mi id="S3.Ex42.m1.1.1.1.3" mathvariant="normal" xref="S3.Ex42.m1.1.1.1.3.cmml">Δ</mi><mo id="S3.Ex42.m1.1.1.1.2" xref="S3.Ex42.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex42.m1.1.1.1.1.1" xref="S3.Ex42.m1.1.1.1.1.1.1.cmml"><mo id="S3.Ex42.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex42.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex42.m1.1.1.1.1.1.1" xref="S3.Ex42.m1.1.1.1.1.1.1.cmml"><mi id="S3.Ex42.m1.1.1.1.1.1.1.2" xref="S3.Ex42.m1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S3.Ex42.m1.1.1.1.1.1.1.1" xref="S3.Ex42.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex42.m1.1.1.1.1.1.1.3" xref="S3.Ex42.m1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S3.Ex42.m1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex42.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex42.m1.1.1.1.2a" lspace="0.337em" xref="S3.Ex42.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex42.m1.1.1.1.4" xref="S3.Ex42.m1.1.1.1.4.cmml"><mover id="S3.Ex42.m1.1.1.1.4.1" xref="S3.Ex42.m1.1.1.1.4.1.cmml"><mo id="S3.Ex42.m1.1.1.1.4.1.2" movablelimits="false" rspace="0.167em" xref="S3.Ex42.m1.1.1.1.4.1.2.cmml">⟶</mo><mi id="S3.Ex42.m1.1.1.1.4.1.3" xref="S3.Ex42.m1.1.1.1.4.1.3.cmml">χ</mi></mover><mrow id="S3.Ex42.m1.1.1.1.4.2" xref="S3.Ex42.m1.1.1.1.4.2.cmml"><mi id="S3.Ex42.m1.1.1.1.4.2.2" xref="S3.Ex42.m1.1.1.1.4.2.2.cmml">𝔉</mi><mo id="S3.Ex42.m1.1.1.1.4.2.1" xref="S3.Ex42.m1.1.1.1.4.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex42.m1.1.1.1.4.2.3" xref="S3.Ex42.m1.1.1.1.4.2.3.cmml">𝒞</mi><mo id="S3.Ex42.m1.1.1.1.4.2.1a" lspace="0.337em" xref="S3.Ex42.m1.1.1.1.4.2.1.cmml">⁢</mo><mrow id="S3.Ex42.m1.1.1.1.4.2.4" xref="S3.Ex42.m1.1.1.1.4.2.4.cmml"><mover id="S3.Ex42.m1.1.1.1.4.2.4.1" xref="S3.Ex42.m1.1.1.1.4.2.4.1.cmml"><mo id="S3.Ex42.m1.1.1.1.4.2.4.1.2" movablelimits="false" rspace="0.167em" xref="S3.Ex42.m1.1.1.1.4.2.4.1.2.cmml">⟶</mo><mi id="S3.Ex42.m1.1.1.1.4.2.4.1.3" xref="S3.Ex42.m1.1.1.1.4.2.4.1.3.cmml">M</mi></mover><mrow id="S3.Ex42.m1.1.1.1.4.2.4.2" xref="S3.Ex42.m1.1.1.1.4.2.4.2.cmml"><mi id="S3.Ex42.m1.1.1.1.4.2.4.2.2" xref="S3.Ex42.m1.1.1.1.4.2.4.2.2.cmml">R</mi><mo id="S3.Ex42.m1.1.1.1.4.2.4.2.1" xref="S3.Ex42.m1.1.1.1.4.2.4.2.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S3.Ex42.m1.1.1.1.4.2.4.2.3" xref="S3.Ex42.m1.1.1.1.4.2.4.2.3a.cmml">-Mod</mtext></mrow></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex42.m1.1b"><apply id="S3.Ex42.m1.1.1.cmml" xref="S3.Ex42.m1.1.1"><ci id="S3.Ex42.m1.1.1.2.cmml" xref="S3.Ex42.m1.1.1.2">:</ci><ci id="S3.Ex42.m1.1.1.3.cmml" xref="S3.Ex42.m1.1.1.3">ℳ</ci><apply id="S3.Ex42.m1.1.1.1.cmml" xref="S3.Ex42.m1.1.1.1"><times id="S3.Ex42.m1.1.1.1.2.cmml" xref="S3.Ex42.m1.1.1.1.2"></times><ci id="S3.Ex42.m1.1.1.1.3.cmml" xref="S3.Ex42.m1.1.1.1.3">Δ</ci><apply id="S3.Ex42.m1.1.1.1.1.1.1.cmml" xref="S3.Ex42.m1.1.1.1.1.1"><times id="S3.Ex42.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex42.m1.1.1.1.1.1.1.1"></times><ci id="S3.Ex42.m1.1.1.1.1.1.1.2.cmml" xref="S3.Ex42.m1.1.1.1.1.1.1.2">𝑁</ci><ci id="S3.Ex42.m1.1.1.1.1.1.1.3.cmml" xref="S3.Ex42.m1.1.1.1.1.1.1.3">𝒞</ci></apply><apply id="S3.Ex42.m1.1.1.1.4.cmml" xref="S3.Ex42.m1.1.1.1.4"><apply id="S3.Ex42.m1.1.1.1.4.1.cmml" xref="S3.Ex42.m1.1.1.1.4.1"><csymbol cd="ambiguous" id="S3.Ex42.m1.1.1.1.4.1.1.cmml" xref="S3.Ex42.m1.1.1.1.4.1">superscript</csymbol><ci id="S3.Ex42.m1.1.1.1.4.1.2.cmml" xref="S3.Ex42.m1.1.1.1.4.1.2">⟶</ci><ci id="S3.Ex42.m1.1.1.1.4.1.3.cmml" xref="S3.Ex42.m1.1.1.1.4.1.3">𝜒</ci></apply><apply id="S3.Ex42.m1.1.1.1.4.2.cmml" xref="S3.Ex42.m1.1.1.1.4.2"><times id="S3.Ex42.m1.1.1.1.4.2.1.cmml" xref="S3.Ex42.m1.1.1.1.4.2.1"></times><ci id="S3.Ex42.m1.1.1.1.4.2.2.cmml" xref="S3.Ex42.m1.1.1.1.4.2.2">𝔉</ci><ci id="S3.Ex42.m1.1.1.1.4.2.3.cmml" xref="S3.Ex42.m1.1.1.1.4.2.3">𝒞</ci><apply id="S3.Ex42.m1.1.1.1.4.2.4.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4"><apply id="S3.Ex42.m1.1.1.1.4.2.4.1.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.1"><csymbol cd="ambiguous" id="S3.Ex42.m1.1.1.1.4.2.4.1.1.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.1">superscript</csymbol><ci id="S3.Ex42.m1.1.1.1.4.2.4.1.2.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.1.2">⟶</ci><ci id="S3.Ex42.m1.1.1.1.4.2.4.1.3.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.1.3">𝑀</ci></apply><apply id="S3.Ex42.m1.1.1.1.4.2.4.2.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.2"><times id="S3.Ex42.m1.1.1.1.4.2.4.2.1.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.2.1"></times><ci id="S3.Ex42.m1.1.1.1.4.2.4.2.2.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.2.2">𝑅</ci><ci id="S3.Ex42.m1.1.1.1.4.2.4.2.3a.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.2.3"><mtext class="ltx_mathvariant_italic" id="S3.Ex42.m1.1.1.1.4.2.4.2.3.cmml" xref="S3.Ex42.m1.1.1.1.4.2.4.2.3">-Mod</mtext></ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex42.m1.1c">\mathcal{M}:\Delta(N\mathcal{C})\smash{\,\mathop{\longrightarrow}\limits^{\chi% }\,}\mathfrak{F}\mathcal{C}\smash{\,\mathop{\longrightarrow}\limits^{M}\,}R% \text{-Mod}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex42.m1.1d">caligraphic_M : roman_Δ ( italic_N caligraphic_C ) ⟶ start_POSTSUPERSCRIPT italic_χ end_POSTSUPERSCRIPT fraktur_F caligraphic_C ⟶ start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT italic_R -Mod</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem7.p1.3"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem7.p1.3.1">for some natural system <math alttext="M" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.3.1.m1.1"><semantics id="S3.Thmtheorem7.p1.3.1.m1.1a"><mi id="S3.Thmtheorem7.p1.3.1.m1.1.1" xref="S3.Thmtheorem7.p1.3.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.3.1.m1.1b"><ci id="S3.Thmtheorem7.p1.3.1.m1.1.1.cmml" xref="S3.Thmtheorem7.p1.3.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.3.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.3.1.m1.1d">italic_M</annotation></semantics></math>, then we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex43"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{*}_{Th}(\mathcal{C};\mathcal{M})\cong H^{*}_{BW}(\mathcal{C};M)." class="ltx_Math" display="block" id="S3.Ex43.m1.5"><semantics id="S3.Ex43.m1.5a"><mrow id="S3.Ex43.m1.5.5.1" xref="S3.Ex43.m1.5.5.1.1.cmml"><mrow id="S3.Ex43.m1.5.5.1.1" xref="S3.Ex43.m1.5.5.1.1.cmml"><mrow id="S3.Ex43.m1.5.5.1.1.2" xref="S3.Ex43.m1.5.5.1.1.2.cmml"><msubsup id="S3.Ex43.m1.5.5.1.1.2.2" xref="S3.Ex43.m1.5.5.1.1.2.2.cmml"><mi id="S3.Ex43.m1.5.5.1.1.2.2.2.2" xref="S3.Ex43.m1.5.5.1.1.2.2.2.2.cmml">H</mi><mrow id="S3.Ex43.m1.5.5.1.1.2.2.3" xref="S3.Ex43.m1.5.5.1.1.2.2.3.cmml"><mi id="S3.Ex43.m1.5.5.1.1.2.2.3.2" xref="S3.Ex43.m1.5.5.1.1.2.2.3.2.cmml">T</mi><mo id="S3.Ex43.m1.5.5.1.1.2.2.3.1" xref="S3.Ex43.m1.5.5.1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.Ex43.m1.5.5.1.1.2.2.3.3" xref="S3.Ex43.m1.5.5.1.1.2.2.3.3.cmml">h</mi></mrow><mo id="S3.Ex43.m1.5.5.1.1.2.2.2.3" xref="S3.Ex43.m1.5.5.1.1.2.2.2.3.cmml">∗</mo></msubsup><mo id="S3.Ex43.m1.5.5.1.1.2.1" xref="S3.Ex43.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S3.Ex43.m1.5.5.1.1.2.3.2" xref="S3.Ex43.m1.5.5.1.1.2.3.1.cmml"><mo id="S3.Ex43.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S3.Ex43.m1.5.5.1.1.2.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex43.m1.1.1" xref="S3.Ex43.m1.1.1.cmml">𝒞</mi><mo id="S3.Ex43.m1.5.5.1.1.2.3.2.2" xref="S3.Ex43.m1.5.5.1.1.2.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex43.m1.2.2" xref="S3.Ex43.m1.2.2.cmml">ℳ</mi><mo id="S3.Ex43.m1.5.5.1.1.2.3.2.3" stretchy="false" xref="S3.Ex43.m1.5.5.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex43.m1.5.5.1.1.1" xref="S3.Ex43.m1.5.5.1.1.1.cmml">≅</mo><mrow id="S3.Ex43.m1.5.5.1.1.3" xref="S3.Ex43.m1.5.5.1.1.3.cmml"><msubsup id="S3.Ex43.m1.5.5.1.1.3.2" xref="S3.Ex43.m1.5.5.1.1.3.2.cmml"><mi id="S3.Ex43.m1.5.5.1.1.3.2.2.2" xref="S3.Ex43.m1.5.5.1.1.3.2.2.2.cmml">H</mi><mrow id="S3.Ex43.m1.5.5.1.1.3.2.3" xref="S3.Ex43.m1.5.5.1.1.3.2.3.cmml"><mi id="S3.Ex43.m1.5.5.1.1.3.2.3.2" xref="S3.Ex43.m1.5.5.1.1.3.2.3.2.cmml">B</mi><mo id="S3.Ex43.m1.5.5.1.1.3.2.3.1" xref="S3.Ex43.m1.5.5.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.Ex43.m1.5.5.1.1.3.2.3.3" xref="S3.Ex43.m1.5.5.1.1.3.2.3.3.cmml">W</mi></mrow><mo id="S3.Ex43.m1.5.5.1.1.3.2.2.3" xref="S3.Ex43.m1.5.5.1.1.3.2.2.3.cmml">∗</mo></msubsup><mo id="S3.Ex43.m1.5.5.1.1.3.1" xref="S3.Ex43.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex43.m1.5.5.1.1.3.3.2" xref="S3.Ex43.m1.5.5.1.1.3.3.1.cmml"><mo id="S3.Ex43.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S3.Ex43.m1.5.5.1.1.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex43.m1.3.3" xref="S3.Ex43.m1.3.3.cmml">𝒞</mi><mo id="S3.Ex43.m1.5.5.1.1.3.3.2.2" xref="S3.Ex43.m1.5.5.1.1.3.3.1.cmml">;</mo><mi id="S3.Ex43.m1.4.4" xref="S3.Ex43.m1.4.4.cmml">M</mi><mo id="S3.Ex43.m1.5.5.1.1.3.3.2.3" stretchy="false" xref="S3.Ex43.m1.5.5.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex43.m1.5.5.1.2" lspace="0em" xref="S3.Ex43.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex43.m1.5b"><apply id="S3.Ex43.m1.5.5.1.1.cmml" xref="S3.Ex43.m1.5.5.1"><approx id="S3.Ex43.m1.5.5.1.1.1.cmml" xref="S3.Ex43.m1.5.5.1.1.1"></approx><apply id="S3.Ex43.m1.5.5.1.1.2.cmml" xref="S3.Ex43.m1.5.5.1.1.2"><times id="S3.Ex43.m1.5.5.1.1.2.1.cmml" xref="S3.Ex43.m1.5.5.1.1.2.1"></times><apply id="S3.Ex43.m1.5.5.1.1.2.2.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex43.m1.5.5.1.1.2.2.1.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2">subscript</csymbol><apply id="S3.Ex43.m1.5.5.1.1.2.2.2.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex43.m1.5.5.1.1.2.2.2.1.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2">superscript</csymbol><ci id="S3.Ex43.m1.5.5.1.1.2.2.2.2.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2.2.2">𝐻</ci><times id="S3.Ex43.m1.5.5.1.1.2.2.2.3.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2.2.3"></times></apply><apply id="S3.Ex43.m1.5.5.1.1.2.2.3.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2.3"><times id="S3.Ex43.m1.5.5.1.1.2.2.3.1.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2.3.1"></times><ci id="S3.Ex43.m1.5.5.1.1.2.2.3.2.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2.3.2">𝑇</ci><ci id="S3.Ex43.m1.5.5.1.1.2.2.3.3.cmml" xref="S3.Ex43.m1.5.5.1.1.2.2.3.3">ℎ</ci></apply></apply><list id="S3.Ex43.m1.5.5.1.1.2.3.1.cmml" xref="S3.Ex43.m1.5.5.1.1.2.3.2"><ci id="S3.Ex43.m1.1.1.cmml" xref="S3.Ex43.m1.1.1">𝒞</ci><ci id="S3.Ex43.m1.2.2.cmml" xref="S3.Ex43.m1.2.2">ℳ</ci></list></apply><apply id="S3.Ex43.m1.5.5.1.1.3.cmml" xref="S3.Ex43.m1.5.5.1.1.3"><times id="S3.Ex43.m1.5.5.1.1.3.1.cmml" xref="S3.Ex43.m1.5.5.1.1.3.1"></times><apply id="S3.Ex43.m1.5.5.1.1.3.2.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2"><csymbol cd="ambiguous" id="S3.Ex43.m1.5.5.1.1.3.2.1.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2">subscript</csymbol><apply id="S3.Ex43.m1.5.5.1.1.3.2.2.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2"><csymbol cd="ambiguous" id="S3.Ex43.m1.5.5.1.1.3.2.2.1.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2">superscript</csymbol><ci id="S3.Ex43.m1.5.5.1.1.3.2.2.2.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2.2.2">𝐻</ci><times id="S3.Ex43.m1.5.5.1.1.3.2.2.3.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2.2.3"></times></apply><apply id="S3.Ex43.m1.5.5.1.1.3.2.3.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2.3"><times id="S3.Ex43.m1.5.5.1.1.3.2.3.1.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2.3.1"></times><ci id="S3.Ex43.m1.5.5.1.1.3.2.3.2.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2.3.2">𝐵</ci><ci id="S3.Ex43.m1.5.5.1.1.3.2.3.3.cmml" xref="S3.Ex43.m1.5.5.1.1.3.2.3.3">𝑊</ci></apply></apply><list id="S3.Ex43.m1.5.5.1.1.3.3.1.cmml" xref="S3.Ex43.m1.5.5.1.1.3.3.2"><ci id="S3.Ex43.m1.3.3.cmml" xref="S3.Ex43.m1.3.3">𝒞</ci><ci id="S3.Ex43.m1.4.4.cmml" xref="S3.Ex43.m1.4.4">𝑀</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex43.m1.5c">H^{*}_{Th}(\mathcal{C};\mathcal{M})\cong H^{*}_{BW}(\mathcal{C};M).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex43.m1.5d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; caligraphic_M ) ≅ italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_W end_POSTSUBSCRIPT ( caligraphic_C ; italic_M ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4. </span>Bisimplicial objects and the Dold-Puppe theorem</h2> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1. </span>Homotopy colimits</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.8">Let <math alttext="\Delta" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" mathvariant="normal" xref="S4.SS1.p1.1.m1.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">\Delta</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">roman_Δ</annotation></semantics></math> denote the simplex category. A <em class="ltx_emph ltx_font_italic" id="S4.SS1.p1.8.1">bisimplicial object</em> in the category <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><ci id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">caligraphic_C</annotation></semantics></math> is a functor <math alttext="X:\Delta^{op}\times\Delta^{op}\to\mathcal{C}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.1"><semantics id="S4.SS1.p1.3.m3.1a"><mrow id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml"><mi id="S4.SS1.p1.3.m3.1.1.2" xref="S4.SS1.p1.3.m3.1.1.2.cmml">X</mi><mo id="S4.SS1.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S4.SS1.p1.3.m3.1.1.3" xref="S4.SS1.p1.3.m3.1.1.3.cmml"><mrow id="S4.SS1.p1.3.m3.1.1.3.2" xref="S4.SS1.p1.3.m3.1.1.3.2.cmml"><msup id="S4.SS1.p1.3.m3.1.1.3.2.2" xref="S4.SS1.p1.3.m3.1.1.3.2.2.cmml"><mi id="S4.SS1.p1.3.m3.1.1.3.2.2.2" mathvariant="normal" xref="S4.SS1.p1.3.m3.1.1.3.2.2.2.cmml">Δ</mi><mrow id="S4.SS1.p1.3.m3.1.1.3.2.2.3" xref="S4.SS1.p1.3.m3.1.1.3.2.2.3.cmml"><mi id="S4.SS1.p1.3.m3.1.1.3.2.2.3.2" xref="S4.SS1.p1.3.m3.1.1.3.2.2.3.2.cmml">o</mi><mo id="S4.SS1.p1.3.m3.1.1.3.2.2.3.1" xref="S4.SS1.p1.3.m3.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S4.SS1.p1.3.m3.1.1.3.2.2.3.3" xref="S4.SS1.p1.3.m3.1.1.3.2.2.3.3.cmml">p</mi></mrow></msup><mo id="S4.SS1.p1.3.m3.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.p1.3.m3.1.1.3.2.1.cmml">×</mo><msup id="S4.SS1.p1.3.m3.1.1.3.2.3" xref="S4.SS1.p1.3.m3.1.1.3.2.3.cmml"><mi id="S4.SS1.p1.3.m3.1.1.3.2.3.2" mathvariant="normal" xref="S4.SS1.p1.3.m3.1.1.3.2.3.2.cmml">Δ</mi><mrow id="S4.SS1.p1.3.m3.1.1.3.2.3.3" xref="S4.SS1.p1.3.m3.1.1.3.2.3.3.cmml"><mi id="S4.SS1.p1.3.m3.1.1.3.2.3.3.2" xref="S4.SS1.p1.3.m3.1.1.3.2.3.3.2.cmml">o</mi><mo id="S4.SS1.p1.3.m3.1.1.3.2.3.3.1" xref="S4.SS1.p1.3.m3.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S4.SS1.p1.3.m3.1.1.3.2.3.3.3" xref="S4.SS1.p1.3.m3.1.1.3.2.3.3.3.cmml">p</mi></mrow></msup></mrow><mo id="S4.SS1.p1.3.m3.1.1.3.1" stretchy="false" xref="S4.SS1.p1.3.m3.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.3.m3.1.1.3.3" xref="S4.SS1.p1.3.m3.1.1.3.3.cmml">𝒞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.1b"><apply id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1"><ci id="S4.SS1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1.1">:</ci><ci id="S4.SS1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.p1.3.m3.1.1.2">𝑋</ci><apply id="S4.SS1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3"><ci id="S4.SS1.p1.3.m3.1.1.3.1.cmml" xref="S4.SS1.p1.3.m3.1.1.3.1">→</ci><apply id="S4.SS1.p1.3.m3.1.1.3.2.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2"><times id="S4.SS1.p1.3.m3.1.1.3.2.1.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.1"></times><apply id="S4.SS1.p1.3.m3.1.1.3.2.2.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.1.1.3.2.2.1.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.2">superscript</csymbol><ci id="S4.SS1.p1.3.m3.1.1.3.2.2.2.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.2.2">Δ</ci><apply id="S4.SS1.p1.3.m3.1.1.3.2.2.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.2.3"><times id="S4.SS1.p1.3.m3.1.1.3.2.2.3.1.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.2.3.1"></times><ci id="S4.SS1.p1.3.m3.1.1.3.2.2.3.2.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.2.3.2">𝑜</ci><ci id="S4.SS1.p1.3.m3.1.1.3.2.2.3.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.2.3.3">𝑝</ci></apply></apply><apply id="S4.SS1.p1.3.m3.1.1.3.2.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.1.1.3.2.3.1.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.3">superscript</csymbol><ci id="S4.SS1.p1.3.m3.1.1.3.2.3.2.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.3.2">Δ</ci><apply id="S4.SS1.p1.3.m3.1.1.3.2.3.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.3.3"><times id="S4.SS1.p1.3.m3.1.1.3.2.3.3.1.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.3.3.1"></times><ci id="S4.SS1.p1.3.m3.1.1.3.2.3.3.2.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.3.3.2">𝑜</ci><ci id="S4.SS1.p1.3.m3.1.1.3.2.3.3.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3.2.3.3.3">𝑝</ci></apply></apply></apply><ci id="S4.SS1.p1.3.m3.1.1.3.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">X:\Delta^{op}\times\Delta^{op}\to\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.1d">italic_X : roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT × roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → caligraphic_C</annotation></semantics></math>. We denote the object <math alttext="X([p]\times[q])" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.3"><semantics id="S4.SS1.p1.4.m4.3a"><mrow id="S4.SS1.p1.4.m4.3.3" xref="S4.SS1.p1.4.m4.3.3.cmml"><mi id="S4.SS1.p1.4.m4.3.3.3" xref="S4.SS1.p1.4.m4.3.3.3.cmml">X</mi><mo id="S4.SS1.p1.4.m4.3.3.2" xref="S4.SS1.p1.4.m4.3.3.2.cmml">⁢</mo><mrow id="S4.SS1.p1.4.m4.3.3.1.1" xref="S4.SS1.p1.4.m4.3.3.1.1.1.cmml"><mo id="S4.SS1.p1.4.m4.3.3.1.1.2" stretchy="false" xref="S4.SS1.p1.4.m4.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS1.p1.4.m4.3.3.1.1.1" xref="S4.SS1.p1.4.m4.3.3.1.1.1.cmml"><mrow id="S4.SS1.p1.4.m4.3.3.1.1.1.2.2" xref="S4.SS1.p1.4.m4.3.3.1.1.1.2.1.cmml"><mo id="S4.SS1.p1.4.m4.3.3.1.1.1.2.2.1" stretchy="false" xref="S4.SS1.p1.4.m4.3.3.1.1.1.2.1.1.cmml">[</mo><mi id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml">p</mi><mo id="S4.SS1.p1.4.m4.3.3.1.1.1.2.2.2" rspace="0.055em" stretchy="false" xref="S4.SS1.p1.4.m4.3.3.1.1.1.2.1.1.cmml">]</mo></mrow><mo id="S4.SS1.p1.4.m4.3.3.1.1.1.1" rspace="0.222em" xref="S4.SS1.p1.4.m4.3.3.1.1.1.1.cmml">×</mo><mrow id="S4.SS1.p1.4.m4.3.3.1.1.1.3.2" xref="S4.SS1.p1.4.m4.3.3.1.1.1.3.1.cmml"><mo id="S4.SS1.p1.4.m4.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p1.4.m4.3.3.1.1.1.3.1.1.cmml">[</mo><mi id="S4.SS1.p1.4.m4.2.2" xref="S4.SS1.p1.4.m4.2.2.cmml">q</mi><mo id="S4.SS1.p1.4.m4.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p1.4.m4.3.3.1.1.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="S4.SS1.p1.4.m4.3.3.1.1.3" stretchy="false" xref="S4.SS1.p1.4.m4.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.3b"><apply id="S4.SS1.p1.4.m4.3.3.cmml" xref="S4.SS1.p1.4.m4.3.3"><times id="S4.SS1.p1.4.m4.3.3.2.cmml" xref="S4.SS1.p1.4.m4.3.3.2"></times><ci id="S4.SS1.p1.4.m4.3.3.3.cmml" xref="S4.SS1.p1.4.m4.3.3.3">𝑋</ci><apply id="S4.SS1.p1.4.m4.3.3.1.1.1.cmml" xref="S4.SS1.p1.4.m4.3.3.1.1"><times id="S4.SS1.p1.4.m4.3.3.1.1.1.1.cmml" xref="S4.SS1.p1.4.m4.3.3.1.1.1.1"></times><apply id="S4.SS1.p1.4.m4.3.3.1.1.1.2.1.cmml" xref="S4.SS1.p1.4.m4.3.3.1.1.1.2.2"><csymbol cd="latexml" id="S4.SS1.p1.4.m4.3.3.1.1.1.2.1.1.cmml" xref="S4.SS1.p1.4.m4.3.3.1.1.1.2.2.1">delimited-[]</csymbol><ci id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1">𝑝</ci></apply><apply id="S4.SS1.p1.4.m4.3.3.1.1.1.3.1.cmml" xref="S4.SS1.p1.4.m4.3.3.1.1.1.3.2"><csymbol cd="latexml" id="S4.SS1.p1.4.m4.3.3.1.1.1.3.1.1.cmml" xref="S4.SS1.p1.4.m4.3.3.1.1.1.3.2.1">delimited-[]</csymbol><ci id="S4.SS1.p1.4.m4.2.2.cmml" xref="S4.SS1.p1.4.m4.2.2">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.3c">X([p]\times[q])</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.3d">italic_X ( [ italic_p ] × [ italic_q ] )</annotation></semantics></math> by <math alttext="X_{p,q}" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m5.2"><semantics id="S4.SS1.p1.5.m5.2a"><msub id="S4.SS1.p1.5.m5.2.3" xref="S4.SS1.p1.5.m5.2.3.cmml"><mi id="S4.SS1.p1.5.m5.2.3.2" xref="S4.SS1.p1.5.m5.2.3.2.cmml">X</mi><mrow id="S4.SS1.p1.5.m5.2.2.2.4" xref="S4.SS1.p1.5.m5.2.2.2.3.cmml"><mi id="S4.SS1.p1.5.m5.1.1.1.1" xref="S4.SS1.p1.5.m5.1.1.1.1.cmml">p</mi><mo id="S4.SS1.p1.5.m5.2.2.2.4.1" xref="S4.SS1.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p1.5.m5.2.2.2.2" xref="S4.SS1.p1.5.m5.2.2.2.2.cmml">q</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m5.2b"><apply id="S4.SS1.p1.5.m5.2.3.cmml" xref="S4.SS1.p1.5.m5.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.5.m5.2.3.1.cmml" xref="S4.SS1.p1.5.m5.2.3">subscript</csymbol><ci id="S4.SS1.p1.5.m5.2.3.2.cmml" xref="S4.SS1.p1.5.m5.2.3.2">𝑋</ci><list id="S4.SS1.p1.5.m5.2.2.2.3.cmml" xref="S4.SS1.p1.5.m5.2.2.2.4"><ci id="S4.SS1.p1.5.m5.1.1.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1.1.1">𝑝</ci><ci id="S4.SS1.p1.5.m5.2.2.2.2.cmml" xref="S4.SS1.p1.5.m5.2.2.2.2">𝑞</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.2c">X_{p,q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m5.2d">italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT</annotation></semantics></math>. The simplicial objects <math alttext="X_{p,*}" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m6.2"><semantics id="S4.SS1.p1.6.m6.2a"><msub id="S4.SS1.p1.6.m6.2.3" xref="S4.SS1.p1.6.m6.2.3.cmml"><mi id="S4.SS1.p1.6.m6.2.3.2" xref="S4.SS1.p1.6.m6.2.3.2.cmml">X</mi><mrow id="S4.SS1.p1.6.m6.2.2.2.4" xref="S4.SS1.p1.6.m6.2.2.2.3.cmml"><mi id="S4.SS1.p1.6.m6.1.1.1.1" xref="S4.SS1.p1.6.m6.1.1.1.1.cmml">p</mi><mo id="S4.SS1.p1.6.m6.2.2.2.4.1" rspace="0em" xref="S4.SS1.p1.6.m6.2.2.2.3.cmml">,</mo><mo id="S4.SS1.p1.6.m6.2.2.2.2" lspace="0em" xref="S4.SS1.p1.6.m6.2.2.2.2.cmml">∗</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m6.2b"><apply id="S4.SS1.p1.6.m6.2.3.cmml" xref="S4.SS1.p1.6.m6.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.6.m6.2.3.1.cmml" xref="S4.SS1.p1.6.m6.2.3">subscript</csymbol><ci id="S4.SS1.p1.6.m6.2.3.2.cmml" xref="S4.SS1.p1.6.m6.2.3.2">𝑋</ci><list id="S4.SS1.p1.6.m6.2.2.2.3.cmml" xref="S4.SS1.p1.6.m6.2.2.2.4"><ci id="S4.SS1.p1.6.m6.1.1.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1.1.1">𝑝</ci><times id="S4.SS1.p1.6.m6.2.2.2.2.cmml" xref="S4.SS1.p1.6.m6.2.2.2.2"></times></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m6.2c">X_{p,*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m6.2d">italic_X start_POSTSUBSCRIPT italic_p , ∗ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="X_{*,q}" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m7.2"><semantics id="S4.SS1.p1.7.m7.2a"><msub id="S4.SS1.p1.7.m7.2.3" xref="S4.SS1.p1.7.m7.2.3.cmml"><mi id="S4.SS1.p1.7.m7.2.3.2" xref="S4.SS1.p1.7.m7.2.3.2.cmml">X</mi><mrow id="S4.SS1.p1.7.m7.2.2.2.4" xref="S4.SS1.p1.7.m7.2.2.2.3.cmml"><mo id="S4.SS1.p1.7.m7.1.1.1.1" rspace="0em" xref="S4.SS1.p1.7.m7.1.1.1.1.cmml">∗</mo><mo id="S4.SS1.p1.7.m7.2.2.2.4.1" xref="S4.SS1.p1.7.m7.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p1.7.m7.2.2.2.2" xref="S4.SS1.p1.7.m7.2.2.2.2.cmml">q</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m7.2b"><apply id="S4.SS1.p1.7.m7.2.3.cmml" xref="S4.SS1.p1.7.m7.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m7.2.3.1.cmml" xref="S4.SS1.p1.7.m7.2.3">subscript</csymbol><ci id="S4.SS1.p1.7.m7.2.3.2.cmml" xref="S4.SS1.p1.7.m7.2.3.2">𝑋</ci><list id="S4.SS1.p1.7.m7.2.2.2.3.cmml" xref="S4.SS1.p1.7.m7.2.2.2.4"><times id="S4.SS1.p1.7.m7.1.1.1.1.cmml" xref="S4.SS1.p1.7.m7.1.1.1.1"></times><ci id="S4.SS1.p1.7.m7.2.2.2.2.cmml" xref="S4.SS1.p1.7.m7.2.2.2.2">𝑞</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m7.2c">X_{*,q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m7.2d">italic_X start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT</annotation></semantics></math> are called vertical and horizontal simplicial objects associated to <math alttext="X" class="ltx_Math" display="inline" id="S4.SS1.p1.8.m8.1"><semantics id="S4.SS1.p1.8.m8.1a"><mi id="S4.SS1.p1.8.m8.1.1" xref="S4.SS1.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.8.m8.1b"><ci id="S4.SS1.p1.8.m8.1.1.cmml" xref="S4.SS1.p1.8.m8.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m8.1d">italic_X</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S4.SS1.p1.8.2">diagonal of a bisimplicial object</em> is the simplicial object defined by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex44"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathrm{diag}X:\Delta^{op}\to\Delta^{op}\times\Delta^{op}\to Sets" class="ltx_Math" display="block" id="S4.Ex44.m1.1"><semantics id="S4.Ex44.m1.1a"><mrow id="S4.Ex44.m1.1.1" xref="S4.Ex44.m1.1.1.cmml"><mrow id="S4.Ex44.m1.1.1.2" xref="S4.Ex44.m1.1.1.2.cmml"><mi id="S4.Ex44.m1.1.1.2.2" xref="S4.Ex44.m1.1.1.2.2.cmml">diag</mi><mo id="S4.Ex44.m1.1.1.2.1" xref="S4.Ex44.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.Ex44.m1.1.1.2.3" xref="S4.Ex44.m1.1.1.2.3.cmml">X</mi></mrow><mo id="S4.Ex44.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Ex44.m1.1.1.1.cmml">:</mo><mrow id="S4.Ex44.m1.1.1.3" xref="S4.Ex44.m1.1.1.3.cmml"><msup id="S4.Ex44.m1.1.1.3.2" xref="S4.Ex44.m1.1.1.3.2.cmml"><mi id="S4.Ex44.m1.1.1.3.2.2" mathvariant="normal" xref="S4.Ex44.m1.1.1.3.2.2.cmml">Δ</mi><mrow id="S4.Ex44.m1.1.1.3.2.3" xref="S4.Ex44.m1.1.1.3.2.3.cmml"><mi id="S4.Ex44.m1.1.1.3.2.3.2" xref="S4.Ex44.m1.1.1.3.2.3.2.cmml">o</mi><mo id="S4.Ex44.m1.1.1.3.2.3.1" xref="S4.Ex44.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S4.Ex44.m1.1.1.3.2.3.3" xref="S4.Ex44.m1.1.1.3.2.3.3.cmml">p</mi></mrow></msup><mo id="S4.Ex44.m1.1.1.3.3" stretchy="false" xref="S4.Ex44.m1.1.1.3.3.cmml">→</mo><mrow id="S4.Ex44.m1.1.1.3.4" xref="S4.Ex44.m1.1.1.3.4.cmml"><msup id="S4.Ex44.m1.1.1.3.4.2" xref="S4.Ex44.m1.1.1.3.4.2.cmml"><mi id="S4.Ex44.m1.1.1.3.4.2.2" mathvariant="normal" xref="S4.Ex44.m1.1.1.3.4.2.2.cmml">Δ</mi><mrow id="S4.Ex44.m1.1.1.3.4.2.3" xref="S4.Ex44.m1.1.1.3.4.2.3.cmml"><mi id="S4.Ex44.m1.1.1.3.4.2.3.2" xref="S4.Ex44.m1.1.1.3.4.2.3.2.cmml">o</mi><mo id="S4.Ex44.m1.1.1.3.4.2.3.1" xref="S4.Ex44.m1.1.1.3.4.2.3.1.cmml">⁢</mo><mi id="S4.Ex44.m1.1.1.3.4.2.3.3" xref="S4.Ex44.m1.1.1.3.4.2.3.3.cmml">p</mi></mrow></msup><mo id="S4.Ex44.m1.1.1.3.4.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex44.m1.1.1.3.4.1.cmml">×</mo><msup id="S4.Ex44.m1.1.1.3.4.3" xref="S4.Ex44.m1.1.1.3.4.3.cmml"><mi id="S4.Ex44.m1.1.1.3.4.3.2" mathvariant="normal" xref="S4.Ex44.m1.1.1.3.4.3.2.cmml">Δ</mi><mrow id="S4.Ex44.m1.1.1.3.4.3.3" xref="S4.Ex44.m1.1.1.3.4.3.3.cmml"><mi id="S4.Ex44.m1.1.1.3.4.3.3.2" xref="S4.Ex44.m1.1.1.3.4.3.3.2.cmml">o</mi><mo id="S4.Ex44.m1.1.1.3.4.3.3.1" xref="S4.Ex44.m1.1.1.3.4.3.3.1.cmml">⁢</mo><mi id="S4.Ex44.m1.1.1.3.4.3.3.3" xref="S4.Ex44.m1.1.1.3.4.3.3.3.cmml">p</mi></mrow></msup></mrow><mo id="S4.Ex44.m1.1.1.3.5" stretchy="false" xref="S4.Ex44.m1.1.1.3.5.cmml">→</mo><mrow id="S4.Ex44.m1.1.1.3.6" xref="S4.Ex44.m1.1.1.3.6.cmml"><mi id="S4.Ex44.m1.1.1.3.6.2" xref="S4.Ex44.m1.1.1.3.6.2.cmml">S</mi><mo id="S4.Ex44.m1.1.1.3.6.1" xref="S4.Ex44.m1.1.1.3.6.1.cmml">⁢</mo><mi id="S4.Ex44.m1.1.1.3.6.3" xref="S4.Ex44.m1.1.1.3.6.3.cmml">e</mi><mo id="S4.Ex44.m1.1.1.3.6.1a" xref="S4.Ex44.m1.1.1.3.6.1.cmml">⁢</mo><mi id="S4.Ex44.m1.1.1.3.6.4" xref="S4.Ex44.m1.1.1.3.6.4.cmml">t</mi><mo id="S4.Ex44.m1.1.1.3.6.1b" xref="S4.Ex44.m1.1.1.3.6.1.cmml">⁢</mo><mi id="S4.Ex44.m1.1.1.3.6.5" xref="S4.Ex44.m1.1.1.3.6.5.cmml">s</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex44.m1.1b"><apply id="S4.Ex44.m1.1.1.cmml" xref="S4.Ex44.m1.1.1"><ci id="S4.Ex44.m1.1.1.1.cmml" xref="S4.Ex44.m1.1.1.1">:</ci><apply id="S4.Ex44.m1.1.1.2.cmml" xref="S4.Ex44.m1.1.1.2"><times id="S4.Ex44.m1.1.1.2.1.cmml" xref="S4.Ex44.m1.1.1.2.1"></times><ci id="S4.Ex44.m1.1.1.2.2.cmml" xref="S4.Ex44.m1.1.1.2.2">diag</ci><ci id="S4.Ex44.m1.1.1.2.3.cmml" xref="S4.Ex44.m1.1.1.2.3">𝑋</ci></apply><apply id="S4.Ex44.m1.1.1.3.cmml" xref="S4.Ex44.m1.1.1.3"><and id="S4.Ex44.m1.1.1.3a.cmml" xref="S4.Ex44.m1.1.1.3"></and><apply id="S4.Ex44.m1.1.1.3b.cmml" xref="S4.Ex44.m1.1.1.3"><ci id="S4.Ex44.m1.1.1.3.3.cmml" xref="S4.Ex44.m1.1.1.3.3">→</ci><apply id="S4.Ex44.m1.1.1.3.2.cmml" xref="S4.Ex44.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.Ex44.m1.1.1.3.2.1.cmml" xref="S4.Ex44.m1.1.1.3.2">superscript</csymbol><ci id="S4.Ex44.m1.1.1.3.2.2.cmml" xref="S4.Ex44.m1.1.1.3.2.2">Δ</ci><apply id="S4.Ex44.m1.1.1.3.2.3.cmml" xref="S4.Ex44.m1.1.1.3.2.3"><times id="S4.Ex44.m1.1.1.3.2.3.1.cmml" xref="S4.Ex44.m1.1.1.3.2.3.1"></times><ci id="S4.Ex44.m1.1.1.3.2.3.2.cmml" xref="S4.Ex44.m1.1.1.3.2.3.2">𝑜</ci><ci id="S4.Ex44.m1.1.1.3.2.3.3.cmml" xref="S4.Ex44.m1.1.1.3.2.3.3">𝑝</ci></apply></apply><apply id="S4.Ex44.m1.1.1.3.4.cmml" xref="S4.Ex44.m1.1.1.3.4"><times id="S4.Ex44.m1.1.1.3.4.1.cmml" xref="S4.Ex44.m1.1.1.3.4.1"></times><apply id="S4.Ex44.m1.1.1.3.4.2.cmml" xref="S4.Ex44.m1.1.1.3.4.2"><csymbol cd="ambiguous" id="S4.Ex44.m1.1.1.3.4.2.1.cmml" xref="S4.Ex44.m1.1.1.3.4.2">superscript</csymbol><ci id="S4.Ex44.m1.1.1.3.4.2.2.cmml" xref="S4.Ex44.m1.1.1.3.4.2.2">Δ</ci><apply id="S4.Ex44.m1.1.1.3.4.2.3.cmml" xref="S4.Ex44.m1.1.1.3.4.2.3"><times id="S4.Ex44.m1.1.1.3.4.2.3.1.cmml" xref="S4.Ex44.m1.1.1.3.4.2.3.1"></times><ci id="S4.Ex44.m1.1.1.3.4.2.3.2.cmml" xref="S4.Ex44.m1.1.1.3.4.2.3.2">𝑜</ci><ci id="S4.Ex44.m1.1.1.3.4.2.3.3.cmml" xref="S4.Ex44.m1.1.1.3.4.2.3.3">𝑝</ci></apply></apply><apply id="S4.Ex44.m1.1.1.3.4.3.cmml" xref="S4.Ex44.m1.1.1.3.4.3"><csymbol cd="ambiguous" id="S4.Ex44.m1.1.1.3.4.3.1.cmml" xref="S4.Ex44.m1.1.1.3.4.3">superscript</csymbol><ci id="S4.Ex44.m1.1.1.3.4.3.2.cmml" xref="S4.Ex44.m1.1.1.3.4.3.2">Δ</ci><apply id="S4.Ex44.m1.1.1.3.4.3.3.cmml" xref="S4.Ex44.m1.1.1.3.4.3.3"><times id="S4.Ex44.m1.1.1.3.4.3.3.1.cmml" xref="S4.Ex44.m1.1.1.3.4.3.3.1"></times><ci id="S4.Ex44.m1.1.1.3.4.3.3.2.cmml" xref="S4.Ex44.m1.1.1.3.4.3.3.2">𝑜</ci><ci id="S4.Ex44.m1.1.1.3.4.3.3.3.cmml" xref="S4.Ex44.m1.1.1.3.4.3.3.3">𝑝</ci></apply></apply></apply></apply><apply id="S4.Ex44.m1.1.1.3c.cmml" xref="S4.Ex44.m1.1.1.3"><ci id="S4.Ex44.m1.1.1.3.5.cmml" xref="S4.Ex44.m1.1.1.3.5">→</ci><share href="https://arxiv.org/html/2503.14659v1#S4.Ex44.m1.1.1.3.4.cmml" id="S4.Ex44.m1.1.1.3d.cmml" xref="S4.Ex44.m1.1.1.3"></share><apply id="S4.Ex44.m1.1.1.3.6.cmml" xref="S4.Ex44.m1.1.1.3.6"><times id="S4.Ex44.m1.1.1.3.6.1.cmml" xref="S4.Ex44.m1.1.1.3.6.1"></times><ci id="S4.Ex44.m1.1.1.3.6.2.cmml" xref="S4.Ex44.m1.1.1.3.6.2">𝑆</ci><ci id="S4.Ex44.m1.1.1.3.6.3.cmml" xref="S4.Ex44.m1.1.1.3.6.3">𝑒</ci><ci id="S4.Ex44.m1.1.1.3.6.4.cmml" xref="S4.Ex44.m1.1.1.3.6.4">𝑡</ci><ci id="S4.Ex44.m1.1.1.3.6.5.cmml" xref="S4.Ex44.m1.1.1.3.6.5">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex44.m1.1c">\mathrm{diag}X:\Delta^{op}\to\Delta^{op}\times\Delta^{op}\to Sets</annotation><annotation encoding="application/x-llamapun" id="S4.Ex44.m1.1d">roman_diag italic_X : roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT × roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → italic_S italic_e italic_t italic_s</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.9">where the first functor is the diagonal functor defined by <math alttext="[n]\to([n],[n])" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m1.5"><semantics id="S4.SS1.p1.9.m1.5a"><mrow id="S4.SS1.p1.9.m1.5.5" xref="S4.SS1.p1.9.m1.5.5.cmml"><mrow id="S4.SS1.p1.9.m1.5.5.4.2" xref="S4.SS1.p1.9.m1.5.5.4.1.cmml"><mo id="S4.SS1.p1.9.m1.5.5.4.2.1" stretchy="false" xref="S4.SS1.p1.9.m1.5.5.4.1.1.cmml">[</mo><mi id="S4.SS1.p1.9.m1.1.1" xref="S4.SS1.p1.9.m1.1.1.cmml">n</mi><mo id="S4.SS1.p1.9.m1.5.5.4.2.2" stretchy="false" xref="S4.SS1.p1.9.m1.5.5.4.1.1.cmml">]</mo></mrow><mo id="S4.SS1.p1.9.m1.5.5.3" stretchy="false" xref="S4.SS1.p1.9.m1.5.5.3.cmml">→</mo><mrow id="S4.SS1.p1.9.m1.5.5.2.2" xref="S4.SS1.p1.9.m1.5.5.2.3.cmml"><mo id="S4.SS1.p1.9.m1.5.5.2.2.3" stretchy="false" xref="S4.SS1.p1.9.m1.5.5.2.3.cmml">(</mo><mrow id="S4.SS1.p1.9.m1.4.4.1.1.1.2" xref="S4.SS1.p1.9.m1.4.4.1.1.1.1.cmml"><mo id="S4.SS1.p1.9.m1.4.4.1.1.1.2.1" stretchy="false" xref="S4.SS1.p1.9.m1.4.4.1.1.1.1.1.cmml">[</mo><mi id="S4.SS1.p1.9.m1.2.2" xref="S4.SS1.p1.9.m1.2.2.cmml">n</mi><mo id="S4.SS1.p1.9.m1.4.4.1.1.1.2.2" stretchy="false" xref="S4.SS1.p1.9.m1.4.4.1.1.1.1.1.cmml">]</mo></mrow><mo id="S4.SS1.p1.9.m1.5.5.2.2.4" xref="S4.SS1.p1.9.m1.5.5.2.3.cmml">,</mo><mrow id="S4.SS1.p1.9.m1.5.5.2.2.2.2" xref="S4.SS1.p1.9.m1.5.5.2.2.2.1.cmml"><mo id="S4.SS1.p1.9.m1.5.5.2.2.2.2.1" stretchy="false" xref="S4.SS1.p1.9.m1.5.5.2.2.2.1.1.cmml">[</mo><mi id="S4.SS1.p1.9.m1.3.3" xref="S4.SS1.p1.9.m1.3.3.cmml">n</mi><mo id="S4.SS1.p1.9.m1.5.5.2.2.2.2.2" stretchy="false" xref="S4.SS1.p1.9.m1.5.5.2.2.2.1.1.cmml">]</mo></mrow><mo id="S4.SS1.p1.9.m1.5.5.2.2.5" stretchy="false" xref="S4.SS1.p1.9.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m1.5b"><apply id="S4.SS1.p1.9.m1.5.5.cmml" xref="S4.SS1.p1.9.m1.5.5"><ci id="S4.SS1.p1.9.m1.5.5.3.cmml" xref="S4.SS1.p1.9.m1.5.5.3">→</ci><apply id="S4.SS1.p1.9.m1.5.5.4.1.cmml" xref="S4.SS1.p1.9.m1.5.5.4.2"><csymbol cd="latexml" id="S4.SS1.p1.9.m1.5.5.4.1.1.cmml" xref="S4.SS1.p1.9.m1.5.5.4.2.1">delimited-[]</csymbol><ci id="S4.SS1.p1.9.m1.1.1.cmml" xref="S4.SS1.p1.9.m1.1.1">𝑛</ci></apply><interval closure="open" id="S4.SS1.p1.9.m1.5.5.2.3.cmml" xref="S4.SS1.p1.9.m1.5.5.2.2"><apply id="S4.SS1.p1.9.m1.4.4.1.1.1.1.cmml" xref="S4.SS1.p1.9.m1.4.4.1.1.1.2"><csymbol cd="latexml" id="S4.SS1.p1.9.m1.4.4.1.1.1.1.1.cmml" xref="S4.SS1.p1.9.m1.4.4.1.1.1.2.1">delimited-[]</csymbol><ci id="S4.SS1.p1.9.m1.2.2.cmml" xref="S4.SS1.p1.9.m1.2.2">𝑛</ci></apply><apply id="S4.SS1.p1.9.m1.5.5.2.2.2.1.cmml" xref="S4.SS1.p1.9.m1.5.5.2.2.2.2"><csymbol cd="latexml" id="S4.SS1.p1.9.m1.5.5.2.2.2.1.1.cmml" xref="S4.SS1.p1.9.m1.5.5.2.2.2.2.1">delimited-[]</csymbol><ci id="S4.SS1.p1.9.m1.3.3.cmml" xref="S4.SS1.p1.9.m1.3.3">𝑛</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m1.5c">[n]\to([n],[n])</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m1.5d">[ italic_n ] → ( [ italic_n ] , [ italic_n ] )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.4">A <em class="ltx_emph ltx_font_italic" id="S4.SS1.p2.4.1">bisimplicial set</em> is a bisimplicial object in sets. We define homotopy colimits of simplicial sets using the notation and terminology in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib13" title="">13</a>]</cite>. Let <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">caligraphic_D</annotation></semantics></math> be a small category and <math alttext="F:\mathcal{D}\to sSet" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mrow id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><mi id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml">F</mi><mo id="S4.SS1.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.2.m2.1.1.3.2" xref="S4.SS1.p2.2.m2.1.1.3.2.cmml">𝒟</mi><mo id="S4.SS1.p2.2.m2.1.1.3.1" stretchy="false" xref="S4.SS1.p2.2.m2.1.1.3.1.cmml">→</mo><mrow id="S4.SS1.p2.2.m2.1.1.3.3" xref="S4.SS1.p2.2.m2.1.1.3.3.cmml"><mi id="S4.SS1.p2.2.m2.1.1.3.3.2" xref="S4.SS1.p2.2.m2.1.1.3.3.2.cmml">s</mi><mo id="S4.SS1.p2.2.m2.1.1.3.3.1" xref="S4.SS1.p2.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.2.m2.1.1.3.3.3" xref="S4.SS1.p2.2.m2.1.1.3.3.3.cmml">S</mi><mo id="S4.SS1.p2.2.m2.1.1.3.3.1a" xref="S4.SS1.p2.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.2.m2.1.1.3.3.4" xref="S4.SS1.p2.2.m2.1.1.3.3.4.cmml">e</mi><mo id="S4.SS1.p2.2.m2.1.1.3.3.1b" xref="S4.SS1.p2.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.2.m2.1.1.3.3.5" xref="S4.SS1.p2.2.m2.1.1.3.3.5.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><ci id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1.1">:</ci><ci id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2">𝐹</ci><apply id="S4.SS1.p2.2.m2.1.1.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3"><ci id="S4.SS1.p2.2.m2.1.1.3.1.cmml" xref="S4.SS1.p2.2.m2.1.1.3.1">→</ci><ci id="S4.SS1.p2.2.m2.1.1.3.2.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2">𝒟</ci><apply id="S4.SS1.p2.2.m2.1.1.3.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3"><times id="S4.SS1.p2.2.m2.1.1.3.3.1.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3.1"></times><ci id="S4.SS1.p2.2.m2.1.1.3.3.2.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3.2">𝑠</ci><ci id="S4.SS1.p2.2.m2.1.1.3.3.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3.3">𝑆</ci><ci id="S4.SS1.p2.2.m2.1.1.3.3.4.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3.4">𝑒</ci><ci id="S4.SS1.p2.2.m2.1.1.3.3.5.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3.5">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">F:\mathcal{D}\to sSet</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_F : caligraphic_D → italic_s italic_S italic_e italic_t</annotation></semantics></math> a functor. Consider the bisimplicial set <math alttext="N(\mathcal{D};F)" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.2"><semantics id="S4.SS1.p2.3.m3.2a"><mrow id="S4.SS1.p2.3.m3.2.3" xref="S4.SS1.p2.3.m3.2.3.cmml"><mi id="S4.SS1.p2.3.m3.2.3.2" xref="S4.SS1.p2.3.m3.2.3.2.cmml">N</mi><mo id="S4.SS1.p2.3.m3.2.3.1" xref="S4.SS1.p2.3.m3.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.3.m3.2.3.3.2" xref="S4.SS1.p2.3.m3.2.3.3.1.cmml"><mo id="S4.SS1.p2.3.m3.2.3.3.2.1" stretchy="false" xref="S4.SS1.p2.3.m3.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">𝒟</mi><mo id="S4.SS1.p2.3.m3.2.3.3.2.2" xref="S4.SS1.p2.3.m3.2.3.3.1.cmml">;</mo><mi id="S4.SS1.p2.3.m3.2.2" xref="S4.SS1.p2.3.m3.2.2.cmml">F</mi><mo id="S4.SS1.p2.3.m3.2.3.3.2.3" stretchy="false" xref="S4.SS1.p2.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.2b"><apply id="S4.SS1.p2.3.m3.2.3.cmml" xref="S4.SS1.p2.3.m3.2.3"><times id="S4.SS1.p2.3.m3.2.3.1.cmml" xref="S4.SS1.p2.3.m3.2.3.1"></times><ci id="S4.SS1.p2.3.m3.2.3.2.cmml" xref="S4.SS1.p2.3.m3.2.3.2">𝑁</ci><list id="S4.SS1.p2.3.m3.2.3.3.1.cmml" xref="S4.SS1.p2.3.m3.2.3.3.2"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝒟</ci><ci id="S4.SS1.p2.3.m3.2.2.cmml" xref="S4.SS1.p2.3.m3.2.2">𝐹</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.2c">N(\mathcal{D};F)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.2d">italic_N ( caligraphic_D ; italic_F )</annotation></semantics></math> such that for <math alttext="p,q\geq 0" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.2"><semantics id="S4.SS1.p2.4.m4.2a"><mrow id="S4.SS1.p2.4.m4.2.3" xref="S4.SS1.p2.4.m4.2.3.cmml"><mrow id="S4.SS1.p2.4.m4.2.3.2.2" xref="S4.SS1.p2.4.m4.2.3.2.1.cmml"><mi id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">p</mi><mo id="S4.SS1.p2.4.m4.2.3.2.2.1" xref="S4.SS1.p2.4.m4.2.3.2.1.cmml">,</mo><mi id="S4.SS1.p2.4.m4.2.2" xref="S4.SS1.p2.4.m4.2.2.cmml">q</mi></mrow><mo id="S4.SS1.p2.4.m4.2.3.1" xref="S4.SS1.p2.4.m4.2.3.1.cmml">≥</mo><mn id="S4.SS1.p2.4.m4.2.3.3" xref="S4.SS1.p2.4.m4.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.2b"><apply id="S4.SS1.p2.4.m4.2.3.cmml" xref="S4.SS1.p2.4.m4.2.3"><geq id="S4.SS1.p2.4.m4.2.3.1.cmml" xref="S4.SS1.p2.4.m4.2.3.1"></geq><list id="S4.SS1.p2.4.m4.2.3.2.1.cmml" xref="S4.SS1.p2.4.m4.2.3.2.2"><ci id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">𝑝</ci><ci id="S4.SS1.p2.4.m4.2.2.cmml" xref="S4.SS1.p2.4.m4.2.2">𝑞</ci></list><cn id="S4.SS1.p2.4.m4.2.3.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.2c">p,q\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.2d">italic_p , italic_q ≥ 0</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex45"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="N(\mathcal{D};F)_{p,q}=\coprod_{\sigma=(d_{0}\to\dots\to d_{p})\in N\mathcal{D% }_{p}}F(d_{0})_{q}" class="ltx_Math" display="block" id="S4.Ex45.m1.6"><semantics id="S4.Ex45.m1.6a"><mrow id="S4.Ex45.m1.6.6" xref="S4.Ex45.m1.6.6.cmml"><mrow id="S4.Ex45.m1.6.6.3" xref="S4.Ex45.m1.6.6.3.cmml"><mi id="S4.Ex45.m1.6.6.3.2" xref="S4.Ex45.m1.6.6.3.2.cmml">N</mi><mo id="S4.Ex45.m1.6.6.3.1" xref="S4.Ex45.m1.6.6.3.1.cmml">⁢</mo><msub id="S4.Ex45.m1.6.6.3.3" xref="S4.Ex45.m1.6.6.3.3.cmml"><mrow id="S4.Ex45.m1.6.6.3.3.2.2" xref="S4.Ex45.m1.6.6.3.3.2.1.cmml"><mo id="S4.Ex45.m1.6.6.3.3.2.2.1" stretchy="false" xref="S4.Ex45.m1.6.6.3.3.2.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex45.m1.4.4" xref="S4.Ex45.m1.4.4.cmml">𝒟</mi><mo id="S4.Ex45.m1.6.6.3.3.2.2.2" xref="S4.Ex45.m1.6.6.3.3.2.1.cmml">;</mo><mi id="S4.Ex45.m1.5.5" xref="S4.Ex45.m1.5.5.cmml">F</mi><mo id="S4.Ex45.m1.6.6.3.3.2.2.3" stretchy="false" xref="S4.Ex45.m1.6.6.3.3.2.1.cmml">)</mo></mrow><mrow id="S4.Ex45.m1.2.2.2.4" xref="S4.Ex45.m1.2.2.2.3.cmml"><mi id="S4.Ex45.m1.1.1.1.1" xref="S4.Ex45.m1.1.1.1.1.cmml">p</mi><mo id="S4.Ex45.m1.2.2.2.4.1" xref="S4.Ex45.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex45.m1.2.2.2.2" xref="S4.Ex45.m1.2.2.2.2.cmml">q</mi></mrow></msub></mrow><mo id="S4.Ex45.m1.6.6.2" rspace="0.111em" xref="S4.Ex45.m1.6.6.2.cmml">=</mo><mrow id="S4.Ex45.m1.6.6.1" xref="S4.Ex45.m1.6.6.1.cmml"><munder id="S4.Ex45.m1.6.6.1.2" xref="S4.Ex45.m1.6.6.1.2.cmml"><mo id="S4.Ex45.m1.6.6.1.2.2" movablelimits="false" xref="S4.Ex45.m1.6.6.1.2.2.cmml">∐</mo><mrow id="S4.Ex45.m1.3.3.1" xref="S4.Ex45.m1.3.3.1.cmml"><mi id="S4.Ex45.m1.3.3.1.3" xref="S4.Ex45.m1.3.3.1.3.cmml">σ</mi><mo id="S4.Ex45.m1.3.3.1.4" xref="S4.Ex45.m1.3.3.1.4.cmml">=</mo><mrow id="S4.Ex45.m1.3.3.1.1.1" xref="S4.Ex45.m1.3.3.1.1.1.1.cmml"><mo id="S4.Ex45.m1.3.3.1.1.1.2" stretchy="false" xref="S4.Ex45.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.Ex45.m1.3.3.1.1.1.1" xref="S4.Ex45.m1.3.3.1.1.1.1.cmml"><msub id="S4.Ex45.m1.3.3.1.1.1.1.2" xref="S4.Ex45.m1.3.3.1.1.1.1.2.cmml"><mi id="S4.Ex45.m1.3.3.1.1.1.1.2.2" xref="S4.Ex45.m1.3.3.1.1.1.1.2.2.cmml">d</mi><mn id="S4.Ex45.m1.3.3.1.1.1.1.2.3" xref="S4.Ex45.m1.3.3.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S4.Ex45.m1.3.3.1.1.1.1.3" stretchy="false" xref="S4.Ex45.m1.3.3.1.1.1.1.3.cmml">→</mo><mi id="S4.Ex45.m1.3.3.1.1.1.1.4" mathvariant="normal" xref="S4.Ex45.m1.3.3.1.1.1.1.4.cmml">…</mi><mo id="S4.Ex45.m1.3.3.1.1.1.1.5" stretchy="false" xref="S4.Ex45.m1.3.3.1.1.1.1.5.cmml">→</mo><msub id="S4.Ex45.m1.3.3.1.1.1.1.6" xref="S4.Ex45.m1.3.3.1.1.1.1.6.cmml"><mi id="S4.Ex45.m1.3.3.1.1.1.1.6.2" xref="S4.Ex45.m1.3.3.1.1.1.1.6.2.cmml">d</mi><mi id="S4.Ex45.m1.3.3.1.1.1.1.6.3" xref="S4.Ex45.m1.3.3.1.1.1.1.6.3.cmml">p</mi></msub></mrow><mo id="S4.Ex45.m1.3.3.1.1.1.3" stretchy="false" xref="S4.Ex45.m1.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex45.m1.3.3.1.5" xref="S4.Ex45.m1.3.3.1.5.cmml">∈</mo><mrow id="S4.Ex45.m1.3.3.1.6" xref="S4.Ex45.m1.3.3.1.6.cmml"><mi id="S4.Ex45.m1.3.3.1.6.2" xref="S4.Ex45.m1.3.3.1.6.2.cmml">N</mi><mo id="S4.Ex45.m1.3.3.1.6.1" xref="S4.Ex45.m1.3.3.1.6.1.cmml">⁢</mo><msub id="S4.Ex45.m1.3.3.1.6.3" xref="S4.Ex45.m1.3.3.1.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex45.m1.3.3.1.6.3.2" xref="S4.Ex45.m1.3.3.1.6.3.2.cmml">𝒟</mi><mi id="S4.Ex45.m1.3.3.1.6.3.3" xref="S4.Ex45.m1.3.3.1.6.3.3.cmml">p</mi></msub></mrow></mrow></munder><mrow id="S4.Ex45.m1.6.6.1.1" xref="S4.Ex45.m1.6.6.1.1.cmml"><mi id="S4.Ex45.m1.6.6.1.1.3" xref="S4.Ex45.m1.6.6.1.1.3.cmml">F</mi><mo id="S4.Ex45.m1.6.6.1.1.2" xref="S4.Ex45.m1.6.6.1.1.2.cmml">⁢</mo><msub id="S4.Ex45.m1.6.6.1.1.1" xref="S4.Ex45.m1.6.6.1.1.1.cmml"><mrow id="S4.Ex45.m1.6.6.1.1.1.1.1" xref="S4.Ex45.m1.6.6.1.1.1.1.1.1.cmml"><mo 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cd="latexml" id="S4.SS1.p2.5.m1.4.4.2.2.2.2.2.1.cmml" xref="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.1.cmml" xref="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.1.1.cmml" xref="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.1">superscript</csymbol><ci id="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.1.2.cmml" xref="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.1.2">𝑞</ci><ci id="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.1.3.cmml" xref="S4.SS1.p2.5.m1.4.4.2.2.2.2.1.1.3">′</ci></apply></apply></interval><interval closure="open" id="S4.SS1.p2.5.m1.6.6.4.4.3.cmml" xref="S4.SS1.p2.5.m1.6.6.4.4.2"><apply id="S4.SS1.p2.5.m1.5.5.3.3.1.1.1.cmml" xref="S4.SS1.p2.5.m1.5.5.3.3.1.1.2"><csymbol cd="latexml" id="S4.SS1.p2.5.m1.5.5.3.3.1.1.1.1.cmml" xref="S4.SS1.p2.5.m1.5.5.3.3.1.1.2.1">delimited-[]</csymbol><ci id="S4.SS1.p2.5.m1.1.1.cmml" xref="S4.SS1.p2.5.m1.1.1">𝑝</ci></apply><apply id="S4.SS1.p2.5.m1.6.6.4.4.2.2.1.cmml" xref="S4.SS1.p2.5.m1.6.6.4.4.2.2.2"><csymbol cd="latexml" id="S4.SS1.p2.5.m1.6.6.4.4.2.2.1.1.cmml" xref="S4.SS1.p2.5.m1.6.6.4.4.2.2.2.1">delimited-[]</csymbol><ci id="S4.SS1.p2.5.m1.2.2.cmml" xref="S4.SS1.p2.5.m1.2.2">𝑞</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m1.6c">f_{h}\times f_{v}:([p^{\prime}],[q^{\prime}])\to([p],[q])</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m1.6d">italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT : ( [ italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] , [ italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] ) → ( [ italic_p ] , [ italic_q ] )</annotation></semantics></math> is a morphism in <math alttext="\Delta\times\Delta" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m2.1"><semantics id="S4.SS1.p2.6.m2.1a"><mrow id="S4.SS1.p2.6.m2.1.1" xref="S4.SS1.p2.6.m2.1.1.cmml"><mi id="S4.SS1.p2.6.m2.1.1.2" mathvariant="normal" xref="S4.SS1.p2.6.m2.1.1.2.cmml">Δ</mi><mo id="S4.SS1.p2.6.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.p2.6.m2.1.1.1.cmml">×</mo><mi id="S4.SS1.p2.6.m2.1.1.3" mathvariant="normal" xref="S4.SS1.p2.6.m2.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m2.1b"><apply id="S4.SS1.p2.6.m2.1.1.cmml" xref="S4.SS1.p2.6.m2.1.1"><times id="S4.SS1.p2.6.m2.1.1.1.cmml" xref="S4.SS1.p2.6.m2.1.1.1"></times><ci id="S4.SS1.p2.6.m2.1.1.2.cmml" xref="S4.SS1.p2.6.m2.1.1.2">Δ</ci><ci id="S4.SS1.p2.6.m2.1.1.3.cmml" xref="S4.SS1.p2.6.m2.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m2.1c">\Delta\times\Delta</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m2.1d">roman_Δ × roman_Δ</annotation></semantics></math>, then for every <math alttext="(\sigma,\tau)" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m3.2"><semantics id="S4.SS1.p2.7.m3.2a"><mrow id="S4.SS1.p2.7.m3.2.3.2" xref="S4.SS1.p2.7.m3.2.3.1.cmml"><mo id="S4.SS1.p2.7.m3.2.3.2.1" stretchy="false" xref="S4.SS1.p2.7.m3.2.3.1.cmml">(</mo><mi id="S4.SS1.p2.7.m3.1.1" xref="S4.SS1.p2.7.m3.1.1.cmml">σ</mi><mo id="S4.SS1.p2.7.m3.2.3.2.2" xref="S4.SS1.p2.7.m3.2.3.1.cmml">,</mo><mi id="S4.SS1.p2.7.m3.2.2" xref="S4.SS1.p2.7.m3.2.2.cmml">τ</mi><mo id="S4.SS1.p2.7.m3.2.3.2.3" stretchy="false" xref="S4.SS1.p2.7.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m3.2b"><interval closure="open" id="S4.SS1.p2.7.m3.2.3.1.cmml" xref="S4.SS1.p2.7.m3.2.3.2"><ci id="S4.SS1.p2.7.m3.1.1.cmml" xref="S4.SS1.p2.7.m3.1.1">𝜎</ci><ci id="S4.SS1.p2.7.m3.2.2.cmml" xref="S4.SS1.p2.7.m3.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m3.2c">(\sigma,\tau)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m3.2d">( italic_σ , italic_τ )</annotation></semantics></math> with</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex46"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma=(d_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}d_{1}\to% \cdots\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{p}}\,}d_{p})\in N% \mathcal{D}_{p}\quad\text{ and }\quad\tau=(x_{0}\smash{\,\mathop{% \longrightarrow}\limits^{\omega_{1}}\,}x_{1}\to\cdots\smash{\,\mathop{% \longrightarrow}\limits^{\omega_{q}}\,}x_{q})\in F(d_{0})_{q}," class="ltx_Math" display="block" id="S4.Ex46.m1.3"><semantics id="S4.Ex46.m1.3a"><mrow id="S4.Ex46.m1.3.3.1"><mrow id="S4.Ex46.m1.3.3.1.1.2" xref="S4.Ex46.m1.3.3.1.1.3.cmml"><mrow id="S4.Ex46.m1.3.3.1.1.1.1" xref="S4.Ex46.m1.3.3.1.1.1.1.cmml"><mi id="S4.Ex46.m1.3.3.1.1.1.1.3" xref="S4.Ex46.m1.3.3.1.1.1.1.3.cmml">σ</mi><mo id="S4.Ex46.m1.3.3.1.1.1.1.4" xref="S4.Ex46.m1.3.3.1.1.1.1.4.cmml">=</mo><mrow id="S4.Ex46.m1.3.3.1.1.1.1.1.1" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex46.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.cmml"><msub id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.2" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.2.cmml"><mi id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.2.2" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.2.2.cmml">d</mi><mn id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.2.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.2.3.cmml">0</mn></msub><mo id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.1" lspace="0.167em" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.cmml"><mover id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.cmml"><mo id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.2.cmml">⟶</mo><msub id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.3.cmml"><mi id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.3.2" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.3.2.cmml">α</mi><mn id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.3.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.1.3.3.cmml">1</mn></msub></mover><msub id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.2" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.2.cmml"><mi id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.2.2" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.2.2.cmml">d</mi><mn id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.2.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.2.3.2.3.cmml">1</mn></msub></mrow></mrow><mo id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.1" stretchy="false" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.1.cmml">→</mo><mrow id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.2.cmml">⋯</mi><mo id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.1" lspace="0.337em" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.cmml"><mover id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.cmml"><mo id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.2" movablelimits="false" rspace="0.167em" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.2.cmml">⟶</mo><msub id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.3.cmml"><mi id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.3.2" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.3.2.cmml">α</mi><mi id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.3.3" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.1.3.3.cmml">p</mi></msub></mover><msub id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.2" xref="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.2.cmml"><mi id="S4.Ex46.m1.3.3.1.1.1.1.1.1.1.3.3.2.2" 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href="https://arxiv.org/html/2503.14659v1#S4.Ex46.m1.3.3.1.1.2.2.1.cmml" id="S4.Ex46.m1.3.3.1.1.2.2d.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2"></share><apply id="S4.Ex46.m1.3.3.1.1.2.2.2.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2"><times id="S4.Ex46.m1.3.3.1.1.2.2.2.2.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2.2"></times><ci id="S4.Ex46.m1.3.3.1.1.2.2.2.3.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2.3">𝐹</ci><apply id="S4.Ex46.m1.3.3.1.1.2.2.2.1.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2.1"><csymbol cd="ambiguous" id="S4.Ex46.m1.3.3.1.1.2.2.2.1.2.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2.1">subscript</csymbol><apply id="S4.Ex46.m1.3.3.1.1.2.2.2.1.1.1.1.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Ex46.m1.3.3.1.1.2.2.2.1.1.1.1.1.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2.1.1.1">subscript</csymbol><ci id="S4.Ex46.m1.3.3.1.1.2.2.2.1.1.1.1.2.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2.1.1.1.1.2">𝑑</ci><cn id="S4.Ex46.m1.3.3.1.1.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.Ex46.m1.3.3.1.1.2.2.2.1.1.1.1.3">0</cn></apply><ci id="S4.Ex46.m1.3.3.1.1.2.2.2.1.3.cmml" xref="S4.Ex46.m1.3.3.1.1.2.2.2.1.3">𝑞</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex46.m1.3c">\sigma=(d_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}d_{1}\to% \cdots\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{p}}\,}d_{p})\in N% \mathcal{D}_{p}\quad\text{ and }\quad\tau=(x_{0}\smash{\,\mathop{% \longrightarrow}\limits^{\omega_{1}}\,}x_{1}\to\cdots\smash{\,\mathop{% \longrightarrow}\limits^{\omega_{q}}\,}x_{q})\in F(d_{0})_{q},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex46.m1.3d">italic_σ = ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ∈ italic_N caligraphic_D start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and italic_τ = ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → ⋯ ⟶ start_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ∈ italic_F ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.9">we define</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex47"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math 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xref="S4.Ex47.m1.6.6.1.1.4.2.2.2.1.1.2">subscript</csymbol><ci id="S4.Ex47.m1.6.6.1.1.4.2.2.2.1.1.2.2.2.cmml" xref="S4.Ex47.m1.6.6.1.1.4.2.2.2.1.1.2.2.2">𝑓</ci><ci id="S4.Ex47.m1.6.6.1.1.4.2.2.2.1.1.2.2.3.cmml" xref="S4.Ex47.m1.6.6.1.1.4.2.2.2.1.1.2.2.3">𝑣</ci></apply><times id="S4.Ex47.m1.6.6.1.1.4.2.2.2.1.1.2.3.cmml" xref="S4.Ex47.m1.6.6.1.1.4.2.2.2.1.1.2.3"></times></apply><ci id="S4.Ex47.m1.5.5.cmml" xref="S4.Ex47.m1.5.5">𝜏</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex47.m1.6c">(f_{h},f_{v})^{*}(\sigma,\tau)=(f_{h}^{*}(\sigma),F(\alpha_{f_{h}(0)}\cdots% \alpha_{1})(f_{v}^{*}(\tau))).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex47.m1.6d">( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ , italic_τ ) = ( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ ) , italic_F ( italic_α start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ ) ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.8">This definition respects the composition of functions in <math alttext="\Delta\times\Delta" class="ltx_Math" display="inline" id="S4.SS1.p2.8.m1.1"><semantics id="S4.SS1.p2.8.m1.1a"><mrow id="S4.SS1.p2.8.m1.1.1" xref="S4.SS1.p2.8.m1.1.1.cmml"><mi id="S4.SS1.p2.8.m1.1.1.2" mathvariant="normal" xref="S4.SS1.p2.8.m1.1.1.2.cmml">Δ</mi><mo id="S4.SS1.p2.8.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.p2.8.m1.1.1.1.cmml">×</mo><mi id="S4.SS1.p2.8.m1.1.1.3" mathvariant="normal" xref="S4.SS1.p2.8.m1.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.8.m1.1b"><apply id="S4.SS1.p2.8.m1.1.1.cmml" xref="S4.SS1.p2.8.m1.1.1"><times id="S4.SS1.p2.8.m1.1.1.1.cmml" xref="S4.SS1.p2.8.m1.1.1.1"></times><ci id="S4.SS1.p2.8.m1.1.1.2.cmml" xref="S4.SS1.p2.8.m1.1.1.2">Δ</ci><ci id="S4.SS1.p2.8.m1.1.1.3.cmml" xref="S4.SS1.p2.8.m1.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.8.m1.1c">\Delta\times\Delta</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.8.m1.1d">roman_Δ × roman_Δ</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Definition 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.6">Let <math alttext="F:\mathcal{D}\to sSet" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.m1.1"><semantics id="S4.Thmtheorem1.p1.1.m1.1a"><mrow id="S4.Thmtheorem1.p1.1.m1.1.1" xref="S4.Thmtheorem1.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.1.m1.1.1.2" xref="S4.Thmtheorem1.p1.1.m1.1.1.2.cmml">F</mi><mo id="S4.Thmtheorem1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S4.Thmtheorem1.p1.1.m1.1.1.3" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem1.p1.1.m1.1.1.3.2" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.2.cmml">𝒟</mi><mo id="S4.Thmtheorem1.p1.1.m1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.1.cmml">→</mo><mrow id="S4.Thmtheorem1.p1.1.m1.1.1.3.3" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.cmml"><mi id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.2" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.2.cmml">s</mi><mo id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.1" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.3" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.3.cmml">S</mi><mo id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.1a" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.4" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.4.cmml">e</mi><mo id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.1b" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.5" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.5.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.m1.1b"><apply id="S4.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1"><ci id="S4.Thmtheorem1.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.1">:</ci><ci id="S4.Thmtheorem1.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.2">𝐹</ci><apply id="S4.Thmtheorem1.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3"><ci id="S4.Thmtheorem1.p1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.1">→</ci><ci id="S4.Thmtheorem1.p1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.2">𝒟</ci><apply id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3"><times id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.1.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.1"></times><ci id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.2.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.2">𝑠</ci><ci id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.3.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.3">𝑆</ci><ci id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.4.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.4">𝑒</ci><ci id="S4.Thmtheorem1.p1.1.m1.1.1.3.3.5.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1.3.3.5">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.m1.1c">F:\mathcal{D}\to sSet</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.m1.1d">italic_F : caligraphic_D → italic_s italic_S italic_e italic_t</annotation></semantics></math> be a functor. The bisimplicial set <math alttext="N(\mathcal{D};F)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.m2.2"><semantics id="S4.Thmtheorem1.p1.2.m2.2a"><mrow id="S4.Thmtheorem1.p1.2.m2.2.3" xref="S4.Thmtheorem1.p1.2.m2.2.3.cmml"><mi id="S4.Thmtheorem1.p1.2.m2.2.3.2" xref="S4.Thmtheorem1.p1.2.m2.2.3.2.cmml">N</mi><mo id="S4.Thmtheorem1.p1.2.m2.2.3.1" xref="S4.Thmtheorem1.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.2.m2.2.3.3.2" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.1.cmml"><mo id="S4.Thmtheorem1.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem1.p1.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.m2.1.1.cmml">𝒟</mi><mo id="S4.Thmtheorem1.p1.2.m2.2.3.3.2.2" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="S4.Thmtheorem1.p1.2.m2.2.2" xref="S4.Thmtheorem1.p1.2.m2.2.2.cmml">F</mi><mo id="S4.Thmtheorem1.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.m2.2b"><apply id="S4.Thmtheorem1.p1.2.m2.2.3.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3"><times id="S4.Thmtheorem1.p1.2.m2.2.3.1.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.1"></times><ci id="S4.Thmtheorem1.p1.2.m2.2.3.2.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.2">𝑁</ci><list id="S4.Thmtheorem1.p1.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.2"><ci id="S4.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.2.m2.1.1">𝒟</ci><ci id="S4.Thmtheorem1.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.2">𝐹</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.m2.2c">N(\mathcal{D};F)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.m2.2d">italic_N ( caligraphic_D ; italic_F )</annotation></semantics></math> is called the <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem1.p1.6.1">simplicial replacement</em> of <math alttext="F" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.m3.1"><semantics id="S4.Thmtheorem1.p1.3.m3.1a"><mi id="S4.Thmtheorem1.p1.3.m3.1.1" xref="S4.Thmtheorem1.p1.3.m3.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.m3.1b"><ci id="S4.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.m3.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.m3.1d">italic_F</annotation></semantics></math>. We define the <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem1.p1.6.2">homotopy colimit</em> <math alttext="\operatorname*{hocolim}_{\mathcal{D}}F" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.m4.1"><semantics id="S4.Thmtheorem1.p1.4.m4.1a"><mrow id="S4.Thmtheorem1.p1.4.m4.1.1" xref="S4.Thmtheorem1.p1.4.m4.1.1.cmml"><msub id="S4.Thmtheorem1.p1.4.m4.1.1.1" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.cmml"><mo id="S4.Thmtheorem1.p1.4.m4.1.1.1.2" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem1.p1.4.m4.1.1.1.3" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.3.cmml">𝒟</mi></msub><mi id="S4.Thmtheorem1.p1.4.m4.1.1.2" xref="S4.Thmtheorem1.p1.4.m4.1.1.2.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.m4.1b"><apply id="S4.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1"><apply id="S4.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem1.p1.4.m4.1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.2">hocolim</ci><ci id="S4.Thmtheorem1.p1.4.m4.1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.3">𝒟</ci></apply><ci id="S4.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1.2">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.m4.1c">\operatorname*{hocolim}_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.m4.1d">roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> to be the diagonal <math alttext="\mathrm{diag}N(\mathcal{D};F)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.5.m5.2"><semantics id="S4.Thmtheorem1.p1.5.m5.2a"><mrow id="S4.Thmtheorem1.p1.5.m5.2.3" xref="S4.Thmtheorem1.p1.5.m5.2.3.cmml"><mi id="S4.Thmtheorem1.p1.5.m5.2.3.2" xref="S4.Thmtheorem1.p1.5.m5.2.3.2.cmml">diag</mi><mo id="S4.Thmtheorem1.p1.5.m5.2.3.1" xref="S4.Thmtheorem1.p1.5.m5.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem1.p1.5.m5.2.3.3" xref="S4.Thmtheorem1.p1.5.m5.2.3.3.cmml">N</mi><mo id="S4.Thmtheorem1.p1.5.m5.2.3.1a" xref="S4.Thmtheorem1.p1.5.m5.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.5.m5.2.3.4.2" xref="S4.Thmtheorem1.p1.5.m5.2.3.4.1.cmml"><mo id="S4.Thmtheorem1.p1.5.m5.2.3.4.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.5.m5.2.3.4.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem1.p1.5.m5.1.1" xref="S4.Thmtheorem1.p1.5.m5.1.1.cmml">𝒟</mi><mo id="S4.Thmtheorem1.p1.5.m5.2.3.4.2.2" xref="S4.Thmtheorem1.p1.5.m5.2.3.4.1.cmml">;</mo><mi id="S4.Thmtheorem1.p1.5.m5.2.2" xref="S4.Thmtheorem1.p1.5.m5.2.2.cmml">F</mi><mo id="S4.Thmtheorem1.p1.5.m5.2.3.4.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.5.m5.2.3.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.5.m5.2b"><apply id="S4.Thmtheorem1.p1.5.m5.2.3.cmml" xref="S4.Thmtheorem1.p1.5.m5.2.3"><times id="S4.Thmtheorem1.p1.5.m5.2.3.1.cmml" xref="S4.Thmtheorem1.p1.5.m5.2.3.1"></times><ci id="S4.Thmtheorem1.p1.5.m5.2.3.2.cmml" xref="S4.Thmtheorem1.p1.5.m5.2.3.2">diag</ci><ci id="S4.Thmtheorem1.p1.5.m5.2.3.3.cmml" xref="S4.Thmtheorem1.p1.5.m5.2.3.3">𝑁</ci><list id="S4.Thmtheorem1.p1.5.m5.2.3.4.1.cmml" xref="S4.Thmtheorem1.p1.5.m5.2.3.4.2"><ci id="S4.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem1.p1.5.m5.1.1">𝒟</ci><ci id="S4.Thmtheorem1.p1.5.m5.2.2.cmml" xref="S4.Thmtheorem1.p1.5.m5.2.2">𝐹</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.5.m5.2c">\mathrm{diag}N(\mathcal{D};F)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.5.m5.2d">roman_diag italic_N ( caligraphic_D ; italic_F )</annotation></semantics></math> of the simplicial replacement of <math alttext="F" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.6.m6.1"><semantics id="S4.Thmtheorem1.p1.6.m6.1a"><mi id="S4.Thmtheorem1.p1.6.m6.1.1" xref="S4.Thmtheorem1.p1.6.m6.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.6.m6.1b"><ci id="S4.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem1.p1.6.m6.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.6.m6.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.6.m6.1d">italic_F</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib2" title="">2</a>, §12.5.2]</cite>).</p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2. </span>The Dold-Puppe Theorem</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.4">A simplicial object in <math alttext="R" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_R</annotation></semantics></math>-modules is called a <em class="ltx_emph ltx_font_italic" id="S4.SS2.p1.2.1">simplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.SS2.p1.2.1.m1.1"><semantics id="S4.SS2.p1.2.1.m1.1a"><mi id="S4.SS2.p1.2.1.m1.1.1" xref="S4.SS2.p1.2.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.1.m1.1b"><ci id="S4.SS2.p1.2.1.m1.1.1.cmml" xref="S4.SS2.p1.2.1.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.1.m1.1d">italic_R</annotation></semantics></math>-module</em>. A functor <math alttext="A:\Delta^{op}\times\Delta^{op}\to R" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m2.1"><semantics id="S4.SS2.p1.3.m2.1a"><mrow id="S4.SS2.p1.3.m2.1.1" xref="S4.SS2.p1.3.m2.1.1.cmml"><mi id="S4.SS2.p1.3.m2.1.1.2" xref="S4.SS2.p1.3.m2.1.1.2.cmml">A</mi><mo id="S4.SS2.p1.3.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.p1.3.m2.1.1.1.cmml">:</mo><mrow id="S4.SS2.p1.3.m2.1.1.3" xref="S4.SS2.p1.3.m2.1.1.3.cmml"><mrow id="S4.SS2.p1.3.m2.1.1.3.2" xref="S4.SS2.p1.3.m2.1.1.3.2.cmml"><msup id="S4.SS2.p1.3.m2.1.1.3.2.2" xref="S4.SS2.p1.3.m2.1.1.3.2.2.cmml"><mi id="S4.SS2.p1.3.m2.1.1.3.2.2.2" mathvariant="normal" xref="S4.SS2.p1.3.m2.1.1.3.2.2.2.cmml">Δ</mi><mrow id="S4.SS2.p1.3.m2.1.1.3.2.2.3" xref="S4.SS2.p1.3.m2.1.1.3.2.2.3.cmml"><mi id="S4.SS2.p1.3.m2.1.1.3.2.2.3.2" xref="S4.SS2.p1.3.m2.1.1.3.2.2.3.2.cmml">o</mi><mo id="S4.SS2.p1.3.m2.1.1.3.2.2.3.1" xref="S4.SS2.p1.3.m2.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.3.m2.1.1.3.2.2.3.3" xref="S4.SS2.p1.3.m2.1.1.3.2.2.3.3.cmml">p</mi></mrow></msup><mo id="S4.SS2.p1.3.m2.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.p1.3.m2.1.1.3.2.1.cmml">×</mo><msup id="S4.SS2.p1.3.m2.1.1.3.2.3" xref="S4.SS2.p1.3.m2.1.1.3.2.3.cmml"><mi id="S4.SS2.p1.3.m2.1.1.3.2.3.2" mathvariant="normal" xref="S4.SS2.p1.3.m2.1.1.3.2.3.2.cmml">Δ</mi><mrow id="S4.SS2.p1.3.m2.1.1.3.2.3.3" xref="S4.SS2.p1.3.m2.1.1.3.2.3.3.cmml"><mi id="S4.SS2.p1.3.m2.1.1.3.2.3.3.2" xref="S4.SS2.p1.3.m2.1.1.3.2.3.3.2.cmml">o</mi><mo id="S4.SS2.p1.3.m2.1.1.3.2.3.3.1" xref="S4.SS2.p1.3.m2.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.3.m2.1.1.3.2.3.3.3" xref="S4.SS2.p1.3.m2.1.1.3.2.3.3.3.cmml">p</mi></mrow></msup></mrow><mo id="S4.SS2.p1.3.m2.1.1.3.1" stretchy="false" xref="S4.SS2.p1.3.m2.1.1.3.1.cmml">→</mo><mi id="S4.SS2.p1.3.m2.1.1.3.3" xref="S4.SS2.p1.3.m2.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m2.1b"><apply id="S4.SS2.p1.3.m2.1.1.cmml" xref="S4.SS2.p1.3.m2.1.1"><ci id="S4.SS2.p1.3.m2.1.1.1.cmml" xref="S4.SS2.p1.3.m2.1.1.1">:</ci><ci id="S4.SS2.p1.3.m2.1.1.2.cmml" xref="S4.SS2.p1.3.m2.1.1.2">𝐴</ci><apply id="S4.SS2.p1.3.m2.1.1.3.cmml" xref="S4.SS2.p1.3.m2.1.1.3"><ci id="S4.SS2.p1.3.m2.1.1.3.1.cmml" xref="S4.SS2.p1.3.m2.1.1.3.1">→</ci><apply id="S4.SS2.p1.3.m2.1.1.3.2.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2"><times id="S4.SS2.p1.3.m2.1.1.3.2.1.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.1"></times><apply id="S4.SS2.p1.3.m2.1.1.3.2.2.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m2.1.1.3.2.2.1.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.2">superscript</csymbol><ci id="S4.SS2.p1.3.m2.1.1.3.2.2.2.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.2.2">Δ</ci><apply id="S4.SS2.p1.3.m2.1.1.3.2.2.3.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.2.3"><times id="S4.SS2.p1.3.m2.1.1.3.2.2.3.1.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.2.3.1"></times><ci id="S4.SS2.p1.3.m2.1.1.3.2.2.3.2.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.2.3.2">𝑜</ci><ci id="S4.SS2.p1.3.m2.1.1.3.2.2.3.3.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.2.3.3">𝑝</ci></apply></apply><apply id="S4.SS2.p1.3.m2.1.1.3.2.3.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m2.1.1.3.2.3.1.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.3">superscript</csymbol><ci id="S4.SS2.p1.3.m2.1.1.3.2.3.2.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.3.2">Δ</ci><apply id="S4.SS2.p1.3.m2.1.1.3.2.3.3.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.3.3"><times id="S4.SS2.p1.3.m2.1.1.3.2.3.3.1.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.3.3.1"></times><ci id="S4.SS2.p1.3.m2.1.1.3.2.3.3.2.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.3.3.2">𝑜</ci><ci id="S4.SS2.p1.3.m2.1.1.3.2.3.3.3.cmml" xref="S4.SS2.p1.3.m2.1.1.3.2.3.3.3">𝑝</ci></apply></apply></apply><ci id="S4.SS2.p1.3.m2.1.1.3.3.cmml" xref="S4.SS2.p1.3.m2.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m2.1c">A:\Delta^{op}\times\Delta^{op}\to R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m2.1d">italic_A : roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT × roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → italic_R</annotation></semantics></math>-Mod is called a <em class="ltx_emph ltx_font_italic" id="S4.SS2.p1.4.2">bisimplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.SS2.p1.4.2.m1.1"><semantics id="S4.SS2.p1.4.2.m1.1a"><mi id="S4.SS2.p1.4.2.m1.1.1" xref="S4.SS2.p1.4.2.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.2.m1.1b"><ci id="S4.SS2.p1.4.2.m1.1.1.cmml" xref="S4.SS2.p1.4.2.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.2.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.2.m1.1d">italic_R</annotation></semantics></math>-module</em>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.1.1.1">Definition 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.9">The <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem2.p1.9.1">Moore complex</em> of a simplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.m1.1"><semantics id="S4.Thmtheorem2.p1.1.m1.1a"><mi id="S4.Thmtheorem2.p1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.m1.1b"><ci id="S4.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.m1.1d">italic_R</annotation></semantics></math>-module <math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.m2.1a"><mi id="S4.Thmtheorem2.p1.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.m2.1b"><ci id="S4.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.m2.1d">italic_A</annotation></semantics></math> is the chain complex whose <math alttext="n" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.m3.1"><semantics id="S4.Thmtheorem2.p1.3.m3.1a"><mi id="S4.Thmtheorem2.p1.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.m3.1b"><ci id="S4.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.m3.1d">italic_n</annotation></semantics></math>-chains are given by <math alttext="A_{n}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.m4.1"><semantics id="S4.Thmtheorem2.p1.4.m4.1a"><msub id="S4.Thmtheorem2.p1.4.m4.1.1" xref="S4.Thmtheorem2.p1.4.m4.1.1.cmml"><mi id="S4.Thmtheorem2.p1.4.m4.1.1.2" xref="S4.Thmtheorem2.p1.4.m4.1.1.2.cmml">A</mi><mi id="S4.Thmtheorem2.p1.4.m4.1.1.3" xref="S4.Thmtheorem2.p1.4.m4.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.4.m4.1b"><apply id="S4.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.4.m4.1.1.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.4.m4.1.1.2.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.2">𝐴</ci><ci id="S4.Thmtheorem2.p1.4.m4.1.1.3.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.m4.1c">A_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.m4.1d">italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and whose boundary maps <math alttext="\partial_{n}:A_{n}\to A_{n-1}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.5.m5.1"><semantics id="S4.Thmtheorem2.p1.5.m5.1a"><mrow id="S4.Thmtheorem2.p1.5.m5.1.1" xref="S4.Thmtheorem2.p1.5.m5.1.1.cmml"><msub id="S4.Thmtheorem2.p1.5.m5.1.1.2" xref="S4.Thmtheorem2.p1.5.m5.1.1.2.cmml"><mo id="S4.Thmtheorem2.p1.5.m5.1.1.2.2" xref="S4.Thmtheorem2.p1.5.m5.1.1.2.2.cmml">∂</mo><mi id="S4.Thmtheorem2.p1.5.m5.1.1.2.3" xref="S4.Thmtheorem2.p1.5.m5.1.1.2.3.cmml">n</mi></msub><mo id="S4.Thmtheorem2.p1.5.m5.1.1.1" rspace="0.278em" xref="S4.Thmtheorem2.p1.5.m5.1.1.1.cmml">:</mo><mrow id="S4.Thmtheorem2.p1.5.m5.1.1.3" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.cmml"><msub id="S4.Thmtheorem2.p1.5.m5.1.1.3.2" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.2.cmml"><mi id="S4.Thmtheorem2.p1.5.m5.1.1.3.2.2" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.2.2.cmml">A</mi><mi id="S4.Thmtheorem2.p1.5.m5.1.1.3.2.3" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.2.3.cmml">n</mi></msub><mo id="S4.Thmtheorem2.p1.5.m5.1.1.3.1" stretchy="false" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.1.cmml">→</mo><msub id="S4.Thmtheorem2.p1.5.m5.1.1.3.3" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.cmml"><mi id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.2" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.2.cmml">A</mi><mrow id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.cmml"><mi id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.2" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.2.cmml">n</mi><mo id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.1" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.1.cmml">−</mo><mn id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.3" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.5.m5.1b"><apply id="S4.Thmtheorem2.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1"><ci id="S4.Thmtheorem2.p1.5.m5.1.1.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.1">:</ci><apply id="S4.Thmtheorem2.p1.5.m5.1.1.2.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.m5.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.2">subscript</csymbol><partialdiff id="S4.Thmtheorem2.p1.5.m5.1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.2.2"></partialdiff><ci id="S4.Thmtheorem2.p1.5.m5.1.1.2.3.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.2.3">𝑛</ci></apply><apply id="S4.Thmtheorem2.p1.5.m5.1.1.3.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3"><ci id="S4.Thmtheorem2.p1.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.1">→</ci><apply id="S4.Thmtheorem2.p1.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.m5.1.1.3.2.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.5.m5.1.1.3.2.2.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.2.2">𝐴</ci><ci id="S4.Thmtheorem2.p1.5.m5.1.1.3.2.3.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.2.3">𝑛</ci></apply><apply id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.2.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.2">𝐴</ci><apply id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3"><minus id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.1"></minus><ci id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.2.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.2">𝑛</ci><cn id="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.5.m5.1c">\partial_{n}:A_{n}\to A_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.5.m5.1d">∂ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> are defined by <math alttext="\partial_{n}(x)=\sum_{i=0}^{n}(-1)^{i}d_{i}(x)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.6.m6.3"><semantics id="S4.Thmtheorem2.p1.6.m6.3a"><mrow id="S4.Thmtheorem2.p1.6.m6.3.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.cmml"><mrow id="S4.Thmtheorem2.p1.6.m6.3.3.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.cmml"><msub id="S4.Thmtheorem2.p1.6.m6.3.3.3.1" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.1.cmml"><mo id="S4.Thmtheorem2.p1.6.m6.3.3.3.1.2" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.1.2.cmml">∂</mo><mi id="S4.Thmtheorem2.p1.6.m6.3.3.3.1.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.1.3.cmml">n</mi></msub><mrow id="S4.Thmtheorem2.p1.6.m6.3.3.3.2.2" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.cmml"><mo id="S4.Thmtheorem2.p1.6.m6.3.3.3.2.2.1" lspace="0em" stretchy="false" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.cmml">(</mo><mi id="S4.Thmtheorem2.p1.6.m6.1.1" xref="S4.Thmtheorem2.p1.6.m6.1.1.cmml">x</mi><mo id="S4.Thmtheorem2.p1.6.m6.3.3.3.2.2.2" stretchy="false" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p1.6.m6.3.3.2" rspace="0.111em" xref="S4.Thmtheorem2.p1.6.m6.3.3.2.cmml">=</mo><mrow id="S4.Thmtheorem2.p1.6.m6.3.3.1" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.cmml"><msubsup id="S4.Thmtheorem2.p1.6.m6.3.3.1.2" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.cmml"><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.2" rspace="0em" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.2.cmml">∑</mo><mrow id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.cmml"><mi id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.2" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.2.cmml">i</mi><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.1" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.1.cmml">=</mo><mn id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.3.cmml">0</mn></mrow><mi id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.3.cmml">n</mi></msubsup><mrow id="S4.Thmtheorem2.p1.6.m6.3.3.1.1" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.cmml"><msup id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.cmml"><mrow id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1a" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.2" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.3.cmml">i</mi></msup><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.2" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.2.cmml">⁢</mo><msub id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.cmml"><mi id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.2" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.2.cmml">d</mi><mi id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.3" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.3.cmml">i</mi></msub><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.2a" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.4.2" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.cmml"><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.4.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.cmml">(</mo><mi id="S4.Thmtheorem2.p1.6.m6.2.2" xref="S4.Thmtheorem2.p1.6.m6.2.2.cmml">x</mi><mo id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.4.2.2" stretchy="false" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.6.m6.3b"><apply id="S4.Thmtheorem2.p1.6.m6.3.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3"><eq id="S4.Thmtheorem2.p1.6.m6.3.3.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.2"></eq><apply id="S4.Thmtheorem2.p1.6.m6.3.3.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.3"><apply id="S4.Thmtheorem2.p1.6.m6.3.3.3.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.m6.3.3.3.1.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.1">subscript</csymbol><partialdiff id="S4.Thmtheorem2.p1.6.m6.3.3.3.1.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.1.2"></partialdiff><ci id="S4.Thmtheorem2.p1.6.m6.3.3.3.1.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.3.1.3">𝑛</ci></apply><ci id="S4.Thmtheorem2.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.1.1">𝑥</ci></apply><apply id="S4.Thmtheorem2.p1.6.m6.3.3.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1"><apply id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2">superscript</csymbol><apply id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2">subscript</csymbol><sum id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.2"></sum><apply id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3"><eq id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.1"></eq><ci id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.2">𝑖</ci><cn id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.2.3.3">0</cn></apply></apply><ci id="S4.Thmtheorem2.p1.6.m6.3.3.1.2.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.2.3">𝑛</ci></apply><apply id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1"><times id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.2"></times><apply id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1">superscript</csymbol><apply id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1"><minus id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1"></minus><cn id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.1.1.1.2">1</cn></apply><ci id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.1.3">𝑖</ci></apply><apply id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.2">𝑑</ci><ci id="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.3.3.1.1.3.3">𝑖</ci></apply><ci id="S4.Thmtheorem2.p1.6.m6.2.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.2.2">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.6.m6.3c">\partial_{n}(x)=\sum_{i=0}^{n}(-1)^{i}d_{i}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.6.m6.3d">∂ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>. The Moore complex of <math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.7.m7.1"><semantics id="S4.Thmtheorem2.p1.7.m7.1a"><mi id="S4.Thmtheorem2.p1.7.m7.1.1" xref="S4.Thmtheorem2.p1.7.m7.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.7.m7.1b"><ci id="S4.Thmtheorem2.p1.7.m7.1.1.cmml" xref="S4.Thmtheorem2.p1.7.m7.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.7.m7.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.7.m7.1d">italic_A</annotation></semantics></math>, denoted by <math alttext="A_{*}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.8.m8.1"><semantics id="S4.Thmtheorem2.p1.8.m8.1a"><msub id="S4.Thmtheorem2.p1.8.m8.1.1" xref="S4.Thmtheorem2.p1.8.m8.1.1.cmml"><mi id="S4.Thmtheorem2.p1.8.m8.1.1.2" xref="S4.Thmtheorem2.p1.8.m8.1.1.2.cmml">A</mi><mo id="S4.Thmtheorem2.p1.8.m8.1.1.3" xref="S4.Thmtheorem2.p1.8.m8.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.8.m8.1b"><apply id="S4.Thmtheorem2.p1.8.m8.1.1.cmml" xref="S4.Thmtheorem2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.8.m8.1.1.1.cmml" xref="S4.Thmtheorem2.p1.8.m8.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.8.m8.1.1.2.cmml" xref="S4.Thmtheorem2.p1.8.m8.1.1.2">𝐴</ci><times id="S4.Thmtheorem2.p1.8.m8.1.1.3.cmml" xref="S4.Thmtheorem2.p1.8.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.8.m8.1c">A_{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.8.m8.1d">italic_A start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>, is a chain complex and its homology is denoted by <math alttext="H_{*}(A)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.9.m9.1"><semantics id="S4.Thmtheorem2.p1.9.m9.1a"><mrow id="S4.Thmtheorem2.p1.9.m9.1.2" xref="S4.Thmtheorem2.p1.9.m9.1.2.cmml"><msub id="S4.Thmtheorem2.p1.9.m9.1.2.2" xref="S4.Thmtheorem2.p1.9.m9.1.2.2.cmml"><mi id="S4.Thmtheorem2.p1.9.m9.1.2.2.2" xref="S4.Thmtheorem2.p1.9.m9.1.2.2.2.cmml">H</mi><mo id="S4.Thmtheorem2.p1.9.m9.1.2.2.3" xref="S4.Thmtheorem2.p1.9.m9.1.2.2.3.cmml">∗</mo></msub><mo id="S4.Thmtheorem2.p1.9.m9.1.2.1" xref="S4.Thmtheorem2.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.9.m9.1.2.3.2" xref="S4.Thmtheorem2.p1.9.m9.1.2.cmml"><mo id="S4.Thmtheorem2.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.9.m9.1.2.cmml">(</mo><mi id="S4.Thmtheorem2.p1.9.m9.1.1" xref="S4.Thmtheorem2.p1.9.m9.1.1.cmml">A</mi><mo id="S4.Thmtheorem2.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p1.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.9.m9.1b"><apply id="S4.Thmtheorem2.p1.9.m9.1.2.cmml" xref="S4.Thmtheorem2.p1.9.m9.1.2"><times id="S4.Thmtheorem2.p1.9.m9.1.2.1.cmml" xref="S4.Thmtheorem2.p1.9.m9.1.2.1"></times><apply id="S4.Thmtheorem2.p1.9.m9.1.2.2.cmml" xref="S4.Thmtheorem2.p1.9.m9.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.9.m9.1.2.2.1.cmml" xref="S4.Thmtheorem2.p1.9.m9.1.2.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.9.m9.1.2.2.2.cmml" xref="S4.Thmtheorem2.p1.9.m9.1.2.2.2">𝐻</ci><times id="S4.Thmtheorem2.p1.9.m9.1.2.2.3.cmml" xref="S4.Thmtheorem2.p1.9.m9.1.2.2.3"></times></apply><ci id="S4.Thmtheorem2.p1.9.m9.1.1.cmml" xref="S4.Thmtheorem2.p1.9.m9.1.1">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.9.m9.1c">H_{*}(A)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.9.m9.1d">italic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_A )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.3">For a bisimplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_R</annotation></semantics></math>-module <math alttext="A:\Delta^{op}\times\Delta^{op}\to R" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml"><mi id="S4.SS2.p2.2.m2.1.1.2" xref="S4.SS2.p2.2.m2.1.1.2.cmml">A</mi><mo id="S4.SS2.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S4.SS2.p2.2.m2.1.1.3" xref="S4.SS2.p2.2.m2.1.1.3.cmml"><mrow id="S4.SS2.p2.2.m2.1.1.3.2" xref="S4.SS2.p2.2.m2.1.1.3.2.cmml"><msup id="S4.SS2.p2.2.m2.1.1.3.2.2" xref="S4.SS2.p2.2.m2.1.1.3.2.2.cmml"><mi id="S4.SS2.p2.2.m2.1.1.3.2.2.2" mathvariant="normal" xref="S4.SS2.p2.2.m2.1.1.3.2.2.2.cmml">Δ</mi><mrow id="S4.SS2.p2.2.m2.1.1.3.2.2.3" xref="S4.SS2.p2.2.m2.1.1.3.2.2.3.cmml"><mi id="S4.SS2.p2.2.m2.1.1.3.2.2.3.2" xref="S4.SS2.p2.2.m2.1.1.3.2.2.3.2.cmml">o</mi><mo id="S4.SS2.p2.2.m2.1.1.3.2.2.3.1" xref="S4.SS2.p2.2.m2.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S4.SS2.p2.2.m2.1.1.3.2.2.3.3" xref="S4.SS2.p2.2.m2.1.1.3.2.2.3.3.cmml">p</mi></mrow></msup><mo id="S4.SS2.p2.2.m2.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.p2.2.m2.1.1.3.2.1.cmml">×</mo><msup id="S4.SS2.p2.2.m2.1.1.3.2.3" xref="S4.SS2.p2.2.m2.1.1.3.2.3.cmml"><mi id="S4.SS2.p2.2.m2.1.1.3.2.3.2" mathvariant="normal" xref="S4.SS2.p2.2.m2.1.1.3.2.3.2.cmml">Δ</mi><mrow id="S4.SS2.p2.2.m2.1.1.3.2.3.3" xref="S4.SS2.p2.2.m2.1.1.3.2.3.3.cmml"><mi id="S4.SS2.p2.2.m2.1.1.3.2.3.3.2" xref="S4.SS2.p2.2.m2.1.1.3.2.3.3.2.cmml">o</mi><mo id="S4.SS2.p2.2.m2.1.1.3.2.3.3.1" xref="S4.SS2.p2.2.m2.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S4.SS2.p2.2.m2.1.1.3.2.3.3.3" xref="S4.SS2.p2.2.m2.1.1.3.2.3.3.3.cmml">p</mi></mrow></msup></mrow><mo id="S4.SS2.p2.2.m2.1.1.3.1" stretchy="false" xref="S4.SS2.p2.2.m2.1.1.3.1.cmml">→</mo><mi id="S4.SS2.p2.2.m2.1.1.3.3" xref="S4.SS2.p2.2.m2.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1"><ci id="S4.SS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1.1">:</ci><ci id="S4.SS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2">𝐴</ci><apply id="S4.SS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3"><ci id="S4.SS2.p2.2.m2.1.1.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.1">→</ci><apply id="S4.SS2.p2.2.m2.1.1.3.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2"><times id="S4.SS2.p2.2.m2.1.1.3.2.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.1"></times><apply id="S4.SS2.p2.2.m2.1.1.3.2.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.3.2.2.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.2">superscript</csymbol><ci id="S4.SS2.p2.2.m2.1.1.3.2.2.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.2.2">Δ</ci><apply id="S4.SS2.p2.2.m2.1.1.3.2.2.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.2.3"><times id="S4.SS2.p2.2.m2.1.1.3.2.2.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.2.3.1"></times><ci id="S4.SS2.p2.2.m2.1.1.3.2.2.3.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.2.3.2">𝑜</ci><ci id="S4.SS2.p2.2.m2.1.1.3.2.2.3.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.2.3.3">𝑝</ci></apply></apply><apply id="S4.SS2.p2.2.m2.1.1.3.2.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.3.2.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3">superscript</csymbol><ci id="S4.SS2.p2.2.m2.1.1.3.2.3.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3.2">Δ</ci><apply id="S4.SS2.p2.2.m2.1.1.3.2.3.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3.3"><times id="S4.SS2.p2.2.m2.1.1.3.2.3.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3.3.1"></times><ci id="S4.SS2.p2.2.m2.1.1.3.2.3.3.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3.3.2">𝑜</ci><ci id="S4.SS2.p2.2.m2.1.1.3.2.3.3.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3.3.3">𝑝</ci></apply></apply></apply><ci id="S4.SS2.p2.2.m2.1.1.3.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">A:\Delta^{op}\times\Delta^{op}\to R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_A : roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT × roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → italic_R</annotation></semantics></math>-Mod, we define the associated double complex <math alttext="A_{*,*}" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.2"><semantics id="S4.SS2.p2.3.m3.2a"><msub id="S4.SS2.p2.3.m3.2.3" xref="S4.SS2.p2.3.m3.2.3.cmml"><mi id="S4.SS2.p2.3.m3.2.3.2" xref="S4.SS2.p2.3.m3.2.3.2.cmml">A</mi><mrow id="S4.SS2.p2.3.m3.2.2.2.4" xref="S4.SS2.p2.3.m3.2.2.2.3.cmml"><mo id="S4.SS2.p2.3.m3.1.1.1.1" rspace="0em" xref="S4.SS2.p2.3.m3.1.1.1.1.cmml">∗</mo><mo id="S4.SS2.p2.3.m3.2.2.2.4.1" rspace="0em" xref="S4.SS2.p2.3.m3.2.2.2.3.cmml">,</mo><mo id="S4.SS2.p2.3.m3.2.2.2.2" lspace="0em" xref="S4.SS2.p2.3.m3.2.2.2.2.cmml">∗</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.2b"><apply id="S4.SS2.p2.3.m3.2.3.cmml" xref="S4.SS2.p2.3.m3.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.2.3.1.cmml" xref="S4.SS2.p2.3.m3.2.3">subscript</csymbol><ci id="S4.SS2.p2.3.m3.2.3.2.cmml" xref="S4.SS2.p2.3.m3.2.3.2">𝐴</ci><list id="S4.SS2.p2.3.m3.2.2.2.3.cmml" xref="S4.SS2.p2.3.m3.2.2.2.4"><times id="S4.SS2.p2.3.m3.1.1.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1.1.1"></times><times id="S4.SS2.p2.3.m3.2.2.2.2.cmml" xref="S4.SS2.p2.3.m3.2.2.2.2"></times></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.2c">A_{*,*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.2d">italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT</annotation></semantics></math> to be the double complex with horizontal and vertical boundary maps</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex48"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\partial^{h}_{p}=\sum_{i=0}^{p}(-1)^{i}d_{i}^{h}\ \text{ and }\ \partial^{v}_{% q}=\sum_{j=1}^{q}(-1)^{j}d_{j}^{v}." class="ltx_Math" display="block" id="S4.Ex48.m1.1"><semantics id="S4.Ex48.m1.1a"><mrow id="S4.Ex48.m1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.cmml"><mrow id="S4.Ex48.m1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.cmml"><msubsup id="S4.Ex48.m1.1.1.1.1.4" xref="S4.Ex48.m1.1.1.1.1.4.cmml"><mo id="S4.Ex48.m1.1.1.1.1.4.2.2" xref="S4.Ex48.m1.1.1.1.1.4.2.2.cmml">∂</mo><mi id="S4.Ex48.m1.1.1.1.1.4.3" xref="S4.Ex48.m1.1.1.1.1.4.3.cmml">p</mi><mi id="S4.Ex48.m1.1.1.1.1.4.2.3" xref="S4.Ex48.m1.1.1.1.1.4.2.3.cmml">h</mi></msubsup><mo id="S4.Ex48.m1.1.1.1.1.5" lspace="0.278em" rspace="0.111em" xref="S4.Ex48.m1.1.1.1.1.5.cmml">=</mo><mrow id="S4.Ex48.m1.1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.1.cmml"><munderover id="S4.Ex48.m1.1.1.1.1.1.2" xref="S4.Ex48.m1.1.1.1.1.1.2.cmml"><mo id="S4.Ex48.m1.1.1.1.1.1.2.2.2" movablelimits="false" rspace="0em" xref="S4.Ex48.m1.1.1.1.1.1.2.2.2.cmml">∑</mo><mrow id="S4.Ex48.m1.1.1.1.1.1.2.2.3" xref="S4.Ex48.m1.1.1.1.1.1.2.2.3.cmml"><mi id="S4.Ex48.m1.1.1.1.1.1.2.2.3.2" xref="S4.Ex48.m1.1.1.1.1.1.2.2.3.2.cmml">i</mi><mo id="S4.Ex48.m1.1.1.1.1.1.2.2.3.1" xref="S4.Ex48.m1.1.1.1.1.1.2.2.3.1.cmml">=</mo><mn id="S4.Ex48.m1.1.1.1.1.1.2.2.3.3" xref="S4.Ex48.m1.1.1.1.1.1.2.2.3.3.cmml">0</mn></mrow><mi id="S4.Ex48.m1.1.1.1.1.1.2.3" xref="S4.Ex48.m1.1.1.1.1.1.2.3.cmml">p</mi></munderover><mrow id="S4.Ex48.m1.1.1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.1.1.cmml"><msup id="S4.Ex48.m1.1.1.1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex48.m1.1.1.1.1.1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1a" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S4.Ex48.m1.1.1.1.1.1.1.1.3" xref="S4.Ex48.m1.1.1.1.1.1.1.1.3.cmml">i</mi></msup><mo id="S4.Ex48.m1.1.1.1.1.1.1.2" xref="S4.Ex48.m1.1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S4.Ex48.m1.1.1.1.1.1.1.3" xref="S4.Ex48.m1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex48.m1.1.1.1.1.1.1.3.2.2" xref="S4.Ex48.m1.1.1.1.1.1.1.3.2.2.cmml">d</mi><mi id="S4.Ex48.m1.1.1.1.1.1.1.3.2.3" xref="S4.Ex48.m1.1.1.1.1.1.1.3.2.3.cmml">i</mi><mi id="S4.Ex48.m1.1.1.1.1.1.1.3.3" xref="S4.Ex48.m1.1.1.1.1.1.1.3.3.cmml">h</mi></msubsup><mo id="S4.Ex48.m1.1.1.1.1.1.1.2a" xref="S4.Ex48.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mtext id="S4.Ex48.m1.1.1.1.1.1.1.4" xref="S4.Ex48.m1.1.1.1.1.1.1.4a.cmml"> and </mtext><mo id="S4.Ex48.m1.1.1.1.1.1.1.2b" lspace="0.500em" xref="S4.Ex48.m1.1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S4.Ex48.m1.1.1.1.1.1.1.5" xref="S4.Ex48.m1.1.1.1.1.1.1.5.cmml"><mo id="S4.Ex48.m1.1.1.1.1.1.1.5.2.2" rspace="0.1389em" xref="S4.Ex48.m1.1.1.1.1.1.1.5.2.2.cmml">∂</mo><mi id="S4.Ex48.m1.1.1.1.1.1.1.5.3" xref="S4.Ex48.m1.1.1.1.1.1.1.5.3.cmml">q</mi><mi id="S4.Ex48.m1.1.1.1.1.1.1.5.2.3" xref="S4.Ex48.m1.1.1.1.1.1.1.5.2.3.cmml">v</mi></msubsup></mrow></mrow><mo id="S4.Ex48.m1.1.1.1.1.6" lspace="0.1389em" rspace="0.111em" xref="S4.Ex48.m1.1.1.1.1.6.cmml">=</mo><mrow id="S4.Ex48.m1.1.1.1.1.2" xref="S4.Ex48.m1.1.1.1.1.2.cmml"><munderover id="S4.Ex48.m1.1.1.1.1.2.2" xref="S4.Ex48.m1.1.1.1.1.2.2.cmml"><mo id="S4.Ex48.m1.1.1.1.1.2.2.2.2" 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id="S4.SS2.p2.4.m1.2a"><msub id="S4.SS2.p2.4.m1.2.3" xref="S4.SS2.p2.4.m1.2.3.cmml"><mi id="S4.SS2.p2.4.m1.2.3.2" xref="S4.SS2.p2.4.m1.2.3.2.cmml">A</mi><mrow id="S4.SS2.p2.4.m1.2.2.2.4" xref="S4.SS2.p2.4.m1.2.2.2.3.cmml"><mo id="S4.SS2.p2.4.m1.1.1.1.1" rspace="0em" xref="S4.SS2.p2.4.m1.1.1.1.1.cmml">∗</mo><mo id="S4.SS2.p2.4.m1.2.2.2.4.1" rspace="0em" xref="S4.SS2.p2.4.m1.2.2.2.3.cmml">,</mo><mo id="S4.SS2.p2.4.m1.2.2.2.2" lspace="0em" xref="S4.SS2.p2.4.m1.2.2.2.2.cmml">∗</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m1.2b"><apply id="S4.SS2.p2.4.m1.2.3.cmml" xref="S4.SS2.p2.4.m1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m1.2.3.1.cmml" xref="S4.SS2.p2.4.m1.2.3">subscript</csymbol><ci id="S4.SS2.p2.4.m1.2.3.2.cmml" xref="S4.SS2.p2.4.m1.2.3.2">𝐴</ci><list id="S4.SS2.p2.4.m1.2.2.2.3.cmml" xref="S4.SS2.p2.4.m1.2.2.2.4"><times id="S4.SS2.p2.4.m1.1.1.1.1.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1"></times><times id="S4.SS2.p2.4.m1.2.2.2.2.cmml" xref="S4.SS2.p2.4.m1.2.2.2.2"></times></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m1.2c">A_{*,*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m1.2d">italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT</annotation></semantics></math> is the chain complex <math alttext="\mathrm{Tot}(A_{*,*})" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m2.3"><semantics id="S4.SS2.p2.5.m2.3a"><mrow id="S4.SS2.p2.5.m2.3.3" xref="S4.SS2.p2.5.m2.3.3.cmml"><mi id="S4.SS2.p2.5.m2.3.3.3" xref="S4.SS2.p2.5.m2.3.3.3.cmml">Tot</mi><mo id="S4.SS2.p2.5.m2.3.3.2" xref="S4.SS2.p2.5.m2.3.3.2.cmml">⁢</mo><mrow id="S4.SS2.p2.5.m2.3.3.1.1" xref="S4.SS2.p2.5.m2.3.3.1.1.1.cmml"><mo id="S4.SS2.p2.5.m2.3.3.1.1.2" stretchy="false" xref="S4.SS2.p2.5.m2.3.3.1.1.1.cmml">(</mo><msub id="S4.SS2.p2.5.m2.3.3.1.1.1" xref="S4.SS2.p2.5.m2.3.3.1.1.1.cmml"><mi id="S4.SS2.p2.5.m2.3.3.1.1.1.2" xref="S4.SS2.p2.5.m2.3.3.1.1.1.2.cmml">A</mi><mrow id="S4.SS2.p2.5.m2.2.2.2.4" xref="S4.SS2.p2.5.m2.2.2.2.3.cmml"><mo id="S4.SS2.p2.5.m2.1.1.1.1" rspace="0em" xref="S4.SS2.p2.5.m2.1.1.1.1.cmml">∗</mo><mo id="S4.SS2.p2.5.m2.2.2.2.4.1" rspace="0em" xref="S4.SS2.p2.5.m2.2.2.2.3.cmml">,</mo><mo id="S4.SS2.p2.5.m2.2.2.2.2" lspace="0em" xref="S4.SS2.p2.5.m2.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S4.SS2.p2.5.m2.3.3.1.1.3" stretchy="false" xref="S4.SS2.p2.5.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m2.3b"><apply id="S4.SS2.p2.5.m2.3.3.cmml" xref="S4.SS2.p2.5.m2.3.3"><times id="S4.SS2.p2.5.m2.3.3.2.cmml" xref="S4.SS2.p2.5.m2.3.3.2"></times><ci id="S4.SS2.p2.5.m2.3.3.3.cmml" xref="S4.SS2.p2.5.m2.3.3.3">Tot</ci><apply id="S4.SS2.p2.5.m2.3.3.1.1.1.cmml" xref="S4.SS2.p2.5.m2.3.3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m2.3.3.1.1.1.1.cmml" xref="S4.SS2.p2.5.m2.3.3.1.1">subscript</csymbol><ci id="S4.SS2.p2.5.m2.3.3.1.1.1.2.cmml" xref="S4.SS2.p2.5.m2.3.3.1.1.1.2">𝐴</ci><list id="S4.SS2.p2.5.m2.2.2.2.3.cmml" xref="S4.SS2.p2.5.m2.2.2.2.4"><times id="S4.SS2.p2.5.m2.1.1.1.1.cmml" xref="S4.SS2.p2.5.m2.1.1.1.1"></times><times id="S4.SS2.p2.5.m2.2.2.2.2.cmml" xref="S4.SS2.p2.5.m2.2.2.2.2"></times></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m2.3c">\mathrm{Tot}(A_{*,*})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m2.3d">roman_Tot ( italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT )</annotation></semantics></math> with</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex49"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathrm{Tot}_{n}(A_{*,*})=\bigoplus_{p+q=n}A_{p,q}" class="ltx_Math" display="block" id="S4.Ex49.m1.5"><semantics id="S4.Ex49.m1.5a"><mrow id="S4.Ex49.m1.5.5" xref="S4.Ex49.m1.5.5.cmml"><mrow id="S4.Ex49.m1.5.5.1" xref="S4.Ex49.m1.5.5.1.cmml"><msub id="S4.Ex49.m1.5.5.1.3" xref="S4.Ex49.m1.5.5.1.3.cmml"><mi id="S4.Ex49.m1.5.5.1.3.2" xref="S4.Ex49.m1.5.5.1.3.2.cmml">Tot</mi><mi id="S4.Ex49.m1.5.5.1.3.3" xref="S4.Ex49.m1.5.5.1.3.3.cmml">n</mi></msub><mo id="S4.Ex49.m1.5.5.1.2" xref="S4.Ex49.m1.5.5.1.2.cmml">⁢</mo><mrow id="S4.Ex49.m1.5.5.1.1.1" xref="S4.Ex49.m1.5.5.1.1.1.1.cmml"><mo id="S4.Ex49.m1.5.5.1.1.1.2" stretchy="false" xref="S4.Ex49.m1.5.5.1.1.1.1.cmml">(</mo><msub id="S4.Ex49.m1.5.5.1.1.1.1" xref="S4.Ex49.m1.5.5.1.1.1.1.cmml"><mi id="S4.Ex49.m1.5.5.1.1.1.1.2" xref="S4.Ex49.m1.5.5.1.1.1.1.2.cmml">A</mi><mrow id="S4.Ex49.m1.2.2.2.4" xref="S4.Ex49.m1.2.2.2.3.cmml"><mo id="S4.Ex49.m1.1.1.1.1" rspace="0em" xref="S4.Ex49.m1.1.1.1.1.cmml">∗</mo><mo id="S4.Ex49.m1.2.2.2.4.1" rspace="0em" xref="S4.Ex49.m1.2.2.2.3.cmml">,</mo><mo id="S4.Ex49.m1.2.2.2.2" lspace="0em" xref="S4.Ex49.m1.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S4.Ex49.m1.5.5.1.1.1.3" stretchy="false" xref="S4.Ex49.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex49.m1.5.5.2" rspace="0.111em" xref="S4.Ex49.m1.5.5.2.cmml">=</mo><mrow id="S4.Ex49.m1.5.5.3" xref="S4.Ex49.m1.5.5.3.cmml"><munder id="S4.Ex49.m1.5.5.3.1" xref="S4.Ex49.m1.5.5.3.1.cmml"><mo id="S4.Ex49.m1.5.5.3.1.2" movablelimits="false" xref="S4.Ex49.m1.5.5.3.1.2.cmml">⨁</mo><mrow id="S4.Ex49.m1.5.5.3.1.3" xref="S4.Ex49.m1.5.5.3.1.3.cmml"><mrow id="S4.Ex49.m1.5.5.3.1.3.2" xref="S4.Ex49.m1.5.5.3.1.3.2.cmml"><mi id="S4.Ex49.m1.5.5.3.1.3.2.2" xref="S4.Ex49.m1.5.5.3.1.3.2.2.cmml">p</mi><mo id="S4.Ex49.m1.5.5.3.1.3.2.1" xref="S4.Ex49.m1.5.5.3.1.3.2.1.cmml">+</mo><mi id="S4.Ex49.m1.5.5.3.1.3.2.3" xref="S4.Ex49.m1.5.5.3.1.3.2.3.cmml">q</mi></mrow><mo id="S4.Ex49.m1.5.5.3.1.3.1" xref="S4.Ex49.m1.5.5.3.1.3.1.cmml">=</mo><mi id="S4.Ex49.m1.5.5.3.1.3.3" xref="S4.Ex49.m1.5.5.3.1.3.3.cmml">n</mi></mrow></munder><msub id="S4.Ex49.m1.5.5.3.2" xref="S4.Ex49.m1.5.5.3.2.cmml"><mi id="S4.Ex49.m1.5.5.3.2.2" xref="S4.Ex49.m1.5.5.3.2.2.cmml">A</mi><mrow id="S4.Ex49.m1.4.4.2.4" xref="S4.Ex49.m1.4.4.2.3.cmml"><mi id="S4.Ex49.m1.3.3.1.1" xref="S4.Ex49.m1.3.3.1.1.cmml">p</mi><mo id="S4.Ex49.m1.4.4.2.4.1" xref="S4.Ex49.m1.4.4.2.3.cmml">,</mo><mi id="S4.Ex49.m1.4.4.2.2" xref="S4.Ex49.m1.4.4.2.2.cmml">q</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex49.m1.5b"><apply id="S4.Ex49.m1.5.5.cmml" xref="S4.Ex49.m1.5.5"><eq id="S4.Ex49.m1.5.5.2.cmml" xref="S4.Ex49.m1.5.5.2"></eq><apply id="S4.Ex49.m1.5.5.1.cmml" xref="S4.Ex49.m1.5.5.1"><times id="S4.Ex49.m1.5.5.1.2.cmml" xref="S4.Ex49.m1.5.5.1.2"></times><apply id="S4.Ex49.m1.5.5.1.3.cmml" xref="S4.Ex49.m1.5.5.1.3"><csymbol cd="ambiguous" id="S4.Ex49.m1.5.5.1.3.1.cmml" xref="S4.Ex49.m1.5.5.1.3">subscript</csymbol><ci id="S4.Ex49.m1.5.5.1.3.2.cmml" xref="S4.Ex49.m1.5.5.1.3.2">Tot</ci><ci id="S4.Ex49.m1.5.5.1.3.3.cmml" xref="S4.Ex49.m1.5.5.1.3.3">𝑛</ci></apply><apply id="S4.Ex49.m1.5.5.1.1.1.1.cmml" xref="S4.Ex49.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S4.Ex49.m1.5.5.1.1.1.1.1.cmml" xref="S4.Ex49.m1.5.5.1.1.1">subscript</csymbol><ci id="S4.Ex49.m1.5.5.1.1.1.1.2.cmml" xref="S4.Ex49.m1.5.5.1.1.1.1.2">𝐴</ci><list id="S4.Ex49.m1.2.2.2.3.cmml" xref="S4.Ex49.m1.2.2.2.4"><times id="S4.Ex49.m1.1.1.1.1.cmml" xref="S4.Ex49.m1.1.1.1.1"></times><times id="S4.Ex49.m1.2.2.2.2.cmml" xref="S4.Ex49.m1.2.2.2.2"></times></list></apply></apply><apply id="S4.Ex49.m1.5.5.3.cmml" xref="S4.Ex49.m1.5.5.3"><apply id="S4.Ex49.m1.5.5.3.1.cmml" xref="S4.Ex49.m1.5.5.3.1"><csymbol cd="ambiguous" id="S4.Ex49.m1.5.5.3.1.1.cmml" xref="S4.Ex49.m1.5.5.3.1">subscript</csymbol><csymbol cd="latexml" id="S4.Ex49.m1.5.5.3.1.2.cmml" xref="S4.Ex49.m1.5.5.3.1.2">direct-sum</csymbol><apply id="S4.Ex49.m1.5.5.3.1.3.cmml" xref="S4.Ex49.m1.5.5.3.1.3"><eq id="S4.Ex49.m1.5.5.3.1.3.1.cmml" xref="S4.Ex49.m1.5.5.3.1.3.1"></eq><apply id="S4.Ex49.m1.5.5.3.1.3.2.cmml" xref="S4.Ex49.m1.5.5.3.1.3.2"><plus id="S4.Ex49.m1.5.5.3.1.3.2.1.cmml" xref="S4.Ex49.m1.5.5.3.1.3.2.1"></plus><ci id="S4.Ex49.m1.5.5.3.1.3.2.2.cmml" xref="S4.Ex49.m1.5.5.3.1.3.2.2">𝑝</ci><ci id="S4.Ex49.m1.5.5.3.1.3.2.3.cmml" xref="S4.Ex49.m1.5.5.3.1.3.2.3">𝑞</ci></apply><ci id="S4.Ex49.m1.5.5.3.1.3.3.cmml" xref="S4.Ex49.m1.5.5.3.1.3.3">𝑛</ci></apply></apply><apply id="S4.Ex49.m1.5.5.3.2.cmml" xref="S4.Ex49.m1.5.5.3.2"><csymbol cd="ambiguous" id="S4.Ex49.m1.5.5.3.2.1.cmml" xref="S4.Ex49.m1.5.5.3.2">subscript</csymbol><ci id="S4.Ex49.m1.5.5.3.2.2.cmml" xref="S4.Ex49.m1.5.5.3.2.2">𝐴</ci><list id="S4.Ex49.m1.4.4.2.3.cmml" xref="S4.Ex49.m1.4.4.2.4"><ci id="S4.Ex49.m1.3.3.1.1.cmml" xref="S4.Ex49.m1.3.3.1.1">𝑝</ci><ci id="S4.Ex49.m1.4.4.2.2.cmml" xref="S4.Ex49.m1.4.4.2.2">𝑞</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex49.m1.5c">\mathrm{Tot}_{n}(A_{*,*})=\bigoplus_{p+q=n}A_{p,q}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex49.m1.5d">roman_Tot start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ) = ⨁ start_POSTSUBSCRIPT italic_p + italic_q = italic_n end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p2.8">where the boundary maps given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex50"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\partial_{n}(x)=\partial^{h}_{p}(x)+(-1)^{q}\partial_{q}^{v}(x)" class="ltx_Math" display="block" id="S4.Ex50.m1.4"><semantics id="S4.Ex50.m1.4a"><mrow id="S4.Ex50.m1.4.4" xref="S4.Ex50.m1.4.4.cmml"><mrow id="S4.Ex50.m1.4.4.3" xref="S4.Ex50.m1.4.4.3.cmml"><msub id="S4.Ex50.m1.4.4.3.1" xref="S4.Ex50.m1.4.4.3.1.cmml"><mo id="S4.Ex50.m1.4.4.3.1.2" xref="S4.Ex50.m1.4.4.3.1.2.cmml">∂</mo><mi id="S4.Ex50.m1.4.4.3.1.3" xref="S4.Ex50.m1.4.4.3.1.3.cmml">n</mi></msub><mrow id="S4.Ex50.m1.4.4.3.2.2" xref="S4.Ex50.m1.4.4.3.cmml"><mo id="S4.Ex50.m1.4.4.3.2.2.1" lspace="0em" stretchy="false" xref="S4.Ex50.m1.4.4.3.cmml">(</mo><mi id="S4.Ex50.m1.1.1" xref="S4.Ex50.m1.1.1.cmml">x</mi><mo id="S4.Ex50.m1.4.4.3.2.2.2" stretchy="false" xref="S4.Ex50.m1.4.4.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex50.m1.4.4.2" rspace="0.1389em" xref="S4.Ex50.m1.4.4.2.cmml">=</mo><mrow id="S4.Ex50.m1.4.4.1" xref="S4.Ex50.m1.4.4.1.cmml"><mrow id="S4.Ex50.m1.4.4.1.3" xref="S4.Ex50.m1.4.4.1.3.cmml"><msubsup id="S4.Ex50.m1.4.4.1.3.1" xref="S4.Ex50.m1.4.4.1.3.1.cmml"><mo id="S4.Ex50.m1.4.4.1.3.1.2.2" lspace="0.1389em" rspace="0em" xref="S4.Ex50.m1.4.4.1.3.1.2.2.cmml">∂</mo><mi id="S4.Ex50.m1.4.4.1.3.1.3" xref="S4.Ex50.m1.4.4.1.3.1.3.cmml">p</mi><mi id="S4.Ex50.m1.4.4.1.3.1.2.3" xref="S4.Ex50.m1.4.4.1.3.1.2.3.cmml">h</mi></msubsup><mrow id="S4.Ex50.m1.4.4.1.3.2.2" xref="S4.Ex50.m1.4.4.1.3.cmml"><mo id="S4.Ex50.m1.4.4.1.3.2.2.1" stretchy="false" xref="S4.Ex50.m1.4.4.1.3.cmml">(</mo><mi id="S4.Ex50.m1.2.2" xref="S4.Ex50.m1.2.2.cmml">x</mi><mo id="S4.Ex50.m1.4.4.1.3.2.2.2" stretchy="false" xref="S4.Ex50.m1.4.4.1.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex50.m1.4.4.1.2" xref="S4.Ex50.m1.4.4.1.2.cmml">+</mo><mrow id="S4.Ex50.m1.4.4.1.1" xref="S4.Ex50.m1.4.4.1.1.cmml"><msup id="S4.Ex50.m1.4.4.1.1.1" xref="S4.Ex50.m1.4.4.1.1.1.cmml"><mrow id="S4.Ex50.m1.4.4.1.1.1.1.1" xref="S4.Ex50.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S4.Ex50.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.Ex50.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex50.m1.4.4.1.1.1.1.1.1" xref="S4.Ex50.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S4.Ex50.m1.4.4.1.1.1.1.1.1a" xref="S4.Ex50.m1.4.4.1.1.1.1.1.1.cmml">−</mo><mn id="S4.Ex50.m1.4.4.1.1.1.1.1.1.2" xref="S4.Ex50.m1.4.4.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Ex50.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="S4.Ex50.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S4.Ex50.m1.4.4.1.1.1.3" xref="S4.Ex50.m1.4.4.1.1.1.3.cmml">q</mi></msup><mo id="S4.Ex50.m1.4.4.1.1.2" lspace="0em" xref="S4.Ex50.m1.4.4.1.1.2.cmml">⁢</mo><mrow id="S4.Ex50.m1.4.4.1.1.3" xref="S4.Ex50.m1.4.4.1.1.3.cmml"><msubsup id="S4.Ex50.m1.4.4.1.1.3.1" xref="S4.Ex50.m1.4.4.1.1.3.1.cmml"><mo id="S4.Ex50.m1.4.4.1.1.3.1.2.2" rspace="0em" xref="S4.Ex50.m1.4.4.1.1.3.1.2.2.cmml">∂</mo><mi id="S4.Ex50.m1.4.4.1.1.3.1.2.3" xref="S4.Ex50.m1.4.4.1.1.3.1.2.3.cmml">q</mi><mi id="S4.Ex50.m1.4.4.1.1.3.1.3" xref="S4.Ex50.m1.4.4.1.1.3.1.3.cmml">v</mi></msubsup><mrow id="S4.Ex50.m1.4.4.1.1.3.2.2" xref="S4.Ex50.m1.4.4.1.1.3.cmml"><mo id="S4.Ex50.m1.4.4.1.1.3.2.2.1" stretchy="false" xref="S4.Ex50.m1.4.4.1.1.3.cmml">(</mo><mi id="S4.Ex50.m1.3.3" xref="S4.Ex50.m1.3.3.cmml">x</mi><mo id="S4.Ex50.m1.4.4.1.1.3.2.2.2" stretchy="false" xref="S4.Ex50.m1.4.4.1.1.3.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" 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id="S4.Ex50.m1.4c">\partial_{n}(x)=\partial^{h}_{p}(x)+(-1)^{q}\partial_{q}^{v}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex50.m1.4d">∂ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) = ∂ start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_x ) + ( - 1 ) start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p2.7">for every <math alttext="x\in X_{p,q}" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m1.2"><semantics id="S4.SS2.p2.6.m1.2a"><mrow id="S4.SS2.p2.6.m1.2.3" xref="S4.SS2.p2.6.m1.2.3.cmml"><mi id="S4.SS2.p2.6.m1.2.3.2" xref="S4.SS2.p2.6.m1.2.3.2.cmml">x</mi><mo id="S4.SS2.p2.6.m1.2.3.1" xref="S4.SS2.p2.6.m1.2.3.1.cmml">∈</mo><msub id="S4.SS2.p2.6.m1.2.3.3" xref="S4.SS2.p2.6.m1.2.3.3.cmml"><mi id="S4.SS2.p2.6.m1.2.3.3.2" xref="S4.SS2.p2.6.m1.2.3.3.2.cmml">X</mi><mrow id="S4.SS2.p2.6.m1.2.2.2.4" xref="S4.SS2.p2.6.m1.2.2.2.3.cmml"><mi id="S4.SS2.p2.6.m1.1.1.1.1" xref="S4.SS2.p2.6.m1.1.1.1.1.cmml">p</mi><mo id="S4.SS2.p2.6.m1.2.2.2.4.1" xref="S4.SS2.p2.6.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.6.m1.2.2.2.2" xref="S4.SS2.p2.6.m1.2.2.2.2.cmml">q</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m1.2b"><apply id="S4.SS2.p2.6.m1.2.3.cmml" xref="S4.SS2.p2.6.m1.2.3"><in id="S4.SS2.p2.6.m1.2.3.1.cmml" xref="S4.SS2.p2.6.m1.2.3.1"></in><ci id="S4.SS2.p2.6.m1.2.3.2.cmml" xref="S4.SS2.p2.6.m1.2.3.2">𝑥</ci><apply id="S4.SS2.p2.6.m1.2.3.3.cmml" xref="S4.SS2.p2.6.m1.2.3.3"><csymbol cd="ambiguous" id="S4.SS2.p2.6.m1.2.3.3.1.cmml" xref="S4.SS2.p2.6.m1.2.3.3">subscript</csymbol><ci id="S4.SS2.p2.6.m1.2.3.3.2.cmml" xref="S4.SS2.p2.6.m1.2.3.3.2">𝑋</ci><list id="S4.SS2.p2.6.m1.2.2.2.3.cmml" xref="S4.SS2.p2.6.m1.2.2.2.4"><ci id="S4.SS2.p2.6.m1.1.1.1.1.cmml" xref="S4.SS2.p2.6.m1.1.1.1.1">𝑝</ci><ci id="S4.SS2.p2.6.m1.2.2.2.2.cmml" xref="S4.SS2.p2.6.m1.2.2.2.2">𝑞</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m1.2c">x\in X_{p,q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m1.2d">italic_x ∈ italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="p+q=n" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m2.1"><semantics id="S4.SS2.p2.7.m2.1a"><mrow id="S4.SS2.p2.7.m2.1.1" xref="S4.SS2.p2.7.m2.1.1.cmml"><mrow id="S4.SS2.p2.7.m2.1.1.2" xref="S4.SS2.p2.7.m2.1.1.2.cmml"><mi id="S4.SS2.p2.7.m2.1.1.2.2" xref="S4.SS2.p2.7.m2.1.1.2.2.cmml">p</mi><mo id="S4.SS2.p2.7.m2.1.1.2.1" xref="S4.SS2.p2.7.m2.1.1.2.1.cmml">+</mo><mi id="S4.SS2.p2.7.m2.1.1.2.3" xref="S4.SS2.p2.7.m2.1.1.2.3.cmml">q</mi></mrow><mo id="S4.SS2.p2.7.m2.1.1.1" xref="S4.SS2.p2.7.m2.1.1.1.cmml">=</mo><mi id="S4.SS2.p2.7.m2.1.1.3" xref="S4.SS2.p2.7.m2.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m2.1b"><apply id="S4.SS2.p2.7.m2.1.1.cmml" xref="S4.SS2.p2.7.m2.1.1"><eq id="S4.SS2.p2.7.m2.1.1.1.cmml" xref="S4.SS2.p2.7.m2.1.1.1"></eq><apply id="S4.SS2.p2.7.m2.1.1.2.cmml" xref="S4.SS2.p2.7.m2.1.1.2"><plus id="S4.SS2.p2.7.m2.1.1.2.1.cmml" xref="S4.SS2.p2.7.m2.1.1.2.1"></plus><ci id="S4.SS2.p2.7.m2.1.1.2.2.cmml" xref="S4.SS2.p2.7.m2.1.1.2.2">𝑝</ci><ci id="S4.SS2.p2.7.m2.1.1.2.3.cmml" xref="S4.SS2.p2.7.m2.1.1.2.3">𝑞</ci></apply><ci id="S4.SS2.p2.7.m2.1.1.3.cmml" xref="S4.SS2.p2.7.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m2.1c">p+q=n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m2.1d">italic_p + italic_q = italic_n</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.5">Another way to get a chain complex from a bisimplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mi id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><ci id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">italic_R</annotation></semantics></math>-module <math alttext="A" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.1"><semantics id="S4.SS2.p3.2.m2.1a"><mi id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><ci id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">italic_A</annotation></semantics></math> is to consider the simplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m3.1"><semantics id="S4.SS2.p3.3.m3.1a"><mi id="S4.SS2.p3.3.m3.1.1" xref="S4.SS2.p3.3.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m3.1b"><ci id="S4.SS2.p3.3.m3.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m3.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.m3.1d">italic_R</annotation></semantics></math>-module <math alttext="\mathrm{diag}A" class="ltx_Math" display="inline" id="S4.SS2.p3.4.m4.1"><semantics id="S4.SS2.p3.4.m4.1a"><mrow id="S4.SS2.p3.4.m4.1.1" xref="S4.SS2.p3.4.m4.1.1.cmml"><mi id="S4.SS2.p3.4.m4.1.1.2" xref="S4.SS2.p3.4.m4.1.1.2.cmml">diag</mi><mo id="S4.SS2.p3.4.m4.1.1.1" xref="S4.SS2.p3.4.m4.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p3.4.m4.1.1.3" xref="S4.SS2.p3.4.m4.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.4.m4.1b"><apply id="S4.SS2.p3.4.m4.1.1.cmml" xref="S4.SS2.p3.4.m4.1.1"><times id="S4.SS2.p3.4.m4.1.1.1.cmml" xref="S4.SS2.p3.4.m4.1.1.1"></times><ci id="S4.SS2.p3.4.m4.1.1.2.cmml" xref="S4.SS2.p3.4.m4.1.1.2">diag</ci><ci id="S4.SS2.p3.4.m4.1.1.3.cmml" xref="S4.SS2.p3.4.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.4.m4.1c">\mathrm{diag}A</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.4.m4.1d">roman_diag italic_A</annotation></semantics></math> and take its Moore complex <math alttext="(\mathrm{diag}A)_{*}" class="ltx_Math" display="inline" id="S4.SS2.p3.5.m5.1"><semantics id="S4.SS2.p3.5.m5.1a"><msub id="S4.SS2.p3.5.m5.1.1" xref="S4.SS2.p3.5.m5.1.1.cmml"><mrow id="S4.SS2.p3.5.m5.1.1.1.1" xref="S4.SS2.p3.5.m5.1.1.1.1.1.cmml"><mo id="S4.SS2.p3.5.m5.1.1.1.1.2" stretchy="false" xref="S4.SS2.p3.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p3.5.m5.1.1.1.1.1" xref="S4.SS2.p3.5.m5.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.5.m5.1.1.1.1.1.2" xref="S4.SS2.p3.5.m5.1.1.1.1.1.2.cmml">diag</mi><mo id="S4.SS2.p3.5.m5.1.1.1.1.1.1" xref="S4.SS2.p3.5.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p3.5.m5.1.1.1.1.1.3" xref="S4.SS2.p3.5.m5.1.1.1.1.1.3.cmml">A</mi></mrow><mo id="S4.SS2.p3.5.m5.1.1.1.1.3" stretchy="false" xref="S4.SS2.p3.5.m5.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.p3.5.m5.1.1.3" xref="S4.SS2.p3.5.m5.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.5.m5.1b"><apply id="S4.SS2.p3.5.m5.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.5.m5.1.1.2.cmml" xref="S4.SS2.p3.5.m5.1.1">subscript</csymbol><apply id="S4.SS2.p3.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1.1.1"><times id="S4.SS2.p3.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1.1.1.1.1"></times><ci id="S4.SS2.p3.5.m5.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.5.m5.1.1.1.1.1.2">diag</ci><ci id="S4.SS2.p3.5.m5.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.5.m5.1.1.1.1.1.3">𝐴</ci></apply><times id="S4.SS2.p3.5.m5.1.1.3.cmml" xref="S4.SS2.p3.5.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.5.m5.1c">(\mathrm{diag}A)_{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.5.m5.1d">( roman_diag italic_A ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>. By the Dold-Puppe theorem, these two different constructions give chain complexes that are homotopic.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.1.1.1">Theorem 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.2.2"> </span>(Dold-Puppe Theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib4" title="">4</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.4.4">Let <math alttext="A:\Delta^{op}\times\Delta^{op}\to R" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem3.p1.1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">A</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S4.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.cmml"><mrow id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml"><msup id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.2" mathvariant="normal" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.2.cmml">Δ</mi><mrow id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.2.cmml">o</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.3.cmml">p</mi></mrow></msup><mo id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.1.cmml">×</mo><msup id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.2" mathvariant="normal" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.2.cmml">Δ</mi><mrow id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.2.cmml">o</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.3.cmml">p</mi></mrow></msup></mrow><mo id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.1.m1.1b"><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1"><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1">:</ci><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2">𝐴</ci><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3"><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.1">→</ci><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2"><times id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.1"></times><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2">superscript</csymbol><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.2">Δ</ci><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3"><times id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.1"></times><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.2">𝑜</ci><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.2.3.3">𝑝</ci></apply></apply><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3">superscript</csymbol><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.2">Δ</ci><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3"><times id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.1"></times><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.2">𝑜</ci><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.3.3.3">𝑝</ci></apply></apply></apply><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.1c">A:\Delta^{op}\times\Delta^{op}\to R</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.1d">italic_A : roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT × roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → italic_R</annotation></semantics></math>-Mod be a bisimplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.1"><semantics id="S4.Thmtheorem3.p1.2.2.m2.1a"><mi id="S4.Thmtheorem3.p1.2.2.m2.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.2.m2.1b"><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.1d">italic_R</annotation></semantics></math>-module. Then the inclusion map <math alttext="(\mathrm{diag}A)_{*}\to\mathrm{Tot}(A_{*,*})" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.3.3.m3.4"><semantics id="S4.Thmtheorem3.p1.3.3.m3.4a"><mrow id="S4.Thmtheorem3.p1.3.3.m3.4.4" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.cmml"><msub id="S4.Thmtheorem3.p1.3.3.m3.3.3.1" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.cmml"><mrow id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.2" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.2.cmml">diag</mi><mo id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.1.cmml">⁢</mo><mi id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.3" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.3.cmml">A</mi></mrow><mo id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.3" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.3.cmml">∗</mo></msub><mo id="S4.Thmtheorem3.p1.3.3.m3.4.4.3" stretchy="false" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.3.cmml">→</mo><mrow id="S4.Thmtheorem3.p1.3.3.m3.4.4.2" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.cmml"><mi id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.3" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.3.cmml">Tot</mi><mo id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.2" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.cmml"><mo id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.2" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.2.cmml">A</mi><mrow id="S4.Thmtheorem3.p1.3.3.m3.2.2.2.4" xref="S4.Thmtheorem3.p1.3.3.m3.2.2.2.3.cmml"><mo id="S4.Thmtheorem3.p1.3.3.m3.1.1.1.1" rspace="0em" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.1.1.cmml">∗</mo><mo id="S4.Thmtheorem3.p1.3.3.m3.2.2.2.4.1" rspace="0em" xref="S4.Thmtheorem3.p1.3.3.m3.2.2.2.3.cmml">,</mo><mo id="S4.Thmtheorem3.p1.3.3.m3.2.2.2.2" lspace="0em" xref="S4.Thmtheorem3.p1.3.3.m3.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.3.3.m3.4b"><apply id="S4.Thmtheorem3.p1.3.3.m3.4.4.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.4.4"><ci id="S4.Thmtheorem3.p1.3.3.m3.4.4.3.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.3">→</ci><apply id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1">subscript</csymbol><apply id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1"><times id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.1"></times><ci id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.2">diag</ci><ci id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.1.1.1.3">𝐴</ci></apply><times id="S4.Thmtheorem3.p1.3.3.m3.3.3.1.3.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.3.3.1.3"></times></apply><apply id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2"><times id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.2"></times><ci id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.3.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.3">Tot</ci><apply id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.4.4.2.1.1.1.2">𝐴</ci><list id="S4.Thmtheorem3.p1.3.3.m3.2.2.2.3.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.2.2.2.4"><times id="S4.Thmtheorem3.p1.3.3.m3.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.1.1"></times><times id="S4.Thmtheorem3.p1.3.3.m3.2.2.2.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.2.2.2.2"></times></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.3.3.m3.4c">(\mathrm{diag}A)_{*}\to\mathrm{Tot}(A_{*,*})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.3.3.m3.4d">( roman_diag italic_A ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → roman_Tot ( italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT )</annotation></semantics></math> is a chain homotopy equivalence. This equivalence is natural with respect to morphisms of bisimplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.4.4.m4.1"><semantics id="S4.Thmtheorem3.p1.4.4.m4.1a"><mi id="S4.Thmtheorem3.p1.4.4.m4.1.1" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.4.4.m4.1b"><ci id="S4.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.4.4.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.4.4.m4.1d">italic_R</annotation></semantics></math>-modules.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.1.p1"> <p class="ltx_p" id="S4.SS2.1.p1.1">See <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib9" title="">9</a>, Chp IV, Thm 2.4]</cite> for a proof. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.1">The homology of the total complex <math alttext="\mathrm{Tot}(A_{*,*})" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.3"><semantics id="S4.SS2.p4.1.m1.3a"><mrow id="S4.SS2.p4.1.m1.3.3" xref="S4.SS2.p4.1.m1.3.3.cmml"><mi id="S4.SS2.p4.1.m1.3.3.3" xref="S4.SS2.p4.1.m1.3.3.3.cmml">Tot</mi><mo id="S4.SS2.p4.1.m1.3.3.2" xref="S4.SS2.p4.1.m1.3.3.2.cmml">⁢</mo><mrow id="S4.SS2.p4.1.m1.3.3.1.1" xref="S4.SS2.p4.1.m1.3.3.1.1.1.cmml"><mo id="S4.SS2.p4.1.m1.3.3.1.1.2" stretchy="false" xref="S4.SS2.p4.1.m1.3.3.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.1.m1.3.3.1.1.1" xref="S4.SS2.p4.1.m1.3.3.1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.3.3.1.1.1.2" xref="S4.SS2.p4.1.m1.3.3.1.1.1.2.cmml">A</mi><mrow id="S4.SS2.p4.1.m1.2.2.2.4" xref="S4.SS2.p4.1.m1.2.2.2.3.cmml"><mo id="S4.SS2.p4.1.m1.1.1.1.1" rspace="0em" xref="S4.SS2.p4.1.m1.1.1.1.1.cmml">∗</mo><mo id="S4.SS2.p4.1.m1.2.2.2.4.1" rspace="0em" xref="S4.SS2.p4.1.m1.2.2.2.3.cmml">,</mo><mo id="S4.SS2.p4.1.m1.2.2.2.2" lspace="0em" xref="S4.SS2.p4.1.m1.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S4.SS2.p4.1.m1.3.3.1.1.3" stretchy="false" xref="S4.SS2.p4.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.3b"><apply id="S4.SS2.p4.1.m1.3.3.cmml" xref="S4.SS2.p4.1.m1.3.3"><times id="S4.SS2.p4.1.m1.3.3.2.cmml" xref="S4.SS2.p4.1.m1.3.3.2"></times><ci id="S4.SS2.p4.1.m1.3.3.3.cmml" xref="S4.SS2.p4.1.m1.3.3.3">Tot</ci><apply id="S4.SS2.p4.1.m1.3.3.1.1.1.cmml" xref="S4.SS2.p4.1.m1.3.3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.3.3.1.1">subscript</csymbol><ci id="S4.SS2.p4.1.m1.3.3.1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.3.3.1.1.1.2">𝐴</ci><list id="S4.SS2.p4.1.m1.2.2.2.3.cmml" xref="S4.SS2.p4.1.m1.2.2.2.4"><times id="S4.SS2.p4.1.m1.1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1"></times><times id="S4.SS2.p4.1.m1.2.2.2.2.cmml" xref="S4.SS2.p4.1.m1.2.2.2.2"></times></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.3c">\mathrm{Tot}(A_{*,*})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.3d">roman_Tot ( italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT )</annotation></semantics></math> can be calculated by spectral sequences associated to the double complex. Depending on the chosen filtration, horizontal or vertical, there are two spectral sequences that converge to the homology of the total complex (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib19" title="">19</a>]</cite>):</p> <table class="ltx_equation ltx_eqn_table" id="S4.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}{}^{I}E^{2}_{p,q}&=H^{h}_{p}(H^{v}_{q}(A_{*,*}))\Rightarrow H_{p+% q}(\mathrm{Tot}(A_{*,*}))\\ {}^{II}E^{2}_{p,q}&=H^{v}_{p}(H^{h}_{q}(A_{*,*}))\Rightarrow H_{p+q}(\mathrm{% Tot}(A_{*,*}))\end{split}" class="ltx_Math" display="block" id="S4.E1.m1.62"><semantics id="S4.E1.m1.62a"><mtable columnspacing="0pt" displaystyle="true" id="S4.E1.m1.62.62.8" rowspacing="0pt" xref="S4.E1.m1.58.58.4.cmml"><mtr id="S4.E1.m1.62.62.8a" xref="S4.E1.m1.58.58.4.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E1.m1.62.62.8b" xref="S4.E1.m1.58.58.4.cmml"><mmultiscripts id="S4.E1.m1.4.4.4.4.4" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.2.2.2.2.2.2" xref="S4.E1.m1.2.2.2.2.2.2.cmml">E</mi><mrow id="S4.E1.m1.4.4.4.4.4.4.1.4" xref="S4.E1.m1.4.4.4.4.4.4.1.3.cmml"><mi id="S4.E1.m1.4.4.4.4.4.4.1.1" xref="S4.E1.m1.4.4.4.4.4.4.1.1.cmml">p</mi><mo id="S4.E1.m1.4.4.4.4.4.4.1.4.1" xref="S4.E1.m1.4.4.4.4.4.4.1.3.cmml">,</mo><mi id="S4.E1.m1.4.4.4.4.4.4.1.2" xref="S4.E1.m1.4.4.4.4.4.4.1.2.cmml">q</mi></mrow><mn id="S4.E1.m1.3.3.3.3.3.3.1" xref="S4.E1.m1.3.3.3.3.3.3.1.cmml">2</mn><mprescripts id="S4.E1.m1.4.4.4.4.4a" xref="S4.E1.m1.58.58.4a.cmml"></mprescripts><mrow id="S4.E1.m1.4.4.4.4.4b" xref="S4.E1.m1.58.58.4a.cmml"></mrow><mi id="S4.E1.m1.1.1.1.1.1.1.1" xref="S4.E1.m1.1.1.1.1.1.1.1.cmml">I</mi></mmultiscripts></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E1.m1.62.62.8c" xref="S4.E1.m1.58.58.4.cmml"><mrow id="S4.E1.m1.60.60.6.56.29.25" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.60.60.6.56.29.25.27" xref="S4.E1.m1.58.58.4a.cmml"></mi><mo id="S4.E1.m1.5.5.5.5.1.1" xref="S4.E1.m1.5.5.5.5.1.1.cmml">=</mo><mrow id="S4.E1.m1.59.59.5.55.28.24.24" xref="S4.E1.m1.58.58.4.cmml"><msubsup id="S4.E1.m1.59.59.5.55.28.24.24.3" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.6.6.6.6.2.2" xref="S4.E1.m1.6.6.6.6.2.2.cmml">H</mi><mi id="S4.E1.m1.8.8.8.8.4.4.1" xref="S4.E1.m1.8.8.8.8.4.4.1.cmml">p</mi><mi id="S4.E1.m1.7.7.7.7.3.3.1" xref="S4.E1.m1.7.7.7.7.3.3.1.cmml">h</mi></msubsup><mo id="S4.E1.m1.59.59.5.55.28.24.24.2" xref="S4.E1.m1.58.58.4a.cmml">⁢</mo><mrow id="S4.E1.m1.59.59.5.55.28.24.24.1.1" xref="S4.E1.m1.58.58.4.cmml"><mo id="S4.E1.m1.9.9.9.9.5.5" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">(</mo><mrow id="S4.E1.m1.59.59.5.55.28.24.24.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><msubsup id="S4.E1.m1.59.59.5.55.28.24.24.1.1.1.3" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.10.10.10.10.6.6" xref="S4.E1.m1.10.10.10.10.6.6.cmml">H</mi><mi id="S4.E1.m1.12.12.12.12.8.8.1" xref="S4.E1.m1.12.12.12.12.8.8.1.cmml">q</mi><mi id="S4.E1.m1.11.11.11.11.7.7.1" xref="S4.E1.m1.11.11.11.11.7.7.1.cmml">v</mi></msubsup><mo id="S4.E1.m1.59.59.5.55.28.24.24.1.1.1.2" xref="S4.E1.m1.58.58.4a.cmml">⁢</mo><mrow id="S4.E1.m1.59.59.5.55.28.24.24.1.1.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><mo id="S4.E1.m1.13.13.13.13.9.9" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">(</mo><msub id="S4.E1.m1.59.59.5.55.28.24.24.1.1.1.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.14.14.14.14.10.10" xref="S4.E1.m1.14.14.14.14.10.10.cmml">A</mi><mrow id="S4.E1.m1.15.15.15.15.11.11.1.4" xref="S4.E1.m1.15.15.15.15.11.11.1.3.cmml"><mo id="S4.E1.m1.15.15.15.15.11.11.1.1" rspace="0em" xref="S4.E1.m1.15.15.15.15.11.11.1.1.cmml">∗</mo><mo id="S4.E1.m1.15.15.15.15.11.11.1.4.1" rspace="0em" xref="S4.E1.m1.15.15.15.15.11.11.1.3.cmml">,</mo><mo id="S4.E1.m1.15.15.15.15.11.11.1.2" lspace="0em" xref="S4.E1.m1.15.15.15.15.11.11.1.2.cmml">∗</mo></mrow></msub><mo id="S4.E1.m1.16.16.16.16.12.12" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.17.17.17.17.13.13" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.18.18.18.18.14.14" stretchy="false" xref="S4.E1.m1.18.18.18.18.14.14.cmml">⇒</mo><mrow id="S4.E1.m1.60.60.6.56.29.25.25" xref="S4.E1.m1.58.58.4.cmml"><msub id="S4.E1.m1.60.60.6.56.29.25.25.3" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.19.19.19.19.15.15" xref="S4.E1.m1.19.19.19.19.15.15.cmml">H</mi><mrow id="S4.E1.m1.20.20.20.20.16.16.1" xref="S4.E1.m1.20.20.20.20.16.16.1.cmml"><mi id="S4.E1.m1.20.20.20.20.16.16.1.2" xref="S4.E1.m1.20.20.20.20.16.16.1.2.cmml">p</mi><mo id="S4.E1.m1.20.20.20.20.16.16.1.1" xref="S4.E1.m1.20.20.20.20.16.16.1.1.cmml">+</mo><mi id="S4.E1.m1.20.20.20.20.16.16.1.3" xref="S4.E1.m1.20.20.20.20.16.16.1.3.cmml">q</mi></mrow></msub><mo id="S4.E1.m1.60.60.6.56.29.25.25.2" xref="S4.E1.m1.58.58.4a.cmml">⁢</mo><mrow id="S4.E1.m1.60.60.6.56.29.25.25.1.1" xref="S4.E1.m1.58.58.4.cmml"><mo id="S4.E1.m1.21.21.21.21.17.17" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">(</mo><mrow id="S4.E1.m1.60.60.6.56.29.25.25.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.22.22.22.22.18.18" xref="S4.E1.m1.22.22.22.22.18.18.cmml">Tot</mi><mo id="S4.E1.m1.60.60.6.56.29.25.25.1.1.1.2" xref="S4.E1.m1.58.58.4a.cmml">⁢</mo><mrow id="S4.E1.m1.60.60.6.56.29.25.25.1.1.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><mo id="S4.E1.m1.23.23.23.23.19.19" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">(</mo><msub id="S4.E1.m1.60.60.6.56.29.25.25.1.1.1.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.24.24.24.24.20.20" xref="S4.E1.m1.24.24.24.24.20.20.cmml">A</mi><mrow id="S4.E1.m1.25.25.25.25.21.21.1.4" xref="S4.E1.m1.25.25.25.25.21.21.1.3.cmml"><mo id="S4.E1.m1.25.25.25.25.21.21.1.1" rspace="0em" xref="S4.E1.m1.25.25.25.25.21.21.1.1.cmml">∗</mo><mo id="S4.E1.m1.25.25.25.25.21.21.1.4.1" rspace="0em" xref="S4.E1.m1.25.25.25.25.21.21.1.3.cmml">,</mo><mo id="S4.E1.m1.25.25.25.25.21.21.1.2" lspace="0em" xref="S4.E1.m1.25.25.25.25.21.21.1.2.cmml">∗</mo></mrow></msub><mo id="S4.E1.m1.26.26.26.26.22.22" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.27.27.27.27.23.23" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E1.m1.62.62.8d" xref="S4.E1.m1.58.58.4.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E1.m1.62.62.8e" xref="S4.E1.m1.58.58.4.cmml"><mmultiscripts id="S4.E1.m1.31.31.31.4.4" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.29.29.29.2.2.2" xref="S4.E1.m1.29.29.29.2.2.2.cmml">E</mi><mrow id="S4.E1.m1.31.31.31.4.4.4.1.4" xref="S4.E1.m1.31.31.31.4.4.4.1.3.cmml"><mi id="S4.E1.m1.31.31.31.4.4.4.1.1" xref="S4.E1.m1.31.31.31.4.4.4.1.1.cmml">p</mi><mo id="S4.E1.m1.31.31.31.4.4.4.1.4.1" xref="S4.E1.m1.31.31.31.4.4.4.1.3.cmml">,</mo><mi id="S4.E1.m1.31.31.31.4.4.4.1.2" xref="S4.E1.m1.31.31.31.4.4.4.1.2.cmml">q</mi></mrow><mn id="S4.E1.m1.30.30.30.3.3.3.1" xref="S4.E1.m1.30.30.30.3.3.3.1.cmml">2</mn><mprescripts id="S4.E1.m1.31.31.31.4.4a" xref="S4.E1.m1.58.58.4a.cmml"></mprescripts><mrow id="S4.E1.m1.31.31.31.4.4b" xref="S4.E1.m1.58.58.4a.cmml"></mrow><mrow id="S4.E1.m1.28.28.28.1.1.1.1" xref="S4.E1.m1.28.28.28.1.1.1.1.cmml"><mi id="S4.E1.m1.28.28.28.1.1.1.1.2" xref="S4.E1.m1.28.28.28.1.1.1.1.2.cmml">I</mi><mo id="S4.E1.m1.28.28.28.1.1.1.1.1" xref="S4.E1.m1.28.28.28.1.1.1.1.1.cmml">⁢</mo><mi id="S4.E1.m1.28.28.28.1.1.1.1.3" xref="S4.E1.m1.28.28.28.1.1.1.1.3.cmml">I</mi></mrow></mmultiscripts></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E1.m1.62.62.8f" xref="S4.E1.m1.58.58.4.cmml"><mrow id="S4.E1.m1.62.62.8.58.29.25" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.62.62.8.58.29.25.27" xref="S4.E1.m1.58.58.4a.cmml"></mi><mo id="S4.E1.m1.32.32.32.5.1.1" xref="S4.E1.m1.32.32.32.5.1.1.cmml">=</mo><mrow id="S4.E1.m1.61.61.7.57.28.24.24" xref="S4.E1.m1.58.58.4.cmml"><msubsup id="S4.E1.m1.61.61.7.57.28.24.24.3" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.33.33.33.6.2.2" xref="S4.E1.m1.33.33.33.6.2.2.cmml">H</mi><mi id="S4.E1.m1.35.35.35.8.4.4.1" xref="S4.E1.m1.35.35.35.8.4.4.1.cmml">p</mi><mi id="S4.E1.m1.34.34.34.7.3.3.1" xref="S4.E1.m1.34.34.34.7.3.3.1.cmml">v</mi></msubsup><mo id="S4.E1.m1.61.61.7.57.28.24.24.2" xref="S4.E1.m1.58.58.4a.cmml">⁢</mo><mrow id="S4.E1.m1.61.61.7.57.28.24.24.1.1" xref="S4.E1.m1.58.58.4.cmml"><mo id="S4.E1.m1.36.36.36.9.5.5" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">(</mo><mrow id="S4.E1.m1.61.61.7.57.28.24.24.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><msubsup id="S4.E1.m1.61.61.7.57.28.24.24.1.1.1.3" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.37.37.37.10.6.6" xref="S4.E1.m1.37.37.37.10.6.6.cmml">H</mi><mi id="S4.E1.m1.39.39.39.12.8.8.1" xref="S4.E1.m1.39.39.39.12.8.8.1.cmml">q</mi><mi id="S4.E1.m1.38.38.38.11.7.7.1" xref="S4.E1.m1.38.38.38.11.7.7.1.cmml">h</mi></msubsup><mo id="S4.E1.m1.61.61.7.57.28.24.24.1.1.1.2" xref="S4.E1.m1.58.58.4a.cmml">⁢</mo><mrow id="S4.E1.m1.61.61.7.57.28.24.24.1.1.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><mo id="S4.E1.m1.40.40.40.13.9.9" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">(</mo><msub id="S4.E1.m1.61.61.7.57.28.24.24.1.1.1.1.1.1" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.41.41.41.14.10.10" xref="S4.E1.m1.41.41.41.14.10.10.cmml">A</mi><mrow id="S4.E1.m1.42.42.42.15.11.11.1.4" xref="S4.E1.m1.42.42.42.15.11.11.1.3.cmml"><mo id="S4.E1.m1.42.42.42.15.11.11.1.1" rspace="0em" xref="S4.E1.m1.42.42.42.15.11.11.1.1.cmml">∗</mo><mo id="S4.E1.m1.42.42.42.15.11.11.1.4.1" rspace="0em" xref="S4.E1.m1.42.42.42.15.11.11.1.3.cmml">,</mo><mo id="S4.E1.m1.42.42.42.15.11.11.1.2" lspace="0em" xref="S4.E1.m1.42.42.42.15.11.11.1.2.cmml">∗</mo></mrow></msub><mo id="S4.E1.m1.43.43.43.16.12.12" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.44.44.44.17.13.13" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.45.45.45.18.14.14" stretchy="false" xref="S4.E1.m1.45.45.45.18.14.14.cmml">⇒</mo><mrow id="S4.E1.m1.62.62.8.58.29.25.25" xref="S4.E1.m1.58.58.4.cmml"><msub id="S4.E1.m1.62.62.8.58.29.25.25.3" xref="S4.E1.m1.58.58.4.cmml"><mi id="S4.E1.m1.46.46.46.19.15.15" xref="S4.E1.m1.46.46.46.19.15.15.cmml">H</mi><mrow id="S4.E1.m1.47.47.47.20.16.16.1" xref="S4.E1.m1.47.47.47.20.16.16.1.cmml"><mi id="S4.E1.m1.47.47.47.20.16.16.1.2" xref="S4.E1.m1.47.47.47.20.16.16.1.2.cmml">p</mi><mo id="S4.E1.m1.47.47.47.20.16.16.1.1" xref="S4.E1.m1.47.47.47.20.16.16.1.1.cmml">+</mo><mi id="S4.E1.m1.47.47.47.20.16.16.1.3" xref="S4.E1.m1.47.47.47.20.16.16.1.3.cmml">q</mi></mrow></msub><mo id="S4.E1.m1.62.62.8.58.29.25.25.2" xref="S4.E1.m1.58.58.4a.cmml">⁢</mo><mrow id="S4.E1.m1.62.62.8.58.29.25.25.1.1" xref="S4.E1.m1.58.58.4.cmml"><mo id="S4.E1.m1.48.48.48.21.17.17" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">(</mo><mrow 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id="S4.E1.m1.53.53.53.26.22.22" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.54.54.54.27.23.23" stretchy="false" xref="S4.E1.m1.58.58.4a.cmml">)</mo></mrow></mrow></mrow></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S4.E1.m1.62b"><apply id="S4.E1.m1.58.58.4.cmml" xref="S4.E1.m1.62.62.8"><and id="S4.E1.m1.58.58.4a.cmml" xref="S4.E1.m1.4.4.4.4.4a"></and><apply id="S4.E1.m1.58.58.4b.cmml" xref="S4.E1.m1.62.62.8"><eq id="S4.E1.m1.5.5.5.5.1.1.cmml" xref="S4.E1.m1.5.5.5.5.1.1"></eq><apply id="S4.E1.m1.58.58.4.6.cmml" xref="S4.E1.m1.62.62.8"><csymbol cd="ambiguous" id="S4.E1.m1.58.58.4.6.1.cmml" xref="S4.E1.m1.4.4.4.4.4a">superscript</csymbol><apply id="S4.E1.m1.58.58.4.6.2.cmml" xref="S4.E1.m1.62.62.8"><csymbol cd="ambiguous" id="S4.E1.m1.58.58.4.6.2.1.cmml" xref="S4.E1.m1.4.4.4.4.4a">subscript</csymbol><apply id="S4.E1.m1.58.58.4.6.2.2.cmml" xref="S4.E1.m1.62.62.8"><csymbol cd="ambiguous" id="S4.E1.m1.58.58.4.6.2.2.1.cmml" 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id="S4.E1.m1.29.29.29.2.2.2.cmml" xref="S4.E1.m1.29.29.29.2.2.2">𝐸</ci><cn id="S4.E1.m1.30.30.30.3.3.3.1.cmml" type="integer" xref="S4.E1.m1.30.30.30.3.3.3.1">2</cn></apply><list id="S4.E1.m1.31.31.31.4.4.4.1.3.cmml" xref="S4.E1.m1.31.31.31.4.4.4.1.4"><ci id="S4.E1.m1.31.31.31.4.4.4.1.1.cmml" xref="S4.E1.m1.31.31.31.4.4.4.1.1">𝑝</ci><ci id="S4.E1.m1.31.31.31.4.4.4.1.2.cmml" xref="S4.E1.m1.31.31.31.4.4.4.1.2">𝑞</ci></list></apply><apply id="S4.E1.m1.28.28.28.1.1.1.1.cmml" xref="S4.E1.m1.28.28.28.1.1.1.1"><times id="S4.E1.m1.28.28.28.1.1.1.1.1.cmml" xref="S4.E1.m1.28.28.28.1.1.1.1.1"></times><ci id="S4.E1.m1.28.28.28.1.1.1.1.2.cmml" xref="S4.E1.m1.28.28.28.1.1.1.1.2">𝐼</ci><ci id="S4.E1.m1.28.28.28.1.1.1.1.3.cmml" xref="S4.E1.m1.28.28.28.1.1.1.1.3">𝐼</ci></apply></apply></apply></apply><apply id="S4.E1.m1.58.58.4e.cmml" xref="S4.E1.m1.62.62.8"><eq id="S4.E1.m1.32.32.32.5.1.1.cmml" xref="S4.E1.m1.32.32.32.5.1.1"></eq><share 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id="S4.E1.m1.57.57.3.3.1.1.1.3.cmml" xref="S4.E1.m1.62.62.8"><csymbol cd="ambiguous" id="S4.E1.m1.57.57.3.3.1.1.1.3.1.cmml" xref="S4.E1.m1.62.62.8">subscript</csymbol><apply id="S4.E1.m1.57.57.3.3.1.1.1.3.2.cmml" xref="S4.E1.m1.62.62.8"><csymbol cd="ambiguous" id="S4.E1.m1.57.57.3.3.1.1.1.3.2.1.cmml" xref="S4.E1.m1.62.62.8">superscript</csymbol><ci id="S4.E1.m1.37.37.37.10.6.6.cmml" xref="S4.E1.m1.37.37.37.10.6.6">𝐻</ci><ci id="S4.E1.m1.38.38.38.11.7.7.1.cmml" xref="S4.E1.m1.38.38.38.11.7.7.1">ℎ</ci></apply><ci id="S4.E1.m1.39.39.39.12.8.8.1.cmml" xref="S4.E1.m1.39.39.39.12.8.8.1">𝑞</ci></apply><apply id="S4.E1.m1.57.57.3.3.1.1.1.1.1.1.cmml" xref="S4.E1.m1.62.62.8"><csymbol cd="ambiguous" id="S4.E1.m1.57.57.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E1.m1.62.62.8">subscript</csymbol><ci id="S4.E1.m1.41.41.41.14.10.10.cmml" xref="S4.E1.m1.41.41.41.14.10.10">𝐴</ci><list id="S4.E1.m1.42.42.42.15.11.11.1.3.cmml" xref="S4.E1.m1.42.42.42.15.11.11.1.4"><times id="S4.E1.m1.42.42.42.15.11.11.1.1.cmml" xref="S4.E1.m1.42.42.42.15.11.11.1.1"></times><times id="S4.E1.m1.42.42.42.15.11.11.1.2.cmml" xref="S4.E1.m1.42.42.42.15.11.11.1.2"></times></list></apply></apply></apply></apply><apply id="S4.E1.m1.58.58.4g.cmml" xref="S4.E1.m1.62.62.8"><ci id="S4.E1.m1.45.45.45.18.14.14.cmml" xref="S4.E1.m1.45.45.45.18.14.14">⇒</ci><share href="https://arxiv.org/html/2503.14659v1#S4.E1.m1.57.57.3.3.cmml" id="S4.E1.m1.58.58.4h.cmml" xref="S4.E1.m1.4.4.4.4.4a"></share><apply id="S4.E1.m1.58.58.4.4.cmml" xref="S4.E1.m1.62.62.8"><times id="S4.E1.m1.58.58.4.4.2.cmml" xref="S4.E1.m1.4.4.4.4.4a"></times><apply id="S4.E1.m1.58.58.4.4.3.cmml" xref="S4.E1.m1.62.62.8"><csymbol cd="ambiguous" id="S4.E1.m1.58.58.4.4.3.1.cmml" xref="S4.E1.m1.4.4.4.4.4a">subscript</csymbol><ci id="S4.E1.m1.46.46.46.19.15.15.cmml" xref="S4.E1.m1.46.46.46.19.15.15">𝐻</ci><apply id="S4.E1.m1.47.47.47.20.16.16.1.cmml" xref="S4.E1.m1.47.47.47.20.16.16.1"><plus id="S4.E1.m1.47.47.47.20.16.16.1.1.cmml" xref="S4.E1.m1.47.47.47.20.16.16.1.1"></plus><ci id="S4.E1.m1.47.47.47.20.16.16.1.2.cmml" xref="S4.E1.m1.47.47.47.20.16.16.1.2">𝑝</ci><ci id="S4.E1.m1.47.47.47.20.16.16.1.3.cmml" xref="S4.E1.m1.47.47.47.20.16.16.1.3">𝑞</ci></apply></apply><apply id="S4.E1.m1.58.58.4.4.1.1.1.cmml" xref="S4.E1.m1.62.62.8"><times id="S4.E1.m1.58.58.4.4.1.1.1.2.cmml" xref="S4.E1.m1.62.62.8"></times><ci id="S4.E1.m1.49.49.49.22.18.18.cmml" xref="S4.E1.m1.49.49.49.22.18.18">Tot</ci><apply id="S4.E1.m1.58.58.4.4.1.1.1.1.1.1.cmml" xref="S4.E1.m1.62.62.8"><csymbol cd="ambiguous" id="S4.E1.m1.58.58.4.4.1.1.1.1.1.1.1.cmml" xref="S4.E1.m1.62.62.8">subscript</csymbol><ci id="S4.E1.m1.51.51.51.24.20.20.cmml" xref="S4.E1.m1.51.51.51.24.20.20">𝐴</ci><list id="S4.E1.m1.52.52.52.25.21.21.1.3.cmml" xref="S4.E1.m1.52.52.52.25.21.21.1.4"><times id="S4.E1.m1.52.52.52.25.21.21.1.1.cmml" xref="S4.E1.m1.52.52.52.25.21.21.1.1"></times><times id="S4.E1.m1.52.52.52.25.21.21.1.2.cmml" xref="S4.E1.m1.52.52.52.25.21.21.1.2"></times></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m1.62c">\begin{split}{}^{I}E^{2}_{p,q}&=H^{h}_{p}(H^{v}_{q}(A_{*,*}))\Rightarrow H_{p+% q}(\mathrm{Tot}(A_{*,*}))\\ {}^{II}E^{2}_{p,q}&=H^{v}_{p}(H^{h}_{q}(A_{*,*}))\Rightarrow H_{p+q}(\mathrm{% Tot}(A_{*,*}))\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.62d">start_ROW start_CELL start_FLOATSUPERSCRIPT italic_I end_FLOATSUPERSCRIPT italic_E start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT end_CELL start_CELL = italic_H start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_H start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ) ) ⇒ italic_H start_POSTSUBSCRIPT italic_p + italic_q end_POSTSUBSCRIPT ( roman_Tot ( italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ) ) end_CELL end_ROW start_ROW start_CELL start_FLOATSUPERSCRIPT italic_I italic_I end_FLOATSUPERSCRIPT italic_E start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT end_CELL start_CELL = italic_H start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_H start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ) ) ⇒ italic_H start_POSTSUBSCRIPT italic_p + italic_q end_POSTSUBSCRIPT ( roman_Tot ( italic_A start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ) ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p4.2">These spectral sequences are natural with respect to morphisms between double complexes. As a consequence of this naturality, and by applying the Dold-Puppe theorem, we obtain the following.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.1.1.1">Proposition 4.4</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem4.p1.2.2">Let <math alttext="f:A\to B" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.1.m1.1"><semantics id="S4.Thmtheorem4.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem4.p1.1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S4.Thmtheorem4.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S4.Thmtheorem4.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml">A</mi><mo id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.1.m1.1b"><apply id="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1"><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.1">:</ci><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3"><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.1">→</ci><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2">𝐴</ci><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.1.m1.1c">f:A\to B</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.1.m1.1d">italic_f : italic_A → italic_B</annotation></semantics></math> be a morphism of bisimplicial <math alttext="R" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.2.m2.1"><semantics id="S4.Thmtheorem4.p1.2.2.m2.1a"><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.2.m2.1b"><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.2.m2.1d">italic_R</annotation></semantics></math>-modules.</span></p> <ol class="ltx_enumerate" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.3.1">If for every </span><math alttext="q\geq 0" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.1"><semantics id="S4.I1.i1.p1.1.m1.1a"><mrow id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml"><mi id="S4.I1.i1.p1.1.m1.1.1.2" xref="S4.I1.i1.p1.1.m1.1.1.2.cmml">q</mi><mo id="S4.I1.i1.p1.1.m1.1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S4.I1.i1.p1.1.m1.1.1.3" xref="S4.I1.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.1b"><apply id="S4.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1"><geq id="S4.I1.i1.p1.1.m1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.1"></geq><ci id="S4.I1.i1.p1.1.m1.1.1.2.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2">𝑞</ci><cn id="S4.I1.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I1.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.1c">q\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.1d">italic_q ≥ 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.3.2">, the induced map </span><math alttext="f_{*}:H^{h}_{*}(A_{*,q})\to H^{h}(B_{*,q})" class="ltx_Math" display="inline" id="S4.I1.i1.p1.2.m2.6"><semantics id="S4.I1.i1.p1.2.m2.6a"><mrow id="S4.I1.i1.p1.2.m2.6.6" xref="S4.I1.i1.p1.2.m2.6.6.cmml"><msub id="S4.I1.i1.p1.2.m2.6.6.4" xref="S4.I1.i1.p1.2.m2.6.6.4.cmml"><mi id="S4.I1.i1.p1.2.m2.6.6.4.2" xref="S4.I1.i1.p1.2.m2.6.6.4.2.cmml">f</mi><mo id="S4.I1.i1.p1.2.m2.6.6.4.3" xref="S4.I1.i1.p1.2.m2.6.6.4.3.cmml">∗</mo></msub><mo 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xref="S4.I1.i1.p1.2.m2.5.5.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.2.m2.5.5.1.1.3.2.1.cmml" xref="S4.I1.i1.p1.2.m2.5.5.1.1.3">superscript</csymbol><ci id="S4.I1.i1.p1.2.m2.5.5.1.1.3.2.2.cmml" xref="S4.I1.i1.p1.2.m2.5.5.1.1.3.2.2">𝐻</ci><ci id="S4.I1.i1.p1.2.m2.5.5.1.1.3.2.3.cmml" xref="S4.I1.i1.p1.2.m2.5.5.1.1.3.2.3">ℎ</ci></apply><times id="S4.I1.i1.p1.2.m2.5.5.1.1.3.3.cmml" xref="S4.I1.i1.p1.2.m2.5.5.1.1.3.3"></times></apply><apply id="S4.I1.i1.p1.2.m2.5.5.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.2.m2.5.5.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.5.5.1.1.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.2.m2.5.5.1.1.1.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.5.5.1.1.1.1.1.2">𝐴</ci><list id="S4.I1.i1.p1.2.m2.2.2.2.3.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.4"><times id="S4.I1.i1.p1.2.m2.1.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1.1"></times><ci id="S4.I1.i1.p1.2.m2.2.2.2.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.2">𝑞</ci></list></apply></apply><apply id="S4.I1.i1.p1.2.m2.6.6.2.2.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2"><times id="S4.I1.i1.p1.2.m2.6.6.2.2.2.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2.2"></times><apply id="S4.I1.i1.p1.2.m2.6.6.2.2.3.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.2.m2.6.6.2.2.3.1.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2.3">superscript</csymbol><ci id="S4.I1.i1.p1.2.m2.6.6.2.2.3.2.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2.3.2">𝐻</ci><ci id="S4.I1.i1.p1.2.m2.6.6.2.2.3.3.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2.3.3">ℎ</ci></apply><apply id="S4.I1.i1.p1.2.m2.6.6.2.2.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.2.m2.6.6.2.2.1.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.2.m2.6.6.2.2.1.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.6.6.2.2.1.1.1.2">𝐵</ci><list id="S4.I1.i1.p1.2.m2.4.4.2.3.cmml" xref="S4.I1.i1.p1.2.m2.4.4.2.4"><times id="S4.I1.i1.p1.2.m2.3.3.1.1.cmml" xref="S4.I1.i1.p1.2.m2.3.3.1.1"></times><ci id="S4.I1.i1.p1.2.m2.4.4.2.2.cmml" xref="S4.I1.i1.p1.2.m2.4.4.2.2">𝑞</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.2.m2.6c">f_{*}:H^{h}_{*}(A_{*,q})\to H^{h}(B_{*,q})</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.2.m2.6d">italic_f start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : italic_H start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ) → italic_H start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ( italic_B start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.3.3"> is an isomorphism, then </span><math alttext="(\mathrm{diag}f)_{*}:H_{*}(\mathrm{diag}A)\to H_{*}(\mathrm{diag}B)" class="ltx_Math" display="inline" id="S4.I1.i1.p1.3.m3.3"><semantics id="S4.I1.i1.p1.3.m3.3a"><mrow id="S4.I1.i1.p1.3.m3.3.3" xref="S4.I1.i1.p1.3.m3.3.3.cmml"><msub id="S4.I1.i1.p1.3.m3.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.cmml"><mrow id="S4.I1.i1.p1.3.m3.1.1.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.I1.i1.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.2" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.3" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.3.cmml">f</mi></mrow><mo id="S4.I1.i1.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.I1.i1.p1.3.m3.1.1.1.3" xref="S4.I1.i1.p1.3.m3.1.1.1.3.cmml">∗</mo></msub><mo id="S4.I1.i1.p1.3.m3.3.3.4" lspace="0.278em" rspace="0.278em" xref="S4.I1.i1.p1.3.m3.3.3.4.cmml">:</mo><mrow id="S4.I1.i1.p1.3.m3.3.3.3" xref="S4.I1.i1.p1.3.m3.3.3.3.cmml"><mrow id="S4.I1.i1.p1.3.m3.2.2.2.1" xref="S4.I1.i1.p1.3.m3.2.2.2.1.cmml"><msub id="S4.I1.i1.p1.3.m3.2.2.2.1.3" xref="S4.I1.i1.p1.3.m3.2.2.2.1.3.cmml"><mi id="S4.I1.i1.p1.3.m3.2.2.2.1.3.2" xref="S4.I1.i1.p1.3.m3.2.2.2.1.3.2.cmml">H</mi><mo id="S4.I1.i1.p1.3.m3.2.2.2.1.3.3" xref="S4.I1.i1.p1.3.m3.2.2.2.1.3.3.cmml">∗</mo></msub><mo id="S4.I1.i1.p1.3.m3.2.2.2.1.2" xref="S4.I1.i1.p1.3.m3.2.2.2.1.2.cmml">⁢</mo><mrow id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.cmml"><mo id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.2" stretchy="false" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.cmml"><mi id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.2" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.2.cmml">diag</mi><mo id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.1" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.3" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.3.cmml">A</mi></mrow><mo id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.3" stretchy="false" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i1.p1.3.m3.3.3.3.3" stretchy="false" xref="S4.I1.i1.p1.3.m3.3.3.3.3.cmml">→</mo><mrow id="S4.I1.i1.p1.3.m3.3.3.3.2" xref="S4.I1.i1.p1.3.m3.3.3.3.2.cmml"><msub id="S4.I1.i1.p1.3.m3.3.3.3.2.3" xref="S4.I1.i1.p1.3.m3.3.3.3.2.3.cmml"><mi id="S4.I1.i1.p1.3.m3.3.3.3.2.3.2" xref="S4.I1.i1.p1.3.m3.3.3.3.2.3.2.cmml">H</mi><mo id="S4.I1.i1.p1.3.m3.3.3.3.2.3.3" xref="S4.I1.i1.p1.3.m3.3.3.3.2.3.3.cmml">∗</mo></msub><mo id="S4.I1.i1.p1.3.m3.3.3.3.2.2" xref="S4.I1.i1.p1.3.m3.3.3.3.2.2.cmml">⁢</mo><mrow id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.cmml"><mo id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.2" stretchy="false" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.cmml">(</mo><mrow id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.cmml"><mi id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.2" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.2.cmml">diag</mi><mo id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.1" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.1.cmml">⁢</mo><mi id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.3" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.3.cmml">B</mi></mrow><mo id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.3" stretchy="false" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.3.m3.3b"><apply id="S4.I1.i1.p1.3.m3.3.3.cmml" xref="S4.I1.i1.p1.3.m3.3.3"><ci id="S4.I1.i1.p1.3.m3.3.3.4.cmml" xref="S4.I1.i1.p1.3.m3.3.3.4">:</ci><apply id="S4.I1.i1.p1.3.m3.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.3.m3.1.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1">subscript</csymbol><apply id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1"><times id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.1"></times><ci id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.2">diag</ci><ci id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.3">𝑓</ci></apply><times id="S4.I1.i1.p1.3.m3.1.1.1.3.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.3"></times></apply><apply id="S4.I1.i1.p1.3.m3.3.3.3.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3"><ci id="S4.I1.i1.p1.3.m3.3.3.3.3.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.3">→</ci><apply id="S4.I1.i1.p1.3.m3.2.2.2.1.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1"><times id="S4.I1.i1.p1.3.m3.2.2.2.1.2.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.2"></times><apply id="S4.I1.i1.p1.3.m3.2.2.2.1.3.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.3.m3.2.2.2.1.3.1.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.3">subscript</csymbol><ci id="S4.I1.i1.p1.3.m3.2.2.2.1.3.2.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.3.2">𝐻</ci><times id="S4.I1.i1.p1.3.m3.2.2.2.1.3.3.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.3.3"></times></apply><apply id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1"><times id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.1"></times><ci id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.2">diag</ci><ci id="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.3.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.1.1.1.1.3">𝐴</ci></apply></apply><apply id="S4.I1.i1.p1.3.m3.3.3.3.2.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2"><times id="S4.I1.i1.p1.3.m3.3.3.3.2.2.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.2"></times><apply id="S4.I1.i1.p1.3.m3.3.3.3.2.3.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.3.m3.3.3.3.2.3.1.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.3">subscript</csymbol><ci id="S4.I1.i1.p1.3.m3.3.3.3.2.3.2.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.3.2">𝐻</ci><times id="S4.I1.i1.p1.3.m3.3.3.3.2.3.3.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.3.3"></times></apply><apply id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1"><times id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.1"></times><ci id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.2">diag</ci><ci id="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.3.cmml" xref="S4.I1.i1.p1.3.m3.3.3.3.2.1.1.1.3">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.3.m3.3c">(\mathrm{diag}f)_{*}:H_{*}(\mathrm{diag}A)\to H_{*}(\mathrm{diag}B)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.3.m3.3d">( roman_diag italic_f ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : italic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( roman_diag italic_A ) → italic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( roman_diag italic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.3.4"> is an isomorphism.</span></p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.3.1">If for every </span><math alttext="p\geq 0" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.1"><semantics id="S4.I1.i2.p1.1.m1.1a"><mrow id="S4.I1.i2.p1.1.m1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.cmml"><mi id="S4.I1.i2.p1.1.m1.1.1.2" xref="S4.I1.i2.p1.1.m1.1.1.2.cmml">p</mi><mo id="S4.I1.i2.p1.1.m1.1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S4.I1.i2.p1.1.m1.1.1.3" xref="S4.I1.i2.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.1.m1.1b"><apply id="S4.I1.i2.p1.1.m1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1"><geq id="S4.I1.i2.p1.1.m1.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.1"></geq><ci id="S4.I1.i2.p1.1.m1.1.1.2.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2">𝑝</ci><cn id="S4.I1.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I1.i2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.1c">p\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.1d">italic_p ≥ 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.3.2">, the induced map </span><math alttext="f_{*}:H^{v}_{*}(A_{p,*})\to H^{v}(B_{p,*})" class="ltx_Math" display="inline" id="S4.I1.i2.p1.2.m2.6"><semantics id="S4.I1.i2.p1.2.m2.6a"><mrow id="S4.I1.i2.p1.2.m2.6.6" xref="S4.I1.i2.p1.2.m2.6.6.cmml"><msub id="S4.I1.i2.p1.2.m2.6.6.4" xref="S4.I1.i2.p1.2.m2.6.6.4.cmml"><mi id="S4.I1.i2.p1.2.m2.6.6.4.2" xref="S4.I1.i2.p1.2.m2.6.6.4.2.cmml">f</mi><mo id="S4.I1.i2.p1.2.m2.6.6.4.3" xref="S4.I1.i2.p1.2.m2.6.6.4.3.cmml">∗</mo></msub><mo id="S4.I1.i2.p1.2.m2.6.6.3" lspace="0.278em" rspace="0.278em" xref="S4.I1.i2.p1.2.m2.6.6.3.cmml">:</mo><mrow id="S4.I1.i2.p1.2.m2.6.6.2" xref="S4.I1.i2.p1.2.m2.6.6.2.cmml"><mrow id="S4.I1.i2.p1.2.m2.5.5.1.1" xref="S4.I1.i2.p1.2.m2.5.5.1.1.cmml"><msubsup 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id="S4.I1.i2.p1.3.m3.3.3.3.2.1.1.1.3.cmml" xref="S4.I1.i2.p1.3.m3.3.3.3.2.1.1.1.3">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.3.m3.3c">(\mathrm{diag}f)_{*}:H_{*}(\mathrm{diag}A)\to H_{*}(\mathrm{diag}B)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.3.m3.3d">( roman_diag italic_f ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : italic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( roman_diag italic_A ) → italic_H start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( roman_diag italic_B )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.3.4"> is an isomorphism.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S4.SS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.2.p1"> <p class="ltx_p" id="S4.SS2.2.p1.1">This follows from the spectral sequences above and from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S4.Thmtheorem3" title="Theorem 4.3 (Dold-Puppe Theorem [4]). ‣ 4.2. The Dold-Puppe Theorem ‣ 4. Bisimplicial objects and the Dold-Puppe theorem ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">4.3</span></a>. ∎</p> </div> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5. </span>Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">To prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a>, we will apply the argument used by Cegarra in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#bib.bib3" title="">3</a>]</cite>. We first recall the terminology used in the statement of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a>.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.16">Let <math alttext="\varphi:\mathcal{C}\to\mathcal{D}" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mrow id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml"><mi id="S5.p2.1.m1.1.1.2" xref="S5.p2.1.m1.1.1.2.cmml">φ</mi><mo id="S5.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S5.p2.1.m1.1.1.3" xref="S5.p2.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p2.1.m1.1.1.3.2" xref="S5.p2.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="S5.p2.1.m1.1.1.3.1" stretchy="false" xref="S5.p2.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S5.p2.1.m1.1.1.3.3" xref="S5.p2.1.m1.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><apply id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1"><ci id="S5.p2.1.m1.1.1.1.cmml" xref="S5.p2.1.m1.1.1.1">:</ci><ci id="S5.p2.1.m1.1.1.2.cmml" xref="S5.p2.1.m1.1.1.2">𝜑</ci><apply id="S5.p2.1.m1.1.1.3.cmml" xref="S5.p2.1.m1.1.1.3"><ci id="S5.p2.1.m1.1.1.3.1.cmml" xref="S5.p2.1.m1.1.1.3.1">→</ci><ci id="S5.p2.1.m1.1.1.3.2.cmml" xref="S5.p2.1.m1.1.1.3.2">𝒞</ci><ci id="S5.p2.1.m1.1.1.3.3.cmml" xref="S5.p2.1.m1.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">\varphi:\mathcal{C}\to\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">italic_φ : caligraphic_C → caligraphic_D</annotation></semantics></math> be a functor between two small categories, and <math alttext="R" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mi id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><ci id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">italic_R</annotation></semantics></math> be a commutative ring. We denote by <math alttext="G:=(\varphi/-):\mathcal{D}\to Cat" class="ltx_math_unparsed" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><mrow id="S5.p2.3.m3.1b"><mi id="S5.p2.3.m3.1.1">G</mi><mo id="S5.p2.3.m3.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S5.p2.3.m3.1.3"><mo id="S5.p2.3.m3.1.3.1" stretchy="false">(</mo><mi id="S5.p2.3.m3.1.3.2">φ</mi><mo id="S5.p2.3.m3.1.3.3" rspace="0em">/</mo><mo id="S5.p2.3.m3.1.3.4" lspace="0em" rspace="0em">−</mo><mo id="S5.p2.3.m3.1.3.5" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S5.p2.3.m3.1.4" rspace="0.278em">:</mo><mi class="ltx_font_mathcaligraphic" id="S5.p2.3.m3.1.5">𝒟</mi><mo id="S5.p2.3.m3.1.6" stretchy="false">→</mo><mi id="S5.p2.3.m3.1.7">C</mi><mi id="S5.p2.3.m3.1.8">a</mi><mi id="S5.p2.3.m3.1.9">t</mi></mrow><annotation encoding="application/x-tex" id="S5.p2.3.m3.1c">G:=(\varphi/-):\mathcal{D}\to Cat</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.1d">italic_G := ( italic_φ / - ) : caligraphic_D → italic_C italic_a italic_t</annotation></semantics></math> the functor that sends <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S5.p2.4.m4.1"><semantics id="S5.p2.4.m4.1a"><mrow id="S5.p2.4.m4.1.1" xref="S5.p2.4.m4.1.1.cmml"><mi id="S5.p2.4.m4.1.1.2" xref="S5.p2.4.m4.1.1.2.cmml">d</mi><mo id="S5.p2.4.m4.1.1.1" xref="S5.p2.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S5.p2.4.m4.1.1.3" xref="S5.p2.4.m4.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.4.m4.1b"><apply id="S5.p2.4.m4.1.1.cmml" xref="S5.p2.4.m4.1.1"><in id="S5.p2.4.m4.1.1.1.cmml" xref="S5.p2.4.m4.1.1.1"></in><ci id="S5.p2.4.m4.1.1.2.cmml" xref="S5.p2.4.m4.1.1.2">𝑑</ci><ci id="S5.p2.4.m4.1.1.3.cmml" xref="S5.p2.4.m4.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.1d">italic_d ∈ caligraphic_D</annotation></semantics></math> to the comma category <math alttext="\varphi/d" class="ltx_Math" display="inline" id="S5.p2.5.m5.1"><semantics id="S5.p2.5.m5.1a"><mrow id="S5.p2.5.m5.1.1" xref="S5.p2.5.m5.1.1.cmml"><mi id="S5.p2.5.m5.1.1.2" xref="S5.p2.5.m5.1.1.2.cmml">φ</mi><mo id="S5.p2.5.m5.1.1.1" xref="S5.p2.5.m5.1.1.1.cmml">/</mo><mi id="S5.p2.5.m5.1.1.3" xref="S5.p2.5.m5.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.5.m5.1b"><apply id="S5.p2.5.m5.1.1.cmml" xref="S5.p2.5.m5.1.1"><divide id="S5.p2.5.m5.1.1.1.cmml" xref="S5.p2.5.m5.1.1.1"></divide><ci id="S5.p2.5.m5.1.1.2.cmml" xref="S5.p2.5.m5.1.1.2">𝜑</ci><ci id="S5.p2.5.m5.1.1.3.cmml" xref="S5.p2.5.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.5.m5.1c">\varphi/d</annotation><annotation encoding="application/x-llamapun" id="S5.p2.5.m5.1d">italic_φ / italic_d</annotation></semantics></math>. Consider the simplicial replacement <math alttext="N(\mathcal{D};G)" class="ltx_Math" display="inline" id="S5.p2.6.m6.2"><semantics id="S5.p2.6.m6.2a"><mrow id="S5.p2.6.m6.2.3" xref="S5.p2.6.m6.2.3.cmml"><mi id="S5.p2.6.m6.2.3.2" xref="S5.p2.6.m6.2.3.2.cmml">N</mi><mo id="S5.p2.6.m6.2.3.1" xref="S5.p2.6.m6.2.3.1.cmml">⁢</mo><mrow id="S5.p2.6.m6.2.3.3.2" xref="S5.p2.6.m6.2.3.3.1.cmml"><mo id="S5.p2.6.m6.2.3.3.2.1" stretchy="false" xref="S5.p2.6.m6.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.p2.6.m6.1.1" xref="S5.p2.6.m6.1.1.cmml">𝒟</mi><mo id="S5.p2.6.m6.2.3.3.2.2" xref="S5.p2.6.m6.2.3.3.1.cmml">;</mo><mi id="S5.p2.6.m6.2.2" xref="S5.p2.6.m6.2.2.cmml">G</mi><mo id="S5.p2.6.m6.2.3.3.2.3" stretchy="false" xref="S5.p2.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.6.m6.2b"><apply id="S5.p2.6.m6.2.3.cmml" xref="S5.p2.6.m6.2.3"><times id="S5.p2.6.m6.2.3.1.cmml" xref="S5.p2.6.m6.2.3.1"></times><ci id="S5.p2.6.m6.2.3.2.cmml" xref="S5.p2.6.m6.2.3.2">𝑁</ci><list id="S5.p2.6.m6.2.3.3.1.cmml" xref="S5.p2.6.m6.2.3.3.2"><ci id="S5.p2.6.m6.1.1.cmml" xref="S5.p2.6.m6.1.1">𝒟</ci><ci id="S5.p2.6.m6.2.2.cmml" xref="S5.p2.6.m6.2.2">𝐺</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.6.m6.2c">N(\mathcal{D};G)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.6.m6.2d">italic_N ( caligraphic_D ; italic_G )</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S5.p2.7.m7.1"><semantics id="S5.p2.7.m7.1a"><mi id="S5.p2.7.m7.1.1" xref="S5.p2.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.p2.7.m7.1b"><ci id="S5.p2.7.m7.1.1.cmml" xref="S5.p2.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.p2.7.m7.1d">italic_G</annotation></semantics></math>. For each <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S5.p2.8.m8.1"><semantics id="S5.p2.8.m8.1a"><mrow id="S5.p2.8.m8.1.1" xref="S5.p2.8.m8.1.1.cmml"><mi id="S5.p2.8.m8.1.1.2" xref="S5.p2.8.m8.1.1.2.cmml">d</mi><mo id="S5.p2.8.m8.1.1.1" xref="S5.p2.8.m8.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S5.p2.8.m8.1.1.3" xref="S5.p2.8.m8.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.8.m8.1b"><apply id="S5.p2.8.m8.1.1.cmml" xref="S5.p2.8.m8.1.1"><in id="S5.p2.8.m8.1.1.1.cmml" xref="S5.p2.8.m8.1.1.1"></in><ci id="S5.p2.8.m8.1.1.2.cmml" xref="S5.p2.8.m8.1.1.2">𝑑</ci><ci id="S5.p2.8.m8.1.1.3.cmml" xref="S5.p2.8.m8.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.8.m8.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.8.m8.1d">italic_d ∈ caligraphic_D</annotation></semantics></math>, we can identify <math alttext="NG(d)_{q}" class="ltx_Math" display="inline" id="S5.p2.9.m9.1"><semantics id="S5.p2.9.m9.1a"><mrow id="S5.p2.9.m9.1.2" xref="S5.p2.9.m9.1.2.cmml"><mi id="S5.p2.9.m9.1.2.2" xref="S5.p2.9.m9.1.2.2.cmml">N</mi><mo id="S5.p2.9.m9.1.2.1" xref="S5.p2.9.m9.1.2.1.cmml">⁢</mo><mi id="S5.p2.9.m9.1.2.3" xref="S5.p2.9.m9.1.2.3.cmml">G</mi><mo id="S5.p2.9.m9.1.2.1a" xref="S5.p2.9.m9.1.2.1.cmml">⁢</mo><msub id="S5.p2.9.m9.1.2.4" xref="S5.p2.9.m9.1.2.4.cmml"><mrow id="S5.p2.9.m9.1.2.4.2.2" xref="S5.p2.9.m9.1.2.4.cmml"><mo id="S5.p2.9.m9.1.2.4.2.2.1" stretchy="false" xref="S5.p2.9.m9.1.2.4.cmml">(</mo><mi id="S5.p2.9.m9.1.1" xref="S5.p2.9.m9.1.1.cmml">d</mi><mo id="S5.p2.9.m9.1.2.4.2.2.2" stretchy="false" xref="S5.p2.9.m9.1.2.4.cmml">)</mo></mrow><mi id="S5.p2.9.m9.1.2.4.3" xref="S5.p2.9.m9.1.2.4.3.cmml">q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.9.m9.1b"><apply id="S5.p2.9.m9.1.2.cmml" xref="S5.p2.9.m9.1.2"><times id="S5.p2.9.m9.1.2.1.cmml" xref="S5.p2.9.m9.1.2.1"></times><ci id="S5.p2.9.m9.1.2.2.cmml" xref="S5.p2.9.m9.1.2.2">𝑁</ci><ci id="S5.p2.9.m9.1.2.3.cmml" xref="S5.p2.9.m9.1.2.3">𝐺</ci><apply id="S5.p2.9.m9.1.2.4.cmml" xref="S5.p2.9.m9.1.2.4"><csymbol cd="ambiguous" id="S5.p2.9.m9.1.2.4.1.cmml" xref="S5.p2.9.m9.1.2.4">subscript</csymbol><ci id="S5.p2.9.m9.1.1.cmml" xref="S5.p2.9.m9.1.1">𝑑</ci><ci id="S5.p2.9.m9.1.2.4.3.cmml" xref="S5.p2.9.m9.1.2.4.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.9.m9.1c">NG(d)_{q}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.9.m9.1d">italic_N italic_G ( italic_d ) start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> with the set of pairs <math alttext="(\tau,\mu)" class="ltx_Math" display="inline" id="S5.p2.10.m10.2"><semantics id="S5.p2.10.m10.2a"><mrow id="S5.p2.10.m10.2.3.2" xref="S5.p2.10.m10.2.3.1.cmml"><mo id="S5.p2.10.m10.2.3.2.1" stretchy="false" xref="S5.p2.10.m10.2.3.1.cmml">(</mo><mi id="S5.p2.10.m10.1.1" xref="S5.p2.10.m10.1.1.cmml">τ</mi><mo id="S5.p2.10.m10.2.3.2.2" xref="S5.p2.10.m10.2.3.1.cmml">,</mo><mi id="S5.p2.10.m10.2.2" xref="S5.p2.10.m10.2.2.cmml">μ</mi><mo id="S5.p2.10.m10.2.3.2.3" stretchy="false" xref="S5.p2.10.m10.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.10.m10.2b"><interval closure="open" id="S5.p2.10.m10.2.3.1.cmml" xref="S5.p2.10.m10.2.3.2"><ci id="S5.p2.10.m10.1.1.cmml" xref="S5.p2.10.m10.1.1">𝜏</ci><ci id="S5.p2.10.m10.2.2.cmml" xref="S5.p2.10.m10.2.2">𝜇</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.10.m10.2c">(\tau,\mu)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.10.m10.2d">( italic_τ , italic_μ )</annotation></semantics></math> where <math alttext="\tau=(c_{0}\to\cdots\to c_{q})\in N\mathcal{C}_{q}" class="ltx_Math" display="inline" id="S5.p2.11.m11.1"><semantics id="S5.p2.11.m11.1a"><mrow id="S5.p2.11.m11.1.1" xref="S5.p2.11.m11.1.1.cmml"><mi id="S5.p2.11.m11.1.1.3" xref="S5.p2.11.m11.1.1.3.cmml">τ</mi><mo id="S5.p2.11.m11.1.1.4" xref="S5.p2.11.m11.1.1.4.cmml">=</mo><mrow id="S5.p2.11.m11.1.1.1.1" xref="S5.p2.11.m11.1.1.1.1.1.cmml"><mo id="S5.p2.11.m11.1.1.1.1.2" stretchy="false" xref="S5.p2.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S5.p2.11.m11.1.1.1.1.1" xref="S5.p2.11.m11.1.1.1.1.1.cmml"><msub id="S5.p2.11.m11.1.1.1.1.1.2" xref="S5.p2.11.m11.1.1.1.1.1.2.cmml"><mi id="S5.p2.11.m11.1.1.1.1.1.2.2" xref="S5.p2.11.m11.1.1.1.1.1.2.2.cmml">c</mi><mn id="S5.p2.11.m11.1.1.1.1.1.2.3" xref="S5.p2.11.m11.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S5.p2.11.m11.1.1.1.1.1.3" stretchy="false" xref="S5.p2.11.m11.1.1.1.1.1.3.cmml">→</mo><mi id="S5.p2.11.m11.1.1.1.1.1.4" mathvariant="normal" xref="S5.p2.11.m11.1.1.1.1.1.4.cmml">⋯</mi><mo id="S5.p2.11.m11.1.1.1.1.1.5" stretchy="false" xref="S5.p2.11.m11.1.1.1.1.1.5.cmml">→</mo><msub id="S5.p2.11.m11.1.1.1.1.1.6" xref="S5.p2.11.m11.1.1.1.1.1.6.cmml"><mi id="S5.p2.11.m11.1.1.1.1.1.6.2" xref="S5.p2.11.m11.1.1.1.1.1.6.2.cmml">c</mi><mi id="S5.p2.11.m11.1.1.1.1.1.6.3" xref="S5.p2.11.m11.1.1.1.1.1.6.3.cmml">q</mi></msub></mrow><mo id="S5.p2.11.m11.1.1.1.1.3" stretchy="false" xref="S5.p2.11.m11.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.p2.11.m11.1.1.5" xref="S5.p2.11.m11.1.1.5.cmml">∈</mo><mrow id="S5.p2.11.m11.1.1.6" xref="S5.p2.11.m11.1.1.6.cmml"><mi id="S5.p2.11.m11.1.1.6.2" xref="S5.p2.11.m11.1.1.6.2.cmml">N</mi><mo id="S5.p2.11.m11.1.1.6.1" xref="S5.p2.11.m11.1.1.6.1.cmml">⁢</mo><msub id="S5.p2.11.m11.1.1.6.3" xref="S5.p2.11.m11.1.1.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p2.11.m11.1.1.6.3.2" xref="S5.p2.11.m11.1.1.6.3.2.cmml">𝒞</mi><mi id="S5.p2.11.m11.1.1.6.3.3" xref="S5.p2.11.m11.1.1.6.3.3.cmml">q</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.11.m11.1b"><apply id="S5.p2.11.m11.1.1.cmml" xref="S5.p2.11.m11.1.1"><and id="S5.p2.11.m11.1.1a.cmml" xref="S5.p2.11.m11.1.1"></and><apply id="S5.p2.11.m11.1.1b.cmml" xref="S5.p2.11.m11.1.1"><eq id="S5.p2.11.m11.1.1.4.cmml" xref="S5.p2.11.m11.1.1.4"></eq><ci id="S5.p2.11.m11.1.1.3.cmml" xref="S5.p2.11.m11.1.1.3">𝜏</ci><apply id="S5.p2.11.m11.1.1.1.1.1.cmml" xref="S5.p2.11.m11.1.1.1.1"><and id="S5.p2.11.m11.1.1.1.1.1a.cmml" xref="S5.p2.11.m11.1.1.1.1"></and><apply id="S5.p2.11.m11.1.1.1.1.1b.cmml" xref="S5.p2.11.m11.1.1.1.1"><ci id="S5.p2.11.m11.1.1.1.1.1.3.cmml" xref="S5.p2.11.m11.1.1.1.1.1.3">→</ci><apply id="S5.p2.11.m11.1.1.1.1.1.2.cmml" xref="S5.p2.11.m11.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.p2.11.m11.1.1.1.1.1.2.1.cmml" xref="S5.p2.11.m11.1.1.1.1.1.2">subscript</csymbol><ci id="S5.p2.11.m11.1.1.1.1.1.2.2.cmml" xref="S5.p2.11.m11.1.1.1.1.1.2.2">𝑐</ci><cn id="S5.p2.11.m11.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.p2.11.m11.1.1.1.1.1.2.3">0</cn></apply><ci id="S5.p2.11.m11.1.1.1.1.1.4.cmml" xref="S5.p2.11.m11.1.1.1.1.1.4">⋯</ci></apply><apply id="S5.p2.11.m11.1.1.1.1.1c.cmml" xref="S5.p2.11.m11.1.1.1.1"><ci id="S5.p2.11.m11.1.1.1.1.1.5.cmml" xref="S5.p2.11.m11.1.1.1.1.1.5">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.p2.11.m11.1.1.1.1.1.4.cmml" id="S5.p2.11.m11.1.1.1.1.1d.cmml" xref="S5.p2.11.m11.1.1.1.1"></share><apply id="S5.p2.11.m11.1.1.1.1.1.6.cmml" xref="S5.p2.11.m11.1.1.1.1.1.6"><csymbol cd="ambiguous" id="S5.p2.11.m11.1.1.1.1.1.6.1.cmml" xref="S5.p2.11.m11.1.1.1.1.1.6">subscript</csymbol><ci id="S5.p2.11.m11.1.1.1.1.1.6.2.cmml" xref="S5.p2.11.m11.1.1.1.1.1.6.2">𝑐</ci><ci id="S5.p2.11.m11.1.1.1.1.1.6.3.cmml" xref="S5.p2.11.m11.1.1.1.1.1.6.3">𝑞</ci></apply></apply></apply></apply><apply id="S5.p2.11.m11.1.1c.cmml" xref="S5.p2.11.m11.1.1"><in id="S5.p2.11.m11.1.1.5.cmml" xref="S5.p2.11.m11.1.1.5"></in><share href="https://arxiv.org/html/2503.14659v1#S5.p2.11.m11.1.1.1.cmml" id="S5.p2.11.m11.1.1d.cmml" xref="S5.p2.11.m11.1.1"></share><apply id="S5.p2.11.m11.1.1.6.cmml" xref="S5.p2.11.m11.1.1.6"><times id="S5.p2.11.m11.1.1.6.1.cmml" xref="S5.p2.11.m11.1.1.6.1"></times><ci id="S5.p2.11.m11.1.1.6.2.cmml" xref="S5.p2.11.m11.1.1.6.2">𝑁</ci><apply id="S5.p2.11.m11.1.1.6.3.cmml" xref="S5.p2.11.m11.1.1.6.3"><csymbol cd="ambiguous" id="S5.p2.11.m11.1.1.6.3.1.cmml" xref="S5.p2.11.m11.1.1.6.3">subscript</csymbol><ci id="S5.p2.11.m11.1.1.6.3.2.cmml" xref="S5.p2.11.m11.1.1.6.3.2">𝒞</ci><ci id="S5.p2.11.m11.1.1.6.3.3.cmml" xref="S5.p2.11.m11.1.1.6.3.3">𝑞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.11.m11.1c">\tau=(c_{0}\to\cdots\to c_{q})\in N\mathcal{C}_{q}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.11.m11.1d">italic_τ = ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ⋯ → italic_c start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mu:\varphi(c_{q})\to d" class="ltx_Math" display="inline" id="S5.p2.12.m12.1"><semantics id="S5.p2.12.m12.1a"><mrow id="S5.p2.12.m12.1.1" xref="S5.p2.12.m12.1.1.cmml"><mi id="S5.p2.12.m12.1.1.3" xref="S5.p2.12.m12.1.1.3.cmml">μ</mi><mo id="S5.p2.12.m12.1.1.2" lspace="0.278em" rspace="0.278em" xref="S5.p2.12.m12.1.1.2.cmml">:</mo><mrow id="S5.p2.12.m12.1.1.1" xref="S5.p2.12.m12.1.1.1.cmml"><mrow id="S5.p2.12.m12.1.1.1.1" xref="S5.p2.12.m12.1.1.1.1.cmml"><mi id="S5.p2.12.m12.1.1.1.1.3" xref="S5.p2.12.m12.1.1.1.1.3.cmml">φ</mi><mo id="S5.p2.12.m12.1.1.1.1.2" xref="S5.p2.12.m12.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.p2.12.m12.1.1.1.1.1.1" xref="S5.p2.12.m12.1.1.1.1.1.1.1.cmml"><mo id="S5.p2.12.m12.1.1.1.1.1.1.2" stretchy="false" xref="S5.p2.12.m12.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.p2.12.m12.1.1.1.1.1.1.1" xref="S5.p2.12.m12.1.1.1.1.1.1.1.cmml"><mi id="S5.p2.12.m12.1.1.1.1.1.1.1.2" xref="S5.p2.12.m12.1.1.1.1.1.1.1.2.cmml">c</mi><mi id="S5.p2.12.m12.1.1.1.1.1.1.1.3" xref="S5.p2.12.m12.1.1.1.1.1.1.1.3.cmml">q</mi></msub><mo id="S5.p2.12.m12.1.1.1.1.1.1.3" stretchy="false" xref="S5.p2.12.m12.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p2.12.m12.1.1.1.2" stretchy="false" xref="S5.p2.12.m12.1.1.1.2.cmml">→</mo><mi id="S5.p2.12.m12.1.1.1.3" xref="S5.p2.12.m12.1.1.1.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.12.m12.1b"><apply id="S5.p2.12.m12.1.1.cmml" xref="S5.p2.12.m12.1.1"><ci id="S5.p2.12.m12.1.1.2.cmml" xref="S5.p2.12.m12.1.1.2">:</ci><ci id="S5.p2.12.m12.1.1.3.cmml" xref="S5.p2.12.m12.1.1.3">𝜇</ci><apply id="S5.p2.12.m12.1.1.1.cmml" xref="S5.p2.12.m12.1.1.1"><ci id="S5.p2.12.m12.1.1.1.2.cmml" xref="S5.p2.12.m12.1.1.1.2">→</ci><apply id="S5.p2.12.m12.1.1.1.1.cmml" xref="S5.p2.12.m12.1.1.1.1"><times id="S5.p2.12.m12.1.1.1.1.2.cmml" xref="S5.p2.12.m12.1.1.1.1.2"></times><ci id="S5.p2.12.m12.1.1.1.1.3.cmml" xref="S5.p2.12.m12.1.1.1.1.3">𝜑</ci><apply id="S5.p2.12.m12.1.1.1.1.1.1.1.cmml" xref="S5.p2.12.m12.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p2.12.m12.1.1.1.1.1.1.1.1.cmml" xref="S5.p2.12.m12.1.1.1.1.1.1">subscript</csymbol><ci id="S5.p2.12.m12.1.1.1.1.1.1.1.2.cmml" xref="S5.p2.12.m12.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S5.p2.12.m12.1.1.1.1.1.1.1.3.cmml" xref="S5.p2.12.m12.1.1.1.1.1.1.1.3">𝑞</ci></apply></apply><ci id="S5.p2.12.m12.1.1.1.3.cmml" xref="S5.p2.12.m12.1.1.1.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.12.m12.1c">\mu:\varphi(c_{q})\to d</annotation><annotation encoding="application/x-llamapun" id="S5.p2.12.m12.1d">italic_μ : italic_φ ( italic_c start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) → italic_d</annotation></semantics></math> is a morphism in <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S5.p2.13.m13.1"><semantics id="S5.p2.13.m13.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p2.13.m13.1.1" xref="S5.p2.13.m13.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S5.p2.13.m13.1b"><ci id="S5.p2.13.m13.1.1.cmml" xref="S5.p2.13.m13.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.13.m13.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.13.m13.1d">caligraphic_D</annotation></semantics></math>. This identification allows us to write the <math alttext="(p,q)" class="ltx_Math" display="inline" id="S5.p2.14.m14.2"><semantics id="S5.p2.14.m14.2a"><mrow id="S5.p2.14.m14.2.3.2" xref="S5.p2.14.m14.2.3.1.cmml"><mo id="S5.p2.14.m14.2.3.2.1" stretchy="false" xref="S5.p2.14.m14.2.3.1.cmml">(</mo><mi id="S5.p2.14.m14.1.1" xref="S5.p2.14.m14.1.1.cmml">p</mi><mo id="S5.p2.14.m14.2.3.2.2" xref="S5.p2.14.m14.2.3.1.cmml">,</mo><mi id="S5.p2.14.m14.2.2" xref="S5.p2.14.m14.2.2.cmml">q</mi><mo id="S5.p2.14.m14.2.3.2.3" stretchy="false" xref="S5.p2.14.m14.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.14.m14.2b"><interval closure="open" id="S5.p2.14.m14.2.3.1.cmml" xref="S5.p2.14.m14.2.3.2"><ci id="S5.p2.14.m14.1.1.cmml" xref="S5.p2.14.m14.1.1">𝑝</ci><ci id="S5.p2.14.m14.2.2.cmml" xref="S5.p2.14.m14.2.2">𝑞</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.14.m14.2c">(p,q)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.14.m14.2d">( italic_p , italic_q )</annotation></semantics></math>-simplices in <math alttext="N(\mathcal{D};G)" class="ltx_Math" display="inline" id="S5.p2.15.m15.2"><semantics id="S5.p2.15.m15.2a"><mrow id="S5.p2.15.m15.2.3" xref="S5.p2.15.m15.2.3.cmml"><mi id="S5.p2.15.m15.2.3.2" xref="S5.p2.15.m15.2.3.2.cmml">N</mi><mo id="S5.p2.15.m15.2.3.1" xref="S5.p2.15.m15.2.3.1.cmml">⁢</mo><mrow id="S5.p2.15.m15.2.3.3.2" xref="S5.p2.15.m15.2.3.3.1.cmml"><mo id="S5.p2.15.m15.2.3.3.2.1" stretchy="false" xref="S5.p2.15.m15.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.p2.15.m15.1.1" xref="S5.p2.15.m15.1.1.cmml">𝒟</mi><mo id="S5.p2.15.m15.2.3.3.2.2" xref="S5.p2.15.m15.2.3.3.1.cmml">;</mo><mi id="S5.p2.15.m15.2.2" xref="S5.p2.15.m15.2.2.cmml">G</mi><mo id="S5.p2.15.m15.2.3.3.2.3" stretchy="false" xref="S5.p2.15.m15.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.15.m15.2b"><apply id="S5.p2.15.m15.2.3.cmml" xref="S5.p2.15.m15.2.3"><times id="S5.p2.15.m15.2.3.1.cmml" xref="S5.p2.15.m15.2.3.1"></times><ci id="S5.p2.15.m15.2.3.2.cmml" xref="S5.p2.15.m15.2.3.2">𝑁</ci><list id="S5.p2.15.m15.2.3.3.1.cmml" xref="S5.p2.15.m15.2.3.3.2"><ci id="S5.p2.15.m15.1.1.cmml" xref="S5.p2.15.m15.1.1">𝒟</ci><ci id="S5.p2.15.m15.2.2.cmml" xref="S5.p2.15.m15.2.2">𝐺</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.15.m15.2c">N(\mathcal{D};G)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.15.m15.2d">italic_N ( caligraphic_D ; italic_G )</annotation></semantics></math> as the triples <math alttext="(\sigma,\tau,\mu)" class="ltx_Math" display="inline" id="S5.p2.16.m16.3"><semantics id="S5.p2.16.m16.3a"><mrow id="S5.p2.16.m16.3.4.2" xref="S5.p2.16.m16.3.4.1.cmml"><mo id="S5.p2.16.m16.3.4.2.1" stretchy="false" xref="S5.p2.16.m16.3.4.1.cmml">(</mo><mi id="S5.p2.16.m16.1.1" xref="S5.p2.16.m16.1.1.cmml">σ</mi><mo id="S5.p2.16.m16.3.4.2.2" xref="S5.p2.16.m16.3.4.1.cmml">,</mo><mi id="S5.p2.16.m16.2.2" xref="S5.p2.16.m16.2.2.cmml">τ</mi><mo id="S5.p2.16.m16.3.4.2.3" xref="S5.p2.16.m16.3.4.1.cmml">,</mo><mi id="S5.p2.16.m16.3.3" xref="S5.p2.16.m16.3.3.cmml">μ</mi><mo id="S5.p2.16.m16.3.4.2.4" stretchy="false" xref="S5.p2.16.m16.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.16.m16.3b"><vector id="S5.p2.16.m16.3.4.1.cmml" xref="S5.p2.16.m16.3.4.2"><ci id="S5.p2.16.m16.1.1.cmml" xref="S5.p2.16.m16.1.1">𝜎</ci><ci id="S5.p2.16.m16.2.2.cmml" xref="S5.p2.16.m16.2.2">𝜏</ci><ci id="S5.p2.16.m16.3.3.cmml" xref="S5.p2.16.m16.3.3">𝜇</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.16.m16.3c">(\sigma,\tau,\mu)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.16.m16.3d">( italic_σ , italic_τ , italic_μ )</annotation></semantics></math> where</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex51"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma=(d_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{p}}\,}d_{p})\in N\mathcal{D}% _{p},\quad\tau=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\beta_{1}}\,}% \cdots\smash{\,\mathop{\longrightarrow}\limits^{\beta_{q}}\,}c_{q})\in N% \mathcal{C}_{q}" class="ltx_Math" display="block" id="S5.Ex51.m1.2"><semantics id="S5.Ex51.m1.2a"><mrow id="S5.Ex51.m1.2.2.2" xref="S5.Ex51.m1.2.2.3.cmml"><mrow id="S5.Ex51.m1.1.1.1.1" xref="S5.Ex51.m1.1.1.1.1.cmml"><mi id="S5.Ex51.m1.1.1.1.1.3" xref="S5.Ex51.m1.1.1.1.1.3.cmml">σ</mi><mo id="S5.Ex51.m1.1.1.1.1.4" xref="S5.Ex51.m1.1.1.1.1.4.cmml">=</mo><mrow id="S5.Ex51.m1.1.1.1.1.1.1" xref="S5.Ex51.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex51.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex51.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex51.m1.1.1.1.1.1.1.1" xref="S5.Ex51.m1.1.1.1.1.1.1.1.cmml"><msub id="S5.Ex51.m1.1.1.1.1.1.1.1.2" xref="S5.Ex51.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex51.m1.1.1.1.1.1.1.1.2.2" xref="S5.Ex51.m1.1.1.1.1.1.1.1.2.2.cmml">d</mi><mn id="S5.Ex51.m1.1.1.1.1.1.1.1.2.3" xref="S5.Ex51.m1.1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S5.Ex51.m1.1.1.1.1.1.1.1.1" lspace="0.167em" xref="S5.Ex51.m1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.Ex51.m1.1.1.1.1.1.1.1.3" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.cmml"><mover id="S5.Ex51.m1.1.1.1.1.1.1.1.3.1" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.2.cmml">⟶</mo><msub id="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.3" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.3.cmml"><mi id="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.3.2" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.3.2.cmml">α</mi><mn id="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.3.3" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.1.3.3.cmml">1</mn></msub></mover><mrow id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.cmml"><mi id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.2" mathvariant="normal" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.2.cmml">⋯</mi><mo id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.1" lspace="0.337em" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.cmml"><mover id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.cmml"><mo id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.3" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.3.cmml"><mi id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.3.2" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.3.2.cmml">α</mi><mi id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.3.3" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.1.3.3.cmml">p</mi></msub></mover><msub id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.2" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.2.cmml"><mi id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.2.2" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.2.2.cmml">d</mi><mi id="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.2.3" xref="S5.Ex51.m1.1.1.1.1.1.1.1.3.2.3.2.3.cmml">p</mi></msub></mrow></mrow></mrow></mrow><mo id="S5.Ex51.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex51.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex51.m1.1.1.1.1.5" xref="S5.Ex51.m1.1.1.1.1.5.cmml">∈</mo><mrow id="S5.Ex51.m1.1.1.1.1.6" xref="S5.Ex51.m1.1.1.1.1.6.cmml"><mi id="S5.Ex51.m1.1.1.1.1.6.2" xref="S5.Ex51.m1.1.1.1.1.6.2.cmml">N</mi><mo id="S5.Ex51.m1.1.1.1.1.6.1" xref="S5.Ex51.m1.1.1.1.1.6.1.cmml">⁢</mo><msub id="S5.Ex51.m1.1.1.1.1.6.3" xref="S5.Ex51.m1.1.1.1.1.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex51.m1.1.1.1.1.6.3.2" xref="S5.Ex51.m1.1.1.1.1.6.3.2.cmml">𝒟</mi><mi id="S5.Ex51.m1.1.1.1.1.6.3.3" xref="S5.Ex51.m1.1.1.1.1.6.3.3.cmml">p</mi></msub></mrow></mrow><mo id="S5.Ex51.m1.2.2.2.3" rspace="1.167em" xref="S5.Ex51.m1.2.2.3a.cmml">,</mo><mrow id="S5.Ex51.m1.2.2.2.2" xref="S5.Ex51.m1.2.2.2.2.cmml"><mi id="S5.Ex51.m1.2.2.2.2.3" xref="S5.Ex51.m1.2.2.2.2.3.cmml">τ</mi><mo id="S5.Ex51.m1.2.2.2.2.4" xref="S5.Ex51.m1.2.2.2.2.4.cmml">=</mo><mrow id="S5.Ex51.m1.2.2.2.2.1.1" xref="S5.Ex51.m1.2.2.2.2.1.1.1.cmml"><mo id="S5.Ex51.m1.2.2.2.2.1.1.2" stretchy="false" xref="S5.Ex51.m1.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.Ex51.m1.2.2.2.2.1.1.1" xref="S5.Ex51.m1.2.2.2.2.1.1.1.cmml"><msub id="S5.Ex51.m1.2.2.2.2.1.1.1.2" xref="S5.Ex51.m1.2.2.2.2.1.1.1.2.cmml"><mi id="S5.Ex51.m1.2.2.2.2.1.1.1.2.2" xref="S5.Ex51.m1.2.2.2.2.1.1.1.2.2.cmml">c</mi><mn id="S5.Ex51.m1.2.2.2.2.1.1.1.2.3" xref="S5.Ex51.m1.2.2.2.2.1.1.1.2.3.cmml">0</mn></msub><mo id="S5.Ex51.m1.2.2.2.2.1.1.1.1" lspace="0.167em" xref="S5.Ex51.m1.2.2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.Ex51.m1.2.2.2.2.1.1.1.3" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.cmml"><mover id="S5.Ex51.m1.2.2.2.2.1.1.1.3.1" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.cmml"><mo id="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.2.cmml">⟶</mo><msub id="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.3" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.3.cmml"><mi id="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.3.2" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.3.2.cmml">β</mi><mn id="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.3.3" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.1.3.3.cmml">1</mn></msub></mover><mrow id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.cmml"><mi id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.2" mathvariant="normal" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.2.cmml">⋯</mi><mo id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.1" lspace="0.337em" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.cmml"><mover id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.cmml"><mo id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.3" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.3.cmml"><mi id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.3.2" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.3.2.cmml">β</mi><mi id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.3.3" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.1.3.3.cmml">q</mi></msub></mover><msub id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.2" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.2.cmml"><mi id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.2.2" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.2.2.cmml">c</mi><mi id="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.2.3" xref="S5.Ex51.m1.2.2.2.2.1.1.1.3.2.3.2.3.cmml">q</mi></msub></mrow></mrow></mrow></mrow><mo id="S5.Ex51.m1.2.2.2.2.1.1.3" stretchy="false" xref="S5.Ex51.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex51.m1.2.2.2.2.5" xref="S5.Ex51.m1.2.2.2.2.5.cmml">∈</mo><mrow id="S5.Ex51.m1.2.2.2.2.6" xref="S5.Ex51.m1.2.2.2.2.6.cmml"><mi id="S5.Ex51.m1.2.2.2.2.6.2" xref="S5.Ex51.m1.2.2.2.2.6.2.cmml">N</mi><mo 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encoding="application/x-tex" id="S5.Ex51.m1.2c">\sigma=(d_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}\cdots% \smash{\,\mathop{\longrightarrow}\limits^{\alpha_{p}}\,}d_{p})\in N\mathcal{D}% _{p},\quad\tau=(c_{0}\smash{\,\mathop{\longrightarrow}\limits^{\beta_{1}}\,}% \cdots\smash{\,\mathop{\longrightarrow}\limits^{\beta_{q}}\,}c_{q})\in N% \mathcal{C}_{q}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex51.m1.2d">italic_σ = ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ∈ italic_N caligraphic_D start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_τ = ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p2.21">and <math alttext="\mu:\varphi(c_{q})\to d_{0}" class="ltx_Math" display="inline" id="S5.p2.17.m1.1"><semantics id="S5.p2.17.m1.1a"><mrow id="S5.p2.17.m1.1.1" xref="S5.p2.17.m1.1.1.cmml"><mi id="S5.p2.17.m1.1.1.3" xref="S5.p2.17.m1.1.1.3.cmml">μ</mi><mo id="S5.p2.17.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S5.p2.17.m1.1.1.2.cmml">:</mo><mrow id="S5.p2.17.m1.1.1.1" xref="S5.p2.17.m1.1.1.1.cmml"><mrow id="S5.p2.17.m1.1.1.1.1" xref="S5.p2.17.m1.1.1.1.1.cmml"><mi id="S5.p2.17.m1.1.1.1.1.3" xref="S5.p2.17.m1.1.1.1.1.3.cmml">φ</mi><mo id="S5.p2.17.m1.1.1.1.1.2" xref="S5.p2.17.m1.1.1.1.1.2.cmml">⁢</mo><mrow 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xref="S5.p2.17.m1.1.1.2">:</ci><ci id="S5.p2.17.m1.1.1.3.cmml" xref="S5.p2.17.m1.1.1.3">𝜇</ci><apply id="S5.p2.17.m1.1.1.1.cmml" xref="S5.p2.17.m1.1.1.1"><ci id="S5.p2.17.m1.1.1.1.2.cmml" xref="S5.p2.17.m1.1.1.1.2">→</ci><apply id="S5.p2.17.m1.1.1.1.1.cmml" xref="S5.p2.17.m1.1.1.1.1"><times id="S5.p2.17.m1.1.1.1.1.2.cmml" xref="S5.p2.17.m1.1.1.1.1.2"></times><ci id="S5.p2.17.m1.1.1.1.1.3.cmml" xref="S5.p2.17.m1.1.1.1.1.3">𝜑</ci><apply id="S5.p2.17.m1.1.1.1.1.1.1.1.cmml" xref="S5.p2.17.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p2.17.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.p2.17.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.p2.17.m1.1.1.1.1.1.1.1.2.cmml" xref="S5.p2.17.m1.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S5.p2.17.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.p2.17.m1.1.1.1.1.1.1.1.3">𝑞</ci></apply></apply><apply id="S5.p2.17.m1.1.1.1.3.cmml" xref="S5.p2.17.m1.1.1.1.3"><csymbol cd="ambiguous" id="S5.p2.17.m1.1.1.1.3.1.cmml" xref="S5.p2.17.m1.1.1.1.3">subscript</csymbol><ci id="S5.p2.17.m1.1.1.1.3.2.cmml" xref="S5.p2.17.m1.1.1.1.3.2">𝑑</ci><cn id="S5.p2.17.m1.1.1.1.3.3.cmml" type="integer" xref="S5.p2.17.m1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.17.m1.1c">\mu:\varphi(c_{q})\to d_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.17.m1.1d">italic_μ : italic_φ ( italic_c start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) → italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a morphism in <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S5.p2.18.m2.1"><semantics id="S5.p2.18.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p2.18.m2.1.1" xref="S5.p2.18.m2.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S5.p2.18.m2.1b"><ci id="S5.p2.18.m2.1.1.cmml" xref="S5.p2.18.m2.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.18.m2.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.18.m2.1d">caligraphic_D</annotation></semantics></math>. If <math alttext="f_{h}\times f_{v}:([r],[s])\to([p],[q])" class="ltx_Math" display="inline" id="S5.p2.19.m3.8"><semantics id="S5.p2.19.m3.8a"><mrow id="S5.p2.19.m3.8.8" xref="S5.p2.19.m3.8.8.cmml"><mrow id="S5.p2.19.m3.8.8.6" xref="S5.p2.19.m3.8.8.6.cmml"><msub id="S5.p2.19.m3.8.8.6.2" xref="S5.p2.19.m3.8.8.6.2.cmml"><mi id="S5.p2.19.m3.8.8.6.2.2" xref="S5.p2.19.m3.8.8.6.2.2.cmml">f</mi><mi id="S5.p2.19.m3.8.8.6.2.3" xref="S5.p2.19.m3.8.8.6.2.3.cmml">h</mi></msub><mo id="S5.p2.19.m3.8.8.6.1" lspace="0.222em" rspace="0.222em" xref="S5.p2.19.m3.8.8.6.1.cmml">×</mo><msub id="S5.p2.19.m3.8.8.6.3" xref="S5.p2.19.m3.8.8.6.3.cmml"><mi id="S5.p2.19.m3.8.8.6.3.2" xref="S5.p2.19.m3.8.8.6.3.2.cmml">f</mi><mi id="S5.p2.19.m3.8.8.6.3.3" xref="S5.p2.19.m3.8.8.6.3.3.cmml">v</mi></msub></mrow><mo id="S5.p2.19.m3.8.8.5" lspace="0.278em" rspace="0.278em" xref="S5.p2.19.m3.8.8.5.cmml">:</mo><mrow id="S5.p2.19.m3.8.8.4" xref="S5.p2.19.m3.8.8.4.cmml"><mrow id="S5.p2.19.m3.6.6.2.2.2" xref="S5.p2.19.m3.6.6.2.2.3.cmml"><mo id="S5.p2.19.m3.6.6.2.2.2.3" stretchy="false" xref="S5.p2.19.m3.6.6.2.2.3.cmml">(</mo><mrow id="S5.p2.19.m3.5.5.1.1.1.1.2" xref="S5.p2.19.m3.5.5.1.1.1.1.1.cmml"><mo id="S5.p2.19.m3.5.5.1.1.1.1.2.1" stretchy="false" xref="S5.p2.19.m3.5.5.1.1.1.1.1.1.cmml">[</mo><mi id="S5.p2.19.m3.1.1" xref="S5.p2.19.m3.1.1.cmml">r</mi><mo id="S5.p2.19.m3.5.5.1.1.1.1.2.2" stretchy="false" xref="S5.p2.19.m3.5.5.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S5.p2.19.m3.6.6.2.2.2.4" xref="S5.p2.19.m3.6.6.2.2.3.cmml">,</mo><mrow id="S5.p2.19.m3.6.6.2.2.2.2.2" xref="S5.p2.19.m3.6.6.2.2.2.2.1.cmml"><mo id="S5.p2.19.m3.6.6.2.2.2.2.2.1" stretchy="false" xref="S5.p2.19.m3.6.6.2.2.2.2.1.1.cmml">[</mo><mi id="S5.p2.19.m3.2.2" xref="S5.p2.19.m3.2.2.cmml">s</mi><mo id="S5.p2.19.m3.6.6.2.2.2.2.2.2" stretchy="false" xref="S5.p2.19.m3.6.6.2.2.2.2.1.1.cmml">]</mo></mrow><mo id="S5.p2.19.m3.6.6.2.2.2.5" stretchy="false" xref="S5.p2.19.m3.6.6.2.2.3.cmml">)</mo></mrow><mo id="S5.p2.19.m3.8.8.4.5" stretchy="false" xref="S5.p2.19.m3.8.8.4.5.cmml">→</mo><mrow id="S5.p2.19.m3.8.8.4.4.2" xref="S5.p2.19.m3.8.8.4.4.3.cmml"><mo id="S5.p2.19.m3.8.8.4.4.2.3" stretchy="false" xref="S5.p2.19.m3.8.8.4.4.3.cmml">(</mo><mrow id="S5.p2.19.m3.7.7.3.3.1.1.2" xref="S5.p2.19.m3.7.7.3.3.1.1.1.cmml"><mo id="S5.p2.19.m3.7.7.3.3.1.1.2.1" stretchy="false" xref="S5.p2.19.m3.7.7.3.3.1.1.1.1.cmml">[</mo><mi id="S5.p2.19.m3.3.3" xref="S5.p2.19.m3.3.3.cmml">p</mi><mo id="S5.p2.19.m3.7.7.3.3.1.1.2.2" stretchy="false" xref="S5.p2.19.m3.7.7.3.3.1.1.1.1.cmml">]</mo></mrow><mo id="S5.p2.19.m3.8.8.4.4.2.4" xref="S5.p2.19.m3.8.8.4.4.3.cmml">,</mo><mrow id="S5.p2.19.m3.8.8.4.4.2.2.2" xref="S5.p2.19.m3.8.8.4.4.2.2.1.cmml"><mo id="S5.p2.19.m3.8.8.4.4.2.2.2.1" stretchy="false" xref="S5.p2.19.m3.8.8.4.4.2.2.1.1.cmml">[</mo><mi id="S5.p2.19.m3.4.4" xref="S5.p2.19.m3.4.4.cmml">q</mi><mo id="S5.p2.19.m3.8.8.4.4.2.2.2.2" stretchy="false" xref="S5.p2.19.m3.8.8.4.4.2.2.1.1.cmml">]</mo></mrow><mo id="S5.p2.19.m3.8.8.4.4.2.5" stretchy="false" xref="S5.p2.19.m3.8.8.4.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.19.m3.8b"><apply id="S5.p2.19.m3.8.8.cmml" xref="S5.p2.19.m3.8.8"><ci id="S5.p2.19.m3.8.8.5.cmml" xref="S5.p2.19.m3.8.8.5">:</ci><apply id="S5.p2.19.m3.8.8.6.cmml" xref="S5.p2.19.m3.8.8.6"><times id="S5.p2.19.m3.8.8.6.1.cmml" xref="S5.p2.19.m3.8.8.6.1"></times><apply id="S5.p2.19.m3.8.8.6.2.cmml" xref="S5.p2.19.m3.8.8.6.2"><csymbol cd="ambiguous" id="S5.p2.19.m3.8.8.6.2.1.cmml" xref="S5.p2.19.m3.8.8.6.2">subscript</csymbol><ci id="S5.p2.19.m3.8.8.6.2.2.cmml" xref="S5.p2.19.m3.8.8.6.2.2">𝑓</ci><ci id="S5.p2.19.m3.8.8.6.2.3.cmml" xref="S5.p2.19.m3.8.8.6.2.3">ℎ</ci></apply><apply id="S5.p2.19.m3.8.8.6.3.cmml" xref="S5.p2.19.m3.8.8.6.3"><csymbol cd="ambiguous" id="S5.p2.19.m3.8.8.6.3.1.cmml" xref="S5.p2.19.m3.8.8.6.3">subscript</csymbol><ci id="S5.p2.19.m3.8.8.6.3.2.cmml" xref="S5.p2.19.m3.8.8.6.3.2">𝑓</ci><ci id="S5.p2.19.m3.8.8.6.3.3.cmml" xref="S5.p2.19.m3.8.8.6.3.3">𝑣</ci></apply></apply><apply id="S5.p2.19.m3.8.8.4.cmml" xref="S5.p2.19.m3.8.8.4"><ci id="S5.p2.19.m3.8.8.4.5.cmml" xref="S5.p2.19.m3.8.8.4.5">→</ci><interval closure="open" id="S5.p2.19.m3.6.6.2.2.3.cmml" xref="S5.p2.19.m3.6.6.2.2.2"><apply id="S5.p2.19.m3.5.5.1.1.1.1.1.cmml" xref="S5.p2.19.m3.5.5.1.1.1.1.2"><csymbol cd="latexml" id="S5.p2.19.m3.5.5.1.1.1.1.1.1.cmml" xref="S5.p2.19.m3.5.5.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S5.p2.19.m3.1.1.cmml" xref="S5.p2.19.m3.1.1">𝑟</ci></apply><apply id="S5.p2.19.m3.6.6.2.2.2.2.1.cmml" xref="S5.p2.19.m3.6.6.2.2.2.2.2"><csymbol cd="latexml" id="S5.p2.19.m3.6.6.2.2.2.2.1.1.cmml" xref="S5.p2.19.m3.6.6.2.2.2.2.2.1">delimited-[]</csymbol><ci id="S5.p2.19.m3.2.2.cmml" xref="S5.p2.19.m3.2.2">𝑠</ci></apply></interval><interval closure="open" id="S5.p2.19.m3.8.8.4.4.3.cmml" xref="S5.p2.19.m3.8.8.4.4.2"><apply id="S5.p2.19.m3.7.7.3.3.1.1.1.cmml" xref="S5.p2.19.m3.7.7.3.3.1.1.2"><csymbol cd="latexml" id="S5.p2.19.m3.7.7.3.3.1.1.1.1.cmml" xref="S5.p2.19.m3.7.7.3.3.1.1.2.1">delimited-[]</csymbol><ci id="S5.p2.19.m3.3.3.cmml" xref="S5.p2.19.m3.3.3">𝑝</ci></apply><apply id="S5.p2.19.m3.8.8.4.4.2.2.1.cmml" xref="S5.p2.19.m3.8.8.4.4.2.2.2"><csymbol cd="latexml" id="S5.p2.19.m3.8.8.4.4.2.2.1.1.cmml" xref="S5.p2.19.m3.8.8.4.4.2.2.2.1">delimited-[]</csymbol><ci id="S5.p2.19.m3.4.4.cmml" xref="S5.p2.19.m3.4.4">𝑞</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.19.m3.8c">f_{h}\times f_{v}:([r],[s])\to([p],[q])</annotation><annotation encoding="application/x-llamapun" id="S5.p2.19.m3.8d">italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT : ( [ italic_r ] , [ italic_s ] ) → ( [ italic_p ] , [ italic_q ] )</annotation></semantics></math> is a morphism in <math alttext="\Delta\times\Delta" class="ltx_Math" display="inline" id="S5.p2.20.m4.1"><semantics id="S5.p2.20.m4.1a"><mrow id="S5.p2.20.m4.1.1" xref="S5.p2.20.m4.1.1.cmml"><mi id="S5.p2.20.m4.1.1.2" mathvariant="normal" xref="S5.p2.20.m4.1.1.2.cmml">Δ</mi><mo id="S5.p2.20.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.p2.20.m4.1.1.1.cmml">×</mo><mi id="S5.p2.20.m4.1.1.3" mathvariant="normal" xref="S5.p2.20.m4.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.20.m4.1b"><apply id="S5.p2.20.m4.1.1.cmml" xref="S5.p2.20.m4.1.1"><times id="S5.p2.20.m4.1.1.1.cmml" xref="S5.p2.20.m4.1.1.1"></times><ci id="S5.p2.20.m4.1.1.2.cmml" xref="S5.p2.20.m4.1.1.2">Δ</ci><ci id="S5.p2.20.m4.1.1.3.cmml" xref="S5.p2.20.m4.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.20.m4.1c">\Delta\times\Delta</annotation><annotation encoding="application/x-llamapun" id="S5.p2.20.m4.1d">roman_Δ × roman_Δ</annotation></semantics></math>, then the induced map <math alttext="(f_{h}\times f_{v})^{*}" class="ltx_Math" display="inline" id="S5.p2.21.m5.1"><semantics id="S5.p2.21.m5.1a"><msup id="S5.p2.21.m5.1.1" xref="S5.p2.21.m5.1.1.cmml"><mrow id="S5.p2.21.m5.1.1.1.1" xref="S5.p2.21.m5.1.1.1.1.1.cmml"><mo id="S5.p2.21.m5.1.1.1.1.2" stretchy="false" xref="S5.p2.21.m5.1.1.1.1.1.cmml">(</mo><mrow id="S5.p2.21.m5.1.1.1.1.1" xref="S5.p2.21.m5.1.1.1.1.1.cmml"><msub id="S5.p2.21.m5.1.1.1.1.1.2" xref="S5.p2.21.m5.1.1.1.1.1.2.cmml"><mi id="S5.p2.21.m5.1.1.1.1.1.2.2" xref="S5.p2.21.m5.1.1.1.1.1.2.2.cmml">f</mi><mi id="S5.p2.21.m5.1.1.1.1.1.2.3" xref="S5.p2.21.m5.1.1.1.1.1.2.3.cmml">h</mi></msub><mo id="S5.p2.21.m5.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.p2.21.m5.1.1.1.1.1.1.cmml">×</mo><msub id="S5.p2.21.m5.1.1.1.1.1.3" xref="S5.p2.21.m5.1.1.1.1.1.3.cmml"><mi id="S5.p2.21.m5.1.1.1.1.1.3.2" xref="S5.p2.21.m5.1.1.1.1.1.3.2.cmml">f</mi><mi id="S5.p2.21.m5.1.1.1.1.1.3.3" xref="S5.p2.21.m5.1.1.1.1.1.3.3.cmml">v</mi></msub></mrow><mo id="S5.p2.21.m5.1.1.1.1.3" stretchy="false" xref="S5.p2.21.m5.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.p2.21.m5.1.1.3" xref="S5.p2.21.m5.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.p2.21.m5.1b"><apply id="S5.p2.21.m5.1.1.cmml" xref="S5.p2.21.m5.1.1"><csymbol cd="ambiguous" id="S5.p2.21.m5.1.1.2.cmml" xref="S5.p2.21.m5.1.1">superscript</csymbol><apply id="S5.p2.21.m5.1.1.1.1.1.cmml" xref="S5.p2.21.m5.1.1.1.1"><times id="S5.p2.21.m5.1.1.1.1.1.1.cmml" xref="S5.p2.21.m5.1.1.1.1.1.1"></times><apply id="S5.p2.21.m5.1.1.1.1.1.2.cmml" xref="S5.p2.21.m5.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.p2.21.m5.1.1.1.1.1.2.1.cmml" xref="S5.p2.21.m5.1.1.1.1.1.2">subscript</csymbol><ci id="S5.p2.21.m5.1.1.1.1.1.2.2.cmml" xref="S5.p2.21.m5.1.1.1.1.1.2.2">𝑓</ci><ci id="S5.p2.21.m5.1.1.1.1.1.2.3.cmml" xref="S5.p2.21.m5.1.1.1.1.1.2.3">ℎ</ci></apply><apply id="S5.p2.21.m5.1.1.1.1.1.3.cmml" xref="S5.p2.21.m5.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.p2.21.m5.1.1.1.1.1.3.1.cmml" xref="S5.p2.21.m5.1.1.1.1.1.3">subscript</csymbol><ci id="S5.p2.21.m5.1.1.1.1.1.3.2.cmml" xref="S5.p2.21.m5.1.1.1.1.1.3.2">𝑓</ci><ci id="S5.p2.21.m5.1.1.1.1.1.3.3.cmml" xref="S5.p2.21.m5.1.1.1.1.1.3.3">𝑣</ci></apply></apply><times id="S5.p2.21.m5.1.1.3.cmml" xref="S5.p2.21.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.21.m5.1c">(f_{h}\times f_{v})^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.21.m5.1d">( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S5.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(f_{h}\times f_{v})^{*}(\sigma,\tau,\mu)=(f_{h}^{*}(\sigma),f_{v}^{*}(\tau),% \mu^{\prime})" class="ltx_Math" display="block" id="S5.E2.m1.9"><semantics id="S5.E2.m1.9a"><mrow id="S5.E2.m1.9.9" xref="S5.E2.m1.9.9.cmml"><mrow id="S5.E2.m1.6.6.1" xref="S5.E2.m1.6.6.1.cmml"><msup id="S5.E2.m1.6.6.1.1" xref="S5.E2.m1.6.6.1.1.cmml"><mrow id="S5.E2.m1.6.6.1.1.1.1" xref="S5.E2.m1.6.6.1.1.1.1.1.cmml"><mo id="S5.E2.m1.6.6.1.1.1.1.2" stretchy="false" xref="S5.E2.m1.6.6.1.1.1.1.1.cmml">(</mo><mrow id="S5.E2.m1.6.6.1.1.1.1.1" xref="S5.E2.m1.6.6.1.1.1.1.1.cmml"><msub id="S5.E2.m1.6.6.1.1.1.1.1.2" xref="S5.E2.m1.6.6.1.1.1.1.1.2.cmml"><mi id="S5.E2.m1.6.6.1.1.1.1.1.2.2" xref="S5.E2.m1.6.6.1.1.1.1.1.2.2.cmml">f</mi><mi id="S5.E2.m1.6.6.1.1.1.1.1.2.3" xref="S5.E2.m1.6.6.1.1.1.1.1.2.3.cmml">h</mi></msub><mo id="S5.E2.m1.6.6.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.E2.m1.6.6.1.1.1.1.1.1.cmml">×</mo><msub id="S5.E2.m1.6.6.1.1.1.1.1.3" xref="S5.E2.m1.6.6.1.1.1.1.1.3.cmml"><mi id="S5.E2.m1.6.6.1.1.1.1.1.3.2" xref="S5.E2.m1.6.6.1.1.1.1.1.3.2.cmml">f</mi><mi id="S5.E2.m1.6.6.1.1.1.1.1.3.3" xref="S5.E2.m1.6.6.1.1.1.1.1.3.3.cmml">v</mi></msub></mrow><mo id="S5.E2.m1.6.6.1.1.1.1.3" stretchy="false" xref="S5.E2.m1.6.6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.E2.m1.6.6.1.1.3" xref="S5.E2.m1.6.6.1.1.3.cmml">∗</mo></msup><mo id="S5.E2.m1.6.6.1.2" xref="S5.E2.m1.6.6.1.2.cmml">⁢</mo><mrow id="S5.E2.m1.6.6.1.3.2" xref="S5.E2.m1.6.6.1.3.1.cmml"><mo id="S5.E2.m1.6.6.1.3.2.1" stretchy="false" xref="S5.E2.m1.6.6.1.3.1.cmml">(</mo><mi id="S5.E2.m1.1.1" xref="S5.E2.m1.1.1.cmml">σ</mi><mo id="S5.E2.m1.6.6.1.3.2.2" xref="S5.E2.m1.6.6.1.3.1.cmml">,</mo><mi id="S5.E2.m1.2.2" xref="S5.E2.m1.2.2.cmml">τ</mi><mo id="S5.E2.m1.6.6.1.3.2.3" xref="S5.E2.m1.6.6.1.3.1.cmml">,</mo><mi id="S5.E2.m1.3.3" xref="S5.E2.m1.3.3.cmml">μ</mi><mo id="S5.E2.m1.6.6.1.3.2.4" stretchy="false" xref="S5.E2.m1.6.6.1.3.1.cmml">)</mo></mrow></mrow><mo id="S5.E2.m1.9.9.5" xref="S5.E2.m1.9.9.5.cmml">=</mo><mrow id="S5.E2.m1.9.9.4.3" xref="S5.E2.m1.9.9.4.4.cmml"><mo id="S5.E2.m1.9.9.4.3.4" stretchy="false" xref="S5.E2.m1.9.9.4.4.cmml">(</mo><mrow id="S5.E2.m1.7.7.2.1.1" xref="S5.E2.m1.7.7.2.1.1.cmml"><msubsup id="S5.E2.m1.7.7.2.1.1.2" xref="S5.E2.m1.7.7.2.1.1.2.cmml"><mi id="S5.E2.m1.7.7.2.1.1.2.2.2" xref="S5.E2.m1.7.7.2.1.1.2.2.2.cmml">f</mi><mi id="S5.E2.m1.7.7.2.1.1.2.2.3" xref="S5.E2.m1.7.7.2.1.1.2.2.3.cmml">h</mi><mo id="S5.E2.m1.7.7.2.1.1.2.3" xref="S5.E2.m1.7.7.2.1.1.2.3.cmml">∗</mo></msubsup><mo id="S5.E2.m1.7.7.2.1.1.1" xref="S5.E2.m1.7.7.2.1.1.1.cmml">⁢</mo><mrow id="S5.E2.m1.7.7.2.1.1.3.2" xref="S5.E2.m1.7.7.2.1.1.cmml"><mo id="S5.E2.m1.7.7.2.1.1.3.2.1" stretchy="false" xref="S5.E2.m1.7.7.2.1.1.cmml">(</mo><mi id="S5.E2.m1.4.4" xref="S5.E2.m1.4.4.cmml">σ</mi><mo id="S5.E2.m1.7.7.2.1.1.3.2.2" stretchy="false" xref="S5.E2.m1.7.7.2.1.1.cmml">)</mo></mrow></mrow><mo id="S5.E2.m1.9.9.4.3.5" xref="S5.E2.m1.9.9.4.4.cmml">,</mo><mrow id="S5.E2.m1.8.8.3.2.2" xref="S5.E2.m1.8.8.3.2.2.cmml"><msubsup id="S5.E2.m1.8.8.3.2.2.2" xref="S5.E2.m1.8.8.3.2.2.2.cmml"><mi id="S5.E2.m1.8.8.3.2.2.2.2.2" xref="S5.E2.m1.8.8.3.2.2.2.2.2.cmml">f</mi><mi id="S5.E2.m1.8.8.3.2.2.2.2.3" xref="S5.E2.m1.8.8.3.2.2.2.2.3.cmml">v</mi><mo id="S5.E2.m1.8.8.3.2.2.2.3" xref="S5.E2.m1.8.8.3.2.2.2.3.cmml">∗</mo></msubsup><mo id="S5.E2.m1.8.8.3.2.2.1" xref="S5.E2.m1.8.8.3.2.2.1.cmml">⁢</mo><mrow id="S5.E2.m1.8.8.3.2.2.3.2" xref="S5.E2.m1.8.8.3.2.2.cmml"><mo id="S5.E2.m1.8.8.3.2.2.3.2.1" stretchy="false" xref="S5.E2.m1.8.8.3.2.2.cmml">(</mo><mi id="S5.E2.m1.5.5" xref="S5.E2.m1.5.5.cmml">τ</mi><mo id="S5.E2.m1.8.8.3.2.2.3.2.2" stretchy="false" xref="S5.E2.m1.8.8.3.2.2.cmml">)</mo></mrow></mrow><mo id="S5.E2.m1.9.9.4.3.6" xref="S5.E2.m1.9.9.4.4.cmml">,</mo><msup id="S5.E2.m1.9.9.4.3.3" xref="S5.E2.m1.9.9.4.3.3.cmml"><mi id="S5.E2.m1.9.9.4.3.3.2" xref="S5.E2.m1.9.9.4.3.3.2.cmml">μ</mi><mo id="S5.E2.m1.9.9.4.3.3.3" xref="S5.E2.m1.9.9.4.3.3.3.cmml">′</mo></msup><mo id="S5.E2.m1.9.9.4.3.7" stretchy="false" xref="S5.E2.m1.9.9.4.4.cmml">)</mo></mrow></mrow><annotation-xml 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xref="S5.E2.m1.6.6.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E2.m1.6.6.1.1.1.1.1.3.1.cmml" xref="S5.E2.m1.6.6.1.1.1.1.1.3">subscript</csymbol><ci id="S5.E2.m1.6.6.1.1.1.1.1.3.2.cmml" xref="S5.E2.m1.6.6.1.1.1.1.1.3.2">𝑓</ci><ci id="S5.E2.m1.6.6.1.1.1.1.1.3.3.cmml" xref="S5.E2.m1.6.6.1.1.1.1.1.3.3">𝑣</ci></apply></apply><times id="S5.E2.m1.6.6.1.1.3.cmml" xref="S5.E2.m1.6.6.1.1.3"></times></apply><vector id="S5.E2.m1.6.6.1.3.1.cmml" xref="S5.E2.m1.6.6.1.3.2"><ci id="S5.E2.m1.1.1.cmml" xref="S5.E2.m1.1.1">𝜎</ci><ci id="S5.E2.m1.2.2.cmml" xref="S5.E2.m1.2.2">𝜏</ci><ci id="S5.E2.m1.3.3.cmml" xref="S5.E2.m1.3.3">𝜇</ci></vector></apply><vector id="S5.E2.m1.9.9.4.4.cmml" xref="S5.E2.m1.9.9.4.3"><apply id="S5.E2.m1.7.7.2.1.1.cmml" xref="S5.E2.m1.7.7.2.1.1"><times id="S5.E2.m1.7.7.2.1.1.1.cmml" xref="S5.E2.m1.7.7.2.1.1.1"></times><apply id="S5.E2.m1.7.7.2.1.1.2.cmml" xref="S5.E2.m1.7.7.2.1.1.2"><csymbol cd="ambiguous" id="S5.E2.m1.7.7.2.1.1.2.1.cmml" xref="S5.E2.m1.7.7.2.1.1.2">superscript</csymbol><apply id="S5.E2.m1.7.7.2.1.1.2.2.cmml" xref="S5.E2.m1.7.7.2.1.1.2"><csymbol cd="ambiguous" id="S5.E2.m1.7.7.2.1.1.2.2.1.cmml" xref="S5.E2.m1.7.7.2.1.1.2">subscript</csymbol><ci id="S5.E2.m1.7.7.2.1.1.2.2.2.cmml" xref="S5.E2.m1.7.7.2.1.1.2.2.2">𝑓</ci><ci id="S5.E2.m1.7.7.2.1.1.2.2.3.cmml" xref="S5.E2.m1.7.7.2.1.1.2.2.3">ℎ</ci></apply><times id="S5.E2.m1.7.7.2.1.1.2.3.cmml" xref="S5.E2.m1.7.7.2.1.1.2.3"></times></apply><ci id="S5.E2.m1.4.4.cmml" xref="S5.E2.m1.4.4">𝜎</ci></apply><apply id="S5.E2.m1.8.8.3.2.2.cmml" xref="S5.E2.m1.8.8.3.2.2"><times id="S5.E2.m1.8.8.3.2.2.1.cmml" xref="S5.E2.m1.8.8.3.2.2.1"></times><apply id="S5.E2.m1.8.8.3.2.2.2.cmml" xref="S5.E2.m1.8.8.3.2.2.2"><csymbol cd="ambiguous" id="S5.E2.m1.8.8.3.2.2.2.1.cmml" xref="S5.E2.m1.8.8.3.2.2.2">superscript</csymbol><apply id="S5.E2.m1.8.8.3.2.2.2.2.cmml" xref="S5.E2.m1.8.8.3.2.2.2"><csymbol cd="ambiguous" id="S5.E2.m1.8.8.3.2.2.2.2.1.cmml" xref="S5.E2.m1.8.8.3.2.2.2">subscript</csymbol><ci id="S5.E2.m1.8.8.3.2.2.2.2.2.cmml" xref="S5.E2.m1.8.8.3.2.2.2.2.2">𝑓</ci><ci id="S5.E2.m1.8.8.3.2.2.2.2.3.cmml" xref="S5.E2.m1.8.8.3.2.2.2.2.3">𝑣</ci></apply><times id="S5.E2.m1.8.8.3.2.2.2.3.cmml" xref="S5.E2.m1.8.8.3.2.2.2.3"></times></apply><ci id="S5.E2.m1.5.5.cmml" xref="S5.E2.m1.5.5">𝜏</ci></apply><apply id="S5.E2.m1.9.9.4.3.3.cmml" xref="S5.E2.m1.9.9.4.3.3"><csymbol cd="ambiguous" id="S5.E2.m1.9.9.4.3.3.1.cmml" xref="S5.E2.m1.9.9.4.3.3">superscript</csymbol><ci id="S5.E2.m1.9.9.4.3.3.2.cmml" xref="S5.E2.m1.9.9.4.3.3.2">𝜇</ci><ci id="S5.E2.m1.9.9.4.3.3.3.cmml" xref="S5.E2.m1.9.9.4.3.3.3">′</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E2.m1.9c">(f_{h}\times f_{v})^{*}(\sigma,\tau,\mu)=(f_{h}^{*}(\sigma),f_{v}^{*}(\tau),% \mu^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.E2.m1.9d">( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ , italic_τ , italic_μ ) = ( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ ) , italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ ) , italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p2.23">where <math alttext="\mu^{\prime}:\varphi(c_{f_{v}(q)})\to d_{f_{h}(0)}" class="ltx_Math" display="inline" id="S5.p2.22.m1.3"><semantics id="S5.p2.22.m1.3a"><mrow id="S5.p2.22.m1.3.3" xref="S5.p2.22.m1.3.3.cmml"><msup id="S5.p2.22.m1.3.3.3" xref="S5.p2.22.m1.3.3.3.cmml"><mi id="S5.p2.22.m1.3.3.3.2" xref="S5.p2.22.m1.3.3.3.2.cmml">μ</mi><mo id="S5.p2.22.m1.3.3.3.3" xref="S5.p2.22.m1.3.3.3.3.cmml">′</mo></msup><mo id="S5.p2.22.m1.3.3.2" lspace="0.278em" rspace="0.278em" xref="S5.p2.22.m1.3.3.2.cmml">:</mo><mrow id="S5.p2.22.m1.3.3.1" xref="S5.p2.22.m1.3.3.1.cmml"><mrow id="S5.p2.22.m1.3.3.1.1" xref="S5.p2.22.m1.3.3.1.1.cmml"><mi id="S5.p2.22.m1.3.3.1.1.3" xref="S5.p2.22.m1.3.3.1.1.3.cmml">φ</mi><mo id="S5.p2.22.m1.3.3.1.1.2" xref="S5.p2.22.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S5.p2.22.m1.3.3.1.1.1.1" xref="S5.p2.22.m1.3.3.1.1.1.1.1.cmml"><mo id="S5.p2.22.m1.3.3.1.1.1.1.2" stretchy="false" xref="S5.p2.22.m1.3.3.1.1.1.1.1.cmml">(</mo><msub id="S5.p2.22.m1.3.3.1.1.1.1.1" xref="S5.p2.22.m1.3.3.1.1.1.1.1.cmml"><mi id="S5.p2.22.m1.3.3.1.1.1.1.1.2" xref="S5.p2.22.m1.3.3.1.1.1.1.1.2.cmml">c</mi><mrow id="S5.p2.22.m1.1.1.1" xref="S5.p2.22.m1.1.1.1.cmml"><msub id="S5.p2.22.m1.1.1.1.3" xref="S5.p2.22.m1.1.1.1.3.cmml"><mi id="S5.p2.22.m1.1.1.1.3.2" xref="S5.p2.22.m1.1.1.1.3.2.cmml">f</mi><mi id="S5.p2.22.m1.1.1.1.3.3" xref="S5.p2.22.m1.1.1.1.3.3.cmml">v</mi></msub><mo id="S5.p2.22.m1.1.1.1.2" xref="S5.p2.22.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.p2.22.m1.1.1.1.4.2" xref="S5.p2.22.m1.1.1.1.cmml"><mo id="S5.p2.22.m1.1.1.1.4.2.1" stretchy="false" xref="S5.p2.22.m1.1.1.1.cmml">(</mo><mi id="S5.p2.22.m1.1.1.1.1" xref="S5.p2.22.m1.1.1.1.1.cmml">q</mi><mo id="S5.p2.22.m1.1.1.1.4.2.2" stretchy="false" xref="S5.p2.22.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.p2.22.m1.3.3.1.1.1.1.3" stretchy="false" xref="S5.p2.22.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p2.22.m1.3.3.1.2" stretchy="false" xref="S5.p2.22.m1.3.3.1.2.cmml">→</mo><msub id="S5.p2.22.m1.3.3.1.3" xref="S5.p2.22.m1.3.3.1.3.cmml"><mi id="S5.p2.22.m1.3.3.1.3.2" xref="S5.p2.22.m1.3.3.1.3.2.cmml">d</mi><mrow id="S5.p2.22.m1.2.2.1" xref="S5.p2.22.m1.2.2.1.cmml"><msub id="S5.p2.22.m1.2.2.1.3" xref="S5.p2.22.m1.2.2.1.3.cmml"><mi id="S5.p2.22.m1.2.2.1.3.2" xref="S5.p2.22.m1.2.2.1.3.2.cmml">f</mi><mi id="S5.p2.22.m1.2.2.1.3.3" xref="S5.p2.22.m1.2.2.1.3.3.cmml">h</mi></msub><mo id="S5.p2.22.m1.2.2.1.2" xref="S5.p2.22.m1.2.2.1.2.cmml">⁢</mo><mrow id="S5.p2.22.m1.2.2.1.4.2" xref="S5.p2.22.m1.2.2.1.cmml"><mo id="S5.p2.22.m1.2.2.1.4.2.1" stretchy="false" xref="S5.p2.22.m1.2.2.1.cmml">(</mo><mn id="S5.p2.22.m1.2.2.1.1" xref="S5.p2.22.m1.2.2.1.1.cmml">0</mn><mo id="S5.p2.22.m1.2.2.1.4.2.2" stretchy="false" xref="S5.p2.22.m1.2.2.1.cmml">)</mo></mrow></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.22.m1.3b"><apply id="S5.p2.22.m1.3.3.cmml" xref="S5.p2.22.m1.3.3"><ci id="S5.p2.22.m1.3.3.2.cmml" xref="S5.p2.22.m1.3.3.2">:</ci><apply id="S5.p2.22.m1.3.3.3.cmml" xref="S5.p2.22.m1.3.3.3"><csymbol cd="ambiguous" id="S5.p2.22.m1.3.3.3.1.cmml" xref="S5.p2.22.m1.3.3.3">superscript</csymbol><ci id="S5.p2.22.m1.3.3.3.2.cmml" xref="S5.p2.22.m1.3.3.3.2">𝜇</ci><ci id="S5.p2.22.m1.3.3.3.3.cmml" xref="S5.p2.22.m1.3.3.3.3">′</ci></apply><apply id="S5.p2.22.m1.3.3.1.cmml" xref="S5.p2.22.m1.3.3.1"><ci id="S5.p2.22.m1.3.3.1.2.cmml" xref="S5.p2.22.m1.3.3.1.2">→</ci><apply id="S5.p2.22.m1.3.3.1.1.cmml" xref="S5.p2.22.m1.3.3.1.1"><times id="S5.p2.22.m1.3.3.1.1.2.cmml" xref="S5.p2.22.m1.3.3.1.1.2"></times><ci id="S5.p2.22.m1.3.3.1.1.3.cmml" xref="S5.p2.22.m1.3.3.1.1.3">𝜑</ci><apply id="S5.p2.22.m1.3.3.1.1.1.1.1.cmml" xref="S5.p2.22.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.p2.22.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.p2.22.m1.3.3.1.1.1.1">subscript</csymbol><ci id="S5.p2.22.m1.3.3.1.1.1.1.1.2.cmml" xref="S5.p2.22.m1.3.3.1.1.1.1.1.2">𝑐</ci><apply id="S5.p2.22.m1.1.1.1.cmml" xref="S5.p2.22.m1.1.1.1"><times id="S5.p2.22.m1.1.1.1.2.cmml" xref="S5.p2.22.m1.1.1.1.2"></times><apply id="S5.p2.22.m1.1.1.1.3.cmml" xref="S5.p2.22.m1.1.1.1.3"><csymbol cd="ambiguous" id="S5.p2.22.m1.1.1.1.3.1.cmml" xref="S5.p2.22.m1.1.1.1.3">subscript</csymbol><ci id="S5.p2.22.m1.1.1.1.3.2.cmml" xref="S5.p2.22.m1.1.1.1.3.2">𝑓</ci><ci id="S5.p2.22.m1.1.1.1.3.3.cmml" xref="S5.p2.22.m1.1.1.1.3.3">𝑣</ci></apply><ci id="S5.p2.22.m1.1.1.1.1.cmml" xref="S5.p2.22.m1.1.1.1.1">𝑞</ci></apply></apply></apply><apply id="S5.p2.22.m1.3.3.1.3.cmml" xref="S5.p2.22.m1.3.3.1.3"><csymbol cd="ambiguous" id="S5.p2.22.m1.3.3.1.3.1.cmml" xref="S5.p2.22.m1.3.3.1.3">subscript</csymbol><ci id="S5.p2.22.m1.3.3.1.3.2.cmml" xref="S5.p2.22.m1.3.3.1.3.2">𝑑</ci><apply id="S5.p2.22.m1.2.2.1.cmml" xref="S5.p2.22.m1.2.2.1"><times id="S5.p2.22.m1.2.2.1.2.cmml" xref="S5.p2.22.m1.2.2.1.2"></times><apply id="S5.p2.22.m1.2.2.1.3.cmml" xref="S5.p2.22.m1.2.2.1.3"><csymbol cd="ambiguous" id="S5.p2.22.m1.2.2.1.3.1.cmml" xref="S5.p2.22.m1.2.2.1.3">subscript</csymbol><ci id="S5.p2.22.m1.2.2.1.3.2.cmml" xref="S5.p2.22.m1.2.2.1.3.2">𝑓</ci><ci id="S5.p2.22.m1.2.2.1.3.3.cmml" xref="S5.p2.22.m1.2.2.1.3.3">ℎ</ci></apply><cn id="S5.p2.22.m1.2.2.1.1.cmml" type="integer" xref="S5.p2.22.m1.2.2.1.1">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.22.m1.3c">\mu^{\prime}:\varphi(c_{f_{v}(q)})\to d_{f_{h}(0)}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.22.m1.3d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_φ ( italic_c start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT ) → italic_d start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT</annotation></semantics></math> is the morphism in <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S5.p2.23.m2.1"><semantics id="S5.p2.23.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p2.23.m2.1.1" xref="S5.p2.23.m2.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S5.p2.23.m2.1b"><ci id="S5.p2.23.m2.1.1.cmml" xref="S5.p2.23.m2.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.23.m2.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.23.m2.1d">caligraphic_D</annotation></semantics></math> defined as the composition</p> <table class="ltx_equation ltx_eqn_table" id="S5.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\varphi(c_{f_{v}(q)})\xrightarrow{\varphi(\beta_{f_{v}(q)+1})}\cdots% \xrightarrow{\varphi(\beta_{q})}\varphi(c_{q})\smash{\,\mathop{\longrightarrow% }\limits^{\mu}\,}d_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}% \cdots\xrightarrow{\alpha_{f_{h}(0)}}d_{f_{h}(0)}." class="ltx_Math" display="block" id="S5.E3.m1.7"><semantics id="S5.E3.m1.7a"><mrow id="S5.E3.m1.7.7.1" xref="S5.E3.m1.7.7.1.1.cmml"><mrow id="S5.E3.m1.7.7.1.1" xref="S5.E3.m1.7.7.1.1.cmml"><mrow id="S5.E3.m1.7.7.1.1.1" xref="S5.E3.m1.7.7.1.1.1.cmml"><mi id="S5.E3.m1.7.7.1.1.1.3" xref="S5.E3.m1.7.7.1.1.1.3.cmml">φ</mi><mo id="S5.E3.m1.7.7.1.1.1.2" xref="S5.E3.m1.7.7.1.1.1.2.cmml">⁢</mo><mrow id="S5.E3.m1.7.7.1.1.1.1.1" xref="S5.E3.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S5.E3.m1.7.7.1.1.1.1.1.2" stretchy="false" xref="S5.E3.m1.7.7.1.1.1.1.1.1.cmml">(</mo><msub id="S5.E3.m1.7.7.1.1.1.1.1.1" xref="S5.E3.m1.7.7.1.1.1.1.1.1.cmml"><mi id="S5.E3.m1.7.7.1.1.1.1.1.1.2" xref="S5.E3.m1.7.7.1.1.1.1.1.1.2.cmml">c</mi><mrow id="S5.E3.m1.1.1.1" xref="S5.E3.m1.1.1.1.cmml"><msub id="S5.E3.m1.1.1.1.3" xref="S5.E3.m1.1.1.1.3.cmml"><mi id="S5.E3.m1.1.1.1.3.2" xref="S5.E3.m1.1.1.1.3.2.cmml">f</mi><mi id="S5.E3.m1.1.1.1.3.3" xref="S5.E3.m1.1.1.1.3.3.cmml">v</mi></msub><mo id="S5.E3.m1.1.1.1.2" xref="S5.E3.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E3.m1.1.1.1.4.2" xref="S5.E3.m1.1.1.1.cmml"><mo id="S5.E3.m1.1.1.1.4.2.1" stretchy="false" xref="S5.E3.m1.1.1.1.cmml">(</mo><mi id="S5.E3.m1.1.1.1.1" xref="S5.E3.m1.1.1.1.1.cmml">q</mi><mo id="S5.E3.m1.1.1.1.4.2.2" stretchy="false" xref="S5.E3.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.E3.m1.7.7.1.1.1.1.1.3" stretchy="false" xref="S5.E3.m1.7.7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mover accent="true" id="S5.E3.m1.3.3" xref="S5.E3.m1.3.3.cmml"><mo id="S5.E3.m1.3.3.3" stretchy="false" xref="S5.E3.m1.3.3.3.cmml">→</mo><mrow id="S5.E3.m1.3.3.2" xref="S5.E3.m1.3.3.2.cmml"><mi id="S5.E3.m1.3.3.2.4" xref="S5.E3.m1.3.3.2.4.cmml">φ</mi><mo id="S5.E3.m1.3.3.2.3" xref="S5.E3.m1.3.3.2.3.cmml">⁢</mo><mrow id="S5.E3.m1.3.3.2.2.1" xref="S5.E3.m1.3.3.2.2.1.1.cmml"><mo id="S5.E3.m1.3.3.2.2.1.2" stretchy="false" xref="S5.E3.m1.3.3.2.2.1.1.cmml">(</mo><msub id="S5.E3.m1.3.3.2.2.1.1" xref="S5.E3.m1.3.3.2.2.1.1.cmml"><mi id="S5.E3.m1.3.3.2.2.1.1.2" xref="S5.E3.m1.3.3.2.2.1.1.2.cmml">β</mi><mrow id="S5.E3.m1.2.2.1.1.1" xref="S5.E3.m1.2.2.1.1.1.cmml"><mrow id="S5.E3.m1.2.2.1.1.1.3" xref="S5.E3.m1.2.2.1.1.1.3.cmml"><msub id="S5.E3.m1.2.2.1.1.1.3.2" xref="S5.E3.m1.2.2.1.1.1.3.2.cmml"><mi id="S5.E3.m1.2.2.1.1.1.3.2.2" xref="S5.E3.m1.2.2.1.1.1.3.2.2.cmml">f</mi><mi id="S5.E3.m1.2.2.1.1.1.3.2.3" xref="S5.E3.m1.2.2.1.1.1.3.2.3.cmml">v</mi></msub><mo id="S5.E3.m1.2.2.1.1.1.3.1" xref="S5.E3.m1.2.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.E3.m1.2.2.1.1.1.3.3.2" xref="S5.E3.m1.2.2.1.1.1.3.cmml"><mo id="S5.E3.m1.2.2.1.1.1.3.3.2.1" stretchy="false" xref="S5.E3.m1.2.2.1.1.1.3.cmml">(</mo><mi id="S5.E3.m1.2.2.1.1.1.1" xref="S5.E3.m1.2.2.1.1.1.1.cmml">q</mi><mo id="S5.E3.m1.2.2.1.1.1.3.3.2.2" stretchy="false" xref="S5.E3.m1.2.2.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S5.E3.m1.2.2.1.1.1.2" xref="S5.E3.m1.2.2.1.1.1.2.cmml">+</mo><mn id="S5.E3.m1.2.2.1.1.1.4" xref="S5.E3.m1.2.2.1.1.1.4.cmml">1</mn></mrow></msub><mo id="S5.E3.m1.3.3.2.2.1.3" stretchy="false" xref="S5.E3.m1.3.3.2.2.1.1.cmml">)</mo></mrow></mrow></mover><mi id="S5.E3.m1.7.7.1.1.4" mathvariant="normal" xref="S5.E3.m1.7.7.1.1.4.cmml">⋯</mi><mover accent="true" id="S5.E3.m1.4.4" xref="S5.E3.m1.4.4.cmml"><mo id="S5.E3.m1.4.4.2" stretchy="false" xref="S5.E3.m1.4.4.2.cmml">→</mo><mrow id="S5.E3.m1.4.4.1" xref="S5.E3.m1.4.4.1.cmml"><mi id="S5.E3.m1.4.4.1.3" xref="S5.E3.m1.4.4.1.3.cmml">φ</mi><mo id="S5.E3.m1.4.4.1.2" xref="S5.E3.m1.4.4.1.2.cmml">⁢</mo><mrow id="S5.E3.m1.4.4.1.1.1" xref="S5.E3.m1.4.4.1.1.1.1.cmml"><mo id="S5.E3.m1.4.4.1.1.1.2" stretchy="false" xref="S5.E3.m1.4.4.1.1.1.1.cmml">(</mo><msub id="S5.E3.m1.4.4.1.1.1.1" xref="S5.E3.m1.4.4.1.1.1.1.cmml"><mi id="S5.E3.m1.4.4.1.1.1.1.2" xref="S5.E3.m1.4.4.1.1.1.1.2.cmml">β</mi><mi id="S5.E3.m1.4.4.1.1.1.1.3" xref="S5.E3.m1.4.4.1.1.1.1.3.cmml">q</mi></msub><mo id="S5.E3.m1.4.4.1.1.1.3" stretchy="false" xref="S5.E3.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow></mover><mrow id="S5.E3.m1.7.7.1.1.2" xref="S5.E3.m1.7.7.1.1.2.cmml"><mi id="S5.E3.m1.7.7.1.1.2.3" xref="S5.E3.m1.7.7.1.1.2.3.cmml">φ</mi><mo id="S5.E3.m1.7.7.1.1.2.2" xref="S5.E3.m1.7.7.1.1.2.2.cmml">⁢</mo><mrow id="S5.E3.m1.7.7.1.1.2.1.1" xref="S5.E3.m1.7.7.1.1.2.1.1.1.cmml"><mo id="S5.E3.m1.7.7.1.1.2.1.1.2" 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xref="S5.E3.m1.5.5.1.1.1.2"></times><apply id="S5.E3.m1.5.5.1.1.1.3.cmml" xref="S5.E3.m1.5.5.1.1.1.3"><csymbol cd="ambiguous" id="S5.E3.m1.5.5.1.1.1.3.1.cmml" xref="S5.E3.m1.5.5.1.1.1.3">subscript</csymbol><ci id="S5.E3.m1.5.5.1.1.1.3.2.cmml" xref="S5.E3.m1.5.5.1.1.1.3.2">𝑓</ci><ci id="S5.E3.m1.5.5.1.1.1.3.3.cmml" xref="S5.E3.m1.5.5.1.1.1.3.3">ℎ</ci></apply><cn id="S5.E3.m1.5.5.1.1.1.1.cmml" type="integer" xref="S5.E3.m1.5.5.1.1.1.1">0</cn></apply></apply><ci id="S5.E3.m1.5.5.2.cmml" xref="S5.E3.m1.5.5.2">→</ci></apply><share href="https://arxiv.org/html/2503.14659v1#S5.E3.m1.7.7.1.1.2.cmml" id="S5.E3.m1.7.7.1.1f.cmml" xref="S5.E3.m1.7.7.1"></share><apply id="S5.E3.m1.7.7.1.1.5.cmml" xref="S5.E3.m1.7.7.1.1.5"><csymbol cd="ambiguous" id="S5.E3.m1.7.7.1.1.5.1.cmml" xref="S5.E3.m1.7.7.1.1.5">subscript</csymbol><ci id="S5.E3.m1.7.7.1.1.5.2.cmml" xref="S5.E3.m1.7.7.1.1.5.2">𝑑</ci><apply id="S5.E3.m1.6.6.1.cmml" xref="S5.E3.m1.6.6.1"><times id="S5.E3.m1.6.6.1.2.cmml" xref="S5.E3.m1.6.6.1.2"></times><apply id="S5.E3.m1.6.6.1.3.cmml" xref="S5.E3.m1.6.6.1.3"><csymbol cd="ambiguous" id="S5.E3.m1.6.6.1.3.1.cmml" xref="S5.E3.m1.6.6.1.3">subscript</csymbol><ci id="S5.E3.m1.6.6.1.3.2.cmml" xref="S5.E3.m1.6.6.1.3.2">𝑓</ci><ci id="S5.E3.m1.6.6.1.3.3.cmml" xref="S5.E3.m1.6.6.1.3.3">ℎ</ci></apply><cn id="S5.E3.m1.6.6.1.1.cmml" type="integer" xref="S5.E3.m1.6.6.1.1">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E3.m1.7c">\varphi(c_{f_{v}(q)})\xrightarrow{\varphi(\beta_{f_{v}(q)+1})}\cdots% \xrightarrow{\varphi(\beta_{q})}\varphi(c_{q})\smash{\,\mathop{\longrightarrow% }\limits^{\mu}\,}d_{0}\smash{\,\mathop{\longrightarrow}\limits^{\alpha_{1}}\,}% \cdots\xrightarrow{\alpha_{f_{h}(0)}}d_{f_{h}(0)}.</annotation><annotation encoding="application/x-llamapun" id="S5.E3.m1.7d">italic_φ ( italic_c start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT ) start_ARROW start_OVERACCENT italic_φ ( italic_β start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_q ) + 1 end_POSTSUBSCRIPT ) end_OVERACCENT → end_ARROW ⋯ start_ARROW start_OVERACCENT italic_φ ( italic_β start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) end_OVERACCENT → end_ARROW italic_φ ( italic_c start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ⟶ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ start_ARROW start_OVERACCENT italic_α start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT end_OVERACCENT → end_ARROW italic_d start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p2.27">This description of induced maps <math alttext="(f_{h}\times f_{v})^{*}" class="ltx_Math" display="inline" id="S5.p2.24.m1.1"><semantics id="S5.p2.24.m1.1a"><msup id="S5.p2.24.m1.1.1" xref="S5.p2.24.m1.1.1.cmml"><mrow id="S5.p2.24.m1.1.1.1.1" xref="S5.p2.24.m1.1.1.1.1.1.cmml"><mo id="S5.p2.24.m1.1.1.1.1.2" stretchy="false" xref="S5.p2.24.m1.1.1.1.1.1.cmml">(</mo><mrow id="S5.p2.24.m1.1.1.1.1.1" xref="S5.p2.24.m1.1.1.1.1.1.cmml"><msub id="S5.p2.24.m1.1.1.1.1.1.2" xref="S5.p2.24.m1.1.1.1.1.1.2.cmml"><mi id="S5.p2.24.m1.1.1.1.1.1.2.2" xref="S5.p2.24.m1.1.1.1.1.1.2.2.cmml">f</mi><mi id="S5.p2.24.m1.1.1.1.1.1.2.3" xref="S5.p2.24.m1.1.1.1.1.1.2.3.cmml">h</mi></msub><mo id="S5.p2.24.m1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.p2.24.m1.1.1.1.1.1.1.cmml">×</mo><msub id="S5.p2.24.m1.1.1.1.1.1.3" xref="S5.p2.24.m1.1.1.1.1.1.3.cmml"><mi id="S5.p2.24.m1.1.1.1.1.1.3.2" xref="S5.p2.24.m1.1.1.1.1.1.3.2.cmml">f</mi><mi id="S5.p2.24.m1.1.1.1.1.1.3.3" xref="S5.p2.24.m1.1.1.1.1.1.3.3.cmml">v</mi></msub></mrow><mo id="S5.p2.24.m1.1.1.1.1.3" stretchy="false" xref="S5.p2.24.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.p2.24.m1.1.1.3" xref="S5.p2.24.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.p2.24.m1.1b"><apply id="S5.p2.24.m1.1.1.cmml" xref="S5.p2.24.m1.1.1"><csymbol cd="ambiguous" id="S5.p2.24.m1.1.1.2.cmml" xref="S5.p2.24.m1.1.1">superscript</csymbol><apply id="S5.p2.24.m1.1.1.1.1.1.cmml" xref="S5.p2.24.m1.1.1.1.1"><times id="S5.p2.24.m1.1.1.1.1.1.1.cmml" xref="S5.p2.24.m1.1.1.1.1.1.1"></times><apply id="S5.p2.24.m1.1.1.1.1.1.2.cmml" xref="S5.p2.24.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.p2.24.m1.1.1.1.1.1.2.1.cmml" xref="S5.p2.24.m1.1.1.1.1.1.2">subscript</csymbol><ci id="S5.p2.24.m1.1.1.1.1.1.2.2.cmml" xref="S5.p2.24.m1.1.1.1.1.1.2.2">𝑓</ci><ci id="S5.p2.24.m1.1.1.1.1.1.2.3.cmml" xref="S5.p2.24.m1.1.1.1.1.1.2.3">ℎ</ci></apply><apply id="S5.p2.24.m1.1.1.1.1.1.3.cmml" xref="S5.p2.24.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.p2.24.m1.1.1.1.1.1.3.1.cmml" xref="S5.p2.24.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.p2.24.m1.1.1.1.1.1.3.2.cmml" xref="S5.p2.24.m1.1.1.1.1.1.3.2">𝑓</ci><ci id="S5.p2.24.m1.1.1.1.1.1.3.3.cmml" xref="S5.p2.24.m1.1.1.1.1.1.3.3">𝑣</ci></apply></apply><times id="S5.p2.24.m1.1.1.3.cmml" xref="S5.p2.24.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.24.m1.1c">(f_{h}\times f_{v})^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.24.m1.1d">( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> follows from the identification of <math alttext="NG(d_{0})_{q}" class="ltx_Math" display="inline" id="S5.p2.25.m2.1"><semantics id="S5.p2.25.m2.1a"><mrow id="S5.p2.25.m2.1.1" xref="S5.p2.25.m2.1.1.cmml"><mi id="S5.p2.25.m2.1.1.3" xref="S5.p2.25.m2.1.1.3.cmml">N</mi><mo id="S5.p2.25.m2.1.1.2" xref="S5.p2.25.m2.1.1.2.cmml">⁢</mo><mi id="S5.p2.25.m2.1.1.4" xref="S5.p2.25.m2.1.1.4.cmml">G</mi><mo id="S5.p2.25.m2.1.1.2a" xref="S5.p2.25.m2.1.1.2.cmml">⁢</mo><msub id="S5.p2.25.m2.1.1.1" xref="S5.p2.25.m2.1.1.1.cmml"><mrow id="S5.p2.25.m2.1.1.1.1.1" xref="S5.p2.25.m2.1.1.1.1.1.1.cmml"><mo id="S5.p2.25.m2.1.1.1.1.1.2" stretchy="false" xref="S5.p2.25.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S5.p2.25.m2.1.1.1.1.1.1" xref="S5.p2.25.m2.1.1.1.1.1.1.cmml"><mi id="S5.p2.25.m2.1.1.1.1.1.1.2" xref="S5.p2.25.m2.1.1.1.1.1.1.2.cmml">d</mi><mn id="S5.p2.25.m2.1.1.1.1.1.1.3" xref="S5.p2.25.m2.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S5.p2.25.m2.1.1.1.1.1.3" stretchy="false" xref="S5.p2.25.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.p2.25.m2.1.1.1.3" xref="S5.p2.25.m2.1.1.1.3.cmml">q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.25.m2.1b"><apply id="S5.p2.25.m2.1.1.cmml" xref="S5.p2.25.m2.1.1"><times id="S5.p2.25.m2.1.1.2.cmml" xref="S5.p2.25.m2.1.1.2"></times><ci id="S5.p2.25.m2.1.1.3.cmml" xref="S5.p2.25.m2.1.1.3">𝑁</ci><ci id="S5.p2.25.m2.1.1.4.cmml" xref="S5.p2.25.m2.1.1.4">𝐺</ci><apply id="S5.p2.25.m2.1.1.1.cmml" xref="S5.p2.25.m2.1.1.1"><csymbol cd="ambiguous" id="S5.p2.25.m2.1.1.1.2.cmml" xref="S5.p2.25.m2.1.1.1">subscript</csymbol><apply id="S5.p2.25.m2.1.1.1.1.1.1.cmml" xref="S5.p2.25.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p2.25.m2.1.1.1.1.1.1.1.cmml" xref="S5.p2.25.m2.1.1.1.1.1">subscript</csymbol><ci id="S5.p2.25.m2.1.1.1.1.1.1.2.cmml" xref="S5.p2.25.m2.1.1.1.1.1.1.2">𝑑</ci><cn id="S5.p2.25.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.p2.25.m2.1.1.1.1.1.1.3">0</cn></apply><ci id="S5.p2.25.m2.1.1.1.3.cmml" xref="S5.p2.25.m2.1.1.1.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.25.m2.1c">NG(d_{0})_{q}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.25.m2.1d">italic_N italic_G ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> with the set of pairs <math alttext="(\tau,\mu:\varphi(c_{q})\to d_{0})" class="ltx_Math" display="inline" id="S5.p2.26.m3.3"><semantics id="S5.p2.26.m3.3a"><mrow id="S5.p2.26.m3.3.3.1" xref="S5.p2.26.m3.3.3.1.1.cmml"><mo id="S5.p2.26.m3.3.3.1.2" stretchy="false" xref="S5.p2.26.m3.3.3.1.1.cmml">(</mo><mrow id="S5.p2.26.m3.3.3.1.1" xref="S5.p2.26.m3.3.3.1.1.cmml"><mrow id="S5.p2.26.m3.3.3.1.1.3.2" xref="S5.p2.26.m3.3.3.1.1.3.1.cmml"><mi id="S5.p2.26.m3.1.1" xref="S5.p2.26.m3.1.1.cmml">τ</mi><mo id="S5.p2.26.m3.3.3.1.1.3.2.1" xref="S5.p2.26.m3.3.3.1.1.3.1.cmml">,</mo><mi id="S5.p2.26.m3.2.2" xref="S5.p2.26.m3.2.2.cmml">μ</mi></mrow><mo id="S5.p2.26.m3.3.3.1.1.2" lspace="0.278em" rspace="0.278em" xref="S5.p2.26.m3.3.3.1.1.2.cmml">:</mo><mrow id="S5.p2.26.m3.3.3.1.1.1" xref="S5.p2.26.m3.3.3.1.1.1.cmml"><mrow id="S5.p2.26.m3.3.3.1.1.1.1" xref="S5.p2.26.m3.3.3.1.1.1.1.cmml"><mi id="S5.p2.26.m3.3.3.1.1.1.1.3" xref="S5.p2.26.m3.3.3.1.1.1.1.3.cmml">φ</mi><mo id="S5.p2.26.m3.3.3.1.1.1.1.2" xref="S5.p2.26.m3.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.p2.26.m3.3.3.1.1.1.1.1.1" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.cmml"><mo id="S5.p2.26.m3.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.p2.26.m3.3.3.1.1.1.1.1.1.1" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.cmml"><mi id="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.2" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.2.cmml">c</mi><mi id="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.3" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.3.cmml">q</mi></msub><mo id="S5.p2.26.m3.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p2.26.m3.3.3.1.1.1.2" stretchy="false" xref="S5.p2.26.m3.3.3.1.1.1.2.cmml">→</mo><msub id="S5.p2.26.m3.3.3.1.1.1.3" xref="S5.p2.26.m3.3.3.1.1.1.3.cmml"><mi id="S5.p2.26.m3.3.3.1.1.1.3.2" xref="S5.p2.26.m3.3.3.1.1.1.3.2.cmml">d</mi><mn id="S5.p2.26.m3.3.3.1.1.1.3.3" xref="S5.p2.26.m3.3.3.1.1.1.3.3.cmml">0</mn></msub></mrow></mrow><mo id="S5.p2.26.m3.3.3.1.3" stretchy="false" xref="S5.p2.26.m3.3.3.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.26.m3.3b"><apply id="S5.p2.26.m3.3.3.1.1.cmml" xref="S5.p2.26.m3.3.3.1"><ci id="S5.p2.26.m3.3.3.1.1.2.cmml" xref="S5.p2.26.m3.3.3.1.1.2">:</ci><list id="S5.p2.26.m3.3.3.1.1.3.1.cmml" xref="S5.p2.26.m3.3.3.1.1.3.2"><ci id="S5.p2.26.m3.1.1.cmml" xref="S5.p2.26.m3.1.1">𝜏</ci><ci id="S5.p2.26.m3.2.2.cmml" xref="S5.p2.26.m3.2.2">𝜇</ci></list><apply id="S5.p2.26.m3.3.3.1.1.1.cmml" xref="S5.p2.26.m3.3.3.1.1.1"><ci id="S5.p2.26.m3.3.3.1.1.1.2.cmml" xref="S5.p2.26.m3.3.3.1.1.1.2">→</ci><apply id="S5.p2.26.m3.3.3.1.1.1.1.cmml" xref="S5.p2.26.m3.3.3.1.1.1.1"><times id="S5.p2.26.m3.3.3.1.1.1.1.2.cmml" xref="S5.p2.26.m3.3.3.1.1.1.1.2"></times><ci id="S5.p2.26.m3.3.3.1.1.1.1.3.cmml" xref="S5.p2.26.m3.3.3.1.1.1.1.3">𝜑</ci><apply id="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.cmml" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.1.cmml" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.2.cmml" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.3.cmml" xref="S5.p2.26.m3.3.3.1.1.1.1.1.1.1.3">𝑞</ci></apply></apply><apply id="S5.p2.26.m3.3.3.1.1.1.3.cmml" xref="S5.p2.26.m3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S5.p2.26.m3.3.3.1.1.1.3.1.cmml" xref="S5.p2.26.m3.3.3.1.1.1.3">subscript</csymbol><ci id="S5.p2.26.m3.3.3.1.1.1.3.2.cmml" xref="S5.p2.26.m3.3.3.1.1.1.3.2">𝑑</ci><cn id="S5.p2.26.m3.3.3.1.1.1.3.3.cmml" type="integer" xref="S5.p2.26.m3.3.3.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.26.m3.3c">(\tau,\mu:\varphi(c_{q})\to d_{0})</annotation><annotation encoding="application/x-llamapun" id="S5.p2.26.m3.3d">( italic_τ , italic_μ : italic_φ ( italic_c start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) → italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>, so it respects the composition of functions in <math alttext="\Delta\times\Delta" class="ltx_Math" display="inline" id="S5.p2.27.m4.1"><semantics id="S5.p2.27.m4.1a"><mrow id="S5.p2.27.m4.1.1" xref="S5.p2.27.m4.1.1.cmml"><mi id="S5.p2.27.m4.1.1.2" mathvariant="normal" xref="S5.p2.27.m4.1.1.2.cmml">Δ</mi><mo id="S5.p2.27.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.p2.27.m4.1.1.1.cmml">×</mo><mi id="S5.p2.27.m4.1.1.3" mathvariant="normal" xref="S5.p2.27.m4.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.27.m4.1b"><apply id="S5.p2.27.m4.1.1.cmml" xref="S5.p2.27.m4.1.1"><times id="S5.p2.27.m4.1.1.1.cmml" xref="S5.p2.27.m4.1.1.1"></times><ci id="S5.p2.27.m4.1.1.2.cmml" xref="S5.p2.27.m4.1.1.2">Δ</ci><ci id="S5.p2.27.m4.1.1.3.cmml" xref="S5.p2.27.m4.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.27.m4.1c">\Delta\times\Delta</annotation><annotation encoding="application/x-llamapun" id="S5.p2.27.m4.1d">roman_Δ × roman_Δ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.6">The homotopy colimit</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex52"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="X:=\operatorname*{hocolim}_{\mathcal{D}}G" class="ltx_Math" display="block" id="S5.Ex52.m1.1"><semantics id="S5.Ex52.m1.1a"><mrow id="S5.Ex52.m1.1.1" xref="S5.Ex52.m1.1.1.cmml"><mi id="S5.Ex52.m1.1.1.2" xref="S5.Ex52.m1.1.1.2.cmml">X</mi><mo id="S5.Ex52.m1.1.1.1" lspace="0.278em" xref="S5.Ex52.m1.1.1.1.cmml">:=</mo><mrow id="S5.Ex52.m1.1.1.3" xref="S5.Ex52.m1.1.1.3.cmml"><munder id="S5.Ex52.m1.1.1.3.1" xref="S5.Ex52.m1.1.1.3.1.cmml"><mo id="S5.Ex52.m1.1.1.3.1.2" lspace="0.111em" rspace="0.167em" xref="S5.Ex52.m1.1.1.3.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex52.m1.1.1.3.1.3" xref="S5.Ex52.m1.1.1.3.1.3.cmml">𝒟</mi></munder><mi id="S5.Ex52.m1.1.1.3.2" xref="S5.Ex52.m1.1.1.3.2.cmml">G</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex52.m1.1b"><apply id="S5.Ex52.m1.1.1.cmml" xref="S5.Ex52.m1.1.1"><csymbol cd="latexml" id="S5.Ex52.m1.1.1.1.cmml" xref="S5.Ex52.m1.1.1.1">assign</csymbol><ci id="S5.Ex52.m1.1.1.2.cmml" xref="S5.Ex52.m1.1.1.2">𝑋</ci><apply id="S5.Ex52.m1.1.1.3.cmml" xref="S5.Ex52.m1.1.1.3"><apply id="S5.Ex52.m1.1.1.3.1.cmml" xref="S5.Ex52.m1.1.1.3.1"><csymbol cd="ambiguous" id="S5.Ex52.m1.1.1.3.1.1.cmml" xref="S5.Ex52.m1.1.1.3.1">subscript</csymbol><ci id="S5.Ex52.m1.1.1.3.1.2.cmml" xref="S5.Ex52.m1.1.1.3.1.2">hocolim</ci><ci id="S5.Ex52.m1.1.1.3.1.3.cmml" xref="S5.Ex52.m1.1.1.3.1.3">𝒟</ci></apply><ci id="S5.Ex52.m1.1.1.3.2.cmml" xref="S5.Ex52.m1.1.1.3.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex52.m1.1c">X:=\operatorname*{hocolim}_{\mathcal{D}}G</annotation><annotation encoding="application/x-llamapun" id="S5.Ex52.m1.1d">italic_X := roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_G</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p3.4">is the diagonal of the bisimplicial set <math alttext="N(\mathcal{D};G)" class="ltx_Math" display="inline" id="S5.p3.1.m1.2"><semantics id="S5.p3.1.m1.2a"><mrow id="S5.p3.1.m1.2.3" xref="S5.p3.1.m1.2.3.cmml"><mi id="S5.p3.1.m1.2.3.2" xref="S5.p3.1.m1.2.3.2.cmml">N</mi><mo id="S5.p3.1.m1.2.3.1" xref="S5.p3.1.m1.2.3.1.cmml">⁢</mo><mrow id="S5.p3.1.m1.2.3.3.2" xref="S5.p3.1.m1.2.3.3.1.cmml"><mo id="S5.p3.1.m1.2.3.3.2.1" stretchy="false" xref="S5.p3.1.m1.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml">𝒟</mi><mo id="S5.p3.1.m1.2.3.3.2.2" xref="S5.p3.1.m1.2.3.3.1.cmml">;</mo><mi id="S5.p3.1.m1.2.2" xref="S5.p3.1.m1.2.2.cmml">G</mi><mo id="S5.p3.1.m1.2.3.3.2.3" stretchy="false" xref="S5.p3.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.2b"><apply id="S5.p3.1.m1.2.3.cmml" xref="S5.p3.1.m1.2.3"><times id="S5.p3.1.m1.2.3.1.cmml" xref="S5.p3.1.m1.2.3.1"></times><ci id="S5.p3.1.m1.2.3.2.cmml" xref="S5.p3.1.m1.2.3.2">𝑁</ci><list id="S5.p3.1.m1.2.3.3.1.cmml" xref="S5.p3.1.m1.2.3.3.2"><ci id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1">𝒟</ci><ci id="S5.p3.1.m1.2.2.cmml" xref="S5.p3.1.m1.2.2">𝐺</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.2c">N(\mathcal{D};G)</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.2d">italic_N ( caligraphic_D ; italic_G )</annotation></semantics></math>. Let <math alttext="\kappa:X\to N\mathcal{C}" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mrow id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml"><mi id="S5.p3.2.m2.1.1.2" xref="S5.p3.2.m2.1.1.2.cmml">κ</mi><mo id="S5.p3.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.p3.2.m2.1.1.1.cmml">:</mo><mrow id="S5.p3.2.m2.1.1.3" xref="S5.p3.2.m2.1.1.3.cmml"><mi id="S5.p3.2.m2.1.1.3.2" xref="S5.p3.2.m2.1.1.3.2.cmml">X</mi><mo id="S5.p3.2.m2.1.1.3.1" stretchy="false" xref="S5.p3.2.m2.1.1.3.1.cmml">→</mo><mrow id="S5.p3.2.m2.1.1.3.3" xref="S5.p3.2.m2.1.1.3.3.cmml"><mi id="S5.p3.2.m2.1.1.3.3.2" xref="S5.p3.2.m2.1.1.3.3.2.cmml">N</mi><mo id="S5.p3.2.m2.1.1.3.3.1" xref="S5.p3.2.m2.1.1.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.p3.2.m2.1.1.3.3.3" xref="S5.p3.2.m2.1.1.3.3.3.cmml">𝒞</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><apply id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1"><ci id="S5.p3.2.m2.1.1.1.cmml" xref="S5.p3.2.m2.1.1.1">:</ci><ci id="S5.p3.2.m2.1.1.2.cmml" xref="S5.p3.2.m2.1.1.2">𝜅</ci><apply id="S5.p3.2.m2.1.1.3.cmml" xref="S5.p3.2.m2.1.1.3"><ci id="S5.p3.2.m2.1.1.3.1.cmml" xref="S5.p3.2.m2.1.1.3.1">→</ci><ci id="S5.p3.2.m2.1.1.3.2.cmml" xref="S5.p3.2.m2.1.1.3.2">𝑋</ci><apply id="S5.p3.2.m2.1.1.3.3.cmml" xref="S5.p3.2.m2.1.1.3.3"><times id="S5.p3.2.m2.1.1.3.3.1.cmml" xref="S5.p3.2.m2.1.1.3.3.1"></times><ci id="S5.p3.2.m2.1.1.3.3.2.cmml" xref="S5.p3.2.m2.1.1.3.3.2">𝑁</ci><ci id="S5.p3.2.m2.1.1.3.3.3.cmml" xref="S5.p3.2.m2.1.1.3.3.3">𝒞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">\kappa:X\to N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">italic_κ : italic_X → italic_N caligraphic_C</annotation></semantics></math> be the simplicial map defined by <math alttext="\kappa(\sigma,\tau,\mu)=\tau" class="ltx_Math" display="inline" id="S5.p3.3.m3.3"><semantics id="S5.p3.3.m3.3a"><mrow id="S5.p3.3.m3.3.4" xref="S5.p3.3.m3.3.4.cmml"><mrow id="S5.p3.3.m3.3.4.2" xref="S5.p3.3.m3.3.4.2.cmml"><mi id="S5.p3.3.m3.3.4.2.2" xref="S5.p3.3.m3.3.4.2.2.cmml">κ</mi><mo id="S5.p3.3.m3.3.4.2.1" xref="S5.p3.3.m3.3.4.2.1.cmml">⁢</mo><mrow id="S5.p3.3.m3.3.4.2.3.2" xref="S5.p3.3.m3.3.4.2.3.1.cmml"><mo id="S5.p3.3.m3.3.4.2.3.2.1" stretchy="false" xref="S5.p3.3.m3.3.4.2.3.1.cmml">(</mo><mi id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml">σ</mi><mo id="S5.p3.3.m3.3.4.2.3.2.2" xref="S5.p3.3.m3.3.4.2.3.1.cmml">,</mo><mi id="S5.p3.3.m3.2.2" xref="S5.p3.3.m3.2.2.cmml">τ</mi><mo id="S5.p3.3.m3.3.4.2.3.2.3" xref="S5.p3.3.m3.3.4.2.3.1.cmml">,</mo><mi id="S5.p3.3.m3.3.3" xref="S5.p3.3.m3.3.3.cmml">μ</mi><mo id="S5.p3.3.m3.3.4.2.3.2.4" stretchy="false" xref="S5.p3.3.m3.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.p3.3.m3.3.4.1" xref="S5.p3.3.m3.3.4.1.cmml">=</mo><mi id="S5.p3.3.m3.3.4.3" xref="S5.p3.3.m3.3.4.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.3b"><apply id="S5.p3.3.m3.3.4.cmml" xref="S5.p3.3.m3.3.4"><eq id="S5.p3.3.m3.3.4.1.cmml" xref="S5.p3.3.m3.3.4.1"></eq><apply id="S5.p3.3.m3.3.4.2.cmml" xref="S5.p3.3.m3.3.4.2"><times id="S5.p3.3.m3.3.4.2.1.cmml" xref="S5.p3.3.m3.3.4.2.1"></times><ci id="S5.p3.3.m3.3.4.2.2.cmml" xref="S5.p3.3.m3.3.4.2.2">𝜅</ci><vector id="S5.p3.3.m3.3.4.2.3.1.cmml" xref="S5.p3.3.m3.3.4.2.3.2"><ci id="S5.p3.3.m3.1.1.cmml" xref="S5.p3.3.m3.1.1">𝜎</ci><ci id="S5.p3.3.m3.2.2.cmml" xref="S5.p3.3.m3.2.2">𝜏</ci><ci id="S5.p3.3.m3.3.3.cmml" xref="S5.p3.3.m3.3.3">𝜇</ci></vector></apply><ci id="S5.p3.3.m3.3.4.3.cmml" xref="S5.p3.3.m3.3.4.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.3c">\kappa(\sigma,\tau,\mu)=\tau</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.3d">italic_κ ( italic_σ , italic_τ , italic_μ ) = italic_τ</annotation></semantics></math>. Our main aim in this section is to prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a>: For any coefficient system <math alttext="\mathcal{M}:\Delta(N\mathcal{C})\to R" class="ltx_Math" display="inline" id="S5.p3.4.m4.1"><semantics id="S5.p3.4.m4.1a"><mrow id="S5.p3.4.m4.1.1" xref="S5.p3.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p3.4.m4.1.1.3" xref="S5.p3.4.m4.1.1.3.cmml">ℳ</mi><mo id="S5.p3.4.m4.1.1.2" lspace="0.278em" rspace="0.278em" xref="S5.p3.4.m4.1.1.2.cmml">:</mo><mrow id="S5.p3.4.m4.1.1.1" xref="S5.p3.4.m4.1.1.1.cmml"><mrow id="S5.p3.4.m4.1.1.1.1" xref="S5.p3.4.m4.1.1.1.1.cmml"><mi id="S5.p3.4.m4.1.1.1.1.3" mathvariant="normal" xref="S5.p3.4.m4.1.1.1.1.3.cmml">Δ</mi><mo id="S5.p3.4.m4.1.1.1.1.2" xref="S5.p3.4.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.p3.4.m4.1.1.1.1.1.1" xref="S5.p3.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S5.p3.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S5.p3.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.p3.4.m4.1.1.1.1.1.1.1" xref="S5.p3.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S5.p3.4.m4.1.1.1.1.1.1.1.2" xref="S5.p3.4.m4.1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S5.p3.4.m4.1.1.1.1.1.1.1.1" xref="S5.p3.4.m4.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.p3.4.m4.1.1.1.1.1.1.1.3" xref="S5.p3.4.m4.1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.p3.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S5.p3.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p3.4.m4.1.1.1.2" stretchy="false" xref="S5.p3.4.m4.1.1.1.2.cmml">→</mo><mi id="S5.p3.4.m4.1.1.1.3" xref="S5.p3.4.m4.1.1.1.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.4.m4.1b"><apply id="S5.p3.4.m4.1.1.cmml" xref="S5.p3.4.m4.1.1"><ci id="S5.p3.4.m4.1.1.2.cmml" xref="S5.p3.4.m4.1.1.2">:</ci><ci id="S5.p3.4.m4.1.1.3.cmml" xref="S5.p3.4.m4.1.1.3">ℳ</ci><apply id="S5.p3.4.m4.1.1.1.cmml" xref="S5.p3.4.m4.1.1.1"><ci id="S5.p3.4.m4.1.1.1.2.cmml" xref="S5.p3.4.m4.1.1.1.2">→</ci><apply id="S5.p3.4.m4.1.1.1.1.cmml" xref="S5.p3.4.m4.1.1.1.1"><times id="S5.p3.4.m4.1.1.1.1.2.cmml" xref="S5.p3.4.m4.1.1.1.1.2"></times><ci id="S5.p3.4.m4.1.1.1.1.3.cmml" xref="S5.p3.4.m4.1.1.1.1.3">Δ</ci><apply id="S5.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.p3.4.m4.1.1.1.1.1.1"><times id="S5.p3.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S5.p3.4.m4.1.1.1.1.1.1.1.1"></times><ci id="S5.p3.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S5.p3.4.m4.1.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.p3.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S5.p3.4.m4.1.1.1.1.1.1.1.3">𝒞</ci></apply></apply><ci id="S5.p3.4.m4.1.1.1.3.cmml" xref="S5.p3.4.m4.1.1.1.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.4.m4.1c">\mathcal{M}:\Delta(N\mathcal{C})\to R</annotation><annotation encoding="application/x-llamapun" id="S5.p3.4.m4.1d">caligraphic_M : roman_Δ ( italic_N caligraphic_C ) → italic_R</annotation></semantics></math>-Mod, the induced map</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex53"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\kappa^{*}:H^{*}_{Th}(\mathcal{C};\mathcal{M})\to H^{*}(X;\kappa^{*}\mathcal{M})" class="ltx_Math" display="block" id="S5.Ex53.m1.4"><semantics id="S5.Ex53.m1.4a"><mrow id="S5.Ex53.m1.4.4" xref="S5.Ex53.m1.4.4.cmml"><msup id="S5.Ex53.m1.4.4.3" xref="S5.Ex53.m1.4.4.3.cmml"><mi id="S5.Ex53.m1.4.4.3.2" xref="S5.Ex53.m1.4.4.3.2.cmml">κ</mi><mo id="S5.Ex53.m1.4.4.3.3" xref="S5.Ex53.m1.4.4.3.3.cmml">∗</mo></msup><mo id="S5.Ex53.m1.4.4.2" lspace="0.278em" rspace="0.278em" xref="S5.Ex53.m1.4.4.2.cmml">:</mo><mrow id="S5.Ex53.m1.4.4.1" xref="S5.Ex53.m1.4.4.1.cmml"><mrow id="S5.Ex53.m1.4.4.1.3" xref="S5.Ex53.m1.4.4.1.3.cmml"><msubsup id="S5.Ex53.m1.4.4.1.3.2" xref="S5.Ex53.m1.4.4.1.3.2.cmml"><mi id="S5.Ex53.m1.4.4.1.3.2.2.2" xref="S5.Ex53.m1.4.4.1.3.2.2.2.cmml">H</mi><mrow id="S5.Ex53.m1.4.4.1.3.2.3" xref="S5.Ex53.m1.4.4.1.3.2.3.cmml"><mi id="S5.Ex53.m1.4.4.1.3.2.3.2" xref="S5.Ex53.m1.4.4.1.3.2.3.2.cmml">T</mi><mo id="S5.Ex53.m1.4.4.1.3.2.3.1" xref="S5.Ex53.m1.4.4.1.3.2.3.1.cmml">⁢</mo><mi id="S5.Ex53.m1.4.4.1.3.2.3.3" xref="S5.Ex53.m1.4.4.1.3.2.3.3.cmml">h</mi></mrow><mo id="S5.Ex53.m1.4.4.1.3.2.2.3" xref="S5.Ex53.m1.4.4.1.3.2.2.3.cmml">∗</mo></msubsup><mo id="S5.Ex53.m1.4.4.1.3.1" xref="S5.Ex53.m1.4.4.1.3.1.cmml">⁢</mo><mrow id="S5.Ex53.m1.4.4.1.3.3.2" xref="S5.Ex53.m1.4.4.1.3.3.1.cmml"><mo id="S5.Ex53.m1.4.4.1.3.3.2.1" stretchy="false" xref="S5.Ex53.m1.4.4.1.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex53.m1.1.1" xref="S5.Ex53.m1.1.1.cmml">𝒞</mi><mo id="S5.Ex53.m1.4.4.1.3.3.2.2" xref="S5.Ex53.m1.4.4.1.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex53.m1.2.2" xref="S5.Ex53.m1.2.2.cmml">ℳ</mi><mo id="S5.Ex53.m1.4.4.1.3.3.2.3" stretchy="false" xref="S5.Ex53.m1.4.4.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex53.m1.4.4.1.2" stretchy="false" xref="S5.Ex53.m1.4.4.1.2.cmml">→</mo><mrow id="S5.Ex53.m1.4.4.1.1" xref="S5.Ex53.m1.4.4.1.1.cmml"><msup id="S5.Ex53.m1.4.4.1.1.3" xref="S5.Ex53.m1.4.4.1.1.3.cmml"><mi id="S5.Ex53.m1.4.4.1.1.3.2" xref="S5.Ex53.m1.4.4.1.1.3.2.cmml">H</mi><mo id="S5.Ex53.m1.4.4.1.1.3.3" xref="S5.Ex53.m1.4.4.1.1.3.3.cmml">∗</mo></msup><mo id="S5.Ex53.m1.4.4.1.1.2" xref="S5.Ex53.m1.4.4.1.1.2.cmml">⁢</mo><mrow id="S5.Ex53.m1.4.4.1.1.1.1" xref="S5.Ex53.m1.4.4.1.1.1.2.cmml"><mo id="S5.Ex53.m1.4.4.1.1.1.1.2" stretchy="false" xref="S5.Ex53.m1.4.4.1.1.1.2.cmml">(</mo><mi id="S5.Ex53.m1.3.3" xref="S5.Ex53.m1.3.3.cmml">X</mi><mo id="S5.Ex53.m1.4.4.1.1.1.1.3" xref="S5.Ex53.m1.4.4.1.1.1.2.cmml">;</mo><mrow id="S5.Ex53.m1.4.4.1.1.1.1.1" xref="S5.Ex53.m1.4.4.1.1.1.1.1.cmml"><msup id="S5.Ex53.m1.4.4.1.1.1.1.1.2" xref="S5.Ex53.m1.4.4.1.1.1.1.1.2.cmml"><mi id="S5.Ex53.m1.4.4.1.1.1.1.1.2.2" xref="S5.Ex53.m1.4.4.1.1.1.1.1.2.2.cmml">κ</mi><mo id="S5.Ex53.m1.4.4.1.1.1.1.1.2.3" xref="S5.Ex53.m1.4.4.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S5.Ex53.m1.4.4.1.1.1.1.1.1" xref="S5.Ex53.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex53.m1.4.4.1.1.1.1.1.3" xref="S5.Ex53.m1.4.4.1.1.1.1.1.3.cmml">ℳ</mi></mrow><mo id="S5.Ex53.m1.4.4.1.1.1.1.4" stretchy="false" xref="S5.Ex53.m1.4.4.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex53.m1.4b"><apply id="S5.Ex53.m1.4.4.cmml" xref="S5.Ex53.m1.4.4"><ci id="S5.Ex53.m1.4.4.2.cmml" xref="S5.Ex53.m1.4.4.2">:</ci><apply id="S5.Ex53.m1.4.4.3.cmml" xref="S5.Ex53.m1.4.4.3"><csymbol cd="ambiguous" id="S5.Ex53.m1.4.4.3.1.cmml" xref="S5.Ex53.m1.4.4.3">superscript</csymbol><ci id="S5.Ex53.m1.4.4.3.2.cmml" xref="S5.Ex53.m1.4.4.3.2">𝜅</ci><times id="S5.Ex53.m1.4.4.3.3.cmml" xref="S5.Ex53.m1.4.4.3.3"></times></apply><apply id="S5.Ex53.m1.4.4.1.cmml" xref="S5.Ex53.m1.4.4.1"><ci id="S5.Ex53.m1.4.4.1.2.cmml" xref="S5.Ex53.m1.4.4.1.2">→</ci><apply id="S5.Ex53.m1.4.4.1.3.cmml" xref="S5.Ex53.m1.4.4.1.3"><times id="S5.Ex53.m1.4.4.1.3.1.cmml" xref="S5.Ex53.m1.4.4.1.3.1"></times><apply id="S5.Ex53.m1.4.4.1.3.2.cmml" xref="S5.Ex53.m1.4.4.1.3.2"><csymbol cd="ambiguous" id="S5.Ex53.m1.4.4.1.3.2.1.cmml" xref="S5.Ex53.m1.4.4.1.3.2">subscript</csymbol><apply id="S5.Ex53.m1.4.4.1.3.2.2.cmml" xref="S5.Ex53.m1.4.4.1.3.2"><csymbol cd="ambiguous" id="S5.Ex53.m1.4.4.1.3.2.2.1.cmml" xref="S5.Ex53.m1.4.4.1.3.2">superscript</csymbol><ci id="S5.Ex53.m1.4.4.1.3.2.2.2.cmml" xref="S5.Ex53.m1.4.4.1.3.2.2.2">𝐻</ci><times id="S5.Ex53.m1.4.4.1.3.2.2.3.cmml" xref="S5.Ex53.m1.4.4.1.3.2.2.3"></times></apply><apply id="S5.Ex53.m1.4.4.1.3.2.3.cmml" xref="S5.Ex53.m1.4.4.1.3.2.3"><times id="S5.Ex53.m1.4.4.1.3.2.3.1.cmml" xref="S5.Ex53.m1.4.4.1.3.2.3.1"></times><ci id="S5.Ex53.m1.4.4.1.3.2.3.2.cmml" xref="S5.Ex53.m1.4.4.1.3.2.3.2">𝑇</ci><ci id="S5.Ex53.m1.4.4.1.3.2.3.3.cmml" xref="S5.Ex53.m1.4.4.1.3.2.3.3">ℎ</ci></apply></apply><list id="S5.Ex53.m1.4.4.1.3.3.1.cmml" xref="S5.Ex53.m1.4.4.1.3.3.2"><ci id="S5.Ex53.m1.1.1.cmml" xref="S5.Ex53.m1.1.1">𝒞</ci><ci id="S5.Ex53.m1.2.2.cmml" xref="S5.Ex53.m1.2.2">ℳ</ci></list></apply><apply id="S5.Ex53.m1.4.4.1.1.cmml" xref="S5.Ex53.m1.4.4.1.1"><times id="S5.Ex53.m1.4.4.1.1.2.cmml" xref="S5.Ex53.m1.4.4.1.1.2"></times><apply id="S5.Ex53.m1.4.4.1.1.3.cmml" xref="S5.Ex53.m1.4.4.1.1.3"><csymbol cd="ambiguous" id="S5.Ex53.m1.4.4.1.1.3.1.cmml" xref="S5.Ex53.m1.4.4.1.1.3">superscript</csymbol><ci id="S5.Ex53.m1.4.4.1.1.3.2.cmml" xref="S5.Ex53.m1.4.4.1.1.3.2">𝐻</ci><times id="S5.Ex53.m1.4.4.1.1.3.3.cmml" xref="S5.Ex53.m1.4.4.1.1.3.3"></times></apply><list id="S5.Ex53.m1.4.4.1.1.1.2.cmml" xref="S5.Ex53.m1.4.4.1.1.1.1"><ci id="S5.Ex53.m1.3.3.cmml" xref="S5.Ex53.m1.3.3">𝑋</ci><apply id="S5.Ex53.m1.4.4.1.1.1.1.1.cmml" xref="S5.Ex53.m1.4.4.1.1.1.1.1"><times id="S5.Ex53.m1.4.4.1.1.1.1.1.1.cmml" xref="S5.Ex53.m1.4.4.1.1.1.1.1.1"></times><apply id="S5.Ex53.m1.4.4.1.1.1.1.1.2.cmml" xref="S5.Ex53.m1.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex53.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S5.Ex53.m1.4.4.1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex53.m1.4.4.1.1.1.1.1.2.2.cmml" xref="S5.Ex53.m1.4.4.1.1.1.1.1.2.2">𝜅</ci><times id="S5.Ex53.m1.4.4.1.1.1.1.1.2.3.cmml" xref="S5.Ex53.m1.4.4.1.1.1.1.1.2.3"></times></apply><ci id="S5.Ex53.m1.4.4.1.1.1.1.1.3.cmml" xref="S5.Ex53.m1.4.4.1.1.1.1.1.3">ℳ</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex53.m1.4c">\kappa^{*}:H^{*}_{Th}(\mathcal{C};\mathcal{M})\to H^{*}(X;\kappa^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex53.m1.4d">italic_κ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; caligraphic_M ) → italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; italic_κ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p3.5">is an isomorphism. For this, we introduce a double complex of projective <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.p3.5.m1.1"><semantics id="S5.p3.5.m1.1a"><mrow id="S5.p3.5.m1.1.1" xref="S5.p3.5.m1.1.1.cmml"><mi id="S5.p3.5.m1.1.1.3" xref="S5.p3.5.m1.1.1.3.cmml">R</mi><mo id="S5.p3.5.m1.1.1.2" xref="S5.p3.5.m1.1.1.2.cmml">⁢</mo><mi id="S5.p3.5.m1.1.1.4" mathvariant="normal" xref="S5.p3.5.m1.1.1.4.cmml">Δ</mi><mo id="S5.p3.5.m1.1.1.2a" xref="S5.p3.5.m1.1.1.2.cmml">⁢</mo><mrow id="S5.p3.5.m1.1.1.1.1" xref="S5.p3.5.m1.1.1.1.1.1.cmml"><mo id="S5.p3.5.m1.1.1.1.1.2" stretchy="false" xref="S5.p3.5.m1.1.1.1.1.1.cmml">(</mo><mrow id="S5.p3.5.m1.1.1.1.1.1" xref="S5.p3.5.m1.1.1.1.1.1.cmml"><mi id="S5.p3.5.m1.1.1.1.1.1.2" xref="S5.p3.5.m1.1.1.1.1.1.2.cmml">N</mi><mo id="S5.p3.5.m1.1.1.1.1.1.1" xref="S5.p3.5.m1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.p3.5.m1.1.1.1.1.1.3" xref="S5.p3.5.m1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.p3.5.m1.1.1.1.1.3" stretchy="false" xref="S5.p3.5.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.5.m1.1b"><apply id="S5.p3.5.m1.1.1.cmml" xref="S5.p3.5.m1.1.1"><times id="S5.p3.5.m1.1.1.2.cmml" xref="S5.p3.5.m1.1.1.2"></times><ci id="S5.p3.5.m1.1.1.3.cmml" xref="S5.p3.5.m1.1.1.3">𝑅</ci><ci id="S5.p3.5.m1.1.1.4.cmml" xref="S5.p3.5.m1.1.1.4">Δ</ci><apply id="S5.p3.5.m1.1.1.1.1.1.cmml" xref="S5.p3.5.m1.1.1.1.1"><times id="S5.p3.5.m1.1.1.1.1.1.1.cmml" xref="S5.p3.5.m1.1.1.1.1.1.1"></times><ci id="S5.p3.5.m1.1.1.1.1.1.2.cmml" xref="S5.p3.5.m1.1.1.1.1.1.2">𝑁</ci><ci id="S5.p3.5.m1.1.1.1.1.1.3.cmml" xref="S5.p3.5.m1.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.5.m1.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.p3.5.m1.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-modules.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S5.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.1.1.1">Definition 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.3">For <math alttext="p,q\geq 0" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.m1.2"><semantics id="S5.Thmtheorem1.p1.1.m1.2a"><mrow id="S5.Thmtheorem1.p1.1.m1.2.3" xref="S5.Thmtheorem1.p1.1.m1.2.3.cmml"><mrow id="S5.Thmtheorem1.p1.1.m1.2.3.2.2" xref="S5.Thmtheorem1.p1.1.m1.2.3.2.1.cmml"><mi id="S5.Thmtheorem1.p1.1.m1.1.1" xref="S5.Thmtheorem1.p1.1.m1.1.1.cmml">p</mi><mo id="S5.Thmtheorem1.p1.1.m1.2.3.2.2.1" xref="S5.Thmtheorem1.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S5.Thmtheorem1.p1.1.m1.2.2" xref="S5.Thmtheorem1.p1.1.m1.2.2.cmml">q</mi></mrow><mo id="S5.Thmtheorem1.p1.1.m1.2.3.1" xref="S5.Thmtheorem1.p1.1.m1.2.3.1.cmml">≥</mo><mn id="S5.Thmtheorem1.p1.1.m1.2.3.3" xref="S5.Thmtheorem1.p1.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.1.m1.2b"><apply id="S5.Thmtheorem1.p1.1.m1.2.3.cmml" xref="S5.Thmtheorem1.p1.1.m1.2.3"><geq id="S5.Thmtheorem1.p1.1.m1.2.3.1.cmml" xref="S5.Thmtheorem1.p1.1.m1.2.3.1"></geq><list id="S5.Thmtheorem1.p1.1.m1.2.3.2.1.cmml" xref="S5.Thmtheorem1.p1.1.m1.2.3.2.2"><ci id="S5.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.m1.1.1">𝑝</ci><ci id="S5.Thmtheorem1.p1.1.m1.2.2.cmml" xref="S5.Thmtheorem1.p1.1.m1.2.2">𝑞</ci></list><cn id="S5.Thmtheorem1.p1.1.m1.2.3.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.1.m1.2c">p,q\geq 0</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.1.m1.2d">italic_p , italic_q ≥ 0</annotation></semantics></math>, let <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.2.m2.1"><semantics id="S5.Thmtheorem1.p1.2.m2.1a"><mi id="S5.Thmtheorem1.p1.2.m2.1.1" xref="S5.Thmtheorem1.p1.2.m2.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.2.m2.1b"><ci id="S5.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem1.p1.2.m2.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.2.m2.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.2.m2.1d">blackboard_P</annotation></semantics></math> denote the bisimplicial <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.3.m3.1"><semantics id="S5.Thmtheorem1.p1.3.m3.1a"><mrow id="S5.Thmtheorem1.p1.3.m3.1.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.cmml"><mi id="S5.Thmtheorem1.p1.3.m3.1.1.3" xref="S5.Thmtheorem1.p1.3.m3.1.1.3.cmml">R</mi><mo id="S5.Thmtheorem1.p1.3.m3.1.1.2" xref="S5.Thmtheorem1.p1.3.m3.1.1.2.cmml">⁢</mo><mi id="S5.Thmtheorem1.p1.3.m3.1.1.4" mathvariant="normal" xref="S5.Thmtheorem1.p1.3.m3.1.1.4.cmml">Δ</mi><mo id="S5.Thmtheorem1.p1.3.m3.1.1.2a" xref="S5.Thmtheorem1.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.3.m3.1.1.1.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.2" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.2.cmml">N</mi><mo id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.3" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.3.m3.1b"><apply id="S5.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1"><times id="S5.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.2"></times><ci id="S5.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.3">𝑅</ci><ci id="S5.Thmtheorem1.p1.3.m3.1.1.4.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.4">Δ</ci><apply id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1"><times id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.1"></times><ci id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.2">𝑁</ci><ci id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.m3.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.m3.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-module defined by</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex54"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbb{P}_{p,q}=\bigoplus_{(\sigma,\tau,\mu)\in N(\mathcal{D};G)_{p,q}}R% \mathrm{Mor}_{\Delta(N\mathcal{C})}(\tau,?)." class="ltx_Math" display="block" id="S5.Ex54.m1.13"><semantics id="S5.Ex54.m1.13a"><mrow id="S5.Ex54.m1.13.13.1" xref="S5.Ex54.m1.13.13.1.1.cmml"><mrow id="S5.Ex54.m1.13.13.1.1" xref="S5.Ex54.m1.13.13.1.1.cmml"><msub id="S5.Ex54.m1.13.13.1.1.2" xref="S5.Ex54.m1.13.13.1.1.2.cmml"><mi id="S5.Ex54.m1.13.13.1.1.2.2" xref="S5.Ex54.m1.13.13.1.1.2.2.cmml">ℙ</mi><mrow id="S5.Ex54.m1.2.2.2.4" xref="S5.Ex54.m1.2.2.2.3.cmml"><mi id="S5.Ex54.m1.1.1.1.1" xref="S5.Ex54.m1.1.1.1.1.cmml">p</mi><mo id="S5.Ex54.m1.2.2.2.4.1" xref="S5.Ex54.m1.2.2.2.3.cmml">,</mo><mi id="S5.Ex54.m1.2.2.2.2" xref="S5.Ex54.m1.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.Ex54.m1.13.13.1.1.1" rspace="0.111em" xref="S5.Ex54.m1.13.13.1.1.1.cmml">=</mo><mrow id="S5.Ex54.m1.13.13.1.1.3" xref="S5.Ex54.m1.13.13.1.1.3.cmml"><munder id="S5.Ex54.m1.13.13.1.1.3.1" xref="S5.Ex54.m1.13.13.1.1.3.1.cmml"><mo id="S5.Ex54.m1.13.13.1.1.3.1.2" movablelimits="false" xref="S5.Ex54.m1.13.13.1.1.3.1.2.cmml">⨁</mo><mrow id="S5.Ex54.m1.9.9.7" xref="S5.Ex54.m1.9.9.7.cmml"><mrow id="S5.Ex54.m1.9.9.7.9.2" xref="S5.Ex54.m1.9.9.7.9.1.cmml"><mo id="S5.Ex54.m1.9.9.7.9.2.1" stretchy="false" xref="S5.Ex54.m1.9.9.7.9.1.cmml">(</mo><mi id="S5.Ex54.m1.5.5.3.3" xref="S5.Ex54.m1.5.5.3.3.cmml">σ</mi><mo id="S5.Ex54.m1.9.9.7.9.2.2" xref="S5.Ex54.m1.9.9.7.9.1.cmml">,</mo><mi id="S5.Ex54.m1.6.6.4.4" xref="S5.Ex54.m1.6.6.4.4.cmml">τ</mi><mo id="S5.Ex54.m1.9.9.7.9.2.3" xref="S5.Ex54.m1.9.9.7.9.1.cmml">,</mo><mi id="S5.Ex54.m1.7.7.5.5" 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xref="S5.Ex54.m1.12.12">?</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex54.m1.13c">\mathbb{P}_{p,q}=\bigoplus_{(\sigma,\tau,\mu)\in N(\mathcal{D};G)_{p,q}}R% \mathrm{Mor}_{\Delta(N\mathcal{C})}(\tau,?).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex54.m1.13d">blackboard_P start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT ( italic_σ , italic_τ , italic_μ ) ∈ italic_N ( caligraphic_D ; italic_G ) start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_R roman_Mor start_POSTSUBSCRIPT roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( italic_τ , ? ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmtheorem1.p1.6">For <math alttext="\theta\in N\mathcal{C}_{n}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.4.m1.1"><semantics id="S5.Thmtheorem1.p1.4.m1.1a"><mrow id="S5.Thmtheorem1.p1.4.m1.1.1" xref="S5.Thmtheorem1.p1.4.m1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.4.m1.1.1.2" xref="S5.Thmtheorem1.p1.4.m1.1.1.2.cmml">θ</mi><mo id="S5.Thmtheorem1.p1.4.m1.1.1.1" xref="S5.Thmtheorem1.p1.4.m1.1.1.1.cmml">∈</mo><mrow id="S5.Thmtheorem1.p1.4.m1.1.1.3" xref="S5.Thmtheorem1.p1.4.m1.1.1.3.cmml"><mi id="S5.Thmtheorem1.p1.4.m1.1.1.3.2" xref="S5.Thmtheorem1.p1.4.m1.1.1.3.2.cmml">N</mi><mo id="S5.Thmtheorem1.p1.4.m1.1.1.3.1" xref="S5.Thmtheorem1.p1.4.m1.1.1.3.1.cmml">⁢</mo><msub id="S5.Thmtheorem1.p1.4.m1.1.1.3.3" xref="S5.Thmtheorem1.p1.4.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem1.p1.4.m1.1.1.3.3.2" xref="S5.Thmtheorem1.p1.4.m1.1.1.3.3.2.cmml">𝒞</mi><mi id="S5.Thmtheorem1.p1.4.m1.1.1.3.3.3" xref="S5.Thmtheorem1.p1.4.m1.1.1.3.3.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.4.m1.1b"><apply id="S5.Thmtheorem1.p1.4.m1.1.1.cmml" 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N\mathcal{C}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.4.m1.1d">italic_θ ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\mathbb{P}_{p,q}(\theta)" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.5.m2.3"><semantics id="S5.Thmtheorem1.p1.5.m2.3a"><mrow id="S5.Thmtheorem1.p1.5.m2.3.4" xref="S5.Thmtheorem1.p1.5.m2.3.4.cmml"><msub id="S5.Thmtheorem1.p1.5.m2.3.4.2" xref="S5.Thmtheorem1.p1.5.m2.3.4.2.cmml"><mi id="S5.Thmtheorem1.p1.5.m2.3.4.2.2" xref="S5.Thmtheorem1.p1.5.m2.3.4.2.2.cmml">ℙ</mi><mrow id="S5.Thmtheorem1.p1.5.m2.2.2.2.4" xref="S5.Thmtheorem1.p1.5.m2.2.2.2.3.cmml"><mi id="S5.Thmtheorem1.p1.5.m2.1.1.1.1" xref="S5.Thmtheorem1.p1.5.m2.1.1.1.1.cmml">p</mi><mo id="S5.Thmtheorem1.p1.5.m2.2.2.2.4.1" xref="S5.Thmtheorem1.p1.5.m2.2.2.2.3.cmml">,</mo><mi id="S5.Thmtheorem1.p1.5.m2.2.2.2.2" xref="S5.Thmtheorem1.p1.5.m2.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.Thmtheorem1.p1.5.m2.3.4.1" xref="S5.Thmtheorem1.p1.5.m2.3.4.1.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.5.m2.3.4.3.2" xref="S5.Thmtheorem1.p1.5.m2.3.4.cmml"><mo id="S5.Thmtheorem1.p1.5.m2.3.4.3.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.5.m2.3.4.cmml">(</mo><mi id="S5.Thmtheorem1.p1.5.m2.3.3" xref="S5.Thmtheorem1.p1.5.m2.3.3.cmml">θ</mi><mo id="S5.Thmtheorem1.p1.5.m2.3.4.3.2.2" stretchy="false" xref="S5.Thmtheorem1.p1.5.m2.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.5.m2.3b"><apply id="S5.Thmtheorem1.p1.5.m2.3.4.cmml" xref="S5.Thmtheorem1.p1.5.m2.3.4"><times id="S5.Thmtheorem1.p1.5.m2.3.4.1.cmml" xref="S5.Thmtheorem1.p1.5.m2.3.4.1"></times><apply id="S5.Thmtheorem1.p1.5.m2.3.4.2.cmml" xref="S5.Thmtheorem1.p1.5.m2.3.4.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.5.m2.3.4.2.1.cmml" xref="S5.Thmtheorem1.p1.5.m2.3.4.2">subscript</csymbol><ci id="S5.Thmtheorem1.p1.5.m2.3.4.2.2.cmml" xref="S5.Thmtheorem1.p1.5.m2.3.4.2.2">ℙ</ci><list id="S5.Thmtheorem1.p1.5.m2.2.2.2.3.cmml" xref="S5.Thmtheorem1.p1.5.m2.2.2.2.4"><ci id="S5.Thmtheorem1.p1.5.m2.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.5.m2.1.1.1.1">𝑝</ci><ci id="S5.Thmtheorem1.p1.5.m2.2.2.2.2.cmml" xref="S5.Thmtheorem1.p1.5.m2.2.2.2.2">𝑞</ci></list></apply><ci id="S5.Thmtheorem1.p1.5.m2.3.3.cmml" xref="S5.Thmtheorem1.p1.5.m2.3.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.5.m2.3c">\mathbb{P}_{p,q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.5.m2.3d">blackboard_P start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> is the free <math alttext="R" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.6.m3.1"><semantics id="S5.Thmtheorem1.p1.6.m3.1a"><mi id="S5.Thmtheorem1.p1.6.m3.1.1" xref="S5.Thmtheorem1.p1.6.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" 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encoding="application/x-tex" id="S5.Ex55.m1.19c">X_{p,q}(\theta)=\{(\sigma,\tau,\mu,f)\,|\,(\sigma,\tau,\mu)\in N(\mathcal{D};G% )_{p.q},f:[q]\to[n],f^{*}(\theta)=\tau\}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex55.m1.19d">italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( italic_θ ) = { ( italic_σ , italic_τ , italic_μ , italic_f ) | ( italic_σ , italic_τ , italic_μ ) ∈ italic_N ( caligraphic_D ; italic_G ) start_POSTSUBSCRIPT italic_p . italic_q end_POSTSUBSCRIPT , italic_f : [ italic_q ] → [ italic_n ] , italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_θ ) = italic_τ } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmtheorem1.p1.10">The vertical and horizontal maps for <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.7.m1.1"><semantics id="S5.Thmtheorem1.p1.7.m1.1a"><mi id="S5.Thmtheorem1.p1.7.m1.1.1" xref="S5.Thmtheorem1.p1.7.m1.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.7.m1.1b"><ci id="S5.Thmtheorem1.p1.7.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.7.m1.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.7.m1.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.7.m1.1d">blackboard_P</annotation></semantics></math> are defined as follows: If <math alttext="f_{h}\times f_{v}:([r],[s])\to([p],[q])" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.8.m2.8"><semantics id="S5.Thmtheorem1.p1.8.m2.8a"><mrow id="S5.Thmtheorem1.p1.8.m2.8.8" xref="S5.Thmtheorem1.p1.8.m2.8.8.cmml"><mrow id="S5.Thmtheorem1.p1.8.m2.8.8.6" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.cmml"><msub id="S5.Thmtheorem1.p1.8.m2.8.8.6.2" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.2.cmml"><mi id="S5.Thmtheorem1.p1.8.m2.8.8.6.2.2" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.2.2.cmml">f</mi><mi 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xref="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.1.1.cmml">[</mo><mi id="S5.Thmtheorem1.p1.8.m2.1.1" xref="S5.Thmtheorem1.p1.8.m2.1.1.cmml">r</mi><mo id="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.2.2" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.4" xref="S5.Thmtheorem1.p1.8.m2.6.6.2.2.3.cmml">,</mo><mrow id="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.2" xref="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.1.cmml"><mo id="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.1.1.cmml">[</mo><mi id="S5.Thmtheorem1.p1.8.m2.2.2" xref="S5.Thmtheorem1.p1.8.m2.2.2.cmml">s</mi><mo id="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.2.2" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.1.1.cmml">]</mo></mrow><mo id="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.5" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.6.6.2.2.3.cmml">)</mo></mrow><mo id="S5.Thmtheorem1.p1.8.m2.8.8.4.5" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.5.cmml">→</mo><mrow id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.3.cmml"><mo id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.3" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.3.cmml">(</mo><mrow id="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.2" xref="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.1.1.cmml">[</mo><mi id="S5.Thmtheorem1.p1.8.m2.3.3" xref="S5.Thmtheorem1.p1.8.m2.3.3.cmml">p</mi><mo id="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.2.2" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.1.1.cmml">]</mo></mrow><mo id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.4" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.3.cmml">,</mo><mrow id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.2" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.1.cmml"><mo id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.1.1.cmml">[</mo><mi id="S5.Thmtheorem1.p1.8.m2.4.4" xref="S5.Thmtheorem1.p1.8.m2.4.4.cmml">q</mi><mo id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.2.2" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.1.1.cmml">]</mo></mrow><mo id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.5" stretchy="false" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.8.m2.8b"><apply id="S5.Thmtheorem1.p1.8.m2.8.8.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8"><ci id="S5.Thmtheorem1.p1.8.m2.8.8.5.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.5">:</ci><apply id="S5.Thmtheorem1.p1.8.m2.8.8.6.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6"><times id="S5.Thmtheorem1.p1.8.m2.8.8.6.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.1"></times><apply id="S5.Thmtheorem1.p1.8.m2.8.8.6.2.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.8.m2.8.8.6.2.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.2">subscript</csymbol><ci id="S5.Thmtheorem1.p1.8.m2.8.8.6.2.2.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.2.2">𝑓</ci><ci id="S5.Thmtheorem1.p1.8.m2.8.8.6.2.3.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.2.3">ℎ</ci></apply><apply id="S5.Thmtheorem1.p1.8.m2.8.8.6.3.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.8.m2.8.8.6.3.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.3">subscript</csymbol><ci id="S5.Thmtheorem1.p1.8.m2.8.8.6.3.2.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.3.2">𝑓</ci><ci id="S5.Thmtheorem1.p1.8.m2.8.8.6.3.3.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.6.3.3">𝑣</ci></apply></apply><apply id="S5.Thmtheorem1.p1.8.m2.8.8.4.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.4"><ci id="S5.Thmtheorem1.p1.8.m2.8.8.4.5.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.5">→</ci><interval closure="open" id="S5.Thmtheorem1.p1.8.m2.6.6.2.2.3.cmml" xref="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2"><apply id="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.2"><csymbol cd="latexml" id="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.5.5.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S5.Thmtheorem1.p1.8.m2.1.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.1.1">𝑟</ci></apply><apply id="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.2"><csymbol cd="latexml" id="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.1.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.6.6.2.2.2.2.2.1">delimited-[]</csymbol><ci id="S5.Thmtheorem1.p1.8.m2.2.2.cmml" xref="S5.Thmtheorem1.p1.8.m2.2.2">𝑠</ci></apply></interval><interval closure="open" id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.3.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2"><apply id="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.2"><csymbol cd="latexml" id="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.7.7.3.3.1.1.2.1">delimited-[]</csymbol><ci id="S5.Thmtheorem1.p1.8.m2.3.3.cmml" xref="S5.Thmtheorem1.p1.8.m2.3.3">𝑝</ci></apply><apply id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.2"><csymbol cd="latexml" id="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.1.1.cmml" xref="S5.Thmtheorem1.p1.8.m2.8.8.4.4.2.2.2.1">delimited-[]</csymbol><ci id="S5.Thmtheorem1.p1.8.m2.4.4.cmml" xref="S5.Thmtheorem1.p1.8.m2.4.4">𝑞</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.8.m2.8c">f_{h}\times f_{v}:([r],[s])\to([p],[q])</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.8.m2.8d">italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT : ( [ italic_r ] , [ italic_s ] ) → ( [ italic_p ] , [ italic_q ] )</annotation></semantics></math> is a morphism in <math alttext="\Delta\times\Delta" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.9.m3.1"><semantics id="S5.Thmtheorem1.p1.9.m3.1a"><mrow id="S5.Thmtheorem1.p1.9.m3.1.1" xref="S5.Thmtheorem1.p1.9.m3.1.1.cmml"><mi id="S5.Thmtheorem1.p1.9.m3.1.1.2" mathvariant="normal" xref="S5.Thmtheorem1.p1.9.m3.1.1.2.cmml">Δ</mi><mo id="S5.Thmtheorem1.p1.9.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmtheorem1.p1.9.m3.1.1.1.cmml">×</mo><mi id="S5.Thmtheorem1.p1.9.m3.1.1.3" mathvariant="normal" xref="S5.Thmtheorem1.p1.9.m3.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.9.m3.1b"><apply id="S5.Thmtheorem1.p1.9.m3.1.1.cmml" xref="S5.Thmtheorem1.p1.9.m3.1.1"><times id="S5.Thmtheorem1.p1.9.m3.1.1.1.cmml" xref="S5.Thmtheorem1.p1.9.m3.1.1.1"></times><ci id="S5.Thmtheorem1.p1.9.m3.1.1.2.cmml" xref="S5.Thmtheorem1.p1.9.m3.1.1.2">Δ</ci><ci id="S5.Thmtheorem1.p1.9.m3.1.1.3.cmml" xref="S5.Thmtheorem1.p1.9.m3.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.9.m3.1c">\Delta\times\Delta</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.9.m3.1d">roman_Δ × roman_Δ</annotation></semantics></math>, then for <math alttext="(\sigma,\tau,\mu,f)\in X_{p,q}(\theta)" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.10.m4.7"><semantics id="S5.Thmtheorem1.p1.10.m4.7a"><mrow id="S5.Thmtheorem1.p1.10.m4.7.8" xref="S5.Thmtheorem1.p1.10.m4.7.8.cmml"><mrow id="S5.Thmtheorem1.p1.10.m4.7.8.2.2" xref="S5.Thmtheorem1.p1.10.m4.7.8.2.1.cmml"><mo id="S5.Thmtheorem1.p1.10.m4.7.8.2.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.10.m4.7.8.2.1.cmml">(</mo><mi id="S5.Thmtheorem1.p1.10.m4.3.3" xref="S5.Thmtheorem1.p1.10.m4.3.3.cmml">σ</mi><mo id="S5.Thmtheorem1.p1.10.m4.7.8.2.2.2" xref="S5.Thmtheorem1.p1.10.m4.7.8.2.1.cmml">,</mo><mi id="S5.Thmtheorem1.p1.10.m4.4.4" xref="S5.Thmtheorem1.p1.10.m4.4.4.cmml">τ</mi><mo id="S5.Thmtheorem1.p1.10.m4.7.8.2.2.3" xref="S5.Thmtheorem1.p1.10.m4.7.8.2.1.cmml">,</mo><mi id="S5.Thmtheorem1.p1.10.m4.5.5" xref="S5.Thmtheorem1.p1.10.m4.5.5.cmml">μ</mi><mo id="S5.Thmtheorem1.p1.10.m4.7.8.2.2.4" xref="S5.Thmtheorem1.p1.10.m4.7.8.2.1.cmml">,</mo><mi id="S5.Thmtheorem1.p1.10.m4.6.6" xref="S5.Thmtheorem1.p1.10.m4.6.6.cmml">f</mi><mo id="S5.Thmtheorem1.p1.10.m4.7.8.2.2.5" stretchy="false" xref="S5.Thmtheorem1.p1.10.m4.7.8.2.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem1.p1.10.m4.7.8.1" xref="S5.Thmtheorem1.p1.10.m4.7.8.1.cmml">∈</mo><mrow id="S5.Thmtheorem1.p1.10.m4.7.8.3" xref="S5.Thmtheorem1.p1.10.m4.7.8.3.cmml"><msub id="S5.Thmtheorem1.p1.10.m4.7.8.3.2" xref="S5.Thmtheorem1.p1.10.m4.7.8.3.2.cmml"><mi id="S5.Thmtheorem1.p1.10.m4.7.8.3.2.2" xref="S5.Thmtheorem1.p1.10.m4.7.8.3.2.2.cmml">X</mi><mrow id="S5.Thmtheorem1.p1.10.m4.2.2.2.4" xref="S5.Thmtheorem1.p1.10.m4.2.2.2.3.cmml"><mi id="S5.Thmtheorem1.p1.10.m4.1.1.1.1" xref="S5.Thmtheorem1.p1.10.m4.1.1.1.1.cmml">p</mi><mo id="S5.Thmtheorem1.p1.10.m4.2.2.2.4.1" xref="S5.Thmtheorem1.p1.10.m4.2.2.2.3.cmml">,</mo><mi 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xref="S5.Thmtheorem1.p1.10.m4.3.3">𝜎</ci><ci id="S5.Thmtheorem1.p1.10.m4.4.4.cmml" xref="S5.Thmtheorem1.p1.10.m4.4.4">𝜏</ci><ci id="S5.Thmtheorem1.p1.10.m4.5.5.cmml" xref="S5.Thmtheorem1.p1.10.m4.5.5">𝜇</ci><ci id="S5.Thmtheorem1.p1.10.m4.6.6.cmml" xref="S5.Thmtheorem1.p1.10.m4.6.6">𝑓</ci></vector><apply id="S5.Thmtheorem1.p1.10.m4.7.8.3.cmml" xref="S5.Thmtheorem1.p1.10.m4.7.8.3"><times id="S5.Thmtheorem1.p1.10.m4.7.8.3.1.cmml" xref="S5.Thmtheorem1.p1.10.m4.7.8.3.1"></times><apply id="S5.Thmtheorem1.p1.10.m4.7.8.3.2.cmml" xref="S5.Thmtheorem1.p1.10.m4.7.8.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.10.m4.7.8.3.2.1.cmml" xref="S5.Thmtheorem1.p1.10.m4.7.8.3.2">subscript</csymbol><ci id="S5.Thmtheorem1.p1.10.m4.7.8.3.2.2.cmml" xref="S5.Thmtheorem1.p1.10.m4.7.8.3.2.2">𝑋</ci><list id="S5.Thmtheorem1.p1.10.m4.2.2.2.3.cmml" xref="S5.Thmtheorem1.p1.10.m4.2.2.2.4"><ci id="S5.Thmtheorem1.p1.10.m4.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.10.m4.1.1.1.1">𝑝</ci><ci id="S5.Thmtheorem1.p1.10.m4.2.2.2.2.cmml" xref="S5.Thmtheorem1.p1.10.m4.2.2.2.2">𝑞</ci></list></apply><ci id="S5.Thmtheorem1.p1.10.m4.7.7.cmml" xref="S5.Thmtheorem1.p1.10.m4.7.7">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.10.m4.7c">(\sigma,\tau,\mu,f)\in X_{p,q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.10.m4.7d">( italic_σ , italic_τ , italic_μ , italic_f ) ∈ italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex56"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(f_{h}\times f_{v})^{*}(\sigma,\tau,\mu,f)=(f_{h}^{*}(\sigma),f_{v}^{*}(\tau),% \mu^{\prime},f\circ f_{v})" class="ltx_Math" display="block" id="S5.Ex56.m1.11"><semantics 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xref="S5.Ex56.m1.11.11.5.4.4.2">𝑓</ci><apply id="S5.Ex56.m1.11.11.5.4.4.3.cmml" xref="S5.Ex56.m1.11.11.5.4.4.3"><csymbol cd="ambiguous" id="S5.Ex56.m1.11.11.5.4.4.3.1.cmml" xref="S5.Ex56.m1.11.11.5.4.4.3">subscript</csymbol><ci id="S5.Ex56.m1.11.11.5.4.4.3.2.cmml" xref="S5.Ex56.m1.11.11.5.4.4.3.2">𝑓</ci><ci id="S5.Ex56.m1.11.11.5.4.4.3.3.cmml" xref="S5.Ex56.m1.11.11.5.4.4.3.3">𝑣</ci></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex56.m1.11c">(f_{h}\times f_{v})^{*}(\sigma,\tau,\mu,f)=(f_{h}^{*}(\sigma),f_{v}^{*}(\tau),% \mu^{\prime},f\circ f_{v})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex56.m1.11d">( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ , italic_τ , italic_μ , italic_f ) = ( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ ) , italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ ) , italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_f ∘ italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmtheorem1.p1.12">where <math alttext="\mu^{\prime}:\varphi(c_{f_{v}(q)})\to d_{f_{h}(0)}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.11.m1.3"><semantics id="S5.Thmtheorem1.p1.11.m1.3a"><mrow id="S5.Thmtheorem1.p1.11.m1.3.3" xref="S5.Thmtheorem1.p1.11.m1.3.3.cmml"><msup id="S5.Thmtheorem1.p1.11.m1.3.3.3" xref="S5.Thmtheorem1.p1.11.m1.3.3.3.cmml"><mi id="S5.Thmtheorem1.p1.11.m1.3.3.3.2" xref="S5.Thmtheorem1.p1.11.m1.3.3.3.2.cmml">μ</mi><mo id="S5.Thmtheorem1.p1.11.m1.3.3.3.3" xref="S5.Thmtheorem1.p1.11.m1.3.3.3.3.cmml">′</mo></msup><mo id="S5.Thmtheorem1.p1.11.m1.3.3.2" lspace="0.278em" rspace="0.278em" xref="S5.Thmtheorem1.p1.11.m1.3.3.2.cmml">:</mo><mrow id="S5.Thmtheorem1.p1.11.m1.3.3.1" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.cmml"><mrow id="S5.Thmtheorem1.p1.11.m1.3.3.1.1" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.cmml"><mi id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.3" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.3.cmml">φ</mi><mo id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.2" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.cmml">(</mo><msub id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.2" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.2.cmml">c</mi><mrow id="S5.Thmtheorem1.p1.11.m1.1.1.1" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.cmml"><msub id="S5.Thmtheorem1.p1.11.m1.1.1.1.3" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.3.cmml"><mi id="S5.Thmtheorem1.p1.11.m1.1.1.1.3.2" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.3.2.cmml">f</mi><mi id="S5.Thmtheorem1.p1.11.m1.1.1.1.3.3" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.3.3.cmml">v</mi></msub><mo id="S5.Thmtheorem1.p1.11.m1.1.1.1.2" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.11.m1.1.1.1.4.2" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.11.m1.1.1.1.4.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.cmml">(</mo><mi id="S5.Thmtheorem1.p1.11.m1.1.1.1.1" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.1.cmml">q</mi><mo id="S5.Thmtheorem1.p1.11.m1.1.1.1.4.2.2" stretchy="false" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem1.p1.11.m1.3.3.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.2.cmml">→</mo><msub id="S5.Thmtheorem1.p1.11.m1.3.3.1.3" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.3.cmml"><mi id="S5.Thmtheorem1.p1.11.m1.3.3.1.3.2" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.3.2.cmml">d</mi><mrow id="S5.Thmtheorem1.p1.11.m1.2.2.1" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.cmml"><msub id="S5.Thmtheorem1.p1.11.m1.2.2.1.3" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.3.cmml"><mi id="S5.Thmtheorem1.p1.11.m1.2.2.1.3.2" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.3.2.cmml">f</mi><mi id="S5.Thmtheorem1.p1.11.m1.2.2.1.3.3" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.3.3.cmml">h</mi></msub><mo id="S5.Thmtheorem1.p1.11.m1.2.2.1.2" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.11.m1.2.2.1.4.2" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.cmml"><mo id="S5.Thmtheorem1.p1.11.m1.2.2.1.4.2.1" stretchy="false" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.cmml">(</mo><mn id="S5.Thmtheorem1.p1.11.m1.2.2.1.1" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.1.cmml">0</mn><mo id="S5.Thmtheorem1.p1.11.m1.2.2.1.4.2.2" stretchy="false" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.cmml">)</mo></mrow></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.11.m1.3b"><apply id="S5.Thmtheorem1.p1.11.m1.3.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3"><ci id="S5.Thmtheorem1.p1.11.m1.3.3.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.2">:</ci><apply id="S5.Thmtheorem1.p1.11.m1.3.3.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.11.m1.3.3.3.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.3">superscript</csymbol><ci id="S5.Thmtheorem1.p1.11.m1.3.3.3.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.3.2">𝜇</ci><ci id="S5.Thmtheorem1.p1.11.m1.3.3.3.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.3.3">′</ci></apply><apply id="S5.Thmtheorem1.p1.11.m1.3.3.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1"><ci id="S5.Thmtheorem1.p1.11.m1.3.3.1.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.2">→</ci><apply id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1"><times id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.2"></times><ci id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.3">𝜑</ci><apply id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.1.1.1.1.2">𝑐</ci><apply id="S5.Thmtheorem1.p1.11.m1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.1.1.1"><times id="S5.Thmtheorem1.p1.11.m1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.2"></times><apply id="S5.Thmtheorem1.p1.11.m1.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.11.m1.1.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem1.p1.11.m1.1.1.1.3.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.3.2">𝑓</ci><ci id="S5.Thmtheorem1.p1.11.m1.1.1.1.3.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.3.3">𝑣</ci></apply><ci id="S5.Thmtheorem1.p1.11.m1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.1.1.1.1">𝑞</ci></apply></apply></apply><apply id="S5.Thmtheorem1.p1.11.m1.3.3.1.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.11.m1.3.3.1.3.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.3">subscript</csymbol><ci id="S5.Thmtheorem1.p1.11.m1.3.3.1.3.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.3.3.1.3.2">𝑑</ci><apply id="S5.Thmtheorem1.p1.11.m1.2.2.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.2.2.1"><times id="S5.Thmtheorem1.p1.11.m1.2.2.1.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.2"></times><apply id="S5.Thmtheorem1.p1.11.m1.2.2.1.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.11.m1.2.2.1.3.1.cmml" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.3">subscript</csymbol><ci id="S5.Thmtheorem1.p1.11.m1.2.2.1.3.2.cmml" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.3.2">𝑓</ci><ci id="S5.Thmtheorem1.p1.11.m1.2.2.1.3.3.cmml" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.3.3">ℎ</ci></apply><cn id="S5.Thmtheorem1.p1.11.m1.2.2.1.1.cmml" type="integer" xref="S5.Thmtheorem1.p1.11.m1.2.2.1.1">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.11.m1.3c">\mu^{\prime}:\varphi(c_{f_{v}(q)})\to d_{f_{h}(0)}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.11.m1.3d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_φ ( italic_c start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT ) → italic_d start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT</annotation></semantics></math> is the morphism in <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.12.m2.1"><semantics id="S5.Thmtheorem1.p1.12.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem1.p1.12.m2.1.1" xref="S5.Thmtheorem1.p1.12.m2.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.12.m2.1b"><ci id="S5.Thmtheorem1.p1.12.m2.1.1.cmml" xref="S5.Thmtheorem1.p1.12.m2.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.12.m2.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.12.m2.1d">caligraphic_D</annotation></semantics></math> defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S5.E3" title="In 5. Proof of Theorem 1.2 ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">3</span></a>).</p> </div> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S5.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.1.1.1">Proposition 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem2.p1.3.3">The chain complex <math alttext="(\mathrm{diag}\mathbb{P})_{*}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.1.m1.1"><semantics id="S5.Thmtheorem2.p1.1.1.m1.1a"><msub id="S5.Thmtheorem2.p1.1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mrow id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.2" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.1.m1.1b"><apply id="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1">subscript</csymbol><apply id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1"><times id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.1"></times><ci id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.2">diag</ci><ci id="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3">ℙ</ci></apply><times id="S5.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.1.m1.1c">(\mathrm{diag}\mathbb{P})_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.1.m1.1d">( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> defines a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.2.m2.1a"><munder accentunder="true" id="S5.Thmtheorem2.p1.2.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">R</mi><mo id="S5.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.2.m2.1b"><apply id="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1"><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1">¯</ci><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.2.m2.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.2.m2.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.3.3.m3.1"><semantics id="S5.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem2.p1.3.3.m3.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mi id="S5.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.3.cmml">R</mi><mo id="S5.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">⁢</mo><mi id="S5.Thmtheorem2.p1.3.3.m3.1.1.4" mathvariant="normal" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.4.cmml">Δ</mi><mo id="S5.Thmtheorem2.p1.3.3.m3.1.1.2a" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2.cmml">N</mi><mo id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.3.m3.1b"><apply id="S5.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1"><times id="S5.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.2"></times><ci id="S5.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.3">𝑅</ci><ci id="S5.Thmtheorem2.p1.3.3.m3.1.1.4.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.4">Δ</ci><apply id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1"><times id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1"></times><ci id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2">𝑁</ci><ci id="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.3.m3.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.3.m3.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-module.</span></p> </div> </div> <div class="ltx_proof" id="S5.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.1.p1"> <p class="ltx_p" id="S5.1.p1.2">Since <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S5.1.p1.1.m1.1"><semantics id="S5.1.p1.1.m1.1a"><mi id="S5.1.p1.1.m1.1.1" xref="S5.1.p1.1.m1.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.1.m1.1b"><ci id="S5.1.p1.1.m1.1.1.cmml" xref="S5.1.p1.1.m1.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.1d">blackboard_P</annotation></semantics></math> consists of projective <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.1.p1.2.m2.1"><semantics id="S5.1.p1.2.m2.1a"><mrow id="S5.1.p1.2.m2.1.1" xref="S5.1.p1.2.m2.1.1.cmml"><mi id="S5.1.p1.2.m2.1.1.3" xref="S5.1.p1.2.m2.1.1.3.cmml">R</mi><mo id="S5.1.p1.2.m2.1.1.2" xref="S5.1.p1.2.m2.1.1.2.cmml">⁢</mo><mi id="S5.1.p1.2.m2.1.1.4" mathvariant="normal" xref="S5.1.p1.2.m2.1.1.4.cmml">Δ</mi><mo id="S5.1.p1.2.m2.1.1.2a" xref="S5.1.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S5.1.p1.2.m2.1.1.1.1" xref="S5.1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S5.1.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S5.1.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.1.p1.2.m2.1.1.1.1.1" xref="S5.1.p1.2.m2.1.1.1.1.1.cmml"><mi id="S5.1.p1.2.m2.1.1.1.1.1.2" xref="S5.1.p1.2.m2.1.1.1.1.1.2.cmml">N</mi><mo id="S5.1.p1.2.m2.1.1.1.1.1.1" xref="S5.1.p1.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.1.p1.2.m2.1.1.1.1.1.3" xref="S5.1.p1.2.m2.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.1.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S5.1.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.2.m2.1b"><apply id="S5.1.p1.2.m2.1.1.cmml" xref="S5.1.p1.2.m2.1.1"><times id="S5.1.p1.2.m2.1.1.2.cmml" xref="S5.1.p1.2.m2.1.1.2"></times><ci id="S5.1.p1.2.m2.1.1.3.cmml" xref="S5.1.p1.2.m2.1.1.3">𝑅</ci><ci id="S5.1.p1.2.m2.1.1.4.cmml" xref="S5.1.p1.2.m2.1.1.4">Δ</ci><apply id="S5.1.p1.2.m2.1.1.1.1.1.cmml" xref="S5.1.p1.2.m2.1.1.1.1"><times id="S5.1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.1.p1.2.m2.1.1.1.1.1.1"></times><ci id="S5.1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S5.1.p1.2.m2.1.1.1.1.1.2">𝑁</ci><ci id="S5.1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S5.1.p1.2.m2.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m2.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m2.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-modules, we only need to show that the sequence</p> <table class="ltx_equation ltx_eqn_table" id="S5.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\cdots\to(\mathrm{diag}\mathbb{P})_{k}\to(\mathrm{diag}\mathbb{P})_{k-1}\to% \cdots\to(\mathrm{diag}\mathbb{P})_{0}\to\underline{R}\to 0" class="ltx_Math" display="block" id="S5.E4.m1.3"><semantics id="S5.E4.m1.3a"><mrow id="S5.E4.m1.3.3" xref="S5.E4.m1.3.3.cmml"><mi id="S5.E4.m1.3.3.5" mathvariant="normal" xref="S5.E4.m1.3.3.5.cmml">⋯</mi><mo id="S5.E4.m1.3.3.6" stretchy="false" xref="S5.E4.m1.3.3.6.cmml">→</mo><msub id="S5.E4.m1.1.1.1" xref="S5.E4.m1.1.1.1.cmml"><mrow id="S5.E4.m1.1.1.1.1.1" xref="S5.E4.m1.1.1.1.1.1.1.cmml"><mo id="S5.E4.m1.1.1.1.1.1.2" stretchy="false" xref="S5.E4.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.E4.m1.1.1.1.1.1.1" xref="S5.E4.m1.1.1.1.1.1.1.cmml"><mi id="S5.E4.m1.1.1.1.1.1.1.2" xref="S5.E4.m1.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.E4.m1.1.1.1.1.1.1.1" xref="S5.E4.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.E4.m1.1.1.1.1.1.1.3" xref="S5.E4.m1.1.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.E4.m1.1.1.1.1.1.3" stretchy="false" xref="S5.E4.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.E4.m1.1.1.1.3" xref="S5.E4.m1.1.1.1.3.cmml">k</mi></msub><mo id="S5.E4.m1.3.3.7" stretchy="false" xref="S5.E4.m1.3.3.7.cmml">→</mo><msub id="S5.E4.m1.2.2.2" xref="S5.E4.m1.2.2.2.cmml"><mrow id="S5.E4.m1.2.2.2.1.1" xref="S5.E4.m1.2.2.2.1.1.1.cmml"><mo id="S5.E4.m1.2.2.2.1.1.2" stretchy="false" xref="S5.E4.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.E4.m1.2.2.2.1.1.1" xref="S5.E4.m1.2.2.2.1.1.1.cmml"><mi id="S5.E4.m1.2.2.2.1.1.1.2" xref="S5.E4.m1.2.2.2.1.1.1.2.cmml">diag</mi><mo id="S5.E4.m1.2.2.2.1.1.1.1" xref="S5.E4.m1.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S5.E4.m1.2.2.2.1.1.1.3" xref="S5.E4.m1.2.2.2.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.E4.m1.2.2.2.1.1.3" stretchy="false" xref="S5.E4.m1.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S5.E4.m1.2.2.2.3" xref="S5.E4.m1.2.2.2.3.cmml"><mi id="S5.E4.m1.2.2.2.3.2" xref="S5.E4.m1.2.2.2.3.2.cmml">k</mi><mo id="S5.E4.m1.2.2.2.3.1" xref="S5.E4.m1.2.2.2.3.1.cmml">−</mo><mn id="S5.E4.m1.2.2.2.3.3" xref="S5.E4.m1.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.E4.m1.3.3.8" stretchy="false" xref="S5.E4.m1.3.3.8.cmml">→</mo><mi id="S5.E4.m1.3.3.9" mathvariant="normal" xref="S5.E4.m1.3.3.9.cmml">⋯</mi><mo id="S5.E4.m1.3.3.10" stretchy="false" xref="S5.E4.m1.3.3.10.cmml">→</mo><msub id="S5.E4.m1.3.3.3" xref="S5.E4.m1.3.3.3.cmml"><mrow id="S5.E4.m1.3.3.3.1.1" xref="S5.E4.m1.3.3.3.1.1.1.cmml"><mo id="S5.E4.m1.3.3.3.1.1.2" stretchy="false" xref="S5.E4.m1.3.3.3.1.1.1.cmml">(</mo><mrow id="S5.E4.m1.3.3.3.1.1.1" xref="S5.E4.m1.3.3.3.1.1.1.cmml"><mi id="S5.E4.m1.3.3.3.1.1.1.2" xref="S5.E4.m1.3.3.3.1.1.1.2.cmml">diag</mi><mo id="S5.E4.m1.3.3.3.1.1.1.1" xref="S5.E4.m1.3.3.3.1.1.1.1.cmml">⁢</mo><mi id="S5.E4.m1.3.3.3.1.1.1.3" xref="S5.E4.m1.3.3.3.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.E4.m1.3.3.3.1.1.3" stretchy="false" xref="S5.E4.m1.3.3.3.1.1.1.cmml">)</mo></mrow><mn id="S5.E4.m1.3.3.3.3" xref="S5.E4.m1.3.3.3.3.cmml">0</mn></msub><mo id="S5.E4.m1.3.3.11" stretchy="false" xref="S5.E4.m1.3.3.11.cmml">→</mo><munder accentunder="true" id="S5.E4.m1.3.3.12" xref="S5.E4.m1.3.3.12.cmml"><mi id="S5.E4.m1.3.3.12.2" xref="S5.E4.m1.3.3.12.2.cmml">R</mi><mo id="S5.E4.m1.3.3.12.1" xref="S5.E4.m1.3.3.12.1.cmml">¯</mo></munder><mo id="S5.E4.m1.3.3.13" stretchy="false" xref="S5.E4.m1.3.3.13.cmml">→</mo><mn id="S5.E4.m1.3.3.14" xref="S5.E4.m1.3.3.14.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.E4.m1.3b"><apply id="S5.E4.m1.3.3.cmml" xref="S5.E4.m1.3.3"><and id="S5.E4.m1.3.3a.cmml" xref="S5.E4.m1.3.3"></and><apply id="S5.E4.m1.3.3b.cmml" xref="S5.E4.m1.3.3"><ci id="S5.E4.m1.3.3.6.cmml" xref="S5.E4.m1.3.3.6">→</ci><ci id="S5.E4.m1.3.3.5.cmml" xref="S5.E4.m1.3.3.5">⋯</ci><apply id="S5.E4.m1.1.1.1.cmml" xref="S5.E4.m1.1.1.1"><csymbol cd="ambiguous" id="S5.E4.m1.1.1.1.2.cmml" xref="S5.E4.m1.1.1.1">subscript</csymbol><apply id="S5.E4.m1.1.1.1.1.1.1.cmml" xref="S5.E4.m1.1.1.1.1.1"><times id="S5.E4.m1.1.1.1.1.1.1.1.cmml" xref="S5.E4.m1.1.1.1.1.1.1.1"></times><ci id="S5.E4.m1.1.1.1.1.1.1.2.cmml" xref="S5.E4.m1.1.1.1.1.1.1.2">diag</ci><ci id="S5.E4.m1.1.1.1.1.1.1.3.cmml" xref="S5.E4.m1.1.1.1.1.1.1.3">ℙ</ci></apply><ci id="S5.E4.m1.1.1.1.3.cmml" xref="S5.E4.m1.1.1.1.3">𝑘</ci></apply></apply><apply id="S5.E4.m1.3.3c.cmml" xref="S5.E4.m1.3.3"><ci id="S5.E4.m1.3.3.7.cmml" xref="S5.E4.m1.3.3.7">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.E4.m1.1.1.1.cmml" id="S5.E4.m1.3.3d.cmml" xref="S5.E4.m1.3.3"></share><apply id="S5.E4.m1.2.2.2.cmml" xref="S5.E4.m1.2.2.2"><csymbol cd="ambiguous" id="S5.E4.m1.2.2.2.2.cmml" xref="S5.E4.m1.2.2.2">subscript</csymbol><apply id="S5.E4.m1.2.2.2.1.1.1.cmml" xref="S5.E4.m1.2.2.2.1.1"><times id="S5.E4.m1.2.2.2.1.1.1.1.cmml" xref="S5.E4.m1.2.2.2.1.1.1.1"></times><ci id="S5.E4.m1.2.2.2.1.1.1.2.cmml" xref="S5.E4.m1.2.2.2.1.1.1.2">diag</ci><ci id="S5.E4.m1.2.2.2.1.1.1.3.cmml" xref="S5.E4.m1.2.2.2.1.1.1.3">ℙ</ci></apply><apply id="S5.E4.m1.2.2.2.3.cmml" xref="S5.E4.m1.2.2.2.3"><minus id="S5.E4.m1.2.2.2.3.1.cmml" xref="S5.E4.m1.2.2.2.3.1"></minus><ci id="S5.E4.m1.2.2.2.3.2.cmml" xref="S5.E4.m1.2.2.2.3.2">𝑘</ci><cn id="S5.E4.m1.2.2.2.3.3.cmml" type="integer" xref="S5.E4.m1.2.2.2.3.3">1</cn></apply></apply></apply><apply id="S5.E4.m1.3.3e.cmml" xref="S5.E4.m1.3.3"><ci id="S5.E4.m1.3.3.8.cmml" xref="S5.E4.m1.3.3.8">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.E4.m1.2.2.2.cmml" id="S5.E4.m1.3.3f.cmml" xref="S5.E4.m1.3.3"></share><ci id="S5.E4.m1.3.3.9.cmml" xref="S5.E4.m1.3.3.9">⋯</ci></apply><apply id="S5.E4.m1.3.3g.cmml" xref="S5.E4.m1.3.3"><ci id="S5.E4.m1.3.3.10.cmml" xref="S5.E4.m1.3.3.10">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.E4.m1.3.3.9.cmml" id="S5.E4.m1.3.3h.cmml" xref="S5.E4.m1.3.3"></share><apply id="S5.E4.m1.3.3.3.cmml" xref="S5.E4.m1.3.3.3"><csymbol cd="ambiguous" id="S5.E4.m1.3.3.3.2.cmml" xref="S5.E4.m1.3.3.3">subscript</csymbol><apply id="S5.E4.m1.3.3.3.1.1.1.cmml" xref="S5.E4.m1.3.3.3.1.1"><times id="S5.E4.m1.3.3.3.1.1.1.1.cmml" xref="S5.E4.m1.3.3.3.1.1.1.1"></times><ci id="S5.E4.m1.3.3.3.1.1.1.2.cmml" xref="S5.E4.m1.3.3.3.1.1.1.2">diag</ci><ci id="S5.E4.m1.3.3.3.1.1.1.3.cmml" xref="S5.E4.m1.3.3.3.1.1.1.3">ℙ</ci></apply><cn id="S5.E4.m1.3.3.3.3.cmml" type="integer" xref="S5.E4.m1.3.3.3.3">0</cn></apply></apply><apply id="S5.E4.m1.3.3i.cmml" xref="S5.E4.m1.3.3"><ci id="S5.E4.m1.3.3.11.cmml" xref="S5.E4.m1.3.3.11">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.E4.m1.3.3.3.cmml" id="S5.E4.m1.3.3j.cmml" xref="S5.E4.m1.3.3"></share><apply id="S5.E4.m1.3.3.12.cmml" xref="S5.E4.m1.3.3.12"><ci id="S5.E4.m1.3.3.12.1.cmml" xref="S5.E4.m1.3.3.12.1">¯</ci><ci id="S5.E4.m1.3.3.12.2.cmml" xref="S5.E4.m1.3.3.12.2">𝑅</ci></apply></apply><apply id="S5.E4.m1.3.3k.cmml" xref="S5.E4.m1.3.3"><ci id="S5.E4.m1.3.3.13.cmml" xref="S5.E4.m1.3.3.13">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.E4.m1.3.3.12.cmml" id="S5.E4.m1.3.3l.cmml" xref="S5.E4.m1.3.3"></share><cn id="S5.E4.m1.3.3.14.cmml" type="integer" xref="S5.E4.m1.3.3.14">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E4.m1.3c">\cdots\to(\mathrm{diag}\mathbb{P})_{k}\to(\mathrm{diag}\mathbb{P})_{k-1}\to% \cdots\to(\mathrm{diag}\mathbb{P})_{0}\to\underline{R}\to 0</annotation><annotation encoding="application/x-llamapun" id="S5.E4.m1.3d">⋯ → ( roman_diag blackboard_P ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT → ( roman_diag blackboard_P ) start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT → ⋯ → ( roman_diag blackboard_P ) start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → under¯ start_ARG italic_R end_ARG → 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.5">is exact. We will do this by relating this chain complex to another complex. Let <math alttext="Q" class="ltx_Math" display="inline" id="S5.1.p1.3.m1.1"><semantics id="S5.1.p1.3.m1.1a"><mi id="S5.1.p1.3.m1.1.1" xref="S5.1.p1.3.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.3.m1.1b"><ci id="S5.1.p1.3.m1.1.1.cmml" xref="S5.1.p1.3.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.3.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.3.m1.1d">italic_Q</annotation></semantics></math> be the simplicial <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.1.p1.4.m2.1"><semantics id="S5.1.p1.4.m2.1a"><mrow id="S5.1.p1.4.m2.1.1" xref="S5.1.p1.4.m2.1.1.cmml"><mi id="S5.1.p1.4.m2.1.1.3" xref="S5.1.p1.4.m2.1.1.3.cmml">R</mi><mo id="S5.1.p1.4.m2.1.1.2" xref="S5.1.p1.4.m2.1.1.2.cmml">⁢</mo><mi id="S5.1.p1.4.m2.1.1.4" mathvariant="normal" xref="S5.1.p1.4.m2.1.1.4.cmml">Δ</mi><mo id="S5.1.p1.4.m2.1.1.2a" xref="S5.1.p1.4.m2.1.1.2.cmml">⁢</mo><mrow id="S5.1.p1.4.m2.1.1.1.1" xref="S5.1.p1.4.m2.1.1.1.1.1.cmml"><mo id="S5.1.p1.4.m2.1.1.1.1.2" stretchy="false" xref="S5.1.p1.4.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.1.p1.4.m2.1.1.1.1.1" xref="S5.1.p1.4.m2.1.1.1.1.1.cmml"><mi id="S5.1.p1.4.m2.1.1.1.1.1.2" xref="S5.1.p1.4.m2.1.1.1.1.1.2.cmml">N</mi><mo id="S5.1.p1.4.m2.1.1.1.1.1.1" xref="S5.1.p1.4.m2.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.1.p1.4.m2.1.1.1.1.1.3" xref="S5.1.p1.4.m2.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.1.p1.4.m2.1.1.1.1.3" stretchy="false" xref="S5.1.p1.4.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.4.m2.1b"><apply id="S5.1.p1.4.m2.1.1.cmml" xref="S5.1.p1.4.m2.1.1"><times id="S5.1.p1.4.m2.1.1.2.cmml" xref="S5.1.p1.4.m2.1.1.2"></times><ci id="S5.1.p1.4.m2.1.1.3.cmml" xref="S5.1.p1.4.m2.1.1.3">𝑅</ci><ci id="S5.1.p1.4.m2.1.1.4.cmml" xref="S5.1.p1.4.m2.1.1.4">Δ</ci><apply id="S5.1.p1.4.m2.1.1.1.1.1.cmml" xref="S5.1.p1.4.m2.1.1.1.1"><times id="S5.1.p1.4.m2.1.1.1.1.1.1.cmml" xref="S5.1.p1.4.m2.1.1.1.1.1.1"></times><ci id="S5.1.p1.4.m2.1.1.1.1.1.2.cmml" xref="S5.1.p1.4.m2.1.1.1.1.1.2">𝑁</ci><ci id="S5.1.p1.4.m2.1.1.1.1.1.3.cmml" xref="S5.1.p1.4.m2.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.4.m2.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.4.m2.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-module such that for each <math alttext="q\geq 0" class="ltx_Math" display="inline" id="S5.1.p1.5.m3.1"><semantics id="S5.1.p1.5.m3.1a"><mrow id="S5.1.p1.5.m3.1.1" xref="S5.1.p1.5.m3.1.1.cmml"><mi id="S5.1.p1.5.m3.1.1.2" xref="S5.1.p1.5.m3.1.1.2.cmml">q</mi><mo id="S5.1.p1.5.m3.1.1.1" xref="S5.1.p1.5.m3.1.1.1.cmml">≥</mo><mn id="S5.1.p1.5.m3.1.1.3" xref="S5.1.p1.5.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.5.m3.1b"><apply id="S5.1.p1.5.m3.1.1.cmml" xref="S5.1.p1.5.m3.1.1"><geq id="S5.1.p1.5.m3.1.1.1.cmml" xref="S5.1.p1.5.m3.1.1.1"></geq><ci id="S5.1.p1.5.m3.1.1.2.cmml" xref="S5.1.p1.5.m3.1.1.2">𝑞</ci><cn id="S5.1.p1.5.m3.1.1.3.cmml" type="integer" xref="S5.1.p1.5.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.5.m3.1c">q\geq 0</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.5.m3.1d">italic_q ≥ 0</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex57"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="Q_{q}=\bigoplus_{\tau\in N\mathcal{C}_{q}}R\mathrm{Mor}_{\Delta(N\mathcal{C})}% (\tau,?)." class="ltx_Math" display="block" id="S5.Ex57.m1.4"><semantics id="S5.Ex57.m1.4a"><mrow id="S5.Ex57.m1.4.4.1" xref="S5.Ex57.m1.4.4.1.1.cmml"><mrow id="S5.Ex57.m1.4.4.1.1" xref="S5.Ex57.m1.4.4.1.1.cmml"><msub id="S5.Ex57.m1.4.4.1.1.2" xref="S5.Ex57.m1.4.4.1.1.2.cmml"><mi id="S5.Ex57.m1.4.4.1.1.2.2" xref="S5.Ex57.m1.4.4.1.1.2.2.cmml">Q</mi><mi id="S5.Ex57.m1.4.4.1.1.2.3" xref="S5.Ex57.m1.4.4.1.1.2.3.cmml">q</mi></msub><mo id="S5.Ex57.m1.4.4.1.1.1" rspace="0.111em" xref="S5.Ex57.m1.4.4.1.1.1.cmml">=</mo><mrow id="S5.Ex57.m1.4.4.1.1.3" xref="S5.Ex57.m1.4.4.1.1.3.cmml"><munder id="S5.Ex57.m1.4.4.1.1.3.1" xref="S5.Ex57.m1.4.4.1.1.3.1.cmml"><mo id="S5.Ex57.m1.4.4.1.1.3.1.2" movablelimits="false" xref="S5.Ex57.m1.4.4.1.1.3.1.2.cmml">⨁</mo><mrow id="S5.Ex57.m1.4.4.1.1.3.1.3" xref="S5.Ex57.m1.4.4.1.1.3.1.3.cmml"><mi id="S5.Ex57.m1.4.4.1.1.3.1.3.2" xref="S5.Ex57.m1.4.4.1.1.3.1.3.2.cmml">τ</mi><mo id="S5.Ex57.m1.4.4.1.1.3.1.3.1" xref="S5.Ex57.m1.4.4.1.1.3.1.3.1.cmml">∈</mo><mrow id="S5.Ex57.m1.4.4.1.1.3.1.3.3" xref="S5.Ex57.m1.4.4.1.1.3.1.3.3.cmml"><mi id="S5.Ex57.m1.4.4.1.1.3.1.3.3.2" xref="S5.Ex57.m1.4.4.1.1.3.1.3.3.2.cmml">N</mi><mo id="S5.Ex57.m1.4.4.1.1.3.1.3.3.1" xref="S5.Ex57.m1.4.4.1.1.3.1.3.3.1.cmml">⁢</mo><msub id="S5.Ex57.m1.4.4.1.1.3.1.3.3.3" xref="S5.Ex57.m1.4.4.1.1.3.1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex57.m1.4.4.1.1.3.1.3.3.3.2" xref="S5.Ex57.m1.4.4.1.1.3.1.3.3.3.2.cmml">𝒞</mi><mi id="S5.Ex57.m1.4.4.1.1.3.1.3.3.3.3" xref="S5.Ex57.m1.4.4.1.1.3.1.3.3.3.3.cmml">q</mi></msub></mrow></mrow></munder><mrow id="S5.Ex57.m1.4.4.1.1.3.2" xref="S5.Ex57.m1.4.4.1.1.3.2.cmml"><mi id="S5.Ex57.m1.4.4.1.1.3.2.2" xref="S5.Ex57.m1.4.4.1.1.3.2.2.cmml">R</mi><mo id="S5.Ex57.m1.4.4.1.1.3.2.1" xref="S5.Ex57.m1.4.4.1.1.3.2.1.cmml">⁢</mo><msub id="S5.Ex57.m1.4.4.1.1.3.2.3" xref="S5.Ex57.m1.4.4.1.1.3.2.3.cmml"><mi id="S5.Ex57.m1.4.4.1.1.3.2.3.2" xref="S5.Ex57.m1.4.4.1.1.3.2.3.2.cmml">Mor</mi><mrow id="S5.Ex57.m1.1.1.1" xref="S5.Ex57.m1.1.1.1.cmml"><mi id="S5.Ex57.m1.1.1.1.3" mathvariant="normal" xref="S5.Ex57.m1.1.1.1.3.cmml">Δ</mi><mo id="S5.Ex57.m1.1.1.1.2" xref="S5.Ex57.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex57.m1.1.1.1.1.1" 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xref="S5.Ex57.m1.4.4.1.1.3.1.3.3.3.3">𝑞</ci></apply></apply></apply></apply><apply id="S5.Ex57.m1.4.4.1.1.3.2.cmml" xref="S5.Ex57.m1.4.4.1.1.3.2"><times id="S5.Ex57.m1.4.4.1.1.3.2.1.cmml" xref="S5.Ex57.m1.4.4.1.1.3.2.1"></times><ci id="S5.Ex57.m1.4.4.1.1.3.2.2.cmml" xref="S5.Ex57.m1.4.4.1.1.3.2.2">𝑅</ci><apply id="S5.Ex57.m1.4.4.1.1.3.2.3.cmml" xref="S5.Ex57.m1.4.4.1.1.3.2.3"><csymbol cd="ambiguous" id="S5.Ex57.m1.4.4.1.1.3.2.3.1.cmml" xref="S5.Ex57.m1.4.4.1.1.3.2.3">subscript</csymbol><ci id="S5.Ex57.m1.4.4.1.1.3.2.3.2.cmml" xref="S5.Ex57.m1.4.4.1.1.3.2.3.2">Mor</ci><apply id="S5.Ex57.m1.1.1.1.cmml" xref="S5.Ex57.m1.1.1.1"><times id="S5.Ex57.m1.1.1.1.2.cmml" xref="S5.Ex57.m1.1.1.1.2"></times><ci id="S5.Ex57.m1.1.1.1.3.cmml" xref="S5.Ex57.m1.1.1.1.3">Δ</ci><apply id="S5.Ex57.m1.1.1.1.1.1.1.cmml" xref="S5.Ex57.m1.1.1.1.1.1"><times id="S5.Ex57.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex57.m1.1.1.1.1.1.1.1"></times><ci id="S5.Ex57.m1.1.1.1.1.1.1.2.cmml" xref="S5.Ex57.m1.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.Ex57.m1.1.1.1.1.1.1.3.cmml" xref="S5.Ex57.m1.1.1.1.1.1.1.3">𝒞</ci></apply></apply></apply><interval closure="open" id="S5.Ex57.m1.4.4.1.1.3.2.4.1.cmml" xref="S5.Ex57.m1.4.4.1.1.3.2.4.2"><ci id="S5.Ex57.m1.2.2.cmml" xref="S5.Ex57.m1.2.2">𝜏</ci><ci id="S5.Ex57.m1.3.3.cmml" xref="S5.Ex57.m1.3.3">?</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex57.m1.4c">Q_{q}=\bigoplus_{\tau\in N\mathcal{C}_{q}}R\mathrm{Mor}_{\Delta(N\mathcal{C})}% (\tau,?).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex57.m1.4d">italic_Q start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT italic_τ ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_R roman_Mor start_POSTSUBSCRIPT roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( italic_τ , ? ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.8">For every simplex <math alttext="\theta\in N\mathcal{C}_{n}" class="ltx_Math" display="inline" id="S5.1.p1.6.m1.1"><semantics id="S5.1.p1.6.m1.1a"><mrow id="S5.1.p1.6.m1.1.1" xref="S5.1.p1.6.m1.1.1.cmml"><mi id="S5.1.p1.6.m1.1.1.2" xref="S5.1.p1.6.m1.1.1.2.cmml">θ</mi><mo id="S5.1.p1.6.m1.1.1.1" xref="S5.1.p1.6.m1.1.1.1.cmml">∈</mo><mrow id="S5.1.p1.6.m1.1.1.3" xref="S5.1.p1.6.m1.1.1.3.cmml"><mi id="S5.1.p1.6.m1.1.1.3.2" xref="S5.1.p1.6.m1.1.1.3.2.cmml">N</mi><mo id="S5.1.p1.6.m1.1.1.3.1" xref="S5.1.p1.6.m1.1.1.3.1.cmml">⁢</mo><msub id="S5.1.p1.6.m1.1.1.3.3" xref="S5.1.p1.6.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.1.p1.6.m1.1.1.3.3.2" xref="S5.1.p1.6.m1.1.1.3.3.2.cmml">𝒞</mi><mi id="S5.1.p1.6.m1.1.1.3.3.3" xref="S5.1.p1.6.m1.1.1.3.3.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.6.m1.1b"><apply id="S5.1.p1.6.m1.1.1.cmml" xref="S5.1.p1.6.m1.1.1"><in id="S5.1.p1.6.m1.1.1.1.cmml" xref="S5.1.p1.6.m1.1.1.1"></in><ci id="S5.1.p1.6.m1.1.1.2.cmml" xref="S5.1.p1.6.m1.1.1.2">𝜃</ci><apply id="S5.1.p1.6.m1.1.1.3.cmml" xref="S5.1.p1.6.m1.1.1.3"><times id="S5.1.p1.6.m1.1.1.3.1.cmml" xref="S5.1.p1.6.m1.1.1.3.1"></times><ci id="S5.1.p1.6.m1.1.1.3.2.cmml" xref="S5.1.p1.6.m1.1.1.3.2">𝑁</ci><apply id="S5.1.p1.6.m1.1.1.3.3.cmml" xref="S5.1.p1.6.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.1.p1.6.m1.1.1.3.3.1.cmml" xref="S5.1.p1.6.m1.1.1.3.3">subscript</csymbol><ci id="S5.1.p1.6.m1.1.1.3.3.2.cmml" xref="S5.1.p1.6.m1.1.1.3.3.2">𝒞</ci><ci id="S5.1.p1.6.m1.1.1.3.3.3.cmml" xref="S5.1.p1.6.m1.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.6.m1.1c">\theta\in N\mathcal{C}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.6.m1.1d">italic_θ ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="Q_{q}(\theta)" class="ltx_Math" display="inline" id="S5.1.p1.7.m2.1"><semantics id="S5.1.p1.7.m2.1a"><mrow id="S5.1.p1.7.m2.1.2" xref="S5.1.p1.7.m2.1.2.cmml"><msub id="S5.1.p1.7.m2.1.2.2" xref="S5.1.p1.7.m2.1.2.2.cmml"><mi id="S5.1.p1.7.m2.1.2.2.2" xref="S5.1.p1.7.m2.1.2.2.2.cmml">Q</mi><mi id="S5.1.p1.7.m2.1.2.2.3" xref="S5.1.p1.7.m2.1.2.2.3.cmml">q</mi></msub><mo id="S5.1.p1.7.m2.1.2.1" xref="S5.1.p1.7.m2.1.2.1.cmml">⁢</mo><mrow id="S5.1.p1.7.m2.1.2.3.2" xref="S5.1.p1.7.m2.1.2.cmml"><mo id="S5.1.p1.7.m2.1.2.3.2.1" stretchy="false" xref="S5.1.p1.7.m2.1.2.cmml">(</mo><mi id="S5.1.p1.7.m2.1.1" xref="S5.1.p1.7.m2.1.1.cmml">θ</mi><mo id="S5.1.p1.7.m2.1.2.3.2.2" stretchy="false" xref="S5.1.p1.7.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.7.m2.1b"><apply id="S5.1.p1.7.m2.1.2.cmml" xref="S5.1.p1.7.m2.1.2"><times id="S5.1.p1.7.m2.1.2.1.cmml" xref="S5.1.p1.7.m2.1.2.1"></times><apply id="S5.1.p1.7.m2.1.2.2.cmml" xref="S5.1.p1.7.m2.1.2.2"><csymbol cd="ambiguous" 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encoding="application/x-llamapun" id="S5.1.p1.8.m3.1d">italic_R</annotation></semantics></math>-module with basis given by</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex58"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="Y_{q}(\theta)=\{(\tau,f)\,|\,\tau\in N\mathcal{C}_{q},f:[q]\to[n]\text{ such % that }f^{*}(\theta)=\tau\}." class="ltx_Math" display="block" id="S5.Ex58.m1.8"><semantics id="S5.Ex58.m1.8a"><mrow id="S5.Ex58.m1.8.8.1" xref="S5.Ex58.m1.8.8.1.1.cmml"><mrow id="S5.Ex58.m1.8.8.1.1" xref="S5.Ex58.m1.8.8.1.1.cmml"><mrow id="S5.Ex58.m1.8.8.1.1.4" xref="S5.Ex58.m1.8.8.1.1.4.cmml"><msub id="S5.Ex58.m1.8.8.1.1.4.2" xref="S5.Ex58.m1.8.8.1.1.4.2.cmml"><mi id="S5.Ex58.m1.8.8.1.1.4.2.2" xref="S5.Ex58.m1.8.8.1.1.4.2.2.cmml">Y</mi><mi id="S5.Ex58.m1.8.8.1.1.4.2.3" xref="S5.Ex58.m1.8.8.1.1.4.2.3.cmml">q</mi></msub><mo id="S5.Ex58.m1.8.8.1.1.4.1" 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id="S5.Ex58.m1.8.8.1.1.2.2.2.3.4.4.2.cmml" xref="S5.Ex58.m1.8.8.1.1.2.2.2.3.4.4.2">𝑓</ci><times id="S5.Ex58.m1.8.8.1.1.2.2.2.3.4.4.3.cmml" xref="S5.Ex58.m1.8.8.1.1.2.2.2.3.4.4.3"></times></apply><ci id="S5.Ex58.m1.7.7.cmml" xref="S5.Ex58.m1.7.7">𝜃</ci></apply></apply><apply id="S5.Ex58.m1.8.8.1.1.2.2.2.3c.cmml" xref="S5.Ex58.m1.8.8.1.1.2.2.2.3"><eq id="S5.Ex58.m1.8.8.1.1.2.2.2.3.5.cmml" xref="S5.Ex58.m1.8.8.1.1.2.2.2.3.5"></eq><share href="https://arxiv.org/html/2503.14659v1#S5.Ex58.m1.8.8.1.1.2.2.2.3.4.cmml" id="S5.Ex58.m1.8.8.1.1.2.2.2.3d.cmml" xref="S5.Ex58.m1.8.8.1.1.2.2.2.3"></share><ci id="S5.Ex58.m1.8.8.1.1.2.2.2.3.6.cmml" xref="S5.Ex58.m1.8.8.1.1.2.2.2.3.6">𝜏</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex58.m1.8c">Y_{q}(\theta)=\{(\tau,f)\,|\,\tau\in N\mathcal{C}_{q},f:[q]\to[n]\text{ such % that }f^{*}(\theta)=\tau\}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex58.m1.8d">italic_Y start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_θ ) = { ( italic_τ , italic_f ) | italic_τ ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , italic_f : [ italic_q ] → [ italic_n ] such that italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_θ ) = italic_τ } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.11">For every morphism <math alttext="g:[q^{\prime}]\to[q]" class="ltx_Math" display="inline" id="S5.1.p1.9.m1.2"><semantics id="S5.1.p1.9.m1.2a"><mrow id="S5.1.p1.9.m1.2.2" xref="S5.1.p1.9.m1.2.2.cmml"><mi id="S5.1.p1.9.m1.2.2.3" xref="S5.1.p1.9.m1.2.2.3.cmml">g</mi><mo id="S5.1.p1.9.m1.2.2.2" lspace="0.278em" rspace="0.278em" xref="S5.1.p1.9.m1.2.2.2.cmml">:</mo><mrow id="S5.1.p1.9.m1.2.2.1" xref="S5.1.p1.9.m1.2.2.1.cmml"><mrow id="S5.1.p1.9.m1.2.2.1.1.1" xref="S5.1.p1.9.m1.2.2.1.1.2.cmml"><mo id="S5.1.p1.9.m1.2.2.1.1.1.2" stretchy="false" xref="S5.1.p1.9.m1.2.2.1.1.2.1.cmml">[</mo><msup id="S5.1.p1.9.m1.2.2.1.1.1.1" xref="S5.1.p1.9.m1.2.2.1.1.1.1.cmml"><mi id="S5.1.p1.9.m1.2.2.1.1.1.1.2" xref="S5.1.p1.9.m1.2.2.1.1.1.1.2.cmml">q</mi><mo id="S5.1.p1.9.m1.2.2.1.1.1.1.3" xref="S5.1.p1.9.m1.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.1.p1.9.m1.2.2.1.1.1.3" stretchy="false" xref="S5.1.p1.9.m1.2.2.1.1.2.1.cmml">]</mo></mrow><mo id="S5.1.p1.9.m1.2.2.1.2" stretchy="false" xref="S5.1.p1.9.m1.2.2.1.2.cmml">→</mo><mrow id="S5.1.p1.9.m1.2.2.1.3.2" xref="S5.1.p1.9.m1.2.2.1.3.1.cmml"><mo id="S5.1.p1.9.m1.2.2.1.3.2.1" stretchy="false" xref="S5.1.p1.9.m1.2.2.1.3.1.1.cmml">[</mo><mi id="S5.1.p1.9.m1.1.1" xref="S5.1.p1.9.m1.1.1.cmml">q</mi><mo id="S5.1.p1.9.m1.2.2.1.3.2.2" stretchy="false" xref="S5.1.p1.9.m1.2.2.1.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.9.m1.2b"><apply id="S5.1.p1.9.m1.2.2.cmml" xref="S5.1.p1.9.m1.2.2"><ci id="S5.1.p1.9.m1.2.2.2.cmml" xref="S5.1.p1.9.m1.2.2.2">:</ci><ci id="S5.1.p1.9.m1.2.2.3.cmml" xref="S5.1.p1.9.m1.2.2.3">𝑔</ci><apply id="S5.1.p1.9.m1.2.2.1.cmml" xref="S5.1.p1.9.m1.2.2.1"><ci id="S5.1.p1.9.m1.2.2.1.2.cmml" xref="S5.1.p1.9.m1.2.2.1.2">→</ci><apply id="S5.1.p1.9.m1.2.2.1.1.2.cmml" xref="S5.1.p1.9.m1.2.2.1.1.1"><csymbol cd="latexml" id="S5.1.p1.9.m1.2.2.1.1.2.1.cmml" xref="S5.1.p1.9.m1.2.2.1.1.1.2">delimited-[]</csymbol><apply id="S5.1.p1.9.m1.2.2.1.1.1.1.cmml" xref="S5.1.p1.9.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.1.p1.9.m1.2.2.1.1.1.1.1.cmml" xref="S5.1.p1.9.m1.2.2.1.1.1.1">superscript</csymbol><ci id="S5.1.p1.9.m1.2.2.1.1.1.1.2.cmml" xref="S5.1.p1.9.m1.2.2.1.1.1.1.2">𝑞</ci><ci id="S5.1.p1.9.m1.2.2.1.1.1.1.3.cmml" xref="S5.1.p1.9.m1.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S5.1.p1.9.m1.2.2.1.3.1.cmml" xref="S5.1.p1.9.m1.2.2.1.3.2"><csymbol cd="latexml" id="S5.1.p1.9.m1.2.2.1.3.1.1.cmml" xref="S5.1.p1.9.m1.2.2.1.3.2.1">delimited-[]</csymbol><ci id="S5.1.p1.9.m1.1.1.cmml" xref="S5.1.p1.9.m1.1.1">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.9.m1.2c">g:[q^{\prime}]\to[q]</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.9.m1.2d">italic_g : [ italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] → [ italic_q ]</annotation></semantics></math> in <math alttext="\Delta" class="ltx_Math" display="inline" id="S5.1.p1.10.m2.1"><semantics id="S5.1.p1.10.m2.1a"><mi id="S5.1.p1.10.m2.1.1" mathvariant="normal" xref="S5.1.p1.10.m2.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.10.m2.1b"><ci id="S5.1.p1.10.m2.1.1.cmml" xref="S5.1.p1.10.m2.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.10.m2.1c">\Delta</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.10.m2.1d">roman_Δ</annotation></semantics></math>, the induced map <math alttext="g^{*}(\theta):Q_{q}(\theta)\to Q_{q^{\prime}}(\theta)" class="ltx_Math" display="inline" 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xref="S5.1.p1.11.m3.3.4.2.1"></times><apply id="S5.1.p1.11.m3.3.4.2.2.cmml" xref="S5.1.p1.11.m3.3.4.2.2"><csymbol cd="ambiguous" id="S5.1.p1.11.m3.3.4.2.2.1.cmml" xref="S5.1.p1.11.m3.3.4.2.2">superscript</csymbol><ci id="S5.1.p1.11.m3.3.4.2.2.2.cmml" xref="S5.1.p1.11.m3.3.4.2.2.2">𝑔</ci><times id="S5.1.p1.11.m3.3.4.2.2.3.cmml" xref="S5.1.p1.11.m3.3.4.2.2.3"></times></apply><ci id="S5.1.p1.11.m3.1.1.cmml" xref="S5.1.p1.11.m3.1.1">𝜃</ci></apply><apply id="S5.1.p1.11.m3.3.4.3.cmml" xref="S5.1.p1.11.m3.3.4.3"><ci id="S5.1.p1.11.m3.3.4.3.1.cmml" xref="S5.1.p1.11.m3.3.4.3.1">→</ci><apply id="S5.1.p1.11.m3.3.4.3.2.cmml" xref="S5.1.p1.11.m3.3.4.3.2"><times id="S5.1.p1.11.m3.3.4.3.2.1.cmml" xref="S5.1.p1.11.m3.3.4.3.2.1"></times><apply id="S5.1.p1.11.m3.3.4.3.2.2.cmml" xref="S5.1.p1.11.m3.3.4.3.2.2"><csymbol cd="ambiguous" id="S5.1.p1.11.m3.3.4.3.2.2.1.cmml" xref="S5.1.p1.11.m3.3.4.3.2.2">subscript</csymbol><ci id="S5.1.p1.11.m3.3.4.3.2.2.2.cmml" xref="S5.1.p1.11.m3.3.4.3.2.2.2">𝑄</ci><ci id="S5.1.p1.11.m3.3.4.3.2.2.3.cmml" xref="S5.1.p1.11.m3.3.4.3.2.2.3">𝑞</ci></apply><ci id="S5.1.p1.11.m3.2.2.cmml" xref="S5.1.p1.11.m3.2.2">𝜃</ci></apply><apply id="S5.1.p1.11.m3.3.4.3.3.cmml" xref="S5.1.p1.11.m3.3.4.3.3"><times id="S5.1.p1.11.m3.3.4.3.3.1.cmml" xref="S5.1.p1.11.m3.3.4.3.3.1"></times><apply id="S5.1.p1.11.m3.3.4.3.3.2.cmml" xref="S5.1.p1.11.m3.3.4.3.3.2"><csymbol cd="ambiguous" id="S5.1.p1.11.m3.3.4.3.3.2.1.cmml" xref="S5.1.p1.11.m3.3.4.3.3.2">subscript</csymbol><ci id="S5.1.p1.11.m3.3.4.3.3.2.2.cmml" xref="S5.1.p1.11.m3.3.4.3.3.2.2">𝑄</ci><apply id="S5.1.p1.11.m3.3.4.3.3.2.3.cmml" xref="S5.1.p1.11.m3.3.4.3.3.2.3"><csymbol cd="ambiguous" id="S5.1.p1.11.m3.3.4.3.3.2.3.1.cmml" xref="S5.1.p1.11.m3.3.4.3.3.2.3">superscript</csymbol><ci id="S5.1.p1.11.m3.3.4.3.3.2.3.2.cmml" xref="S5.1.p1.11.m3.3.4.3.3.2.3.2">𝑞</ci><ci id="S5.1.p1.11.m3.3.4.3.3.2.3.3.cmml" xref="S5.1.p1.11.m3.3.4.3.3.2.3.3">′</ci></apply></apply><ci id="S5.1.p1.11.m3.3.3.cmml" xref="S5.1.p1.11.m3.3.3">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.11.m3.3c">g^{*}(\theta):Q_{q}(\theta)\to Q_{q^{\prime}}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.11.m3.3d">italic_g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_θ ) : italic_Q start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_θ ) → italic_Q start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex59"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g^{*}(\theta)(\tau,f)=(g^{*}\tau,f\circ g)." class="ltx_Math" display="block" id="S5.Ex59.m1.4"><semantics id="S5.Ex59.m1.4a"><mrow id="S5.Ex59.m1.4.4.1" xref="S5.Ex59.m1.4.4.1.1.cmml"><mrow id="S5.Ex59.m1.4.4.1.1" xref="S5.Ex59.m1.4.4.1.1.cmml"><mrow id="S5.Ex59.m1.4.4.1.1.4" xref="S5.Ex59.m1.4.4.1.1.4.cmml"><msup id="S5.Ex59.m1.4.4.1.1.4.2" xref="S5.Ex59.m1.4.4.1.1.4.2.cmml"><mi id="S5.Ex59.m1.4.4.1.1.4.2.2" xref="S5.Ex59.m1.4.4.1.1.4.2.2.cmml">g</mi><mo id="S5.Ex59.m1.4.4.1.1.4.2.3" xref="S5.Ex59.m1.4.4.1.1.4.2.3.cmml">∗</mo></msup><mo id="S5.Ex59.m1.4.4.1.1.4.1" xref="S5.Ex59.m1.4.4.1.1.4.1.cmml">⁢</mo><mrow id="S5.Ex59.m1.4.4.1.1.4.3.2" xref="S5.Ex59.m1.4.4.1.1.4.cmml"><mo id="S5.Ex59.m1.4.4.1.1.4.3.2.1" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.4.cmml">(</mo><mi id="S5.Ex59.m1.1.1" xref="S5.Ex59.m1.1.1.cmml">θ</mi><mo id="S5.Ex59.m1.4.4.1.1.4.3.2.2" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.4.cmml">)</mo></mrow><mo id="S5.Ex59.m1.4.4.1.1.4.1a" xref="S5.Ex59.m1.4.4.1.1.4.1.cmml">⁢</mo><mrow id="S5.Ex59.m1.4.4.1.1.4.4.2" xref="S5.Ex59.m1.4.4.1.1.4.4.1.cmml"><mo id="S5.Ex59.m1.4.4.1.1.4.4.2.1" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.4.4.1.cmml">(</mo><mi id="S5.Ex59.m1.2.2" xref="S5.Ex59.m1.2.2.cmml">τ</mi><mo id="S5.Ex59.m1.4.4.1.1.4.4.2.2" xref="S5.Ex59.m1.4.4.1.1.4.4.1.cmml">,</mo><mi id="S5.Ex59.m1.3.3" xref="S5.Ex59.m1.3.3.cmml">f</mi><mo id="S5.Ex59.m1.4.4.1.1.4.4.2.3" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.4.4.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex59.m1.4.4.1.1.3" xref="S5.Ex59.m1.4.4.1.1.3.cmml">=</mo><mrow id="S5.Ex59.m1.4.4.1.1.2.2" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml"><mo id="S5.Ex59.m1.4.4.1.1.2.2.3" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml">(</mo><mrow id="S5.Ex59.m1.4.4.1.1.1.1.1" xref="S5.Ex59.m1.4.4.1.1.1.1.1.cmml"><msup id="S5.Ex59.m1.4.4.1.1.1.1.1.2" xref="S5.Ex59.m1.4.4.1.1.1.1.1.2.cmml"><mi id="S5.Ex59.m1.4.4.1.1.1.1.1.2.2" xref="S5.Ex59.m1.4.4.1.1.1.1.1.2.2.cmml">g</mi><mo id="S5.Ex59.m1.4.4.1.1.1.1.1.2.3" xref="S5.Ex59.m1.4.4.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S5.Ex59.m1.4.4.1.1.1.1.1.1" xref="S5.Ex59.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex59.m1.4.4.1.1.1.1.1.3" xref="S5.Ex59.m1.4.4.1.1.1.1.1.3.cmml">τ</mi></mrow><mo id="S5.Ex59.m1.4.4.1.1.2.2.4" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml">,</mo><mrow id="S5.Ex59.m1.4.4.1.1.2.2.2" xref="S5.Ex59.m1.4.4.1.1.2.2.2.cmml"><mi id="S5.Ex59.m1.4.4.1.1.2.2.2.2" xref="S5.Ex59.m1.4.4.1.1.2.2.2.2.cmml">f</mi><mo id="S5.Ex59.m1.4.4.1.1.2.2.2.1" lspace="0.222em" rspace="0.222em" xref="S5.Ex59.m1.4.4.1.1.2.2.2.1.cmml">∘</mo><mi id="S5.Ex59.m1.4.4.1.1.2.2.2.3" xref="S5.Ex59.m1.4.4.1.1.2.2.2.3.cmml">g</mi></mrow><mo id="S5.Ex59.m1.4.4.1.1.2.2.5" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex59.m1.4.4.1.2" lspace="0em" xref="S5.Ex59.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex59.m1.4b"><apply id="S5.Ex59.m1.4.4.1.1.cmml" xref="S5.Ex59.m1.4.4.1"><eq id="S5.Ex59.m1.4.4.1.1.3.cmml" xref="S5.Ex59.m1.4.4.1.1.3"></eq><apply id="S5.Ex59.m1.4.4.1.1.4.cmml" xref="S5.Ex59.m1.4.4.1.1.4"><times id="S5.Ex59.m1.4.4.1.1.4.1.cmml" xref="S5.Ex59.m1.4.4.1.1.4.1"></times><apply id="S5.Ex59.m1.4.4.1.1.4.2.cmml" xref="S5.Ex59.m1.4.4.1.1.4.2"><csymbol cd="ambiguous" id="S5.Ex59.m1.4.4.1.1.4.2.1.cmml" xref="S5.Ex59.m1.4.4.1.1.4.2">superscript</csymbol><ci id="S5.Ex59.m1.4.4.1.1.4.2.2.cmml" xref="S5.Ex59.m1.4.4.1.1.4.2.2">𝑔</ci><times id="S5.Ex59.m1.4.4.1.1.4.2.3.cmml" xref="S5.Ex59.m1.4.4.1.1.4.2.3"></times></apply><ci id="S5.Ex59.m1.1.1.cmml" xref="S5.Ex59.m1.1.1">𝜃</ci><interval closure="open" id="S5.Ex59.m1.4.4.1.1.4.4.1.cmml" xref="S5.Ex59.m1.4.4.1.1.4.4.2"><ci id="S5.Ex59.m1.2.2.cmml" xref="S5.Ex59.m1.2.2">𝜏</ci><ci id="S5.Ex59.m1.3.3.cmml" xref="S5.Ex59.m1.3.3">𝑓</ci></interval></apply><interval closure="open" id="S5.Ex59.m1.4.4.1.1.2.3.cmml" xref="S5.Ex59.m1.4.4.1.1.2.2"><apply id="S5.Ex59.m1.4.4.1.1.1.1.1.cmml" xref="S5.Ex59.m1.4.4.1.1.1.1.1"><times id="S5.Ex59.m1.4.4.1.1.1.1.1.1.cmml" xref="S5.Ex59.m1.4.4.1.1.1.1.1.1"></times><apply id="S5.Ex59.m1.4.4.1.1.1.1.1.2.cmml" xref="S5.Ex59.m1.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex59.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S5.Ex59.m1.4.4.1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex59.m1.4.4.1.1.1.1.1.2.2.cmml" xref="S5.Ex59.m1.4.4.1.1.1.1.1.2.2">𝑔</ci><times id="S5.Ex59.m1.4.4.1.1.1.1.1.2.3.cmml" xref="S5.Ex59.m1.4.4.1.1.1.1.1.2.3"></times></apply><ci id="S5.Ex59.m1.4.4.1.1.1.1.1.3.cmml" xref="S5.Ex59.m1.4.4.1.1.1.1.1.3">𝜏</ci></apply><apply id="S5.Ex59.m1.4.4.1.1.2.2.2.cmml" xref="S5.Ex59.m1.4.4.1.1.2.2.2"><compose id="S5.Ex59.m1.4.4.1.1.2.2.2.1.cmml" xref="S5.Ex59.m1.4.4.1.1.2.2.2.1"></compose><ci id="S5.Ex59.m1.4.4.1.1.2.2.2.2.cmml" xref="S5.Ex59.m1.4.4.1.1.2.2.2.2">𝑓</ci><ci id="S5.Ex59.m1.4.4.1.1.2.2.2.3.cmml" xref="S5.Ex59.m1.4.4.1.1.2.2.2.3">𝑔</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex59.m1.4c">g^{*}(\theta)(\tau,f)=(g^{*}\tau,f\circ g).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex59.m1.4d">italic_g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_θ ) ( italic_τ , italic_f ) = ( italic_g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_τ , italic_f ∘ italic_g ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.2.p2"> <p class="ltx_p" id="S5.2.p2.3">Let <math alttext="Q_{*}" class="ltx_Math" display="inline" id="S5.2.p2.1.m1.1"><semantics id="S5.2.p2.1.m1.1a"><msub id="S5.2.p2.1.m1.1.1" xref="S5.2.p2.1.m1.1.1.cmml"><mi id="S5.2.p2.1.m1.1.1.2" xref="S5.2.p2.1.m1.1.1.2.cmml">Q</mi><mo id="S5.2.p2.1.m1.1.1.3" xref="S5.2.p2.1.m1.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S5.2.p2.1.m1.1b"><apply id="S5.2.p2.1.m1.1.1.cmml" xref="S5.2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.1.m1.1.1.1.cmml" xref="S5.2.p2.1.m1.1.1">subscript</csymbol><ci id="S5.2.p2.1.m1.1.1.2.cmml" xref="S5.2.p2.1.m1.1.1.2">𝑄</ci><times id="S5.2.p2.1.m1.1.1.3.cmml" xref="S5.2.p2.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.1.m1.1c">Q_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.1.m1.1d">italic_Q start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> be the Moore complex of the simplicial <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.2.p2.2.m2.1"><semantics id="S5.2.p2.2.m2.1a"><mrow id="S5.2.p2.2.m2.1.1" xref="S5.2.p2.2.m2.1.1.cmml"><mi id="S5.2.p2.2.m2.1.1.3" xref="S5.2.p2.2.m2.1.1.3.cmml">R</mi><mo id="S5.2.p2.2.m2.1.1.2" xref="S5.2.p2.2.m2.1.1.2.cmml">⁢</mo><mi id="S5.2.p2.2.m2.1.1.4" mathvariant="normal" xref="S5.2.p2.2.m2.1.1.4.cmml">Δ</mi><mo id="S5.2.p2.2.m2.1.1.2a" xref="S5.2.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S5.2.p2.2.m2.1.1.1.1" xref="S5.2.p2.2.m2.1.1.1.1.1.cmml"><mo id="S5.2.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S5.2.p2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.2.p2.2.m2.1.1.1.1.1" xref="S5.2.p2.2.m2.1.1.1.1.1.cmml"><mi id="S5.2.p2.2.m2.1.1.1.1.1.2" xref="S5.2.p2.2.m2.1.1.1.1.1.2.cmml">N</mi><mo id="S5.2.p2.2.m2.1.1.1.1.1.1" xref="S5.2.p2.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.2.p2.2.m2.1.1.1.1.1.3" xref="S5.2.p2.2.m2.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.2.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S5.2.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.2.m2.1b"><apply id="S5.2.p2.2.m2.1.1.cmml" xref="S5.2.p2.2.m2.1.1"><times id="S5.2.p2.2.m2.1.1.2.cmml" xref="S5.2.p2.2.m2.1.1.2"></times><ci id="S5.2.p2.2.m2.1.1.3.cmml" xref="S5.2.p2.2.m2.1.1.3">𝑅</ci><ci id="S5.2.p2.2.m2.1.1.4.cmml" xref="S5.2.p2.2.m2.1.1.4">Δ</ci><apply id="S5.2.p2.2.m2.1.1.1.1.1.cmml" xref="S5.2.p2.2.m2.1.1.1.1"><times id="S5.2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S5.2.p2.2.m2.1.1.1.1.1.1"></times><ci id="S5.2.p2.2.m2.1.1.1.1.1.2.cmml" xref="S5.2.p2.2.m2.1.1.1.1.1.2">𝑁</ci><ci id="S5.2.p2.2.m2.1.1.1.1.1.3.cmml" xref="S5.2.p2.2.m2.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.2.m2.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.2.m2.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-module <math alttext="Q" class="ltx_Math" display="inline" id="S5.2.p2.3.m3.1"><semantics id="S5.2.p2.3.m3.1a"><mi id="S5.2.p2.3.m3.1.1" xref="S5.2.p2.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.3.m3.1b"><ci id="S5.2.p2.3.m3.1.1.cmml" xref="S5.2.p2.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.3.m3.1d">italic_Q</annotation></semantics></math>. By the argument used in the proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.Thmtheorem3" title="Proposition 3.3. ‣ 3.2. Cohomology of simplicial sets with general coefficients ‣ 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">3.3</span></a>, the sequence</p> <table class="ltx_equation ltx_eqn_table" id="S5.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{Q}_{*}:\cdots\to Q_{k}\smash{\,\mathop{\longrightarrow}\limits^{% \partial_{k}}\,}Q_{k-1}\to\cdots\to Q_{0}\smash{\,\mathop{\longrightarrow}% \limits^{\epsilon}\,}\underline{R}\to 0" class="ltx_Math" display="block" id="S5.E5.m1.1"><semantics id="S5.E5.m1.1a"><mrow id="S5.E5.m1.1.1" xref="S5.E5.m1.1.1.cmml"><msub id="S5.E5.m1.1.1.2" xref="S5.E5.m1.1.1.2.cmml"><mover accent="true" id="S5.E5.m1.1.1.2.2" xref="S5.E5.m1.1.1.2.2.cmml"><mi id="S5.E5.m1.1.1.2.2.2" xref="S5.E5.m1.1.1.2.2.2.cmml">Q</mi><mo id="S5.E5.m1.1.1.2.2.1" xref="S5.E5.m1.1.1.2.2.1.cmml">~</mo></mover><mo id="S5.E5.m1.1.1.2.3" xref="S5.E5.m1.1.1.2.3.cmml">∗</mo></msub><mo id="S5.E5.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.E5.m1.1.1.1.cmml">:</mo><mrow id="S5.E5.m1.1.1.3" xref="S5.E5.m1.1.1.3.cmml"><mi id="S5.E5.m1.1.1.3.2" mathvariant="normal" xref="S5.E5.m1.1.1.3.2.cmml">⋯</mi><mo id="S5.E5.m1.1.1.3.3" stretchy="false" xref="S5.E5.m1.1.1.3.3.cmml">→</mo><mrow id="S5.E5.m1.1.1.3.4" xref="S5.E5.m1.1.1.3.4.cmml"><msub id="S5.E5.m1.1.1.3.4.2" xref="S5.E5.m1.1.1.3.4.2.cmml"><mi id="S5.E5.m1.1.1.3.4.2.2" xref="S5.E5.m1.1.1.3.4.2.2.cmml">Q</mi><mi id="S5.E5.m1.1.1.3.4.2.3" xref="S5.E5.m1.1.1.3.4.2.3.cmml">k</mi></msub><mo id="S5.E5.m1.1.1.3.4.1" lspace="0.167em" xref="S5.E5.m1.1.1.3.4.1.cmml">⁢</mo><mrow id="S5.E5.m1.1.1.3.4.3" xref="S5.E5.m1.1.1.3.4.3.cmml"><mover id="S5.E5.m1.1.1.3.4.3.1" xref="S5.E5.m1.1.1.3.4.3.1.cmml"><mo id="S5.E5.m1.1.1.3.4.3.1.2" movablelimits="false" rspace="0.167em" xref="S5.E5.m1.1.1.3.4.3.1.2.cmml">⟶</mo><msub id="S5.E5.m1.1.1.3.4.3.1.3" 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id="S5.E5.m1.1.1.3.8.3.1.3.cmml" xref="S5.E5.m1.1.1.3.8.3.1.3">italic-ϵ</ci></apply><apply id="S5.E5.m1.1.1.3.8.3.2.cmml" xref="S5.E5.m1.1.1.3.8.3.2"><ci id="S5.E5.m1.1.1.3.8.3.2.1.cmml" xref="S5.E5.m1.1.1.3.8.3.2.1">¯</ci><ci id="S5.E5.m1.1.1.3.8.3.2.2.cmml" xref="S5.E5.m1.1.1.3.8.3.2.2">𝑅</ci></apply></apply></apply></apply><apply id="S5.E5.m1.1.1.3g.cmml" xref="S5.E5.m1.1.1.3"><ci id="S5.E5.m1.1.1.3.9.cmml" xref="S5.E5.m1.1.1.3.9">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.E5.m1.1.1.3.8.cmml" id="S5.E5.m1.1.1.3h.cmml" xref="S5.E5.m1.1.1.3"></share><cn id="S5.E5.m1.1.1.3.10.cmml" type="integer" xref="S5.E5.m1.1.1.3.10">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E5.m1.1c">\widetilde{Q}_{*}:\cdots\to Q_{k}\smash{\,\mathop{\longrightarrow}\limits^{% \partial_{k}}\,}Q_{k-1}\to\cdots\to Q_{0}\smash{\,\mathop{\longrightarrow}% \limits^{\epsilon}\,}\underline{R}\to 0</annotation><annotation encoding="application/x-llamapun" id="S5.E5.m1.1d">over~ start_ARG italic_Q end_ARG start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : ⋯ → italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT → ⋯ → italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT under¯ start_ARG italic_R end_ARG → 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.2.p2.11">is exact. Let <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S5.2.p2.4.m1.1"><semantics id="S5.2.p2.4.m1.1a"><mi id="S5.2.p2.4.m1.1.1" xref="S5.2.p2.4.m1.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.4.m1.1b"><ci id="S5.2.p2.4.m1.1.1.cmml" xref="S5.2.p2.4.m1.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.4.m1.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.4.m1.1d">blackboard_Q</annotation></semantics></math> be the bisimplicial <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.2.p2.5.m2.1"><semantics id="S5.2.p2.5.m2.1a"><mrow id="S5.2.p2.5.m2.1.1" xref="S5.2.p2.5.m2.1.1.cmml"><mi id="S5.2.p2.5.m2.1.1.3" xref="S5.2.p2.5.m2.1.1.3.cmml">R</mi><mo id="S5.2.p2.5.m2.1.1.2" xref="S5.2.p2.5.m2.1.1.2.cmml">⁢</mo><mi id="S5.2.p2.5.m2.1.1.4" mathvariant="normal" xref="S5.2.p2.5.m2.1.1.4.cmml">Δ</mi><mo id="S5.2.p2.5.m2.1.1.2a" xref="S5.2.p2.5.m2.1.1.2.cmml">⁢</mo><mrow id="S5.2.p2.5.m2.1.1.1.1" xref="S5.2.p2.5.m2.1.1.1.1.1.cmml"><mo id="S5.2.p2.5.m2.1.1.1.1.2" stretchy="false" xref="S5.2.p2.5.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.2.p2.5.m2.1.1.1.1.1" xref="S5.2.p2.5.m2.1.1.1.1.1.cmml"><mi id="S5.2.p2.5.m2.1.1.1.1.1.2" xref="S5.2.p2.5.m2.1.1.1.1.1.2.cmml">N</mi><mo id="S5.2.p2.5.m2.1.1.1.1.1.1" xref="S5.2.p2.5.m2.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.2.p2.5.m2.1.1.1.1.1.3" xref="S5.2.p2.5.m2.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.2.p2.5.m2.1.1.1.1.3" stretchy="false" xref="S5.2.p2.5.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.5.m2.1b"><apply id="S5.2.p2.5.m2.1.1.cmml" xref="S5.2.p2.5.m2.1.1"><times id="S5.2.p2.5.m2.1.1.2.cmml" xref="S5.2.p2.5.m2.1.1.2"></times><ci id="S5.2.p2.5.m2.1.1.3.cmml" xref="S5.2.p2.5.m2.1.1.3">𝑅</ci><ci id="S5.2.p2.5.m2.1.1.4.cmml" xref="S5.2.p2.5.m2.1.1.4">Δ</ci><apply id="S5.2.p2.5.m2.1.1.1.1.1.cmml" xref="S5.2.p2.5.m2.1.1.1.1"><times id="S5.2.p2.5.m2.1.1.1.1.1.1.cmml" xref="S5.2.p2.5.m2.1.1.1.1.1.1"></times><ci id="S5.2.p2.5.m2.1.1.1.1.1.2.cmml" xref="S5.2.p2.5.m2.1.1.1.1.1.2">𝑁</ci><ci id="S5.2.p2.5.m2.1.1.1.1.1.3.cmml" xref="S5.2.p2.5.m2.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.5.m2.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.5.m2.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-module such that <math alttext="\mathbb{Q}_{p,q}=Q_{q}" class="ltx_Math" display="inline" id="S5.2.p2.6.m3.2"><semantics id="S5.2.p2.6.m3.2a"><mrow id="S5.2.p2.6.m3.2.3" xref="S5.2.p2.6.m3.2.3.cmml"><msub id="S5.2.p2.6.m3.2.3.2" xref="S5.2.p2.6.m3.2.3.2.cmml"><mi id="S5.2.p2.6.m3.2.3.2.2" xref="S5.2.p2.6.m3.2.3.2.2.cmml">ℚ</mi><mrow id="S5.2.p2.6.m3.2.2.2.4" xref="S5.2.p2.6.m3.2.2.2.3.cmml"><mi id="S5.2.p2.6.m3.1.1.1.1" xref="S5.2.p2.6.m3.1.1.1.1.cmml">p</mi><mo id="S5.2.p2.6.m3.2.2.2.4.1" xref="S5.2.p2.6.m3.2.2.2.3.cmml">,</mo><mi id="S5.2.p2.6.m3.2.2.2.2" xref="S5.2.p2.6.m3.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.2.p2.6.m3.2.3.1" xref="S5.2.p2.6.m3.2.3.1.cmml">=</mo><msub id="S5.2.p2.6.m3.2.3.3" xref="S5.2.p2.6.m3.2.3.3.cmml"><mi id="S5.2.p2.6.m3.2.3.3.2" xref="S5.2.p2.6.m3.2.3.3.2.cmml">Q</mi><mi id="S5.2.p2.6.m3.2.3.3.3" xref="S5.2.p2.6.m3.2.3.3.3.cmml">q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.6.m3.2b"><apply id="S5.2.p2.6.m3.2.3.cmml" xref="S5.2.p2.6.m3.2.3"><eq id="S5.2.p2.6.m3.2.3.1.cmml" xref="S5.2.p2.6.m3.2.3.1"></eq><apply id="S5.2.p2.6.m3.2.3.2.cmml" xref="S5.2.p2.6.m3.2.3.2"><csymbol cd="ambiguous" id="S5.2.p2.6.m3.2.3.2.1.cmml" xref="S5.2.p2.6.m3.2.3.2">subscript</csymbol><ci id="S5.2.p2.6.m3.2.3.2.2.cmml" xref="S5.2.p2.6.m3.2.3.2.2">ℚ</ci><list id="S5.2.p2.6.m3.2.2.2.3.cmml" xref="S5.2.p2.6.m3.2.2.2.4"><ci id="S5.2.p2.6.m3.1.1.1.1.cmml" xref="S5.2.p2.6.m3.1.1.1.1">𝑝</ci><ci id="S5.2.p2.6.m3.2.2.2.2.cmml" xref="S5.2.p2.6.m3.2.2.2.2">𝑞</ci></list></apply><apply id="S5.2.p2.6.m3.2.3.3.cmml" xref="S5.2.p2.6.m3.2.3.3"><csymbol cd="ambiguous" id="S5.2.p2.6.m3.2.3.3.1.cmml" xref="S5.2.p2.6.m3.2.3.3">subscript</csymbol><ci id="S5.2.p2.6.m3.2.3.3.2.cmml" xref="S5.2.p2.6.m3.2.3.3.2">𝑄</ci><ci id="S5.2.p2.6.m3.2.3.3.3.cmml" xref="S5.2.p2.6.m3.2.3.3.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.6.m3.2c">\mathbb{Q}_{p,q}=Q_{q}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.6.m3.2d">blackboard_Q start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT = italic_Q start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="p,q\geq 0" class="ltx_Math" display="inline" id="S5.2.p2.7.m4.2"><semantics id="S5.2.p2.7.m4.2a"><mrow id="S5.2.p2.7.m4.2.3" xref="S5.2.p2.7.m4.2.3.cmml"><mrow id="S5.2.p2.7.m4.2.3.2.2" xref="S5.2.p2.7.m4.2.3.2.1.cmml"><mi id="S5.2.p2.7.m4.1.1" xref="S5.2.p2.7.m4.1.1.cmml">p</mi><mo id="S5.2.p2.7.m4.2.3.2.2.1" xref="S5.2.p2.7.m4.2.3.2.1.cmml">,</mo><mi id="S5.2.p2.7.m4.2.2" xref="S5.2.p2.7.m4.2.2.cmml">q</mi></mrow><mo id="S5.2.p2.7.m4.2.3.1" xref="S5.2.p2.7.m4.2.3.1.cmml">≥</mo><mn id="S5.2.p2.7.m4.2.3.3" xref="S5.2.p2.7.m4.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.7.m4.2b"><apply id="S5.2.p2.7.m4.2.3.cmml" xref="S5.2.p2.7.m4.2.3"><geq id="S5.2.p2.7.m4.2.3.1.cmml" xref="S5.2.p2.7.m4.2.3.1"></geq><list id="S5.2.p2.7.m4.2.3.2.1.cmml" xref="S5.2.p2.7.m4.2.3.2.2"><ci id="S5.2.p2.7.m4.1.1.cmml" xref="S5.2.p2.7.m4.1.1">𝑝</ci><ci id="S5.2.p2.7.m4.2.2.cmml" xref="S5.2.p2.7.m4.2.2">𝑞</ci></list><cn id="S5.2.p2.7.m4.2.3.3.cmml" type="integer" xref="S5.2.p2.7.m4.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.7.m4.2c">p,q\geq 0</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.7.m4.2d">italic_p , italic_q ≥ 0</annotation></semantics></math>, with vertical boundary and degeneracy maps given by the boundary and degeneracy maps of <math alttext="Q_{*}" class="ltx_Math" display="inline" id="S5.2.p2.8.m5.1"><semantics id="S5.2.p2.8.m5.1a"><msub id="S5.2.p2.8.m5.1.1" xref="S5.2.p2.8.m5.1.1.cmml"><mi id="S5.2.p2.8.m5.1.1.2" xref="S5.2.p2.8.m5.1.1.2.cmml">Q</mi><mo id="S5.2.p2.8.m5.1.1.3" xref="S5.2.p2.8.m5.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S5.2.p2.8.m5.1b"><apply id="S5.2.p2.8.m5.1.1.cmml" xref="S5.2.p2.8.m5.1.1"><csymbol cd="ambiguous" id="S5.2.p2.8.m5.1.1.1.cmml" xref="S5.2.p2.8.m5.1.1">subscript</csymbol><ci id="S5.2.p2.8.m5.1.1.2.cmml" xref="S5.2.p2.8.m5.1.1.2">𝑄</ci><times id="S5.2.p2.8.m5.1.1.3.cmml" xref="S5.2.p2.8.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.8.m5.1c">Q_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.8.m5.1d">italic_Q start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>, and the horizontal boundary and degeneracy maps given as the identity maps. Consider the bisimplicial map <math alttext="\xi:\mathbb{P}\to\mathbb{Q}" class="ltx_Math" display="inline" id="S5.2.p2.9.m6.1"><semantics id="S5.2.p2.9.m6.1a"><mrow id="S5.2.p2.9.m6.1.1" xref="S5.2.p2.9.m6.1.1.cmml"><mi id="S5.2.p2.9.m6.1.1.2" xref="S5.2.p2.9.m6.1.1.2.cmml">ξ</mi><mo id="S5.2.p2.9.m6.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.2.p2.9.m6.1.1.1.cmml">:</mo><mrow id="S5.2.p2.9.m6.1.1.3" xref="S5.2.p2.9.m6.1.1.3.cmml"><mi id="S5.2.p2.9.m6.1.1.3.2" xref="S5.2.p2.9.m6.1.1.3.2.cmml">ℙ</mi><mo id="S5.2.p2.9.m6.1.1.3.1" stretchy="false" xref="S5.2.p2.9.m6.1.1.3.1.cmml">→</mo><mi id="S5.2.p2.9.m6.1.1.3.3" xref="S5.2.p2.9.m6.1.1.3.3.cmml">ℚ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.9.m6.1b"><apply id="S5.2.p2.9.m6.1.1.cmml" xref="S5.2.p2.9.m6.1.1"><ci id="S5.2.p2.9.m6.1.1.1.cmml" xref="S5.2.p2.9.m6.1.1.1">:</ci><ci id="S5.2.p2.9.m6.1.1.2.cmml" xref="S5.2.p2.9.m6.1.1.2">𝜉</ci><apply id="S5.2.p2.9.m6.1.1.3.cmml" xref="S5.2.p2.9.m6.1.1.3"><ci id="S5.2.p2.9.m6.1.1.3.1.cmml" xref="S5.2.p2.9.m6.1.1.3.1">→</ci><ci id="S5.2.p2.9.m6.1.1.3.2.cmml" xref="S5.2.p2.9.m6.1.1.3.2">ℙ</ci><ci id="S5.2.p2.9.m6.1.1.3.3.cmml" xref="S5.2.p2.9.m6.1.1.3.3">ℚ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.9.m6.1c">\xi:\mathbb{P}\to\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.9.m6.1d">italic_ξ : blackboard_P → blackboard_Q</annotation></semantics></math> defined by <math alttext="\xi_{p,q}(\theta)(\sigma,\tau,\mu,f)=(\tau,f)" class="ltx_Math" display="inline" id="S5.2.p2.10.m7.9"><semantics id="S5.2.p2.10.m7.9a"><mrow id="S5.2.p2.10.m7.9.10" xref="S5.2.p2.10.m7.9.10.cmml"><mrow id="S5.2.p2.10.m7.9.10.2" xref="S5.2.p2.10.m7.9.10.2.cmml"><msub id="S5.2.p2.10.m7.9.10.2.2" xref="S5.2.p2.10.m7.9.10.2.2.cmml"><mi id="S5.2.p2.10.m7.9.10.2.2.2" xref="S5.2.p2.10.m7.9.10.2.2.2.cmml">ξ</mi><mrow id="S5.2.p2.10.m7.2.2.2.4" xref="S5.2.p2.10.m7.2.2.2.3.cmml"><mi id="S5.2.p2.10.m7.1.1.1.1" xref="S5.2.p2.10.m7.1.1.1.1.cmml">p</mi><mo id="S5.2.p2.10.m7.2.2.2.4.1" xref="S5.2.p2.10.m7.2.2.2.3.cmml">,</mo><mi id="S5.2.p2.10.m7.2.2.2.2" xref="S5.2.p2.10.m7.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.2.p2.10.m7.9.10.2.1" xref="S5.2.p2.10.m7.9.10.2.1.cmml">⁢</mo><mrow id="S5.2.p2.10.m7.9.10.2.3.2" xref="S5.2.p2.10.m7.9.10.2.cmml"><mo id="S5.2.p2.10.m7.9.10.2.3.2.1" stretchy="false" xref="S5.2.p2.10.m7.9.10.2.cmml">(</mo><mi id="S5.2.p2.10.m7.3.3" xref="S5.2.p2.10.m7.3.3.cmml">θ</mi><mo id="S5.2.p2.10.m7.9.10.2.3.2.2" stretchy="false" xref="S5.2.p2.10.m7.9.10.2.cmml">)</mo></mrow><mo id="S5.2.p2.10.m7.9.10.2.1a" xref="S5.2.p2.10.m7.9.10.2.1.cmml">⁢</mo><mrow id="S5.2.p2.10.m7.9.10.2.4.2" xref="S5.2.p2.10.m7.9.10.2.4.1.cmml"><mo id="S5.2.p2.10.m7.9.10.2.4.2.1" stretchy="false" xref="S5.2.p2.10.m7.9.10.2.4.1.cmml">(</mo><mi id="S5.2.p2.10.m7.4.4" xref="S5.2.p2.10.m7.4.4.cmml">σ</mi><mo id="S5.2.p2.10.m7.9.10.2.4.2.2" xref="S5.2.p2.10.m7.9.10.2.4.1.cmml">,</mo><mi id="S5.2.p2.10.m7.5.5" xref="S5.2.p2.10.m7.5.5.cmml">τ</mi><mo id="S5.2.p2.10.m7.9.10.2.4.2.3" xref="S5.2.p2.10.m7.9.10.2.4.1.cmml">,</mo><mi id="S5.2.p2.10.m7.6.6" xref="S5.2.p2.10.m7.6.6.cmml">μ</mi><mo id="S5.2.p2.10.m7.9.10.2.4.2.4" xref="S5.2.p2.10.m7.9.10.2.4.1.cmml">,</mo><mi id="S5.2.p2.10.m7.7.7" xref="S5.2.p2.10.m7.7.7.cmml">f</mi><mo id="S5.2.p2.10.m7.9.10.2.4.2.5" stretchy="false" xref="S5.2.p2.10.m7.9.10.2.4.1.cmml">)</mo></mrow></mrow><mo id="S5.2.p2.10.m7.9.10.1" xref="S5.2.p2.10.m7.9.10.1.cmml">=</mo><mrow id="S5.2.p2.10.m7.9.10.3.2" xref="S5.2.p2.10.m7.9.10.3.1.cmml"><mo id="S5.2.p2.10.m7.9.10.3.2.1" stretchy="false" xref="S5.2.p2.10.m7.9.10.3.1.cmml">(</mo><mi id="S5.2.p2.10.m7.8.8" xref="S5.2.p2.10.m7.8.8.cmml">τ</mi><mo id="S5.2.p2.10.m7.9.10.3.2.2" xref="S5.2.p2.10.m7.9.10.3.1.cmml">,</mo><mi id="S5.2.p2.10.m7.9.9" xref="S5.2.p2.10.m7.9.9.cmml">f</mi><mo id="S5.2.p2.10.m7.9.10.3.2.3" stretchy="false" xref="S5.2.p2.10.m7.9.10.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.10.m7.9b"><apply id="S5.2.p2.10.m7.9.10.cmml" xref="S5.2.p2.10.m7.9.10"><eq id="S5.2.p2.10.m7.9.10.1.cmml" xref="S5.2.p2.10.m7.9.10.1"></eq><apply id="S5.2.p2.10.m7.9.10.2.cmml" xref="S5.2.p2.10.m7.9.10.2"><times id="S5.2.p2.10.m7.9.10.2.1.cmml" xref="S5.2.p2.10.m7.9.10.2.1"></times><apply id="S5.2.p2.10.m7.9.10.2.2.cmml" xref="S5.2.p2.10.m7.9.10.2.2"><csymbol cd="ambiguous" id="S5.2.p2.10.m7.9.10.2.2.1.cmml" xref="S5.2.p2.10.m7.9.10.2.2">subscript</csymbol><ci id="S5.2.p2.10.m7.9.10.2.2.2.cmml" xref="S5.2.p2.10.m7.9.10.2.2.2">𝜉</ci><list id="S5.2.p2.10.m7.2.2.2.3.cmml" xref="S5.2.p2.10.m7.2.2.2.4"><ci id="S5.2.p2.10.m7.1.1.1.1.cmml" xref="S5.2.p2.10.m7.1.1.1.1">𝑝</ci><ci id="S5.2.p2.10.m7.2.2.2.2.cmml" xref="S5.2.p2.10.m7.2.2.2.2">𝑞</ci></list></apply><ci id="S5.2.p2.10.m7.3.3.cmml" xref="S5.2.p2.10.m7.3.3">𝜃</ci><vector id="S5.2.p2.10.m7.9.10.2.4.1.cmml" xref="S5.2.p2.10.m7.9.10.2.4.2"><ci id="S5.2.p2.10.m7.4.4.cmml" xref="S5.2.p2.10.m7.4.4">𝜎</ci><ci id="S5.2.p2.10.m7.5.5.cmml" xref="S5.2.p2.10.m7.5.5">𝜏</ci><ci id="S5.2.p2.10.m7.6.6.cmml" xref="S5.2.p2.10.m7.6.6">𝜇</ci><ci id="S5.2.p2.10.m7.7.7.cmml" xref="S5.2.p2.10.m7.7.7">𝑓</ci></vector></apply><interval closure="open" id="S5.2.p2.10.m7.9.10.3.1.cmml" xref="S5.2.p2.10.m7.9.10.3.2"><ci id="S5.2.p2.10.m7.8.8.cmml" xref="S5.2.p2.10.m7.8.8">𝜏</ci><ci id="S5.2.p2.10.m7.9.9.cmml" xref="S5.2.p2.10.m7.9.9">𝑓</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.10.m7.9c">\xi_{p,q}(\theta)(\sigma,\tau,\mu,f)=(\tau,f)</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.10.m7.9d">italic_ξ start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( italic_θ ) ( italic_σ , italic_τ , italic_μ , italic_f ) = ( italic_τ , italic_f )</annotation></semantics></math> for every <math alttext="(\sigma,\tau,\mu,f)\in X_{p,q}(\theta)" class="ltx_Math" display="inline" id="S5.2.p2.11.m8.7"><semantics id="S5.2.p2.11.m8.7a"><mrow id="S5.2.p2.11.m8.7.8" xref="S5.2.p2.11.m8.7.8.cmml"><mrow id="S5.2.p2.11.m8.7.8.2.2" xref="S5.2.p2.11.m8.7.8.2.1.cmml"><mo id="S5.2.p2.11.m8.7.8.2.2.1" stretchy="false" xref="S5.2.p2.11.m8.7.8.2.1.cmml">(</mo><mi id="S5.2.p2.11.m8.3.3" xref="S5.2.p2.11.m8.3.3.cmml">σ</mi><mo id="S5.2.p2.11.m8.7.8.2.2.2" xref="S5.2.p2.11.m8.7.8.2.1.cmml">,</mo><mi id="S5.2.p2.11.m8.4.4" xref="S5.2.p2.11.m8.4.4.cmml">τ</mi><mo id="S5.2.p2.11.m8.7.8.2.2.3" xref="S5.2.p2.11.m8.7.8.2.1.cmml">,</mo><mi id="S5.2.p2.11.m8.5.5" xref="S5.2.p2.11.m8.5.5.cmml">μ</mi><mo id="S5.2.p2.11.m8.7.8.2.2.4" xref="S5.2.p2.11.m8.7.8.2.1.cmml">,</mo><mi id="S5.2.p2.11.m8.6.6" xref="S5.2.p2.11.m8.6.6.cmml">f</mi><mo id="S5.2.p2.11.m8.7.8.2.2.5" stretchy="false" xref="S5.2.p2.11.m8.7.8.2.1.cmml">)</mo></mrow><mo id="S5.2.p2.11.m8.7.8.1" xref="S5.2.p2.11.m8.7.8.1.cmml">∈</mo><mrow id="S5.2.p2.11.m8.7.8.3" xref="S5.2.p2.11.m8.7.8.3.cmml"><msub id="S5.2.p2.11.m8.7.8.3.2" xref="S5.2.p2.11.m8.7.8.3.2.cmml"><mi id="S5.2.p2.11.m8.7.8.3.2.2" xref="S5.2.p2.11.m8.7.8.3.2.2.cmml">X</mi><mrow id="S5.2.p2.11.m8.2.2.2.4" xref="S5.2.p2.11.m8.2.2.2.3.cmml"><mi id="S5.2.p2.11.m8.1.1.1.1" xref="S5.2.p2.11.m8.1.1.1.1.cmml">p</mi><mo id="S5.2.p2.11.m8.2.2.2.4.1" xref="S5.2.p2.11.m8.2.2.2.3.cmml">,</mo><mi id="S5.2.p2.11.m8.2.2.2.2" xref="S5.2.p2.11.m8.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.2.p2.11.m8.7.8.3.1" xref="S5.2.p2.11.m8.7.8.3.1.cmml">⁢</mo><mrow id="S5.2.p2.11.m8.7.8.3.3.2" xref="S5.2.p2.11.m8.7.8.3.cmml"><mo id="S5.2.p2.11.m8.7.8.3.3.2.1" stretchy="false" xref="S5.2.p2.11.m8.7.8.3.cmml">(</mo><mi id="S5.2.p2.11.m8.7.7" xref="S5.2.p2.11.m8.7.7.cmml">θ</mi><mo id="S5.2.p2.11.m8.7.8.3.3.2.2" stretchy="false" xref="S5.2.p2.11.m8.7.8.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.11.m8.7b"><apply id="S5.2.p2.11.m8.7.8.cmml" xref="S5.2.p2.11.m8.7.8"><in id="S5.2.p2.11.m8.7.8.1.cmml" xref="S5.2.p2.11.m8.7.8.1"></in><vector id="S5.2.p2.11.m8.7.8.2.1.cmml" xref="S5.2.p2.11.m8.7.8.2.2"><ci id="S5.2.p2.11.m8.3.3.cmml" xref="S5.2.p2.11.m8.3.3">𝜎</ci><ci id="S5.2.p2.11.m8.4.4.cmml" xref="S5.2.p2.11.m8.4.4">𝜏</ci><ci id="S5.2.p2.11.m8.5.5.cmml" xref="S5.2.p2.11.m8.5.5">𝜇</ci><ci id="S5.2.p2.11.m8.6.6.cmml" xref="S5.2.p2.11.m8.6.6">𝑓</ci></vector><apply id="S5.2.p2.11.m8.7.8.3.cmml" xref="S5.2.p2.11.m8.7.8.3"><times id="S5.2.p2.11.m8.7.8.3.1.cmml" xref="S5.2.p2.11.m8.7.8.3.1"></times><apply id="S5.2.p2.11.m8.7.8.3.2.cmml" xref="S5.2.p2.11.m8.7.8.3.2"><csymbol cd="ambiguous" id="S5.2.p2.11.m8.7.8.3.2.1.cmml" xref="S5.2.p2.11.m8.7.8.3.2">subscript</csymbol><ci id="S5.2.p2.11.m8.7.8.3.2.2.cmml" xref="S5.2.p2.11.m8.7.8.3.2.2">𝑋</ci><list id="S5.2.p2.11.m8.2.2.2.3.cmml" xref="S5.2.p2.11.m8.2.2.2.4"><ci id="S5.2.p2.11.m8.1.1.1.1.cmml" xref="S5.2.p2.11.m8.1.1.1.1">𝑝</ci><ci id="S5.2.p2.11.m8.2.2.2.2.cmml" xref="S5.2.p2.11.m8.2.2.2.2">𝑞</ci></list></apply><ci id="S5.2.p2.11.m8.7.7.cmml" xref="S5.2.p2.11.m8.7.7">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.11.m8.7c">(\sigma,\tau,\mu,f)\in X_{p,q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.11.m8.7d">( italic_σ , italic_τ , italic_μ , italic_f ) ∈ italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.3.p3"> <p class="ltx_p" id="S5.3.p3.9">We claim that for a fixed <math alttext="q" class="ltx_Math" display="inline" id="S5.3.p3.1.m1.1"><semantics id="S5.3.p3.1.m1.1a"><mi id="S5.3.p3.1.m1.1.1" xref="S5.3.p3.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.1.m1.1b"><ci id="S5.3.p3.1.m1.1.1.cmml" xref="S5.3.p3.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.1.m1.1d">italic_q</annotation></semantics></math>, the simplicial map <math alttext="\xi_{*,q}(\theta):\mathbb{P}_{*,q}(\theta)\to\mathbb{Q}_{*,q}(\theta)" class="ltx_Math" display="inline" id="S5.3.p3.2.m2.9"><semantics id="S5.3.p3.2.m2.9a"><mrow id="S5.3.p3.2.m2.9.10" xref="S5.3.p3.2.m2.9.10.cmml"><mrow id="S5.3.p3.2.m2.9.10.2" xref="S5.3.p3.2.m2.9.10.2.cmml"><msub id="S5.3.p3.2.m2.9.10.2.2" xref="S5.3.p3.2.m2.9.10.2.2.cmml"><mi id="S5.3.p3.2.m2.9.10.2.2.2" xref="S5.3.p3.2.m2.9.10.2.2.2.cmml">ξ</mi><mrow id="S5.3.p3.2.m2.2.2.2.4" xref="S5.3.p3.2.m2.2.2.2.3.cmml"><mo id="S5.3.p3.2.m2.1.1.1.1" rspace="0em" xref="S5.3.p3.2.m2.1.1.1.1.cmml">∗</mo><mo id="S5.3.p3.2.m2.2.2.2.4.1" xref="S5.3.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S5.3.p3.2.m2.2.2.2.2" xref="S5.3.p3.2.m2.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.3.p3.2.m2.9.10.2.1" xref="S5.3.p3.2.m2.9.10.2.1.cmml">⁢</mo><mrow id="S5.3.p3.2.m2.9.10.2.3.2" xref="S5.3.p3.2.m2.9.10.2.cmml"><mo id="S5.3.p3.2.m2.9.10.2.3.2.1" stretchy="false" xref="S5.3.p3.2.m2.9.10.2.cmml">(</mo><mi id="S5.3.p3.2.m2.7.7" xref="S5.3.p3.2.m2.7.7.cmml">θ</mi><mo id="S5.3.p3.2.m2.9.10.2.3.2.2" rspace="0.278em" stretchy="false" xref="S5.3.p3.2.m2.9.10.2.cmml">)</mo></mrow></mrow><mo id="S5.3.p3.2.m2.9.10.1" rspace="0.278em" xref="S5.3.p3.2.m2.9.10.1.cmml">:</mo><mrow id="S5.3.p3.2.m2.9.10.3" xref="S5.3.p3.2.m2.9.10.3.cmml"><mrow id="S5.3.p3.2.m2.9.10.3.2" xref="S5.3.p3.2.m2.9.10.3.2.cmml"><msub id="S5.3.p3.2.m2.9.10.3.2.2" xref="S5.3.p3.2.m2.9.10.3.2.2.cmml"><mi id="S5.3.p3.2.m2.9.10.3.2.2.2" xref="S5.3.p3.2.m2.9.10.3.2.2.2.cmml">ℙ</mi><mrow id="S5.3.p3.2.m2.4.4.2.4" xref="S5.3.p3.2.m2.4.4.2.3.cmml"><mo id="S5.3.p3.2.m2.3.3.1.1" rspace="0em" xref="S5.3.p3.2.m2.3.3.1.1.cmml">∗</mo><mo id="S5.3.p3.2.m2.4.4.2.4.1" xref="S5.3.p3.2.m2.4.4.2.3.cmml">,</mo><mi id="S5.3.p3.2.m2.4.4.2.2" xref="S5.3.p3.2.m2.4.4.2.2.cmml">q</mi></mrow></msub><mo id="S5.3.p3.2.m2.9.10.3.2.1" xref="S5.3.p3.2.m2.9.10.3.2.1.cmml">⁢</mo><mrow id="S5.3.p3.2.m2.9.10.3.2.3.2" xref="S5.3.p3.2.m2.9.10.3.2.cmml"><mo id="S5.3.p3.2.m2.9.10.3.2.3.2.1" stretchy="false" xref="S5.3.p3.2.m2.9.10.3.2.cmml">(</mo><mi id="S5.3.p3.2.m2.8.8" xref="S5.3.p3.2.m2.8.8.cmml">θ</mi><mo id="S5.3.p3.2.m2.9.10.3.2.3.2.2" stretchy="false" xref="S5.3.p3.2.m2.9.10.3.2.cmml">)</mo></mrow></mrow><mo id="S5.3.p3.2.m2.9.10.3.1" stretchy="false" xref="S5.3.p3.2.m2.9.10.3.1.cmml">→</mo><mrow id="S5.3.p3.2.m2.9.10.3.3" xref="S5.3.p3.2.m2.9.10.3.3.cmml"><msub id="S5.3.p3.2.m2.9.10.3.3.2" xref="S5.3.p3.2.m2.9.10.3.3.2.cmml"><mi id="S5.3.p3.2.m2.9.10.3.3.2.2" xref="S5.3.p3.2.m2.9.10.3.3.2.2.cmml">ℚ</mi><mrow id="S5.3.p3.2.m2.6.6.2.4" xref="S5.3.p3.2.m2.6.6.2.3.cmml"><mo id="S5.3.p3.2.m2.5.5.1.1" rspace="0em" xref="S5.3.p3.2.m2.5.5.1.1.cmml">∗</mo><mo id="S5.3.p3.2.m2.6.6.2.4.1" xref="S5.3.p3.2.m2.6.6.2.3.cmml">,</mo><mi id="S5.3.p3.2.m2.6.6.2.2" xref="S5.3.p3.2.m2.6.6.2.2.cmml">q</mi></mrow></msub><mo id="S5.3.p3.2.m2.9.10.3.3.1" xref="S5.3.p3.2.m2.9.10.3.3.1.cmml">⁢</mo><mrow id="S5.3.p3.2.m2.9.10.3.3.3.2" xref="S5.3.p3.2.m2.9.10.3.3.cmml"><mo id="S5.3.p3.2.m2.9.10.3.3.3.2.1" stretchy="false" xref="S5.3.p3.2.m2.9.10.3.3.cmml">(</mo><mi id="S5.3.p3.2.m2.9.9" xref="S5.3.p3.2.m2.9.9.cmml">θ</mi><mo id="S5.3.p3.2.m2.9.10.3.3.3.2.2" stretchy="false" xref="S5.3.p3.2.m2.9.10.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.2.m2.9b"><apply id="S5.3.p3.2.m2.9.10.cmml" xref="S5.3.p3.2.m2.9.10"><ci id="S5.3.p3.2.m2.9.10.1.cmml" xref="S5.3.p3.2.m2.9.10.1">:</ci><apply id="S5.3.p3.2.m2.9.10.2.cmml" xref="S5.3.p3.2.m2.9.10.2"><times id="S5.3.p3.2.m2.9.10.2.1.cmml" xref="S5.3.p3.2.m2.9.10.2.1"></times><apply id="S5.3.p3.2.m2.9.10.2.2.cmml" xref="S5.3.p3.2.m2.9.10.2.2"><csymbol cd="ambiguous" id="S5.3.p3.2.m2.9.10.2.2.1.cmml" xref="S5.3.p3.2.m2.9.10.2.2">subscript</csymbol><ci id="S5.3.p3.2.m2.9.10.2.2.2.cmml" xref="S5.3.p3.2.m2.9.10.2.2.2">𝜉</ci><list id="S5.3.p3.2.m2.2.2.2.3.cmml" xref="S5.3.p3.2.m2.2.2.2.4"><times id="S5.3.p3.2.m2.1.1.1.1.cmml" xref="S5.3.p3.2.m2.1.1.1.1"></times><ci id="S5.3.p3.2.m2.2.2.2.2.cmml" xref="S5.3.p3.2.m2.2.2.2.2">𝑞</ci></list></apply><ci id="S5.3.p3.2.m2.7.7.cmml" xref="S5.3.p3.2.m2.7.7">𝜃</ci></apply><apply id="S5.3.p3.2.m2.9.10.3.cmml" xref="S5.3.p3.2.m2.9.10.3"><ci id="S5.3.p3.2.m2.9.10.3.1.cmml" xref="S5.3.p3.2.m2.9.10.3.1">→</ci><apply id="S5.3.p3.2.m2.9.10.3.2.cmml" xref="S5.3.p3.2.m2.9.10.3.2"><times id="S5.3.p3.2.m2.9.10.3.2.1.cmml" xref="S5.3.p3.2.m2.9.10.3.2.1"></times><apply id="S5.3.p3.2.m2.9.10.3.2.2.cmml" xref="S5.3.p3.2.m2.9.10.3.2.2"><csymbol cd="ambiguous" id="S5.3.p3.2.m2.9.10.3.2.2.1.cmml" xref="S5.3.p3.2.m2.9.10.3.2.2">subscript</csymbol><ci id="S5.3.p3.2.m2.9.10.3.2.2.2.cmml" xref="S5.3.p3.2.m2.9.10.3.2.2.2">ℙ</ci><list id="S5.3.p3.2.m2.4.4.2.3.cmml" xref="S5.3.p3.2.m2.4.4.2.4"><times id="S5.3.p3.2.m2.3.3.1.1.cmml" xref="S5.3.p3.2.m2.3.3.1.1"></times><ci id="S5.3.p3.2.m2.4.4.2.2.cmml" xref="S5.3.p3.2.m2.4.4.2.2">𝑞</ci></list></apply><ci id="S5.3.p3.2.m2.8.8.cmml" xref="S5.3.p3.2.m2.8.8">𝜃</ci></apply><apply id="S5.3.p3.2.m2.9.10.3.3.cmml" xref="S5.3.p3.2.m2.9.10.3.3"><times id="S5.3.p3.2.m2.9.10.3.3.1.cmml" xref="S5.3.p3.2.m2.9.10.3.3.1"></times><apply id="S5.3.p3.2.m2.9.10.3.3.2.cmml" xref="S5.3.p3.2.m2.9.10.3.3.2"><csymbol cd="ambiguous" id="S5.3.p3.2.m2.9.10.3.3.2.1.cmml" xref="S5.3.p3.2.m2.9.10.3.3.2">subscript</csymbol><ci id="S5.3.p3.2.m2.9.10.3.3.2.2.cmml" xref="S5.3.p3.2.m2.9.10.3.3.2.2">ℚ</ci><list id="S5.3.p3.2.m2.6.6.2.3.cmml" xref="S5.3.p3.2.m2.6.6.2.4"><times id="S5.3.p3.2.m2.5.5.1.1.cmml" xref="S5.3.p3.2.m2.5.5.1.1"></times><ci id="S5.3.p3.2.m2.6.6.2.2.cmml" xref="S5.3.p3.2.m2.6.6.2.2">𝑞</ci></list></apply><ci id="S5.3.p3.2.m2.9.9.cmml" xref="S5.3.p3.2.m2.9.9">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.2.m2.9c">\xi_{*,q}(\theta):\mathbb{P}_{*,q}(\theta)\to\mathbb{Q}_{*,q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.2.m2.9d">italic_ξ start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ ) : blackboard_P start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ ) → blackboard_Q start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> induces a homology isomorphism for every <math alttext="\theta\in\Delta(X)" class="ltx_Math" display="inline" id="S5.3.p3.3.m3.1"><semantics id="S5.3.p3.3.m3.1a"><mrow id="S5.3.p3.3.m3.1.2" xref="S5.3.p3.3.m3.1.2.cmml"><mi id="S5.3.p3.3.m3.1.2.2" xref="S5.3.p3.3.m3.1.2.2.cmml">θ</mi><mo id="S5.3.p3.3.m3.1.2.1" xref="S5.3.p3.3.m3.1.2.1.cmml">∈</mo><mrow id="S5.3.p3.3.m3.1.2.3" xref="S5.3.p3.3.m3.1.2.3.cmml"><mi id="S5.3.p3.3.m3.1.2.3.2" mathvariant="normal" xref="S5.3.p3.3.m3.1.2.3.2.cmml">Δ</mi><mo id="S5.3.p3.3.m3.1.2.3.1" xref="S5.3.p3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.3.p3.3.m3.1.2.3.3.2" xref="S5.3.p3.3.m3.1.2.3.cmml"><mo id="S5.3.p3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.3.p3.3.m3.1.2.3.cmml">(</mo><mi id="S5.3.p3.3.m3.1.1" xref="S5.3.p3.3.m3.1.1.cmml">X</mi><mo id="S5.3.p3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.3.p3.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.3.m3.1b"><apply id="S5.3.p3.3.m3.1.2.cmml" xref="S5.3.p3.3.m3.1.2"><in id="S5.3.p3.3.m3.1.2.1.cmml" xref="S5.3.p3.3.m3.1.2.1"></in><ci id="S5.3.p3.3.m3.1.2.2.cmml" xref="S5.3.p3.3.m3.1.2.2">𝜃</ci><apply id="S5.3.p3.3.m3.1.2.3.cmml" xref="S5.3.p3.3.m3.1.2.3"><times id="S5.3.p3.3.m3.1.2.3.1.cmml" xref="S5.3.p3.3.m3.1.2.3.1"></times><ci id="S5.3.p3.3.m3.1.2.3.2.cmml" xref="S5.3.p3.3.m3.1.2.3.2">Δ</ci><ci id="S5.3.p3.3.m3.1.1.cmml" xref="S5.3.p3.3.m3.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.3.m3.1c">\theta\in\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.3.m3.1d">italic_θ ∈ roman_Δ ( italic_X )</annotation></semantics></math>. Since the simplicial <math alttext="R" class="ltx_Math" display="inline" id="S5.3.p3.4.m4.1"><semantics id="S5.3.p3.4.m4.1a"><mi id="S5.3.p3.4.m4.1.1" xref="S5.3.p3.4.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.4.m4.1b"><ci id="S5.3.p3.4.m4.1.1.cmml" xref="S5.3.p3.4.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.4.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.4.m4.1d">italic_R</annotation></semantics></math>-module <math alttext="\mathbb{Q}_{*,q}(\theta)" class="ltx_Math" display="inline" id="S5.3.p3.5.m5.3"><semantics id="S5.3.p3.5.m5.3a"><mrow id="S5.3.p3.5.m5.3.4" xref="S5.3.p3.5.m5.3.4.cmml"><msub id="S5.3.p3.5.m5.3.4.2" xref="S5.3.p3.5.m5.3.4.2.cmml"><mi id="S5.3.p3.5.m5.3.4.2.2" xref="S5.3.p3.5.m5.3.4.2.2.cmml">ℚ</mi><mrow id="S5.3.p3.5.m5.2.2.2.4" xref="S5.3.p3.5.m5.2.2.2.3.cmml"><mo id="S5.3.p3.5.m5.1.1.1.1" rspace="0em" xref="S5.3.p3.5.m5.1.1.1.1.cmml">∗</mo><mo id="S5.3.p3.5.m5.2.2.2.4.1" xref="S5.3.p3.5.m5.2.2.2.3.cmml">,</mo><mi id="S5.3.p3.5.m5.2.2.2.2" xref="S5.3.p3.5.m5.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.3.p3.5.m5.3.4.1" xref="S5.3.p3.5.m5.3.4.1.cmml">⁢</mo><mrow id="S5.3.p3.5.m5.3.4.3.2" xref="S5.3.p3.5.m5.3.4.cmml"><mo id="S5.3.p3.5.m5.3.4.3.2.1" stretchy="false" xref="S5.3.p3.5.m5.3.4.cmml">(</mo><mi id="S5.3.p3.5.m5.3.3" xref="S5.3.p3.5.m5.3.3.cmml">θ</mi><mo id="S5.3.p3.5.m5.3.4.3.2.2" stretchy="false" xref="S5.3.p3.5.m5.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.5.m5.3b"><apply id="S5.3.p3.5.m5.3.4.cmml" xref="S5.3.p3.5.m5.3.4"><times id="S5.3.p3.5.m5.3.4.1.cmml" xref="S5.3.p3.5.m5.3.4.1"></times><apply id="S5.3.p3.5.m5.3.4.2.cmml" xref="S5.3.p3.5.m5.3.4.2"><csymbol cd="ambiguous" id="S5.3.p3.5.m5.3.4.2.1.cmml" xref="S5.3.p3.5.m5.3.4.2">subscript</csymbol><ci id="S5.3.p3.5.m5.3.4.2.2.cmml" xref="S5.3.p3.5.m5.3.4.2.2">ℚ</ci><list id="S5.3.p3.5.m5.2.2.2.3.cmml" xref="S5.3.p3.5.m5.2.2.2.4"><times id="S5.3.p3.5.m5.1.1.1.1.cmml" xref="S5.3.p3.5.m5.1.1.1.1"></times><ci id="S5.3.p3.5.m5.2.2.2.2.cmml" xref="S5.3.p3.5.m5.2.2.2.2">𝑞</ci></list></apply><ci id="S5.3.p3.5.m5.3.3.cmml" xref="S5.3.p3.5.m5.3.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.5.m5.3c">\mathbb{Q}_{*,q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.5.m5.3d">blackboard_Q start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> is the direct sum of simplicial points indexed by the pairs <math alttext="(\tau,f)" class="ltx_Math" display="inline" id="S5.3.p3.6.m6.2"><semantics id="S5.3.p3.6.m6.2a"><mrow id="S5.3.p3.6.m6.2.3.2" xref="S5.3.p3.6.m6.2.3.1.cmml"><mo id="S5.3.p3.6.m6.2.3.2.1" stretchy="false" xref="S5.3.p3.6.m6.2.3.1.cmml">(</mo><mi id="S5.3.p3.6.m6.1.1" xref="S5.3.p3.6.m6.1.1.cmml">τ</mi><mo id="S5.3.p3.6.m6.2.3.2.2" xref="S5.3.p3.6.m6.2.3.1.cmml">,</mo><mi id="S5.3.p3.6.m6.2.2" xref="S5.3.p3.6.m6.2.2.cmml">f</mi><mo id="S5.3.p3.6.m6.2.3.2.3" stretchy="false" xref="S5.3.p3.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.6.m6.2b"><interval closure="open" id="S5.3.p3.6.m6.2.3.1.cmml" xref="S5.3.p3.6.m6.2.3.2"><ci id="S5.3.p3.6.m6.1.1.cmml" xref="S5.3.p3.6.m6.1.1">𝜏</ci><ci id="S5.3.p3.6.m6.2.2.cmml" xref="S5.3.p3.6.m6.2.2">𝑓</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.6.m6.2c">(\tau,f)</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.6.m6.2d">( italic_τ , italic_f )</annotation></semantics></math>, it is enough to fix a pair <math alttext="(\tau,f)" class="ltx_Math" display="inline" id="S5.3.p3.7.m7.2"><semantics id="S5.3.p3.7.m7.2a"><mrow id="S5.3.p3.7.m7.2.3.2" xref="S5.3.p3.7.m7.2.3.1.cmml"><mo id="S5.3.p3.7.m7.2.3.2.1" stretchy="false" xref="S5.3.p3.7.m7.2.3.1.cmml">(</mo><mi id="S5.3.p3.7.m7.1.1" xref="S5.3.p3.7.m7.1.1.cmml">τ</mi><mo id="S5.3.p3.7.m7.2.3.2.2" xref="S5.3.p3.7.m7.2.3.1.cmml">,</mo><mi id="S5.3.p3.7.m7.2.2" xref="S5.3.p3.7.m7.2.2.cmml">f</mi><mo id="S5.3.p3.7.m7.2.3.2.3" stretchy="false" xref="S5.3.p3.7.m7.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.7.m7.2b"><interval closure="open" id="S5.3.p3.7.m7.2.3.1.cmml" xref="S5.3.p3.7.m7.2.3.2"><ci id="S5.3.p3.7.m7.1.1.cmml" xref="S5.3.p3.7.m7.1.1">𝜏</ci><ci id="S5.3.p3.7.m7.2.2.cmml" xref="S5.3.p3.7.m7.2.2">𝑓</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.7.m7.2c">(\tau,f)</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.7.m7.2d">( italic_τ , italic_f )</annotation></semantics></math>, and show that the simplicial <math alttext="R" class="ltx_Math" display="inline" id="S5.3.p3.8.m8.1"><semantics id="S5.3.p3.8.m8.1a"><mi id="S5.3.p3.8.m8.1.1" xref="S5.3.p3.8.m8.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.8.m8.1b"><ci id="S5.3.p3.8.m8.1.1.cmml" xref="S5.3.p3.8.m8.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.8.m8.1c">R</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.8.m8.1d">italic_R</annotation></semantics></math>-module <math alttext="\xi_{*,q}(\theta)^{-1}(\tau,f)" class="ltx_Math" display="inline" id="S5.3.p3.9.m9.5"><semantics id="S5.3.p3.9.m9.5a"><mrow id="S5.3.p3.9.m9.5.6" xref="S5.3.p3.9.m9.5.6.cmml"><msub id="S5.3.p3.9.m9.5.6.2" xref="S5.3.p3.9.m9.5.6.2.cmml"><mi id="S5.3.p3.9.m9.5.6.2.2" xref="S5.3.p3.9.m9.5.6.2.2.cmml">ξ</mi><mrow id="S5.3.p3.9.m9.2.2.2.4" xref="S5.3.p3.9.m9.2.2.2.3.cmml"><mo id="S5.3.p3.9.m9.1.1.1.1" rspace="0em" xref="S5.3.p3.9.m9.1.1.1.1.cmml">∗</mo><mo id="S5.3.p3.9.m9.2.2.2.4.1" xref="S5.3.p3.9.m9.2.2.2.3.cmml">,</mo><mi id="S5.3.p3.9.m9.2.2.2.2" xref="S5.3.p3.9.m9.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.3.p3.9.m9.5.6.1" xref="S5.3.p3.9.m9.5.6.1.cmml">⁢</mo><msup id="S5.3.p3.9.m9.5.6.3" xref="S5.3.p3.9.m9.5.6.3.cmml"><mrow id="S5.3.p3.9.m9.5.6.3.2.2" xref="S5.3.p3.9.m9.5.6.3.cmml"><mo id="S5.3.p3.9.m9.5.6.3.2.2.1" stretchy="false" xref="S5.3.p3.9.m9.5.6.3.cmml">(</mo><mi id="S5.3.p3.9.m9.3.3" xref="S5.3.p3.9.m9.3.3.cmml">θ</mi><mo id="S5.3.p3.9.m9.5.6.3.2.2.2" stretchy="false" xref="S5.3.p3.9.m9.5.6.3.cmml">)</mo></mrow><mrow id="S5.3.p3.9.m9.5.6.3.3" xref="S5.3.p3.9.m9.5.6.3.3.cmml"><mo id="S5.3.p3.9.m9.5.6.3.3a" xref="S5.3.p3.9.m9.5.6.3.3.cmml">−</mo><mn id="S5.3.p3.9.m9.5.6.3.3.2" xref="S5.3.p3.9.m9.5.6.3.3.2.cmml">1</mn></mrow></msup><mo id="S5.3.p3.9.m9.5.6.1a" xref="S5.3.p3.9.m9.5.6.1.cmml">⁢</mo><mrow id="S5.3.p3.9.m9.5.6.4.2" xref="S5.3.p3.9.m9.5.6.4.1.cmml"><mo id="S5.3.p3.9.m9.5.6.4.2.1" stretchy="false" xref="S5.3.p3.9.m9.5.6.4.1.cmml">(</mo><mi id="S5.3.p3.9.m9.4.4" xref="S5.3.p3.9.m9.4.4.cmml">τ</mi><mo id="S5.3.p3.9.m9.5.6.4.2.2" xref="S5.3.p3.9.m9.5.6.4.1.cmml">,</mo><mi id="S5.3.p3.9.m9.5.5" xref="S5.3.p3.9.m9.5.5.cmml">f</mi><mo id="S5.3.p3.9.m9.5.6.4.2.3" stretchy="false" xref="S5.3.p3.9.m9.5.6.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.9.m9.5b"><apply id="S5.3.p3.9.m9.5.6.cmml" xref="S5.3.p3.9.m9.5.6"><times id="S5.3.p3.9.m9.5.6.1.cmml" xref="S5.3.p3.9.m9.5.6.1"></times><apply id="S5.3.p3.9.m9.5.6.2.cmml" xref="S5.3.p3.9.m9.5.6.2"><csymbol cd="ambiguous" id="S5.3.p3.9.m9.5.6.2.1.cmml" xref="S5.3.p3.9.m9.5.6.2">subscript</csymbol><ci id="S5.3.p3.9.m9.5.6.2.2.cmml" xref="S5.3.p3.9.m9.5.6.2.2">𝜉</ci><list id="S5.3.p3.9.m9.2.2.2.3.cmml" xref="S5.3.p3.9.m9.2.2.2.4"><times id="S5.3.p3.9.m9.1.1.1.1.cmml" xref="S5.3.p3.9.m9.1.1.1.1"></times><ci id="S5.3.p3.9.m9.2.2.2.2.cmml" xref="S5.3.p3.9.m9.2.2.2.2">𝑞</ci></list></apply><apply id="S5.3.p3.9.m9.5.6.3.cmml" xref="S5.3.p3.9.m9.5.6.3"><csymbol cd="ambiguous" id="S5.3.p3.9.m9.5.6.3.1.cmml" xref="S5.3.p3.9.m9.5.6.3">superscript</csymbol><ci id="S5.3.p3.9.m9.3.3.cmml" xref="S5.3.p3.9.m9.3.3">𝜃</ci><apply id="S5.3.p3.9.m9.5.6.3.3.cmml" xref="S5.3.p3.9.m9.5.6.3.3"><minus id="S5.3.p3.9.m9.5.6.3.3.1.cmml" xref="S5.3.p3.9.m9.5.6.3.3"></minus><cn id="S5.3.p3.9.m9.5.6.3.3.2.cmml" type="integer" xref="S5.3.p3.9.m9.5.6.3.3.2">1</cn></apply></apply><interval closure="open" id="S5.3.p3.9.m9.5.6.4.1.cmml" xref="S5.3.p3.9.m9.5.6.4.2"><ci id="S5.3.p3.9.m9.4.4.cmml" xref="S5.3.p3.9.m9.4.4">𝜏</ci><ci id="S5.3.p3.9.m9.5.5.cmml" xref="S5.3.p3.9.m9.5.5">𝑓</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.9.m9.5c">\xi_{*,q}(\theta)^{-1}(\tau,f)</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.9.m9.5d">italic_ξ start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_τ , italic_f )</annotation></semantics></math> has the homology of a point.</p> </div> <div class="ltx_para" id="S5.4.p4"> <p class="ltx_p" id="S5.4.p4.5">The simplicial <math alttext="R" class="ltx_Math" display="inline" id="S5.4.p4.1.m1.1"><semantics id="S5.4.p4.1.m1.1a"><mi id="S5.4.p4.1.m1.1.1" xref="S5.4.p4.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.1.m1.1b"><ci id="S5.4.p4.1.m1.1.1.cmml" xref="S5.4.p4.1.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.1.m1.1d">italic_R</annotation></semantics></math>-module <math alttext="\xi_{*,q}(\theta)^{-1}(\tau,f)" class="ltx_Math" display="inline" id="S5.4.p4.2.m2.5"><semantics id="S5.4.p4.2.m2.5a"><mrow id="S5.4.p4.2.m2.5.6" xref="S5.4.p4.2.m2.5.6.cmml"><msub id="S5.4.p4.2.m2.5.6.2" xref="S5.4.p4.2.m2.5.6.2.cmml"><mi id="S5.4.p4.2.m2.5.6.2.2" xref="S5.4.p4.2.m2.5.6.2.2.cmml">ξ</mi><mrow id="S5.4.p4.2.m2.2.2.2.4" xref="S5.4.p4.2.m2.2.2.2.3.cmml"><mo id="S5.4.p4.2.m2.1.1.1.1" rspace="0em" xref="S5.4.p4.2.m2.1.1.1.1.cmml">∗</mo><mo id="S5.4.p4.2.m2.2.2.2.4.1" xref="S5.4.p4.2.m2.2.2.2.3.cmml">,</mo><mi id="S5.4.p4.2.m2.2.2.2.2" xref="S5.4.p4.2.m2.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.4.p4.2.m2.5.6.1" xref="S5.4.p4.2.m2.5.6.1.cmml">⁢</mo><msup id="S5.4.p4.2.m2.5.6.3" xref="S5.4.p4.2.m2.5.6.3.cmml"><mrow id="S5.4.p4.2.m2.5.6.3.2.2" xref="S5.4.p4.2.m2.5.6.3.cmml"><mo id="S5.4.p4.2.m2.5.6.3.2.2.1" stretchy="false" xref="S5.4.p4.2.m2.5.6.3.cmml">(</mo><mi id="S5.4.p4.2.m2.3.3" xref="S5.4.p4.2.m2.3.3.cmml">θ</mi><mo id="S5.4.p4.2.m2.5.6.3.2.2.2" stretchy="false" xref="S5.4.p4.2.m2.5.6.3.cmml">)</mo></mrow><mrow id="S5.4.p4.2.m2.5.6.3.3" xref="S5.4.p4.2.m2.5.6.3.3.cmml"><mo id="S5.4.p4.2.m2.5.6.3.3a" xref="S5.4.p4.2.m2.5.6.3.3.cmml">−</mo><mn id="S5.4.p4.2.m2.5.6.3.3.2" xref="S5.4.p4.2.m2.5.6.3.3.2.cmml">1</mn></mrow></msup><mo id="S5.4.p4.2.m2.5.6.1a" xref="S5.4.p4.2.m2.5.6.1.cmml">⁢</mo><mrow id="S5.4.p4.2.m2.5.6.4.2" xref="S5.4.p4.2.m2.5.6.4.1.cmml"><mo id="S5.4.p4.2.m2.5.6.4.2.1" stretchy="false" xref="S5.4.p4.2.m2.5.6.4.1.cmml">(</mo><mi id="S5.4.p4.2.m2.4.4" xref="S5.4.p4.2.m2.4.4.cmml">τ</mi><mo id="S5.4.p4.2.m2.5.6.4.2.2" xref="S5.4.p4.2.m2.5.6.4.1.cmml">,</mo><mi id="S5.4.p4.2.m2.5.5" xref="S5.4.p4.2.m2.5.5.cmml">f</mi><mo id="S5.4.p4.2.m2.5.6.4.2.3" stretchy="false" xref="S5.4.p4.2.m2.5.6.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.2.m2.5b"><apply id="S5.4.p4.2.m2.5.6.cmml" xref="S5.4.p4.2.m2.5.6"><times id="S5.4.p4.2.m2.5.6.1.cmml" xref="S5.4.p4.2.m2.5.6.1"></times><apply id="S5.4.p4.2.m2.5.6.2.cmml" xref="S5.4.p4.2.m2.5.6.2"><csymbol cd="ambiguous" id="S5.4.p4.2.m2.5.6.2.1.cmml" xref="S5.4.p4.2.m2.5.6.2">subscript</csymbol><ci id="S5.4.p4.2.m2.5.6.2.2.cmml" xref="S5.4.p4.2.m2.5.6.2.2">𝜉</ci><list id="S5.4.p4.2.m2.2.2.2.3.cmml" xref="S5.4.p4.2.m2.2.2.2.4"><times id="S5.4.p4.2.m2.1.1.1.1.cmml" xref="S5.4.p4.2.m2.1.1.1.1"></times><ci id="S5.4.p4.2.m2.2.2.2.2.cmml" xref="S5.4.p4.2.m2.2.2.2.2">𝑞</ci></list></apply><apply id="S5.4.p4.2.m2.5.6.3.cmml" xref="S5.4.p4.2.m2.5.6.3"><csymbol cd="ambiguous" id="S5.4.p4.2.m2.5.6.3.1.cmml" xref="S5.4.p4.2.m2.5.6.3">superscript</csymbol><ci id="S5.4.p4.2.m2.3.3.cmml" xref="S5.4.p4.2.m2.3.3">𝜃</ci><apply id="S5.4.p4.2.m2.5.6.3.3.cmml" xref="S5.4.p4.2.m2.5.6.3.3"><minus id="S5.4.p4.2.m2.5.6.3.3.1.cmml" xref="S5.4.p4.2.m2.5.6.3.3"></minus><cn id="S5.4.p4.2.m2.5.6.3.3.2.cmml" type="integer" xref="S5.4.p4.2.m2.5.6.3.3.2">1</cn></apply></apply><interval closure="open" id="S5.4.p4.2.m2.5.6.4.1.cmml" xref="S5.4.p4.2.m2.5.6.4.2"><ci id="S5.4.p4.2.m2.4.4.cmml" xref="S5.4.p4.2.m2.4.4">𝜏</ci><ci id="S5.4.p4.2.m2.5.5.cmml" xref="S5.4.p4.2.m2.5.5">𝑓</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.2.m2.5c">\xi_{*,q}(\theta)^{-1}(\tau,f)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.2.m2.5d">italic_ξ start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_τ , italic_f )</annotation></semantics></math> is isomorphic to the linearization of the simplicial set that is isomorphic to the nerve of the comma category <math alttext="\mathcal{D}/\varphi(c_{0})" class="ltx_Math" display="inline" id="S5.4.p4.3.m3.1"><semantics id="S5.4.p4.3.m3.1a"><mrow id="S5.4.p4.3.m3.1.1" xref="S5.4.p4.3.m3.1.1.cmml"><mrow id="S5.4.p4.3.m3.1.1.3" xref="S5.4.p4.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p4.3.m3.1.1.3.2" xref="S5.4.p4.3.m3.1.1.3.2.cmml">𝒟</mi><mo id="S5.4.p4.3.m3.1.1.3.1" xref="S5.4.p4.3.m3.1.1.3.1.cmml">/</mo><mi id="S5.4.p4.3.m3.1.1.3.3" xref="S5.4.p4.3.m3.1.1.3.3.cmml">φ</mi></mrow><mo id="S5.4.p4.3.m3.1.1.2" xref="S5.4.p4.3.m3.1.1.2.cmml">⁢</mo><mrow id="S5.4.p4.3.m3.1.1.1.1" xref="S5.4.p4.3.m3.1.1.1.1.1.cmml"><mo id="S5.4.p4.3.m3.1.1.1.1.2" stretchy="false" xref="S5.4.p4.3.m3.1.1.1.1.1.cmml">(</mo><msub id="S5.4.p4.3.m3.1.1.1.1.1" xref="S5.4.p4.3.m3.1.1.1.1.1.cmml"><mi id="S5.4.p4.3.m3.1.1.1.1.1.2" xref="S5.4.p4.3.m3.1.1.1.1.1.2.cmml">c</mi><mn id="S5.4.p4.3.m3.1.1.1.1.1.3" xref="S5.4.p4.3.m3.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S5.4.p4.3.m3.1.1.1.1.3" stretchy="false" xref="S5.4.p4.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.3.m3.1b"><apply id="S5.4.p4.3.m3.1.1.cmml" xref="S5.4.p4.3.m3.1.1"><times id="S5.4.p4.3.m3.1.1.2.cmml" xref="S5.4.p4.3.m3.1.1.2"></times><apply id="S5.4.p4.3.m3.1.1.3.cmml" xref="S5.4.p4.3.m3.1.1.3"><divide id="S5.4.p4.3.m3.1.1.3.1.cmml" xref="S5.4.p4.3.m3.1.1.3.1"></divide><ci id="S5.4.p4.3.m3.1.1.3.2.cmml" xref="S5.4.p4.3.m3.1.1.3.2">𝒟</ci><ci id="S5.4.p4.3.m3.1.1.3.3.cmml" xref="S5.4.p4.3.m3.1.1.3.3">𝜑</ci></apply><apply id="S5.4.p4.3.m3.1.1.1.1.1.cmml" xref="S5.4.p4.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S5.4.p4.3.m3.1.1.1.1.1.1.cmml" xref="S5.4.p4.3.m3.1.1.1.1">subscript</csymbol><ci id="S5.4.p4.3.m3.1.1.1.1.1.2.cmml" xref="S5.4.p4.3.m3.1.1.1.1.1.2">𝑐</ci><cn id="S5.4.p4.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S5.4.p4.3.m3.1.1.1.1.1.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.3.m3.1c">\mathcal{D}/\varphi(c_{0})</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.3.m3.1d">caligraphic_D / italic_φ ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>. Since this category has a terminal object, the homology of its nerve has the homology of a point. This proves that for every <math alttext="q\geq 0" class="ltx_Math" display="inline" id="S5.4.p4.4.m4.1"><semantics id="S5.4.p4.4.m4.1a"><mrow id="S5.4.p4.4.m4.1.1" xref="S5.4.p4.4.m4.1.1.cmml"><mi id="S5.4.p4.4.m4.1.1.2" xref="S5.4.p4.4.m4.1.1.2.cmml">q</mi><mo id="S5.4.p4.4.m4.1.1.1" xref="S5.4.p4.4.m4.1.1.1.cmml">≥</mo><mn id="S5.4.p4.4.m4.1.1.3" xref="S5.4.p4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.4.m4.1b"><apply id="S5.4.p4.4.m4.1.1.cmml" xref="S5.4.p4.4.m4.1.1"><geq id="S5.4.p4.4.m4.1.1.1.cmml" xref="S5.4.p4.4.m4.1.1.1"></geq><ci id="S5.4.p4.4.m4.1.1.2.cmml" xref="S5.4.p4.4.m4.1.1.2">𝑞</ci><cn id="S5.4.p4.4.m4.1.1.3.cmml" type="integer" xref="S5.4.p4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.4.m4.1c">q\geq 0</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.4.m4.1d">italic_q ≥ 0</annotation></semantics></math>, the simplicial map <math alttext="\xi_{*,q}(\theta):\mathbb{P}_{*,q}(\theta)\to\mathbb{Q}_{*,q}(\theta)" class="ltx_Math" display="inline" id="S5.4.p4.5.m5.9"><semantics id="S5.4.p4.5.m5.9a"><mrow id="S5.4.p4.5.m5.9.10" xref="S5.4.p4.5.m5.9.10.cmml"><mrow id="S5.4.p4.5.m5.9.10.2" xref="S5.4.p4.5.m5.9.10.2.cmml"><msub id="S5.4.p4.5.m5.9.10.2.2" xref="S5.4.p4.5.m5.9.10.2.2.cmml"><mi id="S5.4.p4.5.m5.9.10.2.2.2" xref="S5.4.p4.5.m5.9.10.2.2.2.cmml">ξ</mi><mrow id="S5.4.p4.5.m5.2.2.2.4" xref="S5.4.p4.5.m5.2.2.2.3.cmml"><mo id="S5.4.p4.5.m5.1.1.1.1" rspace="0em" xref="S5.4.p4.5.m5.1.1.1.1.cmml">∗</mo><mo id="S5.4.p4.5.m5.2.2.2.4.1" xref="S5.4.p4.5.m5.2.2.2.3.cmml">,</mo><mi id="S5.4.p4.5.m5.2.2.2.2" xref="S5.4.p4.5.m5.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.4.p4.5.m5.9.10.2.1" xref="S5.4.p4.5.m5.9.10.2.1.cmml">⁢</mo><mrow id="S5.4.p4.5.m5.9.10.2.3.2" xref="S5.4.p4.5.m5.9.10.2.cmml"><mo id="S5.4.p4.5.m5.9.10.2.3.2.1" stretchy="false" xref="S5.4.p4.5.m5.9.10.2.cmml">(</mo><mi id="S5.4.p4.5.m5.7.7" xref="S5.4.p4.5.m5.7.7.cmml">θ</mi><mo id="S5.4.p4.5.m5.9.10.2.3.2.2" rspace="0.278em" stretchy="false" xref="S5.4.p4.5.m5.9.10.2.cmml">)</mo></mrow></mrow><mo id="S5.4.p4.5.m5.9.10.1" rspace="0.278em" xref="S5.4.p4.5.m5.9.10.1.cmml">:</mo><mrow id="S5.4.p4.5.m5.9.10.3" xref="S5.4.p4.5.m5.9.10.3.cmml"><mrow id="S5.4.p4.5.m5.9.10.3.2" xref="S5.4.p4.5.m5.9.10.3.2.cmml"><msub id="S5.4.p4.5.m5.9.10.3.2.2" xref="S5.4.p4.5.m5.9.10.3.2.2.cmml"><mi id="S5.4.p4.5.m5.9.10.3.2.2.2" xref="S5.4.p4.5.m5.9.10.3.2.2.2.cmml">ℙ</mi><mrow id="S5.4.p4.5.m5.4.4.2.4" xref="S5.4.p4.5.m5.4.4.2.3.cmml"><mo id="S5.4.p4.5.m5.3.3.1.1" rspace="0em" xref="S5.4.p4.5.m5.3.3.1.1.cmml">∗</mo><mo id="S5.4.p4.5.m5.4.4.2.4.1" xref="S5.4.p4.5.m5.4.4.2.3.cmml">,</mo><mi id="S5.4.p4.5.m5.4.4.2.2" xref="S5.4.p4.5.m5.4.4.2.2.cmml">q</mi></mrow></msub><mo id="S5.4.p4.5.m5.9.10.3.2.1" xref="S5.4.p4.5.m5.9.10.3.2.1.cmml">⁢</mo><mrow id="S5.4.p4.5.m5.9.10.3.2.3.2" xref="S5.4.p4.5.m5.9.10.3.2.cmml"><mo id="S5.4.p4.5.m5.9.10.3.2.3.2.1" stretchy="false" xref="S5.4.p4.5.m5.9.10.3.2.cmml">(</mo><mi id="S5.4.p4.5.m5.8.8" xref="S5.4.p4.5.m5.8.8.cmml">θ</mi><mo id="S5.4.p4.5.m5.9.10.3.2.3.2.2" stretchy="false" xref="S5.4.p4.5.m5.9.10.3.2.cmml">)</mo></mrow></mrow><mo id="S5.4.p4.5.m5.9.10.3.1" stretchy="false" xref="S5.4.p4.5.m5.9.10.3.1.cmml">→</mo><mrow id="S5.4.p4.5.m5.9.10.3.3" xref="S5.4.p4.5.m5.9.10.3.3.cmml"><msub id="S5.4.p4.5.m5.9.10.3.3.2" xref="S5.4.p4.5.m5.9.10.3.3.2.cmml"><mi id="S5.4.p4.5.m5.9.10.3.3.2.2" xref="S5.4.p4.5.m5.9.10.3.3.2.2.cmml">ℚ</mi><mrow id="S5.4.p4.5.m5.6.6.2.4" xref="S5.4.p4.5.m5.6.6.2.3.cmml"><mo id="S5.4.p4.5.m5.5.5.1.1" rspace="0em" xref="S5.4.p4.5.m5.5.5.1.1.cmml">∗</mo><mo id="S5.4.p4.5.m5.6.6.2.4.1" xref="S5.4.p4.5.m5.6.6.2.3.cmml">,</mo><mi id="S5.4.p4.5.m5.6.6.2.2" xref="S5.4.p4.5.m5.6.6.2.2.cmml">q</mi></mrow></msub><mo id="S5.4.p4.5.m5.9.10.3.3.1" xref="S5.4.p4.5.m5.9.10.3.3.1.cmml">⁢</mo><mrow id="S5.4.p4.5.m5.9.10.3.3.3.2" xref="S5.4.p4.5.m5.9.10.3.3.cmml"><mo id="S5.4.p4.5.m5.9.10.3.3.3.2.1" stretchy="false" xref="S5.4.p4.5.m5.9.10.3.3.cmml">(</mo><mi id="S5.4.p4.5.m5.9.9" xref="S5.4.p4.5.m5.9.9.cmml">θ</mi><mo id="S5.4.p4.5.m5.9.10.3.3.3.2.2" stretchy="false" xref="S5.4.p4.5.m5.9.10.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.5.m5.9b"><apply id="S5.4.p4.5.m5.9.10.cmml" xref="S5.4.p4.5.m5.9.10"><ci id="S5.4.p4.5.m5.9.10.1.cmml" xref="S5.4.p4.5.m5.9.10.1">:</ci><apply id="S5.4.p4.5.m5.9.10.2.cmml" xref="S5.4.p4.5.m5.9.10.2"><times id="S5.4.p4.5.m5.9.10.2.1.cmml" xref="S5.4.p4.5.m5.9.10.2.1"></times><apply id="S5.4.p4.5.m5.9.10.2.2.cmml" xref="S5.4.p4.5.m5.9.10.2.2"><csymbol cd="ambiguous" id="S5.4.p4.5.m5.9.10.2.2.1.cmml" xref="S5.4.p4.5.m5.9.10.2.2">subscript</csymbol><ci id="S5.4.p4.5.m5.9.10.2.2.2.cmml" xref="S5.4.p4.5.m5.9.10.2.2.2">𝜉</ci><list id="S5.4.p4.5.m5.2.2.2.3.cmml" xref="S5.4.p4.5.m5.2.2.2.4"><times id="S5.4.p4.5.m5.1.1.1.1.cmml" xref="S5.4.p4.5.m5.1.1.1.1"></times><ci id="S5.4.p4.5.m5.2.2.2.2.cmml" xref="S5.4.p4.5.m5.2.2.2.2">𝑞</ci></list></apply><ci id="S5.4.p4.5.m5.7.7.cmml" xref="S5.4.p4.5.m5.7.7">𝜃</ci></apply><apply id="S5.4.p4.5.m5.9.10.3.cmml" xref="S5.4.p4.5.m5.9.10.3"><ci id="S5.4.p4.5.m5.9.10.3.1.cmml" xref="S5.4.p4.5.m5.9.10.3.1">→</ci><apply id="S5.4.p4.5.m5.9.10.3.2.cmml" xref="S5.4.p4.5.m5.9.10.3.2"><times id="S5.4.p4.5.m5.9.10.3.2.1.cmml" xref="S5.4.p4.5.m5.9.10.3.2.1"></times><apply id="S5.4.p4.5.m5.9.10.3.2.2.cmml" xref="S5.4.p4.5.m5.9.10.3.2.2"><csymbol cd="ambiguous" id="S5.4.p4.5.m5.9.10.3.2.2.1.cmml" xref="S5.4.p4.5.m5.9.10.3.2.2">subscript</csymbol><ci id="S5.4.p4.5.m5.9.10.3.2.2.2.cmml" xref="S5.4.p4.5.m5.9.10.3.2.2.2">ℙ</ci><list id="S5.4.p4.5.m5.4.4.2.3.cmml" xref="S5.4.p4.5.m5.4.4.2.4"><times id="S5.4.p4.5.m5.3.3.1.1.cmml" xref="S5.4.p4.5.m5.3.3.1.1"></times><ci id="S5.4.p4.5.m5.4.4.2.2.cmml" xref="S5.4.p4.5.m5.4.4.2.2">𝑞</ci></list></apply><ci id="S5.4.p4.5.m5.8.8.cmml" xref="S5.4.p4.5.m5.8.8">𝜃</ci></apply><apply id="S5.4.p4.5.m5.9.10.3.3.cmml" xref="S5.4.p4.5.m5.9.10.3.3"><times id="S5.4.p4.5.m5.9.10.3.3.1.cmml" xref="S5.4.p4.5.m5.9.10.3.3.1"></times><apply id="S5.4.p4.5.m5.9.10.3.3.2.cmml" xref="S5.4.p4.5.m5.9.10.3.3.2"><csymbol cd="ambiguous" id="S5.4.p4.5.m5.9.10.3.3.2.1.cmml" xref="S5.4.p4.5.m5.9.10.3.3.2">subscript</csymbol><ci id="S5.4.p4.5.m5.9.10.3.3.2.2.cmml" xref="S5.4.p4.5.m5.9.10.3.3.2.2">ℚ</ci><list id="S5.4.p4.5.m5.6.6.2.3.cmml" xref="S5.4.p4.5.m5.6.6.2.4"><times id="S5.4.p4.5.m5.5.5.1.1.cmml" xref="S5.4.p4.5.m5.5.5.1.1"></times><ci id="S5.4.p4.5.m5.6.6.2.2.cmml" xref="S5.4.p4.5.m5.6.6.2.2">𝑞</ci></list></apply><ci id="S5.4.p4.5.m5.9.9.cmml" xref="S5.4.p4.5.m5.9.9">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.5.m5.9c">\xi_{*,q}(\theta):\mathbb{P}_{*,q}(\theta)\to\mathbb{Q}_{*,q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.5.m5.9d">italic_ξ start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ ) : blackboard_P start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ ) → blackboard_Q start_POSTSUBSCRIPT ∗ , italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> induces a homology isomorphism.</p> </div> <div class="ltx_para" id="S5.5.p5"> <p class="ltx_p" id="S5.5.p5.7">Applying Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S4.Thmtheorem4" title="Proposition 4.4. ‣ 4.2. The Dold-Puppe Theorem ‣ 4. Bisimplicial objects and the Dold-Puppe theorem ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">4.4</span></a>, we obtain that <math alttext="\mathrm{diag}\,\xi(\theta):\mathrm{diag}\mathbb{P}(\theta)\to\mathrm{diag}% \mathbb{Q}(\theta)" class="ltx_Math" display="inline" id="S5.5.p5.1.m1.3"><semantics id="S5.5.p5.1.m1.3a"><mrow id="S5.5.p5.1.m1.3.4" xref="S5.5.p5.1.m1.3.4.cmml"><mrow id="S5.5.p5.1.m1.3.4.2" xref="S5.5.p5.1.m1.3.4.2.cmml"><mi id="S5.5.p5.1.m1.3.4.2.2" xref="S5.5.p5.1.m1.3.4.2.2.cmml">diag</mi><mo id="S5.5.p5.1.m1.3.4.2.1" lspace="0.170em" xref="S5.5.p5.1.m1.3.4.2.1.cmml">⁢</mo><mi id="S5.5.p5.1.m1.3.4.2.3" xref="S5.5.p5.1.m1.3.4.2.3.cmml">ξ</mi><mo id="S5.5.p5.1.m1.3.4.2.1a" xref="S5.5.p5.1.m1.3.4.2.1.cmml">⁢</mo><mrow id="S5.5.p5.1.m1.3.4.2.4.2" xref="S5.5.p5.1.m1.3.4.2.cmml"><mo id="S5.5.p5.1.m1.3.4.2.4.2.1" stretchy="false" xref="S5.5.p5.1.m1.3.4.2.cmml">(</mo><mi id="S5.5.p5.1.m1.1.1" xref="S5.5.p5.1.m1.1.1.cmml">θ</mi><mo id="S5.5.p5.1.m1.3.4.2.4.2.2" rspace="0.278em" stretchy="false" xref="S5.5.p5.1.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S5.5.p5.1.m1.3.4.1" rspace="0.278em" xref="S5.5.p5.1.m1.3.4.1.cmml">:</mo><mrow id="S5.5.p5.1.m1.3.4.3" xref="S5.5.p5.1.m1.3.4.3.cmml"><mrow id="S5.5.p5.1.m1.3.4.3.2" xref="S5.5.p5.1.m1.3.4.3.2.cmml"><mi id="S5.5.p5.1.m1.3.4.3.2.2" xref="S5.5.p5.1.m1.3.4.3.2.2.cmml">diag</mi><mo id="S5.5.p5.1.m1.3.4.3.2.1" xref="S5.5.p5.1.m1.3.4.3.2.1.cmml">⁢</mo><mi id="S5.5.p5.1.m1.3.4.3.2.3" xref="S5.5.p5.1.m1.3.4.3.2.3.cmml">ℙ</mi><mo id="S5.5.p5.1.m1.3.4.3.2.1a" xref="S5.5.p5.1.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S5.5.p5.1.m1.3.4.3.2.4.2" xref="S5.5.p5.1.m1.3.4.3.2.cmml"><mo id="S5.5.p5.1.m1.3.4.3.2.4.2.1" stretchy="false" xref="S5.5.p5.1.m1.3.4.3.2.cmml">(</mo><mi id="S5.5.p5.1.m1.2.2" xref="S5.5.p5.1.m1.2.2.cmml">θ</mi><mo id="S5.5.p5.1.m1.3.4.3.2.4.2.2" stretchy="false" xref="S5.5.p5.1.m1.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="S5.5.p5.1.m1.3.4.3.1" stretchy="false" xref="S5.5.p5.1.m1.3.4.3.1.cmml">→</mo><mrow id="S5.5.p5.1.m1.3.4.3.3" xref="S5.5.p5.1.m1.3.4.3.3.cmml"><mi id="S5.5.p5.1.m1.3.4.3.3.2" xref="S5.5.p5.1.m1.3.4.3.3.2.cmml">diag</mi><mo id="S5.5.p5.1.m1.3.4.3.3.1" xref="S5.5.p5.1.m1.3.4.3.3.1.cmml">⁢</mo><mi id="S5.5.p5.1.m1.3.4.3.3.3" xref="S5.5.p5.1.m1.3.4.3.3.3.cmml">ℚ</mi><mo id="S5.5.p5.1.m1.3.4.3.3.1a" xref="S5.5.p5.1.m1.3.4.3.3.1.cmml">⁢</mo><mrow id="S5.5.p5.1.m1.3.4.3.3.4.2" xref="S5.5.p5.1.m1.3.4.3.3.cmml"><mo id="S5.5.p5.1.m1.3.4.3.3.4.2.1" stretchy="false" xref="S5.5.p5.1.m1.3.4.3.3.cmml">(</mo><mi id="S5.5.p5.1.m1.3.3" xref="S5.5.p5.1.m1.3.3.cmml">θ</mi><mo id="S5.5.p5.1.m1.3.4.3.3.4.2.2" stretchy="false" xref="S5.5.p5.1.m1.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.1.m1.3b"><apply id="S5.5.p5.1.m1.3.4.cmml" xref="S5.5.p5.1.m1.3.4"><ci id="S5.5.p5.1.m1.3.4.1.cmml" xref="S5.5.p5.1.m1.3.4.1">:</ci><apply id="S5.5.p5.1.m1.3.4.2.cmml" xref="S5.5.p5.1.m1.3.4.2"><times id="S5.5.p5.1.m1.3.4.2.1.cmml" xref="S5.5.p5.1.m1.3.4.2.1"></times><ci id="S5.5.p5.1.m1.3.4.2.2.cmml" xref="S5.5.p5.1.m1.3.4.2.2">diag</ci><ci id="S5.5.p5.1.m1.3.4.2.3.cmml" xref="S5.5.p5.1.m1.3.4.2.3">𝜉</ci><ci id="S5.5.p5.1.m1.1.1.cmml" xref="S5.5.p5.1.m1.1.1">𝜃</ci></apply><apply id="S5.5.p5.1.m1.3.4.3.cmml" xref="S5.5.p5.1.m1.3.4.3"><ci id="S5.5.p5.1.m1.3.4.3.1.cmml" xref="S5.5.p5.1.m1.3.4.3.1">→</ci><apply id="S5.5.p5.1.m1.3.4.3.2.cmml" xref="S5.5.p5.1.m1.3.4.3.2"><times id="S5.5.p5.1.m1.3.4.3.2.1.cmml" xref="S5.5.p5.1.m1.3.4.3.2.1"></times><ci id="S5.5.p5.1.m1.3.4.3.2.2.cmml" xref="S5.5.p5.1.m1.3.4.3.2.2">diag</ci><ci id="S5.5.p5.1.m1.3.4.3.2.3.cmml" xref="S5.5.p5.1.m1.3.4.3.2.3">ℙ</ci><ci id="S5.5.p5.1.m1.2.2.cmml" xref="S5.5.p5.1.m1.2.2">𝜃</ci></apply><apply id="S5.5.p5.1.m1.3.4.3.3.cmml" xref="S5.5.p5.1.m1.3.4.3.3"><times id="S5.5.p5.1.m1.3.4.3.3.1.cmml" xref="S5.5.p5.1.m1.3.4.3.3.1"></times><ci id="S5.5.p5.1.m1.3.4.3.3.2.cmml" xref="S5.5.p5.1.m1.3.4.3.3.2">diag</ci><ci id="S5.5.p5.1.m1.3.4.3.3.3.cmml" xref="S5.5.p5.1.m1.3.4.3.3.3">ℚ</ci><ci id="S5.5.p5.1.m1.3.3.cmml" xref="S5.5.p5.1.m1.3.3">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.1.m1.3c">\mathrm{diag}\,\xi(\theta):\mathrm{diag}\mathbb{P}(\theta)\to\mathrm{diag}% \mathbb{Q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.1.m1.3d">roman_diag italic_ξ ( italic_θ ) : roman_diag blackboard_P ( italic_θ ) → roman_diag blackboard_Q ( italic_θ )</annotation></semantics></math> induces an isomorphism on homology for every <math alttext="\theta\in N\mathcal{C}_{n}" class="ltx_Math" display="inline" id="S5.5.p5.2.m2.1"><semantics id="S5.5.p5.2.m2.1a"><mrow id="S5.5.p5.2.m2.1.1" xref="S5.5.p5.2.m2.1.1.cmml"><mi id="S5.5.p5.2.m2.1.1.2" xref="S5.5.p5.2.m2.1.1.2.cmml">θ</mi><mo id="S5.5.p5.2.m2.1.1.1" xref="S5.5.p5.2.m2.1.1.1.cmml">∈</mo><mrow id="S5.5.p5.2.m2.1.1.3" xref="S5.5.p5.2.m2.1.1.3.cmml"><mi id="S5.5.p5.2.m2.1.1.3.2" xref="S5.5.p5.2.m2.1.1.3.2.cmml">N</mi><mo id="S5.5.p5.2.m2.1.1.3.1" xref="S5.5.p5.2.m2.1.1.3.1.cmml">⁢</mo><msub id="S5.5.p5.2.m2.1.1.3.3" xref="S5.5.p5.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.5.p5.2.m2.1.1.3.3.2" xref="S5.5.p5.2.m2.1.1.3.3.2.cmml">𝒞</mi><mi id="S5.5.p5.2.m2.1.1.3.3.3" xref="S5.5.p5.2.m2.1.1.3.3.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.2.m2.1b"><apply id="S5.5.p5.2.m2.1.1.cmml" xref="S5.5.p5.2.m2.1.1"><in id="S5.5.p5.2.m2.1.1.1.cmml" xref="S5.5.p5.2.m2.1.1.1"></in><ci id="S5.5.p5.2.m2.1.1.2.cmml" xref="S5.5.p5.2.m2.1.1.2">𝜃</ci><apply id="S5.5.p5.2.m2.1.1.3.cmml" xref="S5.5.p5.2.m2.1.1.3"><times id="S5.5.p5.2.m2.1.1.3.1.cmml" xref="S5.5.p5.2.m2.1.1.3.1"></times><ci id="S5.5.p5.2.m2.1.1.3.2.cmml" xref="S5.5.p5.2.m2.1.1.3.2">𝑁</ci><apply id="S5.5.p5.2.m2.1.1.3.3.cmml" xref="S5.5.p5.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.5.p5.2.m2.1.1.3.3.1.cmml" xref="S5.5.p5.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.5.p5.2.m2.1.1.3.3.2.cmml" xref="S5.5.p5.2.m2.1.1.3.3.2">𝒞</ci><ci id="S5.5.p5.2.m2.1.1.3.3.3.cmml" xref="S5.5.p5.2.m2.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.2.m2.1c">\theta\in N\mathcal{C}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.2.m2.1d">italic_θ ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Hence the chain map <math alttext="(\mathrm{diag}\,\xi)_{*}:(\mathrm{diag}\mathbb{P})_{*}\to(\mathrm{diag}\mathbb% {Q})_{*}" class="ltx_Math" display="inline" id="S5.5.p5.3.m3.3"><semantics id="S5.5.p5.3.m3.3a"><mrow id="S5.5.p5.3.m3.3.3" xref="S5.5.p5.3.m3.3.3.cmml"><msub id="S5.5.p5.3.m3.1.1.1" xref="S5.5.p5.3.m3.1.1.1.cmml"><mrow id="S5.5.p5.3.m3.1.1.1.1.1" xref="S5.5.p5.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.5.p5.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.5.p5.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.5.p5.3.m3.1.1.1.1.1.1" xref="S5.5.p5.3.m3.1.1.1.1.1.1.cmml"><mi id="S5.5.p5.3.m3.1.1.1.1.1.1.2" xref="S5.5.p5.3.m3.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.5.p5.3.m3.1.1.1.1.1.1.1" lspace="0.170em" xref="S5.5.p5.3.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.5.p5.3.m3.1.1.1.1.1.1.3" xref="S5.5.p5.3.m3.1.1.1.1.1.1.3.cmml">ξ</mi></mrow><mo id="S5.5.p5.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.5.p5.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.5.p5.3.m3.1.1.1.3" xref="S5.5.p5.3.m3.1.1.1.3.cmml">∗</mo></msub><mo id="S5.5.p5.3.m3.3.3.4" lspace="0.278em" rspace="0.278em" xref="S5.5.p5.3.m3.3.3.4.cmml">:</mo><mrow id="S5.5.p5.3.m3.3.3.3" xref="S5.5.p5.3.m3.3.3.3.cmml"><msub id="S5.5.p5.3.m3.2.2.2.1" xref="S5.5.p5.3.m3.2.2.2.1.cmml"><mrow id="S5.5.p5.3.m3.2.2.2.1.1.1" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.cmml"><mo id="S5.5.p5.3.m3.2.2.2.1.1.1.2" stretchy="false" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.5.p5.3.m3.2.2.2.1.1.1.1" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.cmml"><mi id="S5.5.p5.3.m3.2.2.2.1.1.1.1.2" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.2.cmml">diag</mi><mo id="S5.5.p5.3.m3.2.2.2.1.1.1.1.1" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.5.p5.3.m3.2.2.2.1.1.1.1.3" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.5.p5.3.m3.2.2.2.1.1.1.3" stretchy="false" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S5.5.p5.3.m3.2.2.2.1.3" xref="S5.5.p5.3.m3.2.2.2.1.3.cmml">∗</mo></msub><mo id="S5.5.p5.3.m3.3.3.3.3" stretchy="false" xref="S5.5.p5.3.m3.3.3.3.3.cmml">→</mo><msub id="S5.5.p5.3.m3.3.3.3.2" xref="S5.5.p5.3.m3.3.3.3.2.cmml"><mrow id="S5.5.p5.3.m3.3.3.3.2.1.1" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.cmml"><mo id="S5.5.p5.3.m3.3.3.3.2.1.1.2" stretchy="false" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.cmml">(</mo><mrow id="S5.5.p5.3.m3.3.3.3.2.1.1.1" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.cmml"><mi id="S5.5.p5.3.m3.3.3.3.2.1.1.1.2" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.2.cmml">diag</mi><mo id="S5.5.p5.3.m3.3.3.3.2.1.1.1.1" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.1.cmml">⁢</mo><mi id="S5.5.p5.3.m3.3.3.3.2.1.1.1.3" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.3.cmml">ℚ</mi></mrow><mo id="S5.5.p5.3.m3.3.3.3.2.1.1.3" stretchy="false" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.cmml">)</mo></mrow><mo id="S5.5.p5.3.m3.3.3.3.2.3" xref="S5.5.p5.3.m3.3.3.3.2.3.cmml">∗</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.3.m3.3b"><apply id="S5.5.p5.3.m3.3.3.cmml" xref="S5.5.p5.3.m3.3.3"><ci id="S5.5.p5.3.m3.3.3.4.cmml" xref="S5.5.p5.3.m3.3.3.4">:</ci><apply id="S5.5.p5.3.m3.1.1.1.cmml" xref="S5.5.p5.3.m3.1.1.1"><csymbol cd="ambiguous" id="S5.5.p5.3.m3.1.1.1.2.cmml" xref="S5.5.p5.3.m3.1.1.1">subscript</csymbol><apply id="S5.5.p5.3.m3.1.1.1.1.1.1.cmml" xref="S5.5.p5.3.m3.1.1.1.1.1"><times id="S5.5.p5.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.5.p5.3.m3.1.1.1.1.1.1.1"></times><ci id="S5.5.p5.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.5.p5.3.m3.1.1.1.1.1.1.2">diag</ci><ci id="S5.5.p5.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.5.p5.3.m3.1.1.1.1.1.1.3">𝜉</ci></apply><times id="S5.5.p5.3.m3.1.1.1.3.cmml" xref="S5.5.p5.3.m3.1.1.1.3"></times></apply><apply id="S5.5.p5.3.m3.3.3.3.cmml" xref="S5.5.p5.3.m3.3.3.3"><ci id="S5.5.p5.3.m3.3.3.3.3.cmml" xref="S5.5.p5.3.m3.3.3.3.3">→</ci><apply id="S5.5.p5.3.m3.2.2.2.1.cmml" xref="S5.5.p5.3.m3.2.2.2.1"><csymbol cd="ambiguous" id="S5.5.p5.3.m3.2.2.2.1.2.cmml" xref="S5.5.p5.3.m3.2.2.2.1">subscript</csymbol><apply id="S5.5.p5.3.m3.2.2.2.1.1.1.1.cmml" xref="S5.5.p5.3.m3.2.2.2.1.1.1"><times id="S5.5.p5.3.m3.2.2.2.1.1.1.1.1.cmml" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.1"></times><ci id="S5.5.p5.3.m3.2.2.2.1.1.1.1.2.cmml" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.2">diag</ci><ci id="S5.5.p5.3.m3.2.2.2.1.1.1.1.3.cmml" xref="S5.5.p5.3.m3.2.2.2.1.1.1.1.3">ℙ</ci></apply><times id="S5.5.p5.3.m3.2.2.2.1.3.cmml" xref="S5.5.p5.3.m3.2.2.2.1.3"></times></apply><apply id="S5.5.p5.3.m3.3.3.3.2.cmml" xref="S5.5.p5.3.m3.3.3.3.2"><csymbol cd="ambiguous" id="S5.5.p5.3.m3.3.3.3.2.2.cmml" xref="S5.5.p5.3.m3.3.3.3.2">subscript</csymbol><apply id="S5.5.p5.3.m3.3.3.3.2.1.1.1.cmml" xref="S5.5.p5.3.m3.3.3.3.2.1.1"><times id="S5.5.p5.3.m3.3.3.3.2.1.1.1.1.cmml" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.1"></times><ci id="S5.5.p5.3.m3.3.3.3.2.1.1.1.2.cmml" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.2">diag</ci><ci id="S5.5.p5.3.m3.3.3.3.2.1.1.1.3.cmml" xref="S5.5.p5.3.m3.3.3.3.2.1.1.1.3">ℚ</ci></apply><times id="S5.5.p5.3.m3.3.3.3.2.3.cmml" xref="S5.5.p5.3.m3.3.3.3.2.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.3.m3.3c">(\mathrm{diag}\,\xi)_{*}:(\mathrm{diag}\mathbb{P})_{*}\to(\mathrm{diag}\mathbb% {Q})_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.3.m3.3d">( roman_diag italic_ξ ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : ( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → ( roman_diag blackboard_Q ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> induces a homology isomorphism. Note that <math alttext="(\mathrm{diag}\mathbb{Q})_{*}" class="ltx_Math" display="inline" id="S5.5.p5.4.m4.1"><semantics id="S5.5.p5.4.m4.1a"><msub id="S5.5.p5.4.m4.1.1" xref="S5.5.p5.4.m4.1.1.cmml"><mrow id="S5.5.p5.4.m4.1.1.1.1" xref="S5.5.p5.4.m4.1.1.1.1.1.cmml"><mo id="S5.5.p5.4.m4.1.1.1.1.2" stretchy="false" xref="S5.5.p5.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S5.5.p5.4.m4.1.1.1.1.1" xref="S5.5.p5.4.m4.1.1.1.1.1.cmml"><mi id="S5.5.p5.4.m4.1.1.1.1.1.2" xref="S5.5.p5.4.m4.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.5.p5.4.m4.1.1.1.1.1.1" xref="S5.5.p5.4.m4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.5.p5.4.m4.1.1.1.1.1.3" xref="S5.5.p5.4.m4.1.1.1.1.1.3.cmml">ℚ</mi></mrow><mo id="S5.5.p5.4.m4.1.1.1.1.3" stretchy="false" xref="S5.5.p5.4.m4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.5.p5.4.m4.1.1.3" xref="S5.5.p5.4.m4.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S5.5.p5.4.m4.1b"><apply id="S5.5.p5.4.m4.1.1.cmml" xref="S5.5.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S5.5.p5.4.m4.1.1.2.cmml" xref="S5.5.p5.4.m4.1.1">subscript</csymbol><apply id="S5.5.p5.4.m4.1.1.1.1.1.cmml" xref="S5.5.p5.4.m4.1.1.1.1"><times id="S5.5.p5.4.m4.1.1.1.1.1.1.cmml" xref="S5.5.p5.4.m4.1.1.1.1.1.1"></times><ci id="S5.5.p5.4.m4.1.1.1.1.1.2.cmml" xref="S5.5.p5.4.m4.1.1.1.1.1.2">diag</ci><ci id="S5.5.p5.4.m4.1.1.1.1.1.3.cmml" xref="S5.5.p5.4.m4.1.1.1.1.1.3">ℚ</ci></apply><times id="S5.5.p5.4.m4.1.1.3.cmml" xref="S5.5.p5.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.4.m4.1c">(\mathrm{diag}\mathbb{Q})_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.4.m4.1d">( roman_diag blackboard_Q ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> is isomorphic to the complex <math alttext="Q_{*}" class="ltx_Math" display="inline" id="S5.5.p5.5.m5.1"><semantics id="S5.5.p5.5.m5.1a"><msub id="S5.5.p5.5.m5.1.1" xref="S5.5.p5.5.m5.1.1.cmml"><mi id="S5.5.p5.5.m5.1.1.2" xref="S5.5.p5.5.m5.1.1.2.cmml">Q</mi><mo id="S5.5.p5.5.m5.1.1.3" xref="S5.5.p5.5.m5.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S5.5.p5.5.m5.1b"><apply id="S5.5.p5.5.m5.1.1.cmml" xref="S5.5.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S5.5.p5.5.m5.1.1.1.cmml" xref="S5.5.p5.5.m5.1.1">subscript</csymbol><ci id="S5.5.p5.5.m5.1.1.2.cmml" xref="S5.5.p5.5.m5.1.1.2">𝑄</ci><times id="S5.5.p5.5.m5.1.1.3.cmml" xref="S5.5.p5.5.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.5.m5.1c">Q_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.5.m5.1d">italic_Q start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>, and the augmented complex <math alttext="\widetilde{Q}_{*}" class="ltx_Math" display="inline" id="S5.5.p5.6.m6.1"><semantics id="S5.5.p5.6.m6.1a"><msub id="S5.5.p5.6.m6.1.1" xref="S5.5.p5.6.m6.1.1.cmml"><mover accent="true" id="S5.5.p5.6.m6.1.1.2" xref="S5.5.p5.6.m6.1.1.2.cmml"><mi id="S5.5.p5.6.m6.1.1.2.2" xref="S5.5.p5.6.m6.1.1.2.2.cmml">Q</mi><mo id="S5.5.p5.6.m6.1.1.2.1" xref="S5.5.p5.6.m6.1.1.2.1.cmml">~</mo></mover><mo id="S5.5.p5.6.m6.1.1.3" xref="S5.5.p5.6.m6.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S5.5.p5.6.m6.1b"><apply id="S5.5.p5.6.m6.1.1.cmml" xref="S5.5.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S5.5.p5.6.m6.1.1.1.cmml" xref="S5.5.p5.6.m6.1.1">subscript</csymbol><apply id="S5.5.p5.6.m6.1.1.2.cmml" xref="S5.5.p5.6.m6.1.1.2"><ci id="S5.5.p5.6.m6.1.1.2.1.cmml" xref="S5.5.p5.6.m6.1.1.2.1">~</ci><ci id="S5.5.p5.6.m6.1.1.2.2.cmml" xref="S5.5.p5.6.m6.1.1.2.2">𝑄</ci></apply><times id="S5.5.p5.6.m6.1.1.3.cmml" xref="S5.5.p5.6.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.6.m6.1c">\widetilde{Q}_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.6.m6.1d">over~ start_ARG italic_Q end_ARG start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> is exact. Hence the sequence in <math alttext="(\ref{eqn:P-resolution})" class="ltx_Math" display="inline" id="S5.5.p5.7.m7.1"><semantics id="S5.5.p5.7.m7.1a"><mrow id="S5.5.p5.7.m7.1.2.2" xref="S5.5.p5.7.m7.1.1c.cmml"><mo id="S5.5.p5.7.m7.1.2.2.1" stretchy="false" xref="S5.5.p5.7.m7.1.1c.cmml">(</mo><mtext class="ltx_mathvariant_italic" id="S5.5.p5.7.m7.1.1" xref="S5.5.p5.7.m7.1.1c.cmml"><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2503.14659v1#S5.E4" title="In Proof. ‣ 5. Proof of Theorem 1.2 ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">4</span></a></mtext><mo id="S5.5.p5.7.m7.1.2.2.2" stretchy="false" xref="S5.5.p5.7.m7.1.1c.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.7.m7.1b"><ci id="S5.5.p5.7.m7.1.1c.cmml" xref="S5.5.p5.7.m7.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.5.p5.7.m7.1.1.cmml" xref="S5.5.p5.7.m7.1.2.2"><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2503.14659v1#S5.E4" title="In Proof. ‣ 5. Proof of Theorem 1.2 ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">4</span></a></mtext></ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.7.m7.1c">(\ref{eqn:P-resolution})</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.7.m7.1d">( )</annotation></semantics></math> is also exact. This completes the proof. ∎</p> </div> </div> <div class="ltx_proof" id="S5.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a>.</h6> <div class="ltx_para" id="S5.6.p1"> <p class="ltx_p" id="S5.6.p1.20">By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S5.Thmtheorem2" title="Proposition 5.2. ‣ 5. Proof of Theorem 1.2 ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">5.2</span></a>,</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex60"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(\mathrm{diag}\mathbb{P})_{*}:\cdots\to(\mathrm{diag}\mathbb{P})_{k}\to(% \mathrm{diag}\mathbb{P})_{k-1}\to\cdots\to(\mathrm{diag}\mathbb{P})_{0}\to% \underline{R}\to 0" class="ltx_Math" display="block" id="S5.Ex60.m1.4"><semantics id="S5.Ex60.m1.4a"><mrow id="S5.Ex60.m1.4.4" xref="S5.Ex60.m1.4.4.cmml"><msub id="S5.Ex60.m1.1.1.1" xref="S5.Ex60.m1.1.1.1.cmml"><mrow id="S5.Ex60.m1.1.1.1.1.1" xref="S5.Ex60.m1.1.1.1.1.1.1.cmml"><mo id="S5.Ex60.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex60.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex60.m1.1.1.1.1.1.1" xref="S5.Ex60.m1.1.1.1.1.1.1.cmml"><mi id="S5.Ex60.m1.1.1.1.1.1.1.2" xref="S5.Ex60.m1.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.Ex60.m1.1.1.1.1.1.1.1" xref="S5.Ex60.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex60.m1.1.1.1.1.1.1.3" xref="S5.Ex60.m1.1.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.Ex60.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex60.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex60.m1.1.1.1.3" xref="S5.Ex60.m1.1.1.1.3.cmml">∗</mo></msub><mo id="S5.Ex60.m1.4.4.5" lspace="0.278em" rspace="0.278em" xref="S5.Ex60.m1.4.4.5.cmml">:</mo><mrow id="S5.Ex60.m1.4.4.4" xref="S5.Ex60.m1.4.4.4.cmml"><mi id="S5.Ex60.m1.4.4.4.5" mathvariant="normal" xref="S5.Ex60.m1.4.4.4.5.cmml">⋯</mi><mo id="S5.Ex60.m1.4.4.4.6" stretchy="false" xref="S5.Ex60.m1.4.4.4.6.cmml">→</mo><msub id="S5.Ex60.m1.2.2.2.1" xref="S5.Ex60.m1.2.2.2.1.cmml"><mrow id="S5.Ex60.m1.2.2.2.1.1.1" xref="S5.Ex60.m1.2.2.2.1.1.1.1.cmml"><mo id="S5.Ex60.m1.2.2.2.1.1.1.2" stretchy="false" xref="S5.Ex60.m1.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Ex60.m1.2.2.2.1.1.1.1" xref="S5.Ex60.m1.2.2.2.1.1.1.1.cmml"><mi id="S5.Ex60.m1.2.2.2.1.1.1.1.2" xref="S5.Ex60.m1.2.2.2.1.1.1.1.2.cmml">diag</mi><mo id="S5.Ex60.m1.2.2.2.1.1.1.1.1" xref="S5.Ex60.m1.2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex60.m1.2.2.2.1.1.1.1.3" xref="S5.Ex60.m1.2.2.2.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.Ex60.m1.2.2.2.1.1.1.3" stretchy="false" xref="S5.Ex60.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow><mi id="S5.Ex60.m1.2.2.2.1.3" xref="S5.Ex60.m1.2.2.2.1.3.cmml">k</mi></msub><mo id="S5.Ex60.m1.4.4.4.7" stretchy="false" xref="S5.Ex60.m1.4.4.4.7.cmml">→</mo><msub id="S5.Ex60.m1.3.3.3.2" xref="S5.Ex60.m1.3.3.3.2.cmml"><mrow id="S5.Ex60.m1.3.3.3.2.1.1" xref="S5.Ex60.m1.3.3.3.2.1.1.1.cmml"><mo id="S5.Ex60.m1.3.3.3.2.1.1.2" stretchy="false" xref="S5.Ex60.m1.3.3.3.2.1.1.1.cmml">(</mo><mrow id="S5.Ex60.m1.3.3.3.2.1.1.1" xref="S5.Ex60.m1.3.3.3.2.1.1.1.cmml"><mi id="S5.Ex60.m1.3.3.3.2.1.1.1.2" xref="S5.Ex60.m1.3.3.3.2.1.1.1.2.cmml">diag</mi><mo id="S5.Ex60.m1.3.3.3.2.1.1.1.1" xref="S5.Ex60.m1.3.3.3.2.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex60.m1.3.3.3.2.1.1.1.3" xref="S5.Ex60.m1.3.3.3.2.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.Ex60.m1.3.3.3.2.1.1.3" stretchy="false" xref="S5.Ex60.m1.3.3.3.2.1.1.1.cmml">)</mo></mrow><mrow id="S5.Ex60.m1.3.3.3.2.3" xref="S5.Ex60.m1.3.3.3.2.3.cmml"><mi id="S5.Ex60.m1.3.3.3.2.3.2" xref="S5.Ex60.m1.3.3.3.2.3.2.cmml">k</mi><mo id="S5.Ex60.m1.3.3.3.2.3.1" xref="S5.Ex60.m1.3.3.3.2.3.1.cmml">−</mo><mn id="S5.Ex60.m1.3.3.3.2.3.3" xref="S5.Ex60.m1.3.3.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.Ex60.m1.4.4.4.8" stretchy="false" xref="S5.Ex60.m1.4.4.4.8.cmml">→</mo><mi id="S5.Ex60.m1.4.4.4.9" mathvariant="normal" xref="S5.Ex60.m1.4.4.4.9.cmml">⋯</mi><mo id="S5.Ex60.m1.4.4.4.10" stretchy="false" xref="S5.Ex60.m1.4.4.4.10.cmml">→</mo><msub id="S5.Ex60.m1.4.4.4.3" xref="S5.Ex60.m1.4.4.4.3.cmml"><mrow id="S5.Ex60.m1.4.4.4.3.1.1" xref="S5.Ex60.m1.4.4.4.3.1.1.1.cmml"><mo id="S5.Ex60.m1.4.4.4.3.1.1.2" stretchy="false" xref="S5.Ex60.m1.4.4.4.3.1.1.1.cmml">(</mo><mrow id="S5.Ex60.m1.4.4.4.3.1.1.1" xref="S5.Ex60.m1.4.4.4.3.1.1.1.cmml"><mi id="S5.Ex60.m1.4.4.4.3.1.1.1.2" xref="S5.Ex60.m1.4.4.4.3.1.1.1.2.cmml">diag</mi><mo id="S5.Ex60.m1.4.4.4.3.1.1.1.1" xref="S5.Ex60.m1.4.4.4.3.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex60.m1.4.4.4.3.1.1.1.3" xref="S5.Ex60.m1.4.4.4.3.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.Ex60.m1.4.4.4.3.1.1.3" stretchy="false" xref="S5.Ex60.m1.4.4.4.3.1.1.1.cmml">)</mo></mrow><mn id="S5.Ex60.m1.4.4.4.3.3" xref="S5.Ex60.m1.4.4.4.3.3.cmml">0</mn></msub><mo id="S5.Ex60.m1.4.4.4.11" stretchy="false" xref="S5.Ex60.m1.4.4.4.11.cmml">→</mo><munder accentunder="true" id="S5.Ex60.m1.4.4.4.12" xref="S5.Ex60.m1.4.4.4.12.cmml"><mi id="S5.Ex60.m1.4.4.4.12.2" xref="S5.Ex60.m1.4.4.4.12.2.cmml">R</mi><mo id="S5.Ex60.m1.4.4.4.12.1" xref="S5.Ex60.m1.4.4.4.12.1.cmml">¯</mo></munder><mo id="S5.Ex60.m1.4.4.4.13" stretchy="false" xref="S5.Ex60.m1.4.4.4.13.cmml">→</mo><mn id="S5.Ex60.m1.4.4.4.14" xref="S5.Ex60.m1.4.4.4.14.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex60.m1.4b"><apply id="S5.Ex60.m1.4.4.cmml" xref="S5.Ex60.m1.4.4"><ci id="S5.Ex60.m1.4.4.5.cmml" xref="S5.Ex60.m1.4.4.5">:</ci><apply id="S5.Ex60.m1.1.1.1.cmml" xref="S5.Ex60.m1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex60.m1.1.1.1.2.cmml" xref="S5.Ex60.m1.1.1.1">subscript</csymbol><apply id="S5.Ex60.m1.1.1.1.1.1.1.cmml" xref="S5.Ex60.m1.1.1.1.1.1"><times id="S5.Ex60.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex60.m1.1.1.1.1.1.1.1"></times><ci id="S5.Ex60.m1.1.1.1.1.1.1.2.cmml" xref="S5.Ex60.m1.1.1.1.1.1.1.2">diag</ci><ci id="S5.Ex60.m1.1.1.1.1.1.1.3.cmml" xref="S5.Ex60.m1.1.1.1.1.1.1.3">ℙ</ci></apply><times id="S5.Ex60.m1.1.1.1.3.cmml" xref="S5.Ex60.m1.1.1.1.3"></times></apply><apply id="S5.Ex60.m1.4.4.4.cmml" xref="S5.Ex60.m1.4.4.4"><and id="S5.Ex60.m1.4.4.4a.cmml" xref="S5.Ex60.m1.4.4.4"></and><apply id="S5.Ex60.m1.4.4.4b.cmml" xref="S5.Ex60.m1.4.4.4"><ci id="S5.Ex60.m1.4.4.4.6.cmml" xref="S5.Ex60.m1.4.4.4.6">→</ci><ci id="S5.Ex60.m1.4.4.4.5.cmml" xref="S5.Ex60.m1.4.4.4.5">⋯</ci><apply id="S5.Ex60.m1.2.2.2.1.cmml" xref="S5.Ex60.m1.2.2.2.1"><csymbol cd="ambiguous" id="S5.Ex60.m1.2.2.2.1.2.cmml" xref="S5.Ex60.m1.2.2.2.1">subscript</csymbol><apply id="S5.Ex60.m1.2.2.2.1.1.1.1.cmml" xref="S5.Ex60.m1.2.2.2.1.1.1"><times id="S5.Ex60.m1.2.2.2.1.1.1.1.1.cmml" xref="S5.Ex60.m1.2.2.2.1.1.1.1.1"></times><ci id="S5.Ex60.m1.2.2.2.1.1.1.1.2.cmml" xref="S5.Ex60.m1.2.2.2.1.1.1.1.2">diag</ci><ci id="S5.Ex60.m1.2.2.2.1.1.1.1.3.cmml" xref="S5.Ex60.m1.2.2.2.1.1.1.1.3">ℙ</ci></apply><ci id="S5.Ex60.m1.2.2.2.1.3.cmml" xref="S5.Ex60.m1.2.2.2.1.3">𝑘</ci></apply></apply><apply id="S5.Ex60.m1.4.4.4c.cmml" xref="S5.Ex60.m1.4.4.4"><ci id="S5.Ex60.m1.4.4.4.7.cmml" xref="S5.Ex60.m1.4.4.4.7">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.Ex60.m1.2.2.2.1.cmml" id="S5.Ex60.m1.4.4.4d.cmml" xref="S5.Ex60.m1.4.4.4"></share><apply id="S5.Ex60.m1.3.3.3.2.cmml" xref="S5.Ex60.m1.3.3.3.2"><csymbol cd="ambiguous" id="S5.Ex60.m1.3.3.3.2.2.cmml" xref="S5.Ex60.m1.3.3.3.2">subscript</csymbol><apply id="S5.Ex60.m1.3.3.3.2.1.1.1.cmml" xref="S5.Ex60.m1.3.3.3.2.1.1"><times id="S5.Ex60.m1.3.3.3.2.1.1.1.1.cmml" xref="S5.Ex60.m1.3.3.3.2.1.1.1.1"></times><ci id="S5.Ex60.m1.3.3.3.2.1.1.1.2.cmml" xref="S5.Ex60.m1.3.3.3.2.1.1.1.2">diag</ci><ci id="S5.Ex60.m1.3.3.3.2.1.1.1.3.cmml" xref="S5.Ex60.m1.3.3.3.2.1.1.1.3">ℙ</ci></apply><apply id="S5.Ex60.m1.3.3.3.2.3.cmml" xref="S5.Ex60.m1.3.3.3.2.3"><minus id="S5.Ex60.m1.3.3.3.2.3.1.cmml" xref="S5.Ex60.m1.3.3.3.2.3.1"></minus><ci id="S5.Ex60.m1.3.3.3.2.3.2.cmml" xref="S5.Ex60.m1.3.3.3.2.3.2">𝑘</ci><cn id="S5.Ex60.m1.3.3.3.2.3.3.cmml" type="integer" xref="S5.Ex60.m1.3.3.3.2.3.3">1</cn></apply></apply></apply><apply id="S5.Ex60.m1.4.4.4e.cmml" xref="S5.Ex60.m1.4.4.4"><ci id="S5.Ex60.m1.4.4.4.8.cmml" xref="S5.Ex60.m1.4.4.4.8">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.Ex60.m1.3.3.3.2.cmml" id="S5.Ex60.m1.4.4.4f.cmml" xref="S5.Ex60.m1.4.4.4"></share><ci id="S5.Ex60.m1.4.4.4.9.cmml" xref="S5.Ex60.m1.4.4.4.9">⋯</ci></apply><apply id="S5.Ex60.m1.4.4.4g.cmml" xref="S5.Ex60.m1.4.4.4"><ci id="S5.Ex60.m1.4.4.4.10.cmml" xref="S5.Ex60.m1.4.4.4.10">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.Ex60.m1.4.4.4.9.cmml" id="S5.Ex60.m1.4.4.4h.cmml" xref="S5.Ex60.m1.4.4.4"></share><apply id="S5.Ex60.m1.4.4.4.3.cmml" xref="S5.Ex60.m1.4.4.4.3"><csymbol cd="ambiguous" id="S5.Ex60.m1.4.4.4.3.2.cmml" xref="S5.Ex60.m1.4.4.4.3">subscript</csymbol><apply id="S5.Ex60.m1.4.4.4.3.1.1.1.cmml" xref="S5.Ex60.m1.4.4.4.3.1.1"><times id="S5.Ex60.m1.4.4.4.3.1.1.1.1.cmml" xref="S5.Ex60.m1.4.4.4.3.1.1.1.1"></times><ci id="S5.Ex60.m1.4.4.4.3.1.1.1.2.cmml" xref="S5.Ex60.m1.4.4.4.3.1.1.1.2">diag</ci><ci id="S5.Ex60.m1.4.4.4.3.1.1.1.3.cmml" xref="S5.Ex60.m1.4.4.4.3.1.1.1.3">ℙ</ci></apply><cn id="S5.Ex60.m1.4.4.4.3.3.cmml" type="integer" xref="S5.Ex60.m1.4.4.4.3.3">0</cn></apply></apply><apply id="S5.Ex60.m1.4.4.4i.cmml" xref="S5.Ex60.m1.4.4.4"><ci id="S5.Ex60.m1.4.4.4.11.cmml" xref="S5.Ex60.m1.4.4.4.11">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.Ex60.m1.4.4.4.3.cmml" id="S5.Ex60.m1.4.4.4j.cmml" xref="S5.Ex60.m1.4.4.4"></share><apply id="S5.Ex60.m1.4.4.4.12.cmml" xref="S5.Ex60.m1.4.4.4.12"><ci id="S5.Ex60.m1.4.4.4.12.1.cmml" xref="S5.Ex60.m1.4.4.4.12.1">¯</ci><ci id="S5.Ex60.m1.4.4.4.12.2.cmml" xref="S5.Ex60.m1.4.4.4.12.2">𝑅</ci></apply></apply><apply id="S5.Ex60.m1.4.4.4k.cmml" xref="S5.Ex60.m1.4.4.4"><ci id="S5.Ex60.m1.4.4.4.13.cmml" xref="S5.Ex60.m1.4.4.4.13">→</ci><share href="https://arxiv.org/html/2503.14659v1#S5.Ex60.m1.4.4.4.12.cmml" id="S5.Ex60.m1.4.4.4l.cmml" xref="S5.Ex60.m1.4.4.4"></share><cn id="S5.Ex60.m1.4.4.4.14.cmml" type="integer" xref="S5.Ex60.m1.4.4.4.14">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex60.m1.4c">(\mathrm{diag}\mathbb{P})_{*}:\cdots\to(\mathrm{diag}\mathbb{P})_{k}\to(% \mathrm{diag}\mathbb{P})_{k-1}\to\cdots\to(\mathrm{diag}\mathbb{P})_{0}\to% \underline{R}\to 0</annotation><annotation encoding="application/x-llamapun" id="S5.Ex60.m1.4d">( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : ⋯ → ( roman_diag blackboard_P ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT → ( roman_diag blackboard_P ) start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT → ⋯ → ( roman_diag blackboard_P ) start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → under¯ start_ARG italic_R end_ARG → 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.6.p1.6">is a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S5.6.p1.1.m1.1"><semantics id="S5.6.p1.1.m1.1a"><munder accentunder="true" id="S5.6.p1.1.m1.1.1" xref="S5.6.p1.1.m1.1.1.cmml"><mi id="S5.6.p1.1.m1.1.1.2" xref="S5.6.p1.1.m1.1.1.2.cmml">R</mi><mo id="S5.6.p1.1.m1.1.1.1" xref="S5.6.p1.1.m1.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S5.6.p1.1.m1.1b"><apply id="S5.6.p1.1.m1.1.1.cmml" xref="S5.6.p1.1.m1.1.1"><ci id="S5.6.p1.1.m1.1.1.1.cmml" xref="S5.6.p1.1.m1.1.1.1">¯</ci><ci id="S5.6.p1.1.m1.1.1.2.cmml" xref="S5.6.p1.1.m1.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.1.m1.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.1.m1.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.6.p1.2.m2.1"><semantics id="S5.6.p1.2.m2.1a"><mrow id="S5.6.p1.2.m2.1.1" xref="S5.6.p1.2.m2.1.1.cmml"><mi id="S5.6.p1.2.m2.1.1.3" xref="S5.6.p1.2.m2.1.1.3.cmml">R</mi><mo id="S5.6.p1.2.m2.1.1.2" xref="S5.6.p1.2.m2.1.1.2.cmml">⁢</mo><mi id="S5.6.p1.2.m2.1.1.4" mathvariant="normal" xref="S5.6.p1.2.m2.1.1.4.cmml">Δ</mi><mo id="S5.6.p1.2.m2.1.1.2a" xref="S5.6.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S5.6.p1.2.m2.1.1.1.1" xref="S5.6.p1.2.m2.1.1.1.1.1.cmml"><mo id="S5.6.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S5.6.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.6.p1.2.m2.1.1.1.1.1" xref="S5.6.p1.2.m2.1.1.1.1.1.cmml"><mi id="S5.6.p1.2.m2.1.1.1.1.1.2" xref="S5.6.p1.2.m2.1.1.1.1.1.2.cmml">N</mi><mo id="S5.6.p1.2.m2.1.1.1.1.1.1" xref="S5.6.p1.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.2.m2.1.1.1.1.1.3" xref="S5.6.p1.2.m2.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.6.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S5.6.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.2.m2.1b"><apply id="S5.6.p1.2.m2.1.1.cmml" xref="S5.6.p1.2.m2.1.1"><times id="S5.6.p1.2.m2.1.1.2.cmml" xref="S5.6.p1.2.m2.1.1.2"></times><ci id="S5.6.p1.2.m2.1.1.3.cmml" xref="S5.6.p1.2.m2.1.1.3">𝑅</ci><ci id="S5.6.p1.2.m2.1.1.4.cmml" xref="S5.6.p1.2.m2.1.1.4">Δ</ci><apply id="S5.6.p1.2.m2.1.1.1.1.1.cmml" xref="S5.6.p1.2.m2.1.1.1.1"><times id="S5.6.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.6.p1.2.m2.1.1.1.1.1.1"></times><ci id="S5.6.p1.2.m2.1.1.1.1.1.2.cmml" xref="S5.6.p1.2.m2.1.1.1.1.1.2">𝑁</ci><ci id="S5.6.p1.2.m2.1.1.1.1.1.3.cmml" xref="S5.6.p1.2.m2.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.2.m2.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.2.m2.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-module. Hence for any functor <math alttext="\mathcal{M}:\Delta(N\mathcal{C})\to R" class="ltx_Math" display="inline" id="S5.6.p1.3.m3.1"><semantics id="S5.6.p1.3.m3.1a"><mrow id="S5.6.p1.3.m3.1.1" xref="S5.6.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.3.m3.1.1.3" xref="S5.6.p1.3.m3.1.1.3.cmml">ℳ</mi><mo id="S5.6.p1.3.m3.1.1.2" lspace="0.278em" rspace="0.278em" xref="S5.6.p1.3.m3.1.1.2.cmml">:</mo><mrow id="S5.6.p1.3.m3.1.1.1" xref="S5.6.p1.3.m3.1.1.1.cmml"><mrow id="S5.6.p1.3.m3.1.1.1.1" xref="S5.6.p1.3.m3.1.1.1.1.cmml"><mi id="S5.6.p1.3.m3.1.1.1.1.3" mathvariant="normal" xref="S5.6.p1.3.m3.1.1.1.1.3.cmml">Δ</mi><mo id="S5.6.p1.3.m3.1.1.1.1.2" xref="S5.6.p1.3.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.6.p1.3.m3.1.1.1.1.1.1" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.cmml"><mo id="S5.6.p1.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.6.p1.3.m3.1.1.1.1.1.1.1" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.cmml"><mi id="S5.6.p1.3.m3.1.1.1.1.1.1.1.2" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S5.6.p1.3.m3.1.1.1.1.1.1.1.1" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.3.m3.1.1.1.1.1.1.1.3" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.6.p1.3.m3.1.1.1.1.1.1.3" stretchy="false" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.6.p1.3.m3.1.1.1.2" stretchy="false" xref="S5.6.p1.3.m3.1.1.1.2.cmml">→</mo><mi id="S5.6.p1.3.m3.1.1.1.3" xref="S5.6.p1.3.m3.1.1.1.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.3.m3.1b"><apply id="S5.6.p1.3.m3.1.1.cmml" xref="S5.6.p1.3.m3.1.1"><ci id="S5.6.p1.3.m3.1.1.2.cmml" xref="S5.6.p1.3.m3.1.1.2">:</ci><ci id="S5.6.p1.3.m3.1.1.3.cmml" xref="S5.6.p1.3.m3.1.1.3">ℳ</ci><apply id="S5.6.p1.3.m3.1.1.1.cmml" xref="S5.6.p1.3.m3.1.1.1"><ci id="S5.6.p1.3.m3.1.1.1.2.cmml" xref="S5.6.p1.3.m3.1.1.1.2">→</ci><apply id="S5.6.p1.3.m3.1.1.1.1.cmml" xref="S5.6.p1.3.m3.1.1.1.1"><times id="S5.6.p1.3.m3.1.1.1.1.2.cmml" xref="S5.6.p1.3.m3.1.1.1.1.2"></times><ci id="S5.6.p1.3.m3.1.1.1.1.3.cmml" xref="S5.6.p1.3.m3.1.1.1.1.3">Δ</ci><apply id="S5.6.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.6.p1.3.m3.1.1.1.1.1.1"><times id="S5.6.p1.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.1"></times><ci id="S5.6.p1.3.m3.1.1.1.1.1.1.1.2.cmml" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.6.p1.3.m3.1.1.1.1.1.1.1.3.cmml" xref="S5.6.p1.3.m3.1.1.1.1.1.1.1.3">𝒞</ci></apply></apply><ci id="S5.6.p1.3.m3.1.1.1.3.cmml" xref="S5.6.p1.3.m3.1.1.1.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.3.m3.1c">\mathcal{M}:\Delta(N\mathcal{C})\to R</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.3.m3.1d">caligraphic_M : roman_Δ ( italic_N caligraphic_C ) → italic_R</annotation></semantics></math>-mod, the cohomology of the cochain complex <math alttext="\mathrm{Hom}_{R\Delta(N\mathcal{C})}((\mathrm{diag}\mathbb{P})_{*},\mathcal{M})" class="ltx_Math" display="inline" id="S5.6.p1.4.m4.3"><semantics id="S5.6.p1.4.m4.3a"><mrow id="S5.6.p1.4.m4.3.3" xref="S5.6.p1.4.m4.3.3.cmml"><msub id="S5.6.p1.4.m4.3.3.3" xref="S5.6.p1.4.m4.3.3.3.cmml"><mi id="S5.6.p1.4.m4.3.3.3.2" xref="S5.6.p1.4.m4.3.3.3.2.cmml">Hom</mi><mrow id="S5.6.p1.4.m4.1.1.1" xref="S5.6.p1.4.m4.1.1.1.cmml"><mi id="S5.6.p1.4.m4.1.1.1.3" xref="S5.6.p1.4.m4.1.1.1.3.cmml">R</mi><mo id="S5.6.p1.4.m4.1.1.1.2" xref="S5.6.p1.4.m4.1.1.1.2.cmml">⁢</mo><mi id="S5.6.p1.4.m4.1.1.1.4" mathvariant="normal" xref="S5.6.p1.4.m4.1.1.1.4.cmml">Δ</mi><mo id="S5.6.p1.4.m4.1.1.1.2a" xref="S5.6.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.6.p1.4.m4.1.1.1.1.1" xref="S5.6.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.6.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.6.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.6.p1.4.m4.1.1.1.1.1.1" xref="S5.6.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.6.p1.4.m4.1.1.1.1.1.1.2" xref="S5.6.p1.4.m4.1.1.1.1.1.1.2.cmml">N</mi><mo id="S5.6.p1.4.m4.1.1.1.1.1.1.1" xref="S5.6.p1.4.m4.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.4.m4.1.1.1.1.1.1.3" xref="S5.6.p1.4.m4.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.6.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.6.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.6.p1.4.m4.3.3.2" xref="S5.6.p1.4.m4.3.3.2.cmml">⁢</mo><mrow id="S5.6.p1.4.m4.3.3.1.1" xref="S5.6.p1.4.m4.3.3.1.2.cmml"><mo id="S5.6.p1.4.m4.3.3.1.1.2" stretchy="false" xref="S5.6.p1.4.m4.3.3.1.2.cmml">(</mo><msub id="S5.6.p1.4.m4.3.3.1.1.1" xref="S5.6.p1.4.m4.3.3.1.1.1.cmml"><mrow id="S5.6.p1.4.m4.3.3.1.1.1.1.1" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.cmml"><mo id="S5.6.p1.4.m4.3.3.1.1.1.1.1.2" stretchy="false" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.6.p1.4.m4.3.3.1.1.1.1.1.1" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.cmml"><mi id="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.2" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.1" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.3" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.6.p1.4.m4.3.3.1.1.1.1.1.3" stretchy="false" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.6.p1.4.m4.3.3.1.1.1.3" xref="S5.6.p1.4.m4.3.3.1.1.1.3.cmml">∗</mo></msub><mo id="S5.6.p1.4.m4.3.3.1.1.3" xref="S5.6.p1.4.m4.3.3.1.2.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.4.m4.2.2" xref="S5.6.p1.4.m4.2.2.cmml">ℳ</mi><mo id="S5.6.p1.4.m4.3.3.1.1.4" stretchy="false" xref="S5.6.p1.4.m4.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.4.m4.3b"><apply id="S5.6.p1.4.m4.3.3.cmml" xref="S5.6.p1.4.m4.3.3"><times id="S5.6.p1.4.m4.3.3.2.cmml" xref="S5.6.p1.4.m4.3.3.2"></times><apply id="S5.6.p1.4.m4.3.3.3.cmml" xref="S5.6.p1.4.m4.3.3.3"><csymbol cd="ambiguous" id="S5.6.p1.4.m4.3.3.3.1.cmml" xref="S5.6.p1.4.m4.3.3.3">subscript</csymbol><ci id="S5.6.p1.4.m4.3.3.3.2.cmml" xref="S5.6.p1.4.m4.3.3.3.2">Hom</ci><apply id="S5.6.p1.4.m4.1.1.1.cmml" xref="S5.6.p1.4.m4.1.1.1"><times id="S5.6.p1.4.m4.1.1.1.2.cmml" xref="S5.6.p1.4.m4.1.1.1.2"></times><ci id="S5.6.p1.4.m4.1.1.1.3.cmml" xref="S5.6.p1.4.m4.1.1.1.3">𝑅</ci><ci id="S5.6.p1.4.m4.1.1.1.4.cmml" xref="S5.6.p1.4.m4.1.1.1.4">Δ</ci><apply id="S5.6.p1.4.m4.1.1.1.1.1.1.cmml" xref="S5.6.p1.4.m4.1.1.1.1.1"><times id="S5.6.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.6.p1.4.m4.1.1.1.1.1.1.1"></times><ci id="S5.6.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.6.p1.4.m4.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.6.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.6.p1.4.m4.1.1.1.1.1.1.3">𝒞</ci></apply></apply></apply><interval closure="open" id="S5.6.p1.4.m4.3.3.1.2.cmml" xref="S5.6.p1.4.m4.3.3.1.1"><apply id="S5.6.p1.4.m4.3.3.1.1.1.cmml" xref="S5.6.p1.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.6.p1.4.m4.3.3.1.1.1.2.cmml" xref="S5.6.p1.4.m4.3.3.1.1.1">subscript</csymbol><apply id="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.cmml" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1"><times id="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.1.cmml" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.1"></times><ci id="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.2.cmml" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.2">diag</ci><ci id="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.3.cmml" xref="S5.6.p1.4.m4.3.3.1.1.1.1.1.1.3">ℙ</ci></apply><times id="S5.6.p1.4.m4.3.3.1.1.1.3.cmml" xref="S5.6.p1.4.m4.3.3.1.1.1.3"></times></apply><ci id="S5.6.p1.4.m4.2.2.cmml" xref="S5.6.p1.4.m4.2.2">ℳ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.4.m4.3c">\mathrm{Hom}_{R\Delta(N\mathcal{C})}((\mathrm{diag}\mathbb{P})_{*},\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.4.m4.3d">roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( ( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , caligraphic_M )</annotation></semantics></math> is isomorphic to the Thomason cohomology <math alttext="H^{*}_{Th}(\mathcal{C};\mathcal{M})" class="ltx_Math" display="inline" id="S5.6.p1.5.m5.2"><semantics id="S5.6.p1.5.m5.2a"><mrow id="S5.6.p1.5.m5.2.3" xref="S5.6.p1.5.m5.2.3.cmml"><msubsup id="S5.6.p1.5.m5.2.3.2" xref="S5.6.p1.5.m5.2.3.2.cmml"><mi id="S5.6.p1.5.m5.2.3.2.2.2" xref="S5.6.p1.5.m5.2.3.2.2.2.cmml">H</mi><mrow id="S5.6.p1.5.m5.2.3.2.3" xref="S5.6.p1.5.m5.2.3.2.3.cmml"><mi id="S5.6.p1.5.m5.2.3.2.3.2" xref="S5.6.p1.5.m5.2.3.2.3.2.cmml">T</mi><mo id="S5.6.p1.5.m5.2.3.2.3.1" xref="S5.6.p1.5.m5.2.3.2.3.1.cmml">⁢</mo><mi id="S5.6.p1.5.m5.2.3.2.3.3" xref="S5.6.p1.5.m5.2.3.2.3.3.cmml">h</mi></mrow><mo id="S5.6.p1.5.m5.2.3.2.2.3" xref="S5.6.p1.5.m5.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S5.6.p1.5.m5.2.3.1" xref="S5.6.p1.5.m5.2.3.1.cmml">⁢</mo><mrow id="S5.6.p1.5.m5.2.3.3.2" xref="S5.6.p1.5.m5.2.3.3.1.cmml"><mo id="S5.6.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S5.6.p1.5.m5.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.5.m5.1.1" xref="S5.6.p1.5.m5.1.1.cmml">𝒞</mi><mo id="S5.6.p1.5.m5.2.3.3.2.2" xref="S5.6.p1.5.m5.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.5.m5.2.2" xref="S5.6.p1.5.m5.2.2.cmml">ℳ</mi><mo id="S5.6.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S5.6.p1.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.5.m5.2b"><apply id="S5.6.p1.5.m5.2.3.cmml" xref="S5.6.p1.5.m5.2.3"><times id="S5.6.p1.5.m5.2.3.1.cmml" xref="S5.6.p1.5.m5.2.3.1"></times><apply id="S5.6.p1.5.m5.2.3.2.cmml" xref="S5.6.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="S5.6.p1.5.m5.2.3.2.1.cmml" xref="S5.6.p1.5.m5.2.3.2">subscript</csymbol><apply id="S5.6.p1.5.m5.2.3.2.2.cmml" xref="S5.6.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="S5.6.p1.5.m5.2.3.2.2.1.cmml" xref="S5.6.p1.5.m5.2.3.2">superscript</csymbol><ci id="S5.6.p1.5.m5.2.3.2.2.2.cmml" xref="S5.6.p1.5.m5.2.3.2.2.2">𝐻</ci><times id="S5.6.p1.5.m5.2.3.2.2.3.cmml" xref="S5.6.p1.5.m5.2.3.2.2.3"></times></apply><apply id="S5.6.p1.5.m5.2.3.2.3.cmml" xref="S5.6.p1.5.m5.2.3.2.3"><times id="S5.6.p1.5.m5.2.3.2.3.1.cmml" xref="S5.6.p1.5.m5.2.3.2.3.1"></times><ci id="S5.6.p1.5.m5.2.3.2.3.2.cmml" xref="S5.6.p1.5.m5.2.3.2.3.2">𝑇</ci><ci id="S5.6.p1.5.m5.2.3.2.3.3.cmml" xref="S5.6.p1.5.m5.2.3.2.3.3">ℎ</ci></apply></apply><list id="S5.6.p1.5.m5.2.3.3.1.cmml" xref="S5.6.p1.5.m5.2.3.3.2"><ci id="S5.6.p1.5.m5.1.1.cmml" xref="S5.6.p1.5.m5.1.1">𝒞</ci><ci id="S5.6.p1.5.m5.2.2.cmml" xref="S5.6.p1.5.m5.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.5.m5.2c">H^{*}_{Th}(\mathcal{C};\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.5.m5.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; caligraphic_M )</annotation></semantics></math>. Note that for each <math alttext="k\geq 0" class="ltx_Math" display="inline" id="S5.6.p1.6.m6.1"><semantics id="S5.6.p1.6.m6.1a"><mrow id="S5.6.p1.6.m6.1.1" xref="S5.6.p1.6.m6.1.1.cmml"><mi id="S5.6.p1.6.m6.1.1.2" xref="S5.6.p1.6.m6.1.1.2.cmml">k</mi><mo id="S5.6.p1.6.m6.1.1.1" xref="S5.6.p1.6.m6.1.1.1.cmml">≥</mo><mn id="S5.6.p1.6.m6.1.1.3" xref="S5.6.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.6.m6.1b"><apply id="S5.6.p1.6.m6.1.1.cmml" xref="S5.6.p1.6.m6.1.1"><geq id="S5.6.p1.6.m6.1.1.1.cmml" xref="S5.6.p1.6.m6.1.1.1"></geq><ci id="S5.6.p1.6.m6.1.1.2.cmml" xref="S5.6.p1.6.m6.1.1.2">𝑘</ci><cn id="S5.6.p1.6.m6.1.1.3.cmml" type="integer" xref="S5.6.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.6.m6.1c">k\geq 0</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.6.m6.1d">italic_k ≥ 0</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S7.EGx3"> <tbody id="S5.Ex61"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathrm{Hom}_{R\Delta(N\mathcal{C})}((\mathrm{diag}\mathbb{P})_{k% },\mathcal{M})" class="ltx_Math" display="inline" id="S5.Ex61.m1.3"><semantics id="S5.Ex61.m1.3a"><mrow id="S5.Ex61.m1.3.3" xref="S5.Ex61.m1.3.3.cmml"><msub id="S5.Ex61.m1.3.3.3" xref="S5.Ex61.m1.3.3.3.cmml"><mi id="S5.Ex61.m1.3.3.3.2" xref="S5.Ex61.m1.3.3.3.2.cmml">Hom</mi><mrow id="S5.Ex61.m1.1.1.1" xref="S5.Ex61.m1.1.1.1.cmml"><mi id="S5.Ex61.m1.1.1.1.3" xref="S5.Ex61.m1.1.1.1.3.cmml">R</mi><mo id="S5.Ex61.m1.1.1.1.2" xref="S5.Ex61.m1.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex61.m1.1.1.1.4" mathvariant="normal" xref="S5.Ex61.m1.1.1.1.4.cmml">Δ</mi><mo id="S5.Ex61.m1.1.1.1.2a" xref="S5.Ex61.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex61.m1.1.1.1.1.1" xref="S5.Ex61.m1.1.1.1.1.1.1.cmml"><mo id="S5.Ex61.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex61.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex61.m1.1.1.1.1.1.1" xref="S5.Ex61.m1.1.1.1.1.1.1.cmml"><mi id="S5.Ex61.m1.1.1.1.1.1.1.2" xref="S5.Ex61.m1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S5.Ex61.m1.1.1.1.1.1.1.1" xref="S5.Ex61.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex61.m1.1.1.1.1.1.1.3" xref="S5.Ex61.m1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.Ex61.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex61.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.Ex61.m1.3.3.2" xref="S5.Ex61.m1.3.3.2.cmml">⁢</mo><mrow id="S5.Ex61.m1.3.3.1.1" xref="S5.Ex61.m1.3.3.1.2.cmml"><mo id="S5.Ex61.m1.3.3.1.1.2" stretchy="false" xref="S5.Ex61.m1.3.3.1.2.cmml">(</mo><msub id="S5.Ex61.m1.3.3.1.1.1" xref="S5.Ex61.m1.3.3.1.1.1.cmml"><mrow id="S5.Ex61.m1.3.3.1.1.1.1.1" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S5.Ex61.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex61.m1.3.3.1.1.1.1.1.1" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S5.Ex61.m1.3.3.1.1.1.1.1.1.2" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.Ex61.m1.3.3.1.1.1.1.1.1.1" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex61.m1.3.3.1.1.1.1.1.1.3" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.Ex61.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.Ex61.m1.3.3.1.1.1.3" xref="S5.Ex61.m1.3.3.1.1.1.3.cmml">k</mi></msub><mo id="S5.Ex61.m1.3.3.1.1.3" xref="S5.Ex61.m1.3.3.1.2.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex61.m1.2.2" xref="S5.Ex61.m1.2.2.cmml">ℳ</mi><mo id="S5.Ex61.m1.3.3.1.1.4" stretchy="false" xref="S5.Ex61.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex61.m1.3b"><apply id="S5.Ex61.m1.3.3.cmml" xref="S5.Ex61.m1.3.3"><times id="S5.Ex61.m1.3.3.2.cmml" xref="S5.Ex61.m1.3.3.2"></times><apply id="S5.Ex61.m1.3.3.3.cmml" xref="S5.Ex61.m1.3.3.3"><csymbol cd="ambiguous" id="S5.Ex61.m1.3.3.3.1.cmml" xref="S5.Ex61.m1.3.3.3">subscript</csymbol><ci id="S5.Ex61.m1.3.3.3.2.cmml" xref="S5.Ex61.m1.3.3.3.2">Hom</ci><apply id="S5.Ex61.m1.1.1.1.cmml" xref="S5.Ex61.m1.1.1.1"><times id="S5.Ex61.m1.1.1.1.2.cmml" xref="S5.Ex61.m1.1.1.1.2"></times><ci id="S5.Ex61.m1.1.1.1.3.cmml" xref="S5.Ex61.m1.1.1.1.3">𝑅</ci><ci id="S5.Ex61.m1.1.1.1.4.cmml" xref="S5.Ex61.m1.1.1.1.4">Δ</ci><apply id="S5.Ex61.m1.1.1.1.1.1.1.cmml" xref="S5.Ex61.m1.1.1.1.1.1"><times id="S5.Ex61.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex61.m1.1.1.1.1.1.1.1"></times><ci id="S5.Ex61.m1.1.1.1.1.1.1.2.cmml" xref="S5.Ex61.m1.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.Ex61.m1.1.1.1.1.1.1.3.cmml" xref="S5.Ex61.m1.1.1.1.1.1.1.3">𝒞</ci></apply></apply></apply><interval closure="open" id="S5.Ex61.m1.3.3.1.2.cmml" xref="S5.Ex61.m1.3.3.1.1"><apply id="S5.Ex61.m1.3.3.1.1.1.cmml" xref="S5.Ex61.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.Ex61.m1.3.3.1.1.1.2.cmml" xref="S5.Ex61.m1.3.3.1.1.1">subscript</csymbol><apply id="S5.Ex61.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.Ex61.m1.3.3.1.1.1.1.1"><times id="S5.Ex61.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.1"></times><ci id="S5.Ex61.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.2">diag</ci><ci id="S5.Ex61.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S5.Ex61.m1.3.3.1.1.1.1.1.1.3">ℙ</ci></apply><ci id="S5.Ex61.m1.3.3.1.1.1.3.cmml" xref="S5.Ex61.m1.3.3.1.1.1.3">𝑘</ci></apply><ci id="S5.Ex61.m1.2.2.cmml" xref="S5.Ex61.m1.2.2">ℳ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex61.m1.3c">\displaystyle\mathrm{Hom}_{R\Delta(N\mathcal{C})}((\mathrm{diag}\mathbb{P})_{k% },\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex61.m1.3d">roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( ( roman_diag blackboard_P ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , caligraphic_M )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\cong\prod_{(\sigma,\tau,\mu)\in N(\mathcal{D};G)_{k,k}}\mathrm{% Hom}_{R\Delta(N\mathcal{C})}(R\mathrm{Mor}_{\Delta(N\mathcal{C})}(\tau,?),% \mathcal{M})" class="ltx_Math" display="inline" id="S5.Ex61.m2.13"><semantics id="S5.Ex61.m2.13a"><mrow id="S5.Ex61.m2.13.13" xref="S5.Ex61.m2.13.13.cmml"><mi id="S5.Ex61.m2.13.13.3" xref="S5.Ex61.m2.13.13.3.cmml"></mi><mo id="S5.Ex61.m2.13.13.2" xref="S5.Ex61.m2.13.13.2.cmml">≅</mo><mrow id="S5.Ex61.m2.13.13.1" xref="S5.Ex61.m2.13.13.1.cmml"><mstyle displaystyle="true" id="S5.Ex61.m2.13.13.1.2" xref="S5.Ex61.m2.13.13.1.2.cmml"><munder id="S5.Ex61.m2.13.13.1.2a" xref="S5.Ex61.m2.13.13.1.2.cmml"><mo id="S5.Ex61.m2.13.13.1.2.2" movablelimits="false" xref="S5.Ex61.m2.13.13.1.2.2.cmml">∏</mo><mrow id="S5.Ex61.m2.7.7.7" xref="S5.Ex61.m2.7.7.7.cmml"><mrow id="S5.Ex61.m2.7.7.7.9.2" xref="S5.Ex61.m2.7.7.7.9.1.cmml"><mo id="S5.Ex61.m2.7.7.7.9.2.1" stretchy="false" xref="S5.Ex61.m2.7.7.7.9.1.cmml">(</mo><mi id="S5.Ex61.m2.3.3.3.3" xref="S5.Ex61.m2.3.3.3.3.cmml">σ</mi><mo id="S5.Ex61.m2.7.7.7.9.2.2" xref="S5.Ex61.m2.7.7.7.9.1.cmml">,</mo><mi id="S5.Ex61.m2.4.4.4.4" xref="S5.Ex61.m2.4.4.4.4.cmml">τ</mi><mo id="S5.Ex61.m2.7.7.7.9.2.3" xref="S5.Ex61.m2.7.7.7.9.1.cmml">,</mo><mi id="S5.Ex61.m2.5.5.5.5" xref="S5.Ex61.m2.5.5.5.5.cmml">μ</mi><mo id="S5.Ex61.m2.7.7.7.9.2.4" stretchy="false" xref="S5.Ex61.m2.7.7.7.9.1.cmml">)</mo></mrow><mo id="S5.Ex61.m2.7.7.7.8" xref="S5.Ex61.m2.7.7.7.8.cmml">∈</mo><mrow id="S5.Ex61.m2.7.7.7.10" xref="S5.Ex61.m2.7.7.7.10.cmml"><mi id="S5.Ex61.m2.7.7.7.10.2" xref="S5.Ex61.m2.7.7.7.10.2.cmml">N</mi><mo id="S5.Ex61.m2.7.7.7.10.1" xref="S5.Ex61.m2.7.7.7.10.1.cmml">⁢</mo><msub id="S5.Ex61.m2.7.7.7.10.3" xref="S5.Ex61.m2.7.7.7.10.3.cmml"><mrow id="S5.Ex61.m2.7.7.7.10.3.2.2" xref="S5.Ex61.m2.7.7.7.10.3.2.1.cmml"><mo id="S5.Ex61.m2.7.7.7.10.3.2.2.1" stretchy="false" xref="S5.Ex61.m2.7.7.7.10.3.2.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex61.m2.6.6.6.6" xref="S5.Ex61.m2.6.6.6.6.cmml">𝒟</mi><mo id="S5.Ex61.m2.7.7.7.10.3.2.2.2" xref="S5.Ex61.m2.7.7.7.10.3.2.1.cmml">;</mo><mi id="S5.Ex61.m2.7.7.7.7" xref="S5.Ex61.m2.7.7.7.7.cmml">G</mi><mo id="S5.Ex61.m2.7.7.7.10.3.2.2.3" stretchy="false" xref="S5.Ex61.m2.7.7.7.10.3.2.1.cmml">)</mo></mrow><mrow id="S5.Ex61.m2.2.2.2.2.2.4" xref="S5.Ex61.m2.2.2.2.2.2.3.cmml"><mi id="S5.Ex61.m2.1.1.1.1.1.1" xref="S5.Ex61.m2.1.1.1.1.1.1.cmml">k</mi><mo id="S5.Ex61.m2.2.2.2.2.2.4.1" xref="S5.Ex61.m2.2.2.2.2.2.3.cmml">,</mo><mi id="S5.Ex61.m2.2.2.2.2.2.2" xref="S5.Ex61.m2.2.2.2.2.2.2.cmml">k</mi></mrow></msub></mrow></mrow></munder></mstyle><mrow id="S5.Ex61.m2.13.13.1.1" xref="S5.Ex61.m2.13.13.1.1.cmml"><msub id="S5.Ex61.m2.13.13.1.1.3" xref="S5.Ex61.m2.13.13.1.1.3.cmml"><mi id="S5.Ex61.m2.13.13.1.1.3.2" xref="S5.Ex61.m2.13.13.1.1.3.2.cmml">Hom</mi><mrow id="S5.Ex61.m2.8.8.1" xref="S5.Ex61.m2.8.8.1.cmml"><mi id="S5.Ex61.m2.8.8.1.3" xref="S5.Ex61.m2.8.8.1.3.cmml">R</mi><mo id="S5.Ex61.m2.8.8.1.2" xref="S5.Ex61.m2.8.8.1.2.cmml">⁢</mo><mi id="S5.Ex61.m2.8.8.1.4" mathvariant="normal" xref="S5.Ex61.m2.8.8.1.4.cmml">Δ</mi><mo id="S5.Ex61.m2.8.8.1.2a" xref="S5.Ex61.m2.8.8.1.2.cmml">⁢</mo><mrow id="S5.Ex61.m2.8.8.1.1.1" xref="S5.Ex61.m2.8.8.1.1.1.1.cmml"><mo id="S5.Ex61.m2.8.8.1.1.1.2" stretchy="false" xref="S5.Ex61.m2.8.8.1.1.1.1.cmml">(</mo><mrow id="S5.Ex61.m2.8.8.1.1.1.1" xref="S5.Ex61.m2.8.8.1.1.1.1.cmml"><mi id="S5.Ex61.m2.8.8.1.1.1.1.2" xref="S5.Ex61.m2.8.8.1.1.1.1.2.cmml">N</mi><mo id="S5.Ex61.m2.8.8.1.1.1.1.1" xref="S5.Ex61.m2.8.8.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex61.m2.8.8.1.1.1.1.3" xref="S5.Ex61.m2.8.8.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.Ex61.m2.8.8.1.1.1.3" stretchy="false" xref="S5.Ex61.m2.8.8.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.Ex61.m2.13.13.1.1.2" xref="S5.Ex61.m2.13.13.1.1.2.cmml">⁢</mo><mrow id="S5.Ex61.m2.13.13.1.1.1.1" xref="S5.Ex61.m2.13.13.1.1.1.2.cmml"><mo id="S5.Ex61.m2.13.13.1.1.1.1.2" stretchy="false" xref="S5.Ex61.m2.13.13.1.1.1.2.cmml">(</mo><mrow id="S5.Ex61.m2.13.13.1.1.1.1.1" xref="S5.Ex61.m2.13.13.1.1.1.1.1.cmml"><mi id="S5.Ex61.m2.13.13.1.1.1.1.1.2" xref="S5.Ex61.m2.13.13.1.1.1.1.1.2.cmml">R</mi><mo id="S5.Ex61.m2.13.13.1.1.1.1.1.1" xref="S5.Ex61.m2.13.13.1.1.1.1.1.1.cmml">⁢</mo><msub id="S5.Ex61.m2.13.13.1.1.1.1.1.3" xref="S5.Ex61.m2.13.13.1.1.1.1.1.3.cmml"><mi id="S5.Ex61.m2.13.13.1.1.1.1.1.3.2" xref="S5.Ex61.m2.13.13.1.1.1.1.1.3.2.cmml">Mor</mi><mrow id="S5.Ex61.m2.9.9.1" xref="S5.Ex61.m2.9.9.1.cmml"><mi id="S5.Ex61.m2.9.9.1.3" mathvariant="normal" xref="S5.Ex61.m2.9.9.1.3.cmml">Δ</mi><mo id="S5.Ex61.m2.9.9.1.2" xref="S5.Ex61.m2.9.9.1.2.cmml">⁢</mo><mrow id="S5.Ex61.m2.9.9.1.1.1" xref="S5.Ex61.m2.9.9.1.1.1.1.cmml"><mo id="S5.Ex61.m2.9.9.1.1.1.2" stretchy="false" xref="S5.Ex61.m2.9.9.1.1.1.1.cmml">(</mo><mrow id="S5.Ex61.m2.9.9.1.1.1.1" 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\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex61.m2.13d">≅ ∏ start_POSTSUBSCRIPT ( italic_σ , italic_τ , italic_μ ) ∈ italic_N ( caligraphic_D ; italic_G ) start_POSTSUBSCRIPT italic_k , italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( italic_R roman_Mor start_POSTSUBSCRIPT roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( italic_τ , ? ) , caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex62"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\cong\prod_{(\sigma,\tau,\mu)\in N(\mathcal{D};G)_{k,k}}\mathcal{% M}(\tau)" class="ltx_Math" display="inline" id="S5.Ex62.m1.8"><semantics 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xref="S5.Ex62.m1.5.5.5.5.cmml">μ</mi><mo id="S5.Ex62.m1.7.7.7.9.2.4" stretchy="false" xref="S5.Ex62.m1.7.7.7.9.1.cmml">)</mo></mrow><mo id="S5.Ex62.m1.7.7.7.8" xref="S5.Ex62.m1.7.7.7.8.cmml">∈</mo><mrow id="S5.Ex62.m1.7.7.7.10" xref="S5.Ex62.m1.7.7.7.10.cmml"><mi id="S5.Ex62.m1.7.7.7.10.2" xref="S5.Ex62.m1.7.7.7.10.2.cmml">N</mi><mo id="S5.Ex62.m1.7.7.7.10.1" xref="S5.Ex62.m1.7.7.7.10.1.cmml">⁢</mo><msub id="S5.Ex62.m1.7.7.7.10.3" xref="S5.Ex62.m1.7.7.7.10.3.cmml"><mrow id="S5.Ex62.m1.7.7.7.10.3.2.2" xref="S5.Ex62.m1.7.7.7.10.3.2.1.cmml"><mo id="S5.Ex62.m1.7.7.7.10.3.2.2.1" stretchy="false" xref="S5.Ex62.m1.7.7.7.10.3.2.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex62.m1.6.6.6.6" xref="S5.Ex62.m1.6.6.6.6.cmml">𝒟</mi><mo id="S5.Ex62.m1.7.7.7.10.3.2.2.2" xref="S5.Ex62.m1.7.7.7.10.3.2.1.cmml">;</mo><mi id="S5.Ex62.m1.7.7.7.7" xref="S5.Ex62.m1.7.7.7.7.cmml">G</mi><mo id="S5.Ex62.m1.7.7.7.10.3.2.2.3" stretchy="false" xref="S5.Ex62.m1.7.7.7.10.3.2.1.cmml">)</mo></mrow><mrow 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xref="S5.Ex62.m1.7.7.7.10.1"></times><ci id="S5.Ex62.m1.7.7.7.10.2.cmml" xref="S5.Ex62.m1.7.7.7.10.2">𝑁</ci><apply id="S5.Ex62.m1.7.7.7.10.3.cmml" xref="S5.Ex62.m1.7.7.7.10.3"><csymbol cd="ambiguous" id="S5.Ex62.m1.7.7.7.10.3.1.cmml" xref="S5.Ex62.m1.7.7.7.10.3">subscript</csymbol><list id="S5.Ex62.m1.7.7.7.10.3.2.1.cmml" xref="S5.Ex62.m1.7.7.7.10.3.2.2"><ci id="S5.Ex62.m1.6.6.6.6.cmml" xref="S5.Ex62.m1.6.6.6.6">𝒟</ci><ci id="S5.Ex62.m1.7.7.7.7.cmml" xref="S5.Ex62.m1.7.7.7.7">𝐺</ci></list><list id="S5.Ex62.m1.2.2.2.2.2.3.cmml" xref="S5.Ex62.m1.2.2.2.2.2.4"><ci id="S5.Ex62.m1.1.1.1.1.1.1.cmml" xref="S5.Ex62.m1.1.1.1.1.1.1">𝑘</ci><ci id="S5.Ex62.m1.2.2.2.2.2.2.cmml" xref="S5.Ex62.m1.2.2.2.2.2.2">𝑘</ci></list></apply></apply></apply></apply><apply id="S5.Ex62.m1.8.9.3.2.cmml" xref="S5.Ex62.m1.8.9.3.2"><times id="S5.Ex62.m1.8.9.3.2.1.cmml" xref="S5.Ex62.m1.8.9.3.2.1"></times><ci id="S5.Ex62.m1.8.9.3.2.2.cmml" xref="S5.Ex62.m1.8.9.3.2.2">ℳ</ci><ci id="S5.Ex62.m1.8.8.cmml" xref="S5.Ex62.m1.8.8">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex62.m1.8c">\displaystyle\cong\prod_{(\sigma,\tau,\mu)\in N(\mathcal{D};G)_{k,k}}\mathcal{% M}(\tau)</annotation><annotation encoding="application/x-llamapun" id="S5.Ex62.m1.8d">≅ ∏ start_POSTSUBSCRIPT ( italic_σ , italic_τ , italic_μ ) ∈ italic_N ( caligraphic_D ; italic_G ) start_POSTSUBSCRIPT italic_k , italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_M ( italic_τ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex63"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\cong C^{k}(\mathrm{diag}N(\mathcal{D};G);\kappa^{*}(\mathcal{M}))." class="ltx_Math" display="inline" id="S5.Ex63.m1.4"><semantics 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xref="S5.Ex63.m1.3.3">ℳ</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex63.m1.4c">\displaystyle\cong C^{k}(\mathrm{diag}N(\mathcal{D};G);\kappa^{*}(\mathcal{M})).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex63.m1.4d">≅ italic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( roman_diag italic_N ( caligraphic_D ; italic_G ) ; italic_κ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_M ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.6.p1.12">Let <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S5.6.p1.7.m1.1"><semantics id="S5.6.p1.7.m1.1a"><mi id="S5.6.p1.7.m1.1.1" xref="S5.6.p1.7.m1.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S5.6.p1.7.m1.1b"><ci id="S5.6.p1.7.m1.1.1.cmml" xref="S5.6.p1.7.m1.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.7.m1.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.7.m1.1d">blackboard_Q</annotation></semantics></math> be the bisimplicial <math alttext="R\Delta(N\mathcal{C})" class="ltx_Math" display="inline" id="S5.6.p1.8.m2.1"><semantics id="S5.6.p1.8.m2.1a"><mrow id="S5.6.p1.8.m2.1.1" xref="S5.6.p1.8.m2.1.1.cmml"><mi id="S5.6.p1.8.m2.1.1.3" xref="S5.6.p1.8.m2.1.1.3.cmml">R</mi><mo id="S5.6.p1.8.m2.1.1.2" xref="S5.6.p1.8.m2.1.1.2.cmml">⁢</mo><mi id="S5.6.p1.8.m2.1.1.4" mathvariant="normal" xref="S5.6.p1.8.m2.1.1.4.cmml">Δ</mi><mo id="S5.6.p1.8.m2.1.1.2a" xref="S5.6.p1.8.m2.1.1.2.cmml">⁢</mo><mrow id="S5.6.p1.8.m2.1.1.1.1" xref="S5.6.p1.8.m2.1.1.1.1.1.cmml"><mo id="S5.6.p1.8.m2.1.1.1.1.2" stretchy="false" xref="S5.6.p1.8.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.6.p1.8.m2.1.1.1.1.1" xref="S5.6.p1.8.m2.1.1.1.1.1.cmml"><mi id="S5.6.p1.8.m2.1.1.1.1.1.2" xref="S5.6.p1.8.m2.1.1.1.1.1.2.cmml">N</mi><mo id="S5.6.p1.8.m2.1.1.1.1.1.1" xref="S5.6.p1.8.m2.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.8.m2.1.1.1.1.1.3" xref="S5.6.p1.8.m2.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.6.p1.8.m2.1.1.1.1.3" stretchy="false" xref="S5.6.p1.8.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.8.m2.1b"><apply id="S5.6.p1.8.m2.1.1.cmml" xref="S5.6.p1.8.m2.1.1"><times id="S5.6.p1.8.m2.1.1.2.cmml" xref="S5.6.p1.8.m2.1.1.2"></times><ci id="S5.6.p1.8.m2.1.1.3.cmml" xref="S5.6.p1.8.m2.1.1.3">𝑅</ci><ci id="S5.6.p1.8.m2.1.1.4.cmml" xref="S5.6.p1.8.m2.1.1.4">Δ</ci><apply id="S5.6.p1.8.m2.1.1.1.1.1.cmml" xref="S5.6.p1.8.m2.1.1.1.1"><times id="S5.6.p1.8.m2.1.1.1.1.1.1.cmml" xref="S5.6.p1.8.m2.1.1.1.1.1.1"></times><ci id="S5.6.p1.8.m2.1.1.1.1.1.2.cmml" xref="S5.6.p1.8.m2.1.1.1.1.1.2">𝑁</ci><ci id="S5.6.p1.8.m2.1.1.1.1.1.3.cmml" xref="S5.6.p1.8.m2.1.1.1.1.1.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.8.m2.1c">R\Delta(N\mathcal{C})</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.8.m2.1d">italic_R roman_Δ ( italic_N caligraphic_C )</annotation></semantics></math>-module introduced in the proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S5.Thmtheorem2" title="Proposition 5.2. ‣ 5. Proof of Theorem 1.2 ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">5.2</span></a>, and let <math alttext="\xi:\mathbb{P}\to\mathbb{Q}" class="ltx_Math" display="inline" id="S5.6.p1.9.m3.1"><semantics id="S5.6.p1.9.m3.1a"><mrow id="S5.6.p1.9.m3.1.1" xref="S5.6.p1.9.m3.1.1.cmml"><mi id="S5.6.p1.9.m3.1.1.2" xref="S5.6.p1.9.m3.1.1.2.cmml">ξ</mi><mo id="S5.6.p1.9.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.6.p1.9.m3.1.1.1.cmml">:</mo><mrow id="S5.6.p1.9.m3.1.1.3" xref="S5.6.p1.9.m3.1.1.3.cmml"><mi id="S5.6.p1.9.m3.1.1.3.2" xref="S5.6.p1.9.m3.1.1.3.2.cmml">ℙ</mi><mo id="S5.6.p1.9.m3.1.1.3.1" stretchy="false" xref="S5.6.p1.9.m3.1.1.3.1.cmml">→</mo><mi id="S5.6.p1.9.m3.1.1.3.3" xref="S5.6.p1.9.m3.1.1.3.3.cmml">ℚ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.9.m3.1b"><apply id="S5.6.p1.9.m3.1.1.cmml" xref="S5.6.p1.9.m3.1.1"><ci id="S5.6.p1.9.m3.1.1.1.cmml" xref="S5.6.p1.9.m3.1.1.1">:</ci><ci id="S5.6.p1.9.m3.1.1.2.cmml" xref="S5.6.p1.9.m3.1.1.2">𝜉</ci><apply id="S5.6.p1.9.m3.1.1.3.cmml" xref="S5.6.p1.9.m3.1.1.3"><ci id="S5.6.p1.9.m3.1.1.3.1.cmml" xref="S5.6.p1.9.m3.1.1.3.1">→</ci><ci id="S5.6.p1.9.m3.1.1.3.2.cmml" xref="S5.6.p1.9.m3.1.1.3.2">ℙ</ci><ci id="S5.6.p1.9.m3.1.1.3.3.cmml" xref="S5.6.p1.9.m3.1.1.3.3">ℚ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.9.m3.1c">\xi:\mathbb{P}\to\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.9.m3.1d">italic_ξ : blackboard_P → blackboard_Q</annotation></semantics></math> be the bisimplicial map defined by <math alttext="\xi_{p,q}(\theta)(\sigma,\tau,\mu,f)=(\tau,f)" class="ltx_Math" display="inline" id="S5.6.p1.10.m4.9"><semantics id="S5.6.p1.10.m4.9a"><mrow id="S5.6.p1.10.m4.9.10" xref="S5.6.p1.10.m4.9.10.cmml"><mrow id="S5.6.p1.10.m4.9.10.2" xref="S5.6.p1.10.m4.9.10.2.cmml"><msub id="S5.6.p1.10.m4.9.10.2.2" xref="S5.6.p1.10.m4.9.10.2.2.cmml"><mi id="S5.6.p1.10.m4.9.10.2.2.2" xref="S5.6.p1.10.m4.9.10.2.2.2.cmml">ξ</mi><mrow id="S5.6.p1.10.m4.2.2.2.4" xref="S5.6.p1.10.m4.2.2.2.3.cmml"><mi id="S5.6.p1.10.m4.1.1.1.1" xref="S5.6.p1.10.m4.1.1.1.1.cmml">p</mi><mo id="S5.6.p1.10.m4.2.2.2.4.1" xref="S5.6.p1.10.m4.2.2.2.3.cmml">,</mo><mi id="S5.6.p1.10.m4.2.2.2.2" xref="S5.6.p1.10.m4.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.6.p1.10.m4.9.10.2.1" xref="S5.6.p1.10.m4.9.10.2.1.cmml">⁢</mo><mrow id="S5.6.p1.10.m4.9.10.2.3.2" xref="S5.6.p1.10.m4.9.10.2.cmml"><mo id="S5.6.p1.10.m4.9.10.2.3.2.1" stretchy="false" xref="S5.6.p1.10.m4.9.10.2.cmml">(</mo><mi id="S5.6.p1.10.m4.3.3" xref="S5.6.p1.10.m4.3.3.cmml">θ</mi><mo id="S5.6.p1.10.m4.9.10.2.3.2.2" stretchy="false" xref="S5.6.p1.10.m4.9.10.2.cmml">)</mo></mrow><mo id="S5.6.p1.10.m4.9.10.2.1a" xref="S5.6.p1.10.m4.9.10.2.1.cmml">⁢</mo><mrow id="S5.6.p1.10.m4.9.10.2.4.2" xref="S5.6.p1.10.m4.9.10.2.4.1.cmml"><mo id="S5.6.p1.10.m4.9.10.2.4.2.1" stretchy="false" xref="S5.6.p1.10.m4.9.10.2.4.1.cmml">(</mo><mi id="S5.6.p1.10.m4.4.4" xref="S5.6.p1.10.m4.4.4.cmml">σ</mi><mo id="S5.6.p1.10.m4.9.10.2.4.2.2" xref="S5.6.p1.10.m4.9.10.2.4.1.cmml">,</mo><mi id="S5.6.p1.10.m4.5.5" xref="S5.6.p1.10.m4.5.5.cmml">τ</mi><mo id="S5.6.p1.10.m4.9.10.2.4.2.3" xref="S5.6.p1.10.m4.9.10.2.4.1.cmml">,</mo><mi id="S5.6.p1.10.m4.6.6" xref="S5.6.p1.10.m4.6.6.cmml">μ</mi><mo id="S5.6.p1.10.m4.9.10.2.4.2.4" xref="S5.6.p1.10.m4.9.10.2.4.1.cmml">,</mo><mi id="S5.6.p1.10.m4.7.7" xref="S5.6.p1.10.m4.7.7.cmml">f</mi><mo id="S5.6.p1.10.m4.9.10.2.4.2.5" stretchy="false" xref="S5.6.p1.10.m4.9.10.2.4.1.cmml">)</mo></mrow></mrow><mo id="S5.6.p1.10.m4.9.10.1" xref="S5.6.p1.10.m4.9.10.1.cmml">=</mo><mrow id="S5.6.p1.10.m4.9.10.3.2" xref="S5.6.p1.10.m4.9.10.3.1.cmml"><mo id="S5.6.p1.10.m4.9.10.3.2.1" stretchy="false" xref="S5.6.p1.10.m4.9.10.3.1.cmml">(</mo><mi id="S5.6.p1.10.m4.8.8" xref="S5.6.p1.10.m4.8.8.cmml">τ</mi><mo id="S5.6.p1.10.m4.9.10.3.2.2" xref="S5.6.p1.10.m4.9.10.3.1.cmml">,</mo><mi id="S5.6.p1.10.m4.9.9" xref="S5.6.p1.10.m4.9.9.cmml">f</mi><mo id="S5.6.p1.10.m4.9.10.3.2.3" stretchy="false" xref="S5.6.p1.10.m4.9.10.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.10.m4.9b"><apply id="S5.6.p1.10.m4.9.10.cmml" xref="S5.6.p1.10.m4.9.10"><eq id="S5.6.p1.10.m4.9.10.1.cmml" xref="S5.6.p1.10.m4.9.10.1"></eq><apply id="S5.6.p1.10.m4.9.10.2.cmml" xref="S5.6.p1.10.m4.9.10.2"><times id="S5.6.p1.10.m4.9.10.2.1.cmml" xref="S5.6.p1.10.m4.9.10.2.1"></times><apply id="S5.6.p1.10.m4.9.10.2.2.cmml" xref="S5.6.p1.10.m4.9.10.2.2"><csymbol cd="ambiguous" id="S5.6.p1.10.m4.9.10.2.2.1.cmml" xref="S5.6.p1.10.m4.9.10.2.2">subscript</csymbol><ci id="S5.6.p1.10.m4.9.10.2.2.2.cmml" xref="S5.6.p1.10.m4.9.10.2.2.2">𝜉</ci><list id="S5.6.p1.10.m4.2.2.2.3.cmml" xref="S5.6.p1.10.m4.2.2.2.4"><ci id="S5.6.p1.10.m4.1.1.1.1.cmml" xref="S5.6.p1.10.m4.1.1.1.1">𝑝</ci><ci id="S5.6.p1.10.m4.2.2.2.2.cmml" xref="S5.6.p1.10.m4.2.2.2.2">𝑞</ci></list></apply><ci id="S5.6.p1.10.m4.3.3.cmml" xref="S5.6.p1.10.m4.3.3">𝜃</ci><vector id="S5.6.p1.10.m4.9.10.2.4.1.cmml" xref="S5.6.p1.10.m4.9.10.2.4.2"><ci id="S5.6.p1.10.m4.4.4.cmml" xref="S5.6.p1.10.m4.4.4">𝜎</ci><ci id="S5.6.p1.10.m4.5.5.cmml" xref="S5.6.p1.10.m4.5.5">𝜏</ci><ci id="S5.6.p1.10.m4.6.6.cmml" xref="S5.6.p1.10.m4.6.6">𝜇</ci><ci id="S5.6.p1.10.m4.7.7.cmml" xref="S5.6.p1.10.m4.7.7">𝑓</ci></vector></apply><interval closure="open" id="S5.6.p1.10.m4.9.10.3.1.cmml" xref="S5.6.p1.10.m4.9.10.3.2"><ci id="S5.6.p1.10.m4.8.8.cmml" xref="S5.6.p1.10.m4.8.8">𝜏</ci><ci id="S5.6.p1.10.m4.9.9.cmml" xref="S5.6.p1.10.m4.9.9">𝑓</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.10.m4.9c">\xi_{p,q}(\theta)(\sigma,\tau,\mu,f)=(\tau,f)</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.10.m4.9d">italic_ξ start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( italic_θ ) ( italic_σ , italic_τ , italic_μ , italic_f ) = ( italic_τ , italic_f )</annotation></semantics></math> for every <math alttext="(\sigma,\tau,\mu,f)\in X_{p,q}(\theta)" class="ltx_Math" display="inline" id="S5.6.p1.11.m5.7"><semantics id="S5.6.p1.11.m5.7a"><mrow id="S5.6.p1.11.m5.7.8" xref="S5.6.p1.11.m5.7.8.cmml"><mrow id="S5.6.p1.11.m5.7.8.2.2" xref="S5.6.p1.11.m5.7.8.2.1.cmml"><mo id="S5.6.p1.11.m5.7.8.2.2.1" stretchy="false" xref="S5.6.p1.11.m5.7.8.2.1.cmml">(</mo><mi id="S5.6.p1.11.m5.3.3" xref="S5.6.p1.11.m5.3.3.cmml">σ</mi><mo id="S5.6.p1.11.m5.7.8.2.2.2" xref="S5.6.p1.11.m5.7.8.2.1.cmml">,</mo><mi id="S5.6.p1.11.m5.4.4" xref="S5.6.p1.11.m5.4.4.cmml">τ</mi><mo id="S5.6.p1.11.m5.7.8.2.2.3" xref="S5.6.p1.11.m5.7.8.2.1.cmml">,</mo><mi id="S5.6.p1.11.m5.5.5" xref="S5.6.p1.11.m5.5.5.cmml">μ</mi><mo id="S5.6.p1.11.m5.7.8.2.2.4" xref="S5.6.p1.11.m5.7.8.2.1.cmml">,</mo><mi id="S5.6.p1.11.m5.6.6" xref="S5.6.p1.11.m5.6.6.cmml">f</mi><mo id="S5.6.p1.11.m5.7.8.2.2.5" stretchy="false" xref="S5.6.p1.11.m5.7.8.2.1.cmml">)</mo></mrow><mo id="S5.6.p1.11.m5.7.8.1" xref="S5.6.p1.11.m5.7.8.1.cmml">∈</mo><mrow id="S5.6.p1.11.m5.7.8.3" xref="S5.6.p1.11.m5.7.8.3.cmml"><msub id="S5.6.p1.11.m5.7.8.3.2" xref="S5.6.p1.11.m5.7.8.3.2.cmml"><mi id="S5.6.p1.11.m5.7.8.3.2.2" xref="S5.6.p1.11.m5.7.8.3.2.2.cmml">X</mi><mrow id="S5.6.p1.11.m5.2.2.2.4" xref="S5.6.p1.11.m5.2.2.2.3.cmml"><mi id="S5.6.p1.11.m5.1.1.1.1" xref="S5.6.p1.11.m5.1.1.1.1.cmml">p</mi><mo id="S5.6.p1.11.m5.2.2.2.4.1" xref="S5.6.p1.11.m5.2.2.2.3.cmml">,</mo><mi id="S5.6.p1.11.m5.2.2.2.2" xref="S5.6.p1.11.m5.2.2.2.2.cmml">q</mi></mrow></msub><mo id="S5.6.p1.11.m5.7.8.3.1" xref="S5.6.p1.11.m5.7.8.3.1.cmml">⁢</mo><mrow id="S5.6.p1.11.m5.7.8.3.3.2" xref="S5.6.p1.11.m5.7.8.3.cmml"><mo id="S5.6.p1.11.m5.7.8.3.3.2.1" stretchy="false" xref="S5.6.p1.11.m5.7.8.3.cmml">(</mo><mi id="S5.6.p1.11.m5.7.7" xref="S5.6.p1.11.m5.7.7.cmml">θ</mi><mo id="S5.6.p1.11.m5.7.8.3.3.2.2" stretchy="false" xref="S5.6.p1.11.m5.7.8.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.11.m5.7b"><apply id="S5.6.p1.11.m5.7.8.cmml" xref="S5.6.p1.11.m5.7.8"><in id="S5.6.p1.11.m5.7.8.1.cmml" xref="S5.6.p1.11.m5.7.8.1"></in><vector id="S5.6.p1.11.m5.7.8.2.1.cmml" xref="S5.6.p1.11.m5.7.8.2.2"><ci id="S5.6.p1.11.m5.3.3.cmml" xref="S5.6.p1.11.m5.3.3">𝜎</ci><ci id="S5.6.p1.11.m5.4.4.cmml" xref="S5.6.p1.11.m5.4.4">𝜏</ci><ci id="S5.6.p1.11.m5.5.5.cmml" xref="S5.6.p1.11.m5.5.5">𝜇</ci><ci id="S5.6.p1.11.m5.6.6.cmml" xref="S5.6.p1.11.m5.6.6">𝑓</ci></vector><apply id="S5.6.p1.11.m5.7.8.3.cmml" xref="S5.6.p1.11.m5.7.8.3"><times id="S5.6.p1.11.m5.7.8.3.1.cmml" xref="S5.6.p1.11.m5.7.8.3.1"></times><apply id="S5.6.p1.11.m5.7.8.3.2.cmml" xref="S5.6.p1.11.m5.7.8.3.2"><csymbol cd="ambiguous" id="S5.6.p1.11.m5.7.8.3.2.1.cmml" xref="S5.6.p1.11.m5.7.8.3.2">subscript</csymbol><ci id="S5.6.p1.11.m5.7.8.3.2.2.cmml" xref="S5.6.p1.11.m5.7.8.3.2.2">𝑋</ci><list id="S5.6.p1.11.m5.2.2.2.3.cmml" xref="S5.6.p1.11.m5.2.2.2.4"><ci id="S5.6.p1.11.m5.1.1.1.1.cmml" xref="S5.6.p1.11.m5.1.1.1.1">𝑝</ci><ci id="S5.6.p1.11.m5.2.2.2.2.cmml" xref="S5.6.p1.11.m5.2.2.2.2">𝑞</ci></list></apply><ci id="S5.6.p1.11.m5.7.7.cmml" xref="S5.6.p1.11.m5.7.7">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.11.m5.7c">(\sigma,\tau,\mu,f)\in X_{p,q}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.11.m5.7d">( italic_σ , italic_τ , italic_μ , italic_f ) ∈ italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math>. There is an isomorphism for <math alttext="\mathrm{diag}\mathbb{Q}" class="ltx_Math" display="inline" id="S5.6.p1.12.m6.1"><semantics id="S5.6.p1.12.m6.1a"><mrow id="S5.6.p1.12.m6.1.1" xref="S5.6.p1.12.m6.1.1.cmml"><mi id="S5.6.p1.12.m6.1.1.2" xref="S5.6.p1.12.m6.1.1.2.cmml">diag</mi><mo id="S5.6.p1.12.m6.1.1.1" xref="S5.6.p1.12.m6.1.1.1.cmml">⁢</mo><mi id="S5.6.p1.12.m6.1.1.3" xref="S5.6.p1.12.m6.1.1.3.cmml">ℚ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.12.m6.1b"><apply id="S5.6.p1.12.m6.1.1.cmml" xref="S5.6.p1.12.m6.1.1"><times id="S5.6.p1.12.m6.1.1.1.cmml" xref="S5.6.p1.12.m6.1.1.1"></times><ci id="S5.6.p1.12.m6.1.1.2.cmml" xref="S5.6.p1.12.m6.1.1.2">diag</ci><ci id="S5.6.p1.12.m6.1.1.3.cmml" xref="S5.6.p1.12.m6.1.1.3">ℚ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.12.m6.1c">\mathrm{diag}\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.12.m6.1d">roman_diag blackboard_Q</annotation></semantics></math> similar to the above isomorphism:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S7.EGx4"> <tbody id="S5.Ex64"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathrm{Hom}_{R\Delta(N\mathcal{C})}((\mathrm{diag}\mathbb{Q})_{k% },\mathcal{M})" class="ltx_Math" display="inline" id="S5.Ex64.m1.3"><semantics id="S5.Ex64.m1.3a"><mrow id="S5.Ex64.m1.3.3" xref="S5.Ex64.m1.3.3.cmml"><msub id="S5.Ex64.m1.3.3.3" xref="S5.Ex64.m1.3.3.3.cmml"><mi id="S5.Ex64.m1.3.3.3.2" xref="S5.Ex64.m1.3.3.3.2.cmml">Hom</mi><mrow id="S5.Ex64.m1.1.1.1" xref="S5.Ex64.m1.1.1.1.cmml"><mi id="S5.Ex64.m1.1.1.1.3" xref="S5.Ex64.m1.1.1.1.3.cmml">R</mi><mo id="S5.Ex64.m1.1.1.1.2" xref="S5.Ex64.m1.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex64.m1.1.1.1.4" mathvariant="normal" xref="S5.Ex64.m1.1.1.1.4.cmml">Δ</mi><mo id="S5.Ex64.m1.1.1.1.2a" xref="S5.Ex64.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex64.m1.1.1.1.1.1" xref="S5.Ex64.m1.1.1.1.1.1.1.cmml"><mo id="S5.Ex64.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex64.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex64.m1.1.1.1.1.1.1" xref="S5.Ex64.m1.1.1.1.1.1.1.cmml"><mi id="S5.Ex64.m1.1.1.1.1.1.1.2" xref="S5.Ex64.m1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S5.Ex64.m1.1.1.1.1.1.1.1" xref="S5.Ex64.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex64.m1.1.1.1.1.1.1.3" xref="S5.Ex64.m1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.Ex64.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex64.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.Ex64.m1.3.3.2" xref="S5.Ex64.m1.3.3.2.cmml">⁢</mo><mrow id="S5.Ex64.m1.3.3.1.1" xref="S5.Ex64.m1.3.3.1.2.cmml"><mo id="S5.Ex64.m1.3.3.1.1.2" stretchy="false" xref="S5.Ex64.m1.3.3.1.2.cmml">(</mo><msub id="S5.Ex64.m1.3.3.1.1.1" xref="S5.Ex64.m1.3.3.1.1.1.cmml"><mrow id="S5.Ex64.m1.3.3.1.1.1.1.1" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S5.Ex64.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex64.m1.3.3.1.1.1.1.1.1" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S5.Ex64.m1.3.3.1.1.1.1.1.1.2" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.Ex64.m1.3.3.1.1.1.1.1.1.1" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex64.m1.3.3.1.1.1.1.1.1.3" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.3.cmml">ℚ</mi></mrow><mo id="S5.Ex64.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.Ex64.m1.3.3.1.1.1.3" xref="S5.Ex64.m1.3.3.1.1.1.3.cmml">k</mi></msub><mo id="S5.Ex64.m1.3.3.1.1.3" xref="S5.Ex64.m1.3.3.1.2.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex64.m1.2.2" xref="S5.Ex64.m1.2.2.cmml">ℳ</mi><mo id="S5.Ex64.m1.3.3.1.1.4" stretchy="false" xref="S5.Ex64.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex64.m1.3b"><apply id="S5.Ex64.m1.3.3.cmml" xref="S5.Ex64.m1.3.3"><times id="S5.Ex64.m1.3.3.2.cmml" xref="S5.Ex64.m1.3.3.2"></times><apply id="S5.Ex64.m1.3.3.3.cmml" xref="S5.Ex64.m1.3.3.3"><csymbol cd="ambiguous" id="S5.Ex64.m1.3.3.3.1.cmml" xref="S5.Ex64.m1.3.3.3">subscript</csymbol><ci id="S5.Ex64.m1.3.3.3.2.cmml" xref="S5.Ex64.m1.3.3.3.2">Hom</ci><apply id="S5.Ex64.m1.1.1.1.cmml" xref="S5.Ex64.m1.1.1.1"><times id="S5.Ex64.m1.1.1.1.2.cmml" xref="S5.Ex64.m1.1.1.1.2"></times><ci id="S5.Ex64.m1.1.1.1.3.cmml" xref="S5.Ex64.m1.1.1.1.3">𝑅</ci><ci id="S5.Ex64.m1.1.1.1.4.cmml" xref="S5.Ex64.m1.1.1.1.4">Δ</ci><apply id="S5.Ex64.m1.1.1.1.1.1.1.cmml" xref="S5.Ex64.m1.1.1.1.1.1"><times id="S5.Ex64.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex64.m1.1.1.1.1.1.1.1"></times><ci id="S5.Ex64.m1.1.1.1.1.1.1.2.cmml" xref="S5.Ex64.m1.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.Ex64.m1.1.1.1.1.1.1.3.cmml" xref="S5.Ex64.m1.1.1.1.1.1.1.3">𝒞</ci></apply></apply></apply><interval closure="open" id="S5.Ex64.m1.3.3.1.2.cmml" xref="S5.Ex64.m1.3.3.1.1"><apply id="S5.Ex64.m1.3.3.1.1.1.cmml" xref="S5.Ex64.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.Ex64.m1.3.3.1.1.1.2.cmml" xref="S5.Ex64.m1.3.3.1.1.1">subscript</csymbol><apply id="S5.Ex64.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.Ex64.m1.3.3.1.1.1.1.1"><times id="S5.Ex64.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.1"></times><ci id="S5.Ex64.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.2">diag</ci><ci id="S5.Ex64.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S5.Ex64.m1.3.3.1.1.1.1.1.1.3">ℚ</ci></apply><ci id="S5.Ex64.m1.3.3.1.1.1.3.cmml" xref="S5.Ex64.m1.3.3.1.1.1.3">𝑘</ci></apply><ci id="S5.Ex64.m1.2.2.cmml" xref="S5.Ex64.m1.2.2">ℳ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex64.m1.3c">\displaystyle\mathrm{Hom}_{R\Delta(N\mathcal{C})}((\mathrm{diag}\mathbb{Q})_{k% },\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex64.m1.3d">roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( ( roman_diag blackboard_Q ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , caligraphic_M )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\cong\prod_{\tau\in N\mathcal{C}_{q}}\mathrm{Hom}_{R\Delta(N% \mathcal{C})}(R\mathrm{Mor}_{\Delta(N\mathcal{C})}(\tau,?),\mathcal{M})" class="ltx_Math" display="inline" id="S5.Ex64.m2.6"><semantics id="S5.Ex64.m2.6a"><mrow id="S5.Ex64.m2.6.6" xref="S5.Ex64.m2.6.6.cmml"><mi id="S5.Ex64.m2.6.6.3" xref="S5.Ex64.m2.6.6.3.cmml"></mi><mo id="S5.Ex64.m2.6.6.2" xref="S5.Ex64.m2.6.6.2.cmml">≅</mo><mrow id="S5.Ex64.m2.6.6.1" xref="S5.Ex64.m2.6.6.1.cmml"><mstyle displaystyle="true" id="S5.Ex64.m2.6.6.1.2" xref="S5.Ex64.m2.6.6.1.2.cmml"><munder id="S5.Ex64.m2.6.6.1.2a" xref="S5.Ex64.m2.6.6.1.2.cmml"><mo id="S5.Ex64.m2.6.6.1.2.2" movablelimits="false" xref="S5.Ex64.m2.6.6.1.2.2.cmml">∏</mo><mrow id="S5.Ex64.m2.6.6.1.2.3" xref="S5.Ex64.m2.6.6.1.2.3.cmml"><mi id="S5.Ex64.m2.6.6.1.2.3.2" xref="S5.Ex64.m2.6.6.1.2.3.2.cmml">τ</mi><mo id="S5.Ex64.m2.6.6.1.2.3.1" xref="S5.Ex64.m2.6.6.1.2.3.1.cmml">∈</mo><mrow id="S5.Ex64.m2.6.6.1.2.3.3" xref="S5.Ex64.m2.6.6.1.2.3.3.cmml"><mi id="S5.Ex64.m2.6.6.1.2.3.3.2" xref="S5.Ex64.m2.6.6.1.2.3.3.2.cmml">N</mi><mo id="S5.Ex64.m2.6.6.1.2.3.3.1" xref="S5.Ex64.m2.6.6.1.2.3.3.1.cmml">⁢</mo><msub id="S5.Ex64.m2.6.6.1.2.3.3.3" xref="S5.Ex64.m2.6.6.1.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex64.m2.6.6.1.2.3.3.3.2" xref="S5.Ex64.m2.6.6.1.2.3.3.3.2.cmml">𝒞</mi><mi id="S5.Ex64.m2.6.6.1.2.3.3.3.3" xref="S5.Ex64.m2.6.6.1.2.3.3.3.3.cmml">q</mi></msub></mrow></mrow></munder></mstyle><mrow id="S5.Ex64.m2.6.6.1.1" xref="S5.Ex64.m2.6.6.1.1.cmml"><msub id="S5.Ex64.m2.6.6.1.1.3" xref="S5.Ex64.m2.6.6.1.1.3.cmml"><mi id="S5.Ex64.m2.6.6.1.1.3.2" xref="S5.Ex64.m2.6.6.1.1.3.2.cmml">Hom</mi><mrow id="S5.Ex64.m2.1.1.1" xref="S5.Ex64.m2.1.1.1.cmml"><mi id="S5.Ex64.m2.1.1.1.3" xref="S5.Ex64.m2.1.1.1.3.cmml">R</mi><mo id="S5.Ex64.m2.1.1.1.2" xref="S5.Ex64.m2.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex64.m2.1.1.1.4" mathvariant="normal" xref="S5.Ex64.m2.1.1.1.4.cmml">Δ</mi><mo id="S5.Ex64.m2.1.1.1.2a" xref="S5.Ex64.m2.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex64.m2.1.1.1.1.1" xref="S5.Ex64.m2.1.1.1.1.1.1.cmml"><mo id="S5.Ex64.m2.1.1.1.1.1.2" stretchy="false" xref="S5.Ex64.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex64.m2.1.1.1.1.1.1" xref="S5.Ex64.m2.1.1.1.1.1.1.cmml"><mi id="S5.Ex64.m2.1.1.1.1.1.1.2" xref="S5.Ex64.m2.1.1.1.1.1.1.2.cmml">N</mi><mo id="S5.Ex64.m2.1.1.1.1.1.1.1" xref="S5.Ex64.m2.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex64.m2.1.1.1.1.1.1.3" xref="S5.Ex64.m2.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.Ex64.m2.1.1.1.1.1.3" stretchy="false" xref="S5.Ex64.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.Ex64.m2.6.6.1.1.2" xref="S5.Ex64.m2.6.6.1.1.2.cmml">⁢</mo><mrow id="S5.Ex64.m2.6.6.1.1.1.1" 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xref="S5.Ex64.m2.6.6.1.1.1.2.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex64.m2.5.5" xref="S5.Ex64.m2.5.5.cmml">ℳ</mi><mo id="S5.Ex64.m2.6.6.1.1.1.1.4" stretchy="false" xref="S5.Ex64.m2.6.6.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex64.m2.6b"><apply id="S5.Ex64.m2.6.6.cmml" xref="S5.Ex64.m2.6.6"><approx id="S5.Ex64.m2.6.6.2.cmml" xref="S5.Ex64.m2.6.6.2"></approx><csymbol cd="latexml" id="S5.Ex64.m2.6.6.3.cmml" xref="S5.Ex64.m2.6.6.3">absent</csymbol><apply id="S5.Ex64.m2.6.6.1.cmml" xref="S5.Ex64.m2.6.6.1"><apply id="S5.Ex64.m2.6.6.1.2.cmml" xref="S5.Ex64.m2.6.6.1.2"><csymbol cd="ambiguous" id="S5.Ex64.m2.6.6.1.2.1.cmml" xref="S5.Ex64.m2.6.6.1.2">subscript</csymbol><csymbol cd="latexml" id="S5.Ex64.m2.6.6.1.2.2.cmml" xref="S5.Ex64.m2.6.6.1.2.2">product</csymbol><apply id="S5.Ex64.m2.6.6.1.2.3.cmml" xref="S5.Ex64.m2.6.6.1.2.3"><in id="S5.Ex64.m2.6.6.1.2.3.1.cmml" xref="S5.Ex64.m2.6.6.1.2.3.1"></in><ci 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id="S5.Ex64.m2.6.6.1.1.3.2.cmml" xref="S5.Ex64.m2.6.6.1.1.3.2">Hom</ci><apply id="S5.Ex64.m2.1.1.1.cmml" xref="S5.Ex64.m2.1.1.1"><times id="S5.Ex64.m2.1.1.1.2.cmml" xref="S5.Ex64.m2.1.1.1.2"></times><ci id="S5.Ex64.m2.1.1.1.3.cmml" xref="S5.Ex64.m2.1.1.1.3">𝑅</ci><ci id="S5.Ex64.m2.1.1.1.4.cmml" xref="S5.Ex64.m2.1.1.1.4">Δ</ci><apply id="S5.Ex64.m2.1.1.1.1.1.1.cmml" xref="S5.Ex64.m2.1.1.1.1.1"><times id="S5.Ex64.m2.1.1.1.1.1.1.1.cmml" xref="S5.Ex64.m2.1.1.1.1.1.1.1"></times><ci id="S5.Ex64.m2.1.1.1.1.1.1.2.cmml" xref="S5.Ex64.m2.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.Ex64.m2.1.1.1.1.1.1.3.cmml" xref="S5.Ex64.m2.1.1.1.1.1.1.3">𝒞</ci></apply></apply></apply><interval closure="open" id="S5.Ex64.m2.6.6.1.1.1.2.cmml" xref="S5.Ex64.m2.6.6.1.1.1.1"><apply id="S5.Ex64.m2.6.6.1.1.1.1.1.cmml" xref="S5.Ex64.m2.6.6.1.1.1.1.1"><times id="S5.Ex64.m2.6.6.1.1.1.1.1.1.cmml" xref="S5.Ex64.m2.6.6.1.1.1.1.1.1"></times><ci id="S5.Ex64.m2.6.6.1.1.1.1.1.2.cmml" xref="S5.Ex64.m2.6.6.1.1.1.1.1.2">𝑅</ci><apply id="S5.Ex64.m2.6.6.1.1.1.1.1.3.cmml" xref="S5.Ex64.m2.6.6.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex64.m2.6.6.1.1.1.1.1.3.1.cmml" xref="S5.Ex64.m2.6.6.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex64.m2.6.6.1.1.1.1.1.3.2.cmml" xref="S5.Ex64.m2.6.6.1.1.1.1.1.3.2">Mor</ci><apply id="S5.Ex64.m2.2.2.1.cmml" xref="S5.Ex64.m2.2.2.1"><times id="S5.Ex64.m2.2.2.1.2.cmml" xref="S5.Ex64.m2.2.2.1.2"></times><ci id="S5.Ex64.m2.2.2.1.3.cmml" xref="S5.Ex64.m2.2.2.1.3">Δ</ci><apply id="S5.Ex64.m2.2.2.1.1.1.1.cmml" xref="S5.Ex64.m2.2.2.1.1.1"><times id="S5.Ex64.m2.2.2.1.1.1.1.1.cmml" xref="S5.Ex64.m2.2.2.1.1.1.1.1"></times><ci id="S5.Ex64.m2.2.2.1.1.1.1.2.cmml" xref="S5.Ex64.m2.2.2.1.1.1.1.2">𝑁</ci><ci id="S5.Ex64.m2.2.2.1.1.1.1.3.cmml" xref="S5.Ex64.m2.2.2.1.1.1.1.3">𝒞</ci></apply></apply></apply><interval closure="open" id="S5.Ex64.m2.6.6.1.1.1.1.1.4.1.cmml" xref="S5.Ex64.m2.6.6.1.1.1.1.1.4.2"><ci id="S5.Ex64.m2.3.3.cmml" xref="S5.Ex64.m2.3.3">𝜏</ci><ci id="S5.Ex64.m2.4.4.cmml" xref="S5.Ex64.m2.4.4">?</ci></interval></apply><ci id="S5.Ex64.m2.5.5.cmml" xref="S5.Ex64.m2.5.5">ℳ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex64.m2.6c">\displaystyle\cong\prod_{\tau\in N\mathcal{C}_{q}}\mathrm{Hom}_{R\Delta(N% \mathcal{C})}(R\mathrm{Mor}_{\Delta(N\mathcal{C})}(\tau,?),\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex64.m2.6d">≅ ∏ start_POSTSUBSCRIPT italic_τ ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( italic_R roman_Mor start_POSTSUBSCRIPT roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( italic_τ , ? ) , caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex65"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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xref="S5.Ex65.m1.2.2">ℳ</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex65.m1.3c">\displaystyle\cong\prod_{\tau\in N\mathcal{C}_{k}}\mathcal{M}(\tau)\cong C^{k}% (N\mathcal{C};\mathcal{M}).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex65.m1.3d">≅ ∏ start_POSTSUBSCRIPT italic_τ ∈ italic_N caligraphic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_M ( italic_τ ) ≅ italic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_N caligraphic_C ; caligraphic_M ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.6.p1.21">These two isomorphisms fit into a commuting diagram</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex66"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell 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id="S5.Ex66.m1.1.1.pic1.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.3.cmml" xref="S5.Ex66.m1.1.1.pic1.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.3"></times></apply><ci id="S5.Ex66.m1.1.1.pic1.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.2.2.cmml" xref="S5.Ex66.m1.1.1.pic1.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.2.2">ℳ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex66.m1.1.1.pic1.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.3c">\textstyle{\mathrm{Hom}_{R\Delta(N\mathcal{C})}((\mathrm{diag}\mathbb{Q})_{*},% \mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \ignorespaces\ignorespaces\ignorespaces}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex66.m1.1.1.pic1.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.m1.3d">roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_N caligraphic_C ) end_POSTSUBSCRIPT ( ( roman_diag blackboard_Q ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , caligraphic_M 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xref="S5.Ex66.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.m1.2.2.1.1.1.2">𝑁</ci><ci id="S5.Ex66.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.m1.2.2.1.1.1.3.cmml" xref="S5.Ex66.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.m1.2.2.1.1.1.3">𝒞</ci></apply><ci id="S5.Ex66.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.Ex66.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.m1.1.1">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex66.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.m1.2c">\textstyle{C^{*}(N\mathcal{C};\mathcal{M})\ignorespaces\ignorespaces% \ignorespaces\ignorespaces}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex66.m1.1.1.pic1.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.m1.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_N caligraphic_C ; caligraphic_M )</annotation></semantics></math></foreignobject></g><path class="droprule" d="M 181.88 -42.2 L 182.43 -42.2" fill="none" stroke="#000000"></path><g transform="translate(182.16,0) translate(0,-28.81) translate(4.15,0) translate(4.15,0) translate(0,-2.73)"><foreignobject height="5.46" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="8.35"><math alttext="\scriptstyle{\kappa^{*}}" class="ltx_Math" display="inline" id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msup id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" mathsize="70%" xref="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">κ</mi><mo id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" mathsize="71%" xref="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">superscript</csymbol><ci id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝜅</ci><times id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\scriptstyle{\kappa^{*}}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex66.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_κ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math></foreignobject></g><g transform="translate(182.16,0) translate(0,-7.61)"><path d="M 0 0 A 13.84 13.84 45 0 0 -2.77 -6.92" fill="none" stroke="#000000"></path><path d="M 0 0 A 13.84 13.84 45 0 1 2.77 -6.92" fill="none" stroke="#000000"></path></g><path class="droprule" d="M 181.88 -42.2 L 182.43 -7.61" fill="none" stroke="#000000"></path><path class="droprule" d="M 181.88 -7.61 L 182.43 -7.61" fill="none" stroke="#000000"></path></g></svg></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.6.p1.19">where the first vertical map is induced by the bisimplicial map <math alttext="\xi:\mathbb{P}\to\mathbb{Q}" class="ltx_Math" display="inline" id="S5.6.p1.13.m1.1"><semantics id="S5.6.p1.13.m1.1a"><mrow id="S5.6.p1.13.m1.1.1" xref="S5.6.p1.13.m1.1.1.cmml"><mi id="S5.6.p1.13.m1.1.1.2" xref="S5.6.p1.13.m1.1.1.2.cmml">ξ</mi><mo id="S5.6.p1.13.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.6.p1.13.m1.1.1.1.cmml">:</mo><mrow id="S5.6.p1.13.m1.1.1.3" xref="S5.6.p1.13.m1.1.1.3.cmml"><mi id="S5.6.p1.13.m1.1.1.3.2" xref="S5.6.p1.13.m1.1.1.3.2.cmml">ℙ</mi><mo id="S5.6.p1.13.m1.1.1.3.1" stretchy="false" xref="S5.6.p1.13.m1.1.1.3.1.cmml">→</mo><mi id="S5.6.p1.13.m1.1.1.3.3" xref="S5.6.p1.13.m1.1.1.3.3.cmml">ℚ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.13.m1.1b"><apply id="S5.6.p1.13.m1.1.1.cmml" xref="S5.6.p1.13.m1.1.1"><ci id="S5.6.p1.13.m1.1.1.1.cmml" xref="S5.6.p1.13.m1.1.1.1">:</ci><ci id="S5.6.p1.13.m1.1.1.2.cmml" xref="S5.6.p1.13.m1.1.1.2">𝜉</ci><apply id="S5.6.p1.13.m1.1.1.3.cmml" xref="S5.6.p1.13.m1.1.1.3"><ci id="S5.6.p1.13.m1.1.1.3.1.cmml" xref="S5.6.p1.13.m1.1.1.3.1">→</ci><ci id="S5.6.p1.13.m1.1.1.3.2.cmml" xref="S5.6.p1.13.m1.1.1.3.2">ℙ</ci><ci id="S5.6.p1.13.m1.1.1.3.3.cmml" xref="S5.6.p1.13.m1.1.1.3.3">ℚ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.13.m1.1c">\xi:\mathbb{P}\to\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.13.m1.1d">italic_ξ : blackboard_P → blackboard_Q</annotation></semantics></math> and the second vertical map is induced by the simplicial map <math alttext="\kappa:N(\mathcal{D};G)\to N\mathcal{C}" class="ltx_Math" display="inline" id="S5.6.p1.14.m2.2"><semantics id="S5.6.p1.14.m2.2a"><mrow id="S5.6.p1.14.m2.2.3" xref="S5.6.p1.14.m2.2.3.cmml"><mi id="S5.6.p1.14.m2.2.3.2" xref="S5.6.p1.14.m2.2.3.2.cmml">κ</mi><mo id="S5.6.p1.14.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="S5.6.p1.14.m2.2.3.1.cmml">:</mo><mrow id="S5.6.p1.14.m2.2.3.3" xref="S5.6.p1.14.m2.2.3.3.cmml"><mrow id="S5.6.p1.14.m2.2.3.3.2" xref="S5.6.p1.14.m2.2.3.3.2.cmml"><mi id="S5.6.p1.14.m2.2.3.3.2.2" xref="S5.6.p1.14.m2.2.3.3.2.2.cmml">N</mi><mo id="S5.6.p1.14.m2.2.3.3.2.1" xref="S5.6.p1.14.m2.2.3.3.2.1.cmml">⁢</mo><mrow id="S5.6.p1.14.m2.2.3.3.2.3.2" xref="S5.6.p1.14.m2.2.3.3.2.3.1.cmml"><mo id="S5.6.p1.14.m2.2.3.3.2.3.2.1" stretchy="false" xref="S5.6.p1.14.m2.2.3.3.2.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.14.m2.1.1" xref="S5.6.p1.14.m2.1.1.cmml">𝒟</mi><mo id="S5.6.p1.14.m2.2.3.3.2.3.2.2" xref="S5.6.p1.14.m2.2.3.3.2.3.1.cmml">;</mo><mi id="S5.6.p1.14.m2.2.2" xref="S5.6.p1.14.m2.2.2.cmml">G</mi><mo id="S5.6.p1.14.m2.2.3.3.2.3.2.3" stretchy="false" xref="S5.6.p1.14.m2.2.3.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.6.p1.14.m2.2.3.3.1" stretchy="false" xref="S5.6.p1.14.m2.2.3.3.1.cmml">→</mo><mrow id="S5.6.p1.14.m2.2.3.3.3" xref="S5.6.p1.14.m2.2.3.3.3.cmml"><mi id="S5.6.p1.14.m2.2.3.3.3.2" xref="S5.6.p1.14.m2.2.3.3.3.2.cmml">N</mi><mo id="S5.6.p1.14.m2.2.3.3.3.1" xref="S5.6.p1.14.m2.2.3.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.6.p1.14.m2.2.3.3.3.3" xref="S5.6.p1.14.m2.2.3.3.3.3.cmml">𝒞</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p1.14.m2.2b"><apply id="S5.6.p1.14.m2.2.3.cmml" xref="S5.6.p1.14.m2.2.3"><ci id="S5.6.p1.14.m2.2.3.1.cmml" xref="S5.6.p1.14.m2.2.3.1">:</ci><ci id="S5.6.p1.14.m2.2.3.2.cmml" xref="S5.6.p1.14.m2.2.3.2">𝜅</ci><apply id="S5.6.p1.14.m2.2.3.3.cmml" xref="S5.6.p1.14.m2.2.3.3"><ci id="S5.6.p1.14.m2.2.3.3.1.cmml" xref="S5.6.p1.14.m2.2.3.3.1">→</ci><apply id="S5.6.p1.14.m2.2.3.3.2.cmml" xref="S5.6.p1.14.m2.2.3.3.2"><times id="S5.6.p1.14.m2.2.3.3.2.1.cmml" xref="S5.6.p1.14.m2.2.3.3.2.1"></times><ci id="S5.6.p1.14.m2.2.3.3.2.2.cmml" xref="S5.6.p1.14.m2.2.3.3.2.2">𝑁</ci><list id="S5.6.p1.14.m2.2.3.3.2.3.1.cmml" xref="S5.6.p1.14.m2.2.3.3.2.3.2"><ci id="S5.6.p1.14.m2.1.1.cmml" xref="S5.6.p1.14.m2.1.1">𝒟</ci><ci id="S5.6.p1.14.m2.2.2.cmml" xref="S5.6.p1.14.m2.2.2">𝐺</ci></list></apply><apply id="S5.6.p1.14.m2.2.3.3.3.cmml" xref="S5.6.p1.14.m2.2.3.3.3"><times id="S5.6.p1.14.m2.2.3.3.3.1.cmml" xref="S5.6.p1.14.m2.2.3.3.3.1"></times><ci id="S5.6.p1.14.m2.2.3.3.3.2.cmml" xref="S5.6.p1.14.m2.2.3.3.3.2">𝑁</ci><ci id="S5.6.p1.14.m2.2.3.3.3.3.cmml" xref="S5.6.p1.14.m2.2.3.3.3.3">𝒞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.14.m2.2c">\kappa:N(\mathcal{D};G)\to N\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.14.m2.2d">italic_κ : italic_N ( caligraphic_D ; italic_G ) → italic_N caligraphic_C</annotation></semantics></math>. Since the augmented complexes for <math alttext="(\mathrm{diag}\mathbb{P})_{*}" class="ltx_Math" display="inline" id="S5.6.p1.15.m3.1"><semantics id="S5.6.p1.15.m3.1a"><msub id="S5.6.p1.15.m3.1.1" xref="S5.6.p1.15.m3.1.1.cmml"><mrow id="S5.6.p1.15.m3.1.1.1.1" xref="S5.6.p1.15.m3.1.1.1.1.1.cmml"><mo id="S5.6.p1.15.m3.1.1.1.1.2" stretchy="false" xref="S5.6.p1.15.m3.1.1.1.1.1.cmml">(</mo><mrow id="S5.6.p1.15.m3.1.1.1.1.1" xref="S5.6.p1.15.m3.1.1.1.1.1.cmml"><mi id="S5.6.p1.15.m3.1.1.1.1.1.2" xref="S5.6.p1.15.m3.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.6.p1.15.m3.1.1.1.1.1.1" xref="S5.6.p1.15.m3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.6.p1.15.m3.1.1.1.1.1.3" xref="S5.6.p1.15.m3.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S5.6.p1.15.m3.1.1.1.1.3" stretchy="false" xref="S5.6.p1.15.m3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.6.p1.15.m3.1.1.3" xref="S5.6.p1.15.m3.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S5.6.p1.15.m3.1b"><apply id="S5.6.p1.15.m3.1.1.cmml" xref="S5.6.p1.15.m3.1.1"><csymbol cd="ambiguous" id="S5.6.p1.15.m3.1.1.2.cmml" xref="S5.6.p1.15.m3.1.1">subscript</csymbol><apply id="S5.6.p1.15.m3.1.1.1.1.1.cmml" xref="S5.6.p1.15.m3.1.1.1.1"><times id="S5.6.p1.15.m3.1.1.1.1.1.1.cmml" xref="S5.6.p1.15.m3.1.1.1.1.1.1"></times><ci id="S5.6.p1.15.m3.1.1.1.1.1.2.cmml" xref="S5.6.p1.15.m3.1.1.1.1.1.2">diag</ci><ci id="S5.6.p1.15.m3.1.1.1.1.1.3.cmml" xref="S5.6.p1.15.m3.1.1.1.1.1.3">ℙ</ci></apply><times id="S5.6.p1.15.m3.1.1.3.cmml" xref="S5.6.p1.15.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.15.m3.1c">(\mathrm{diag}\mathbb{P})_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.15.m3.1d">( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="(\mathrm{diag}\mathbb{Q})_{*}" class="ltx_Math" display="inline" id="S5.6.p1.16.m4.1"><semantics id="S5.6.p1.16.m4.1a"><msub id="S5.6.p1.16.m4.1.1" xref="S5.6.p1.16.m4.1.1.cmml"><mrow id="S5.6.p1.16.m4.1.1.1.1" xref="S5.6.p1.16.m4.1.1.1.1.1.cmml"><mo id="S5.6.p1.16.m4.1.1.1.1.2" stretchy="false" xref="S5.6.p1.16.m4.1.1.1.1.1.cmml">(</mo><mrow id="S5.6.p1.16.m4.1.1.1.1.1" xref="S5.6.p1.16.m4.1.1.1.1.1.cmml"><mi id="S5.6.p1.16.m4.1.1.1.1.1.2" xref="S5.6.p1.16.m4.1.1.1.1.1.2.cmml">diag</mi><mo id="S5.6.p1.16.m4.1.1.1.1.1.1" xref="S5.6.p1.16.m4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.6.p1.16.m4.1.1.1.1.1.3" xref="S5.6.p1.16.m4.1.1.1.1.1.3.cmml">ℚ</mi></mrow><mo id="S5.6.p1.16.m4.1.1.1.1.3" stretchy="false" xref="S5.6.p1.16.m4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.6.p1.16.m4.1.1.3" xref="S5.6.p1.16.m4.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S5.6.p1.16.m4.1b"><apply id="S5.6.p1.16.m4.1.1.cmml" xref="S5.6.p1.16.m4.1.1"><csymbol cd="ambiguous" id="S5.6.p1.16.m4.1.1.2.cmml" xref="S5.6.p1.16.m4.1.1">subscript</csymbol><apply id="S5.6.p1.16.m4.1.1.1.1.1.cmml" xref="S5.6.p1.16.m4.1.1.1.1"><times id="S5.6.p1.16.m4.1.1.1.1.1.1.cmml" xref="S5.6.p1.16.m4.1.1.1.1.1.1"></times><ci id="S5.6.p1.16.m4.1.1.1.1.1.2.cmml" xref="S5.6.p1.16.m4.1.1.1.1.1.2">diag</ci><ci id="S5.6.p1.16.m4.1.1.1.1.1.3.cmml" xref="S5.6.p1.16.m4.1.1.1.1.1.3">ℚ</ci></apply><times id="S5.6.p1.16.m4.1.1.3.cmml" xref="S5.6.p1.16.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.16.m4.1c">(\mathrm{diag}\mathbb{Q})_{*}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.16.m4.1d">( roman_diag blackboard_Q ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> are projective resolutions for the constant functor <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S5.6.p1.17.m5.1"><semantics id="S5.6.p1.17.m5.1a"><munder accentunder="true" id="S5.6.p1.17.m5.1.1" xref="S5.6.p1.17.m5.1.1.cmml"><mi id="S5.6.p1.17.m5.1.1.2" xref="S5.6.p1.17.m5.1.1.2.cmml">R</mi><mo id="S5.6.p1.17.m5.1.1.1" xref="S5.6.p1.17.m5.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S5.6.p1.17.m5.1b"><apply id="S5.6.p1.17.m5.1.1.cmml" xref="S5.6.p1.17.m5.1.1"><ci id="S5.6.p1.17.m5.1.1.1.cmml" xref="S5.6.p1.17.m5.1.1.1">¯</ci><ci id="S5.6.p1.17.m5.1.1.2.cmml" xref="S5.6.p1.17.m5.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.17.m5.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.17.m5.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math>, the first vertical map <math alttext="\xi^{*}" class="ltx_Math" display="inline" id="S5.6.p1.18.m6.1"><semantics id="S5.6.p1.18.m6.1a"><msup id="S5.6.p1.18.m6.1.1" xref="S5.6.p1.18.m6.1.1.cmml"><mi id="S5.6.p1.18.m6.1.1.2" xref="S5.6.p1.18.m6.1.1.2.cmml">ξ</mi><mo id="S5.6.p1.18.m6.1.1.3" xref="S5.6.p1.18.m6.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.6.p1.18.m6.1b"><apply id="S5.6.p1.18.m6.1.1.cmml" xref="S5.6.p1.18.m6.1.1"><csymbol cd="ambiguous" id="S5.6.p1.18.m6.1.1.1.cmml" xref="S5.6.p1.18.m6.1.1">superscript</csymbol><ci id="S5.6.p1.18.m6.1.1.2.cmml" xref="S5.6.p1.18.m6.1.1.2">𝜉</ci><times id="S5.6.p1.18.m6.1.1.3.cmml" xref="S5.6.p1.18.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.18.m6.1c">\xi^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.18.m6.1d">italic_ξ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> induces an isomorphism on cohomology. Hence <math alttext="\kappa^{*}" class="ltx_Math" display="inline" id="S5.6.p1.19.m7.1"><semantics id="S5.6.p1.19.m7.1a"><msup id="S5.6.p1.19.m7.1.1" xref="S5.6.p1.19.m7.1.1.cmml"><mi id="S5.6.p1.19.m7.1.1.2" xref="S5.6.p1.19.m7.1.1.2.cmml">κ</mi><mo id="S5.6.p1.19.m7.1.1.3" xref="S5.6.p1.19.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.6.p1.19.m7.1b"><apply id="S5.6.p1.19.m7.1.1.cmml" xref="S5.6.p1.19.m7.1.1"><csymbol cd="ambiguous" id="S5.6.p1.19.m7.1.1.1.cmml" xref="S5.6.p1.19.m7.1.1">superscript</csymbol><ci id="S5.6.p1.19.m7.1.1.2.cmml" xref="S5.6.p1.19.m7.1.1.2">𝜅</ci><times id="S5.6.p1.19.m7.1.1.3.cmml" xref="S5.6.p1.19.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p1.19.m7.1c">\kappa^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p1.19.m7.1d">italic_κ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> also induces an isomorphism on cohomology. ∎</p> </div> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">By reversing the direction of morphisms and changing the statements accordingly, it is easy to see that the following version of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction and statement of results ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">1.2</span></a> also holds.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.1.1.1">Theorem 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.6"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p1.6.6">Let <math alttext="\varphi:\mathcal{C}\rightarrow\mathcal{D}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S5.Thmtheorem3.p1.1.1.m1.1.1" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">φ</mi><mo id="S5.Thmtheorem3.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.2" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml">𝒞</mi><mo id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.3" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml">𝒟</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.1.m1.1b"><apply id="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1"><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.1">:</ci><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.2">𝜑</ci><apply id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3"><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.1">→</ci><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.2">𝒞</ci><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.3">𝒟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.1.m1.1c">\varphi:\mathcal{C}\rightarrow\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.1.m1.1d">italic_φ : caligraphic_C → caligraphic_D</annotation></semantics></math> be a functor between two small categories and <math alttext="-\backslash\varphi:\mathcal{D}^{op}\rightarrow Cat" class="ltx_math_unparsed" display="inline" id="S5.Thmtheorem3.p1.2.2.m2.2"><semantics id="S5.Thmtheorem3.p1.2.2.m2.2a"><mrow id="S5.Thmtheorem3.p1.2.2.m2.2b"><mo id="S5.Thmtheorem3.p1.2.2.m2.1.1" rspace="0em">−</mo><mo id="S5.Thmtheorem3.p1.2.2.m2.2.2" rspace="0.222em">\</mo><mi id="S5.Thmtheorem3.p1.2.2.m2.2.3">φ</mi><mo id="S5.Thmtheorem3.p1.2.2.m2.2.4" lspace="0.278em" rspace="0.278em">:</mo><msup id="S5.Thmtheorem3.p1.2.2.m2.2.5"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.2.2.m2.2.5.2">𝒟</mi><mrow id="S5.Thmtheorem3.p1.2.2.m2.2.5.3"><mi id="S5.Thmtheorem3.p1.2.2.m2.2.5.3.2">o</mi><mo id="S5.Thmtheorem3.p1.2.2.m2.2.5.3.1">⁢</mo><mi id="S5.Thmtheorem3.p1.2.2.m2.2.5.3.3">p</mi></mrow></msup><mo id="S5.Thmtheorem3.p1.2.2.m2.2.6" stretchy="false">→</mo><mi id="S5.Thmtheorem3.p1.2.2.m2.2.7">C</mi><mi id="S5.Thmtheorem3.p1.2.2.m2.2.8">a</mi><mi id="S5.Thmtheorem3.p1.2.2.m2.2.9">t</mi></mrow><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.2.m2.2c">-\backslash\varphi:\mathcal{D}^{op}\rightarrow Cat</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.2.m2.2d">- \ italic_φ : caligraphic_D start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → italic_C italic_a italic_t</annotation></semantics></math> denote the functor that sends each <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.3.m3.1"><semantics id="S5.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem3.p1.3.3.m3.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">d</mi><mo id="S5.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.3.m3.1b"><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1"><in id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1"></in><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.2">𝑑</ci><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.3.m3.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.3.m3.1d">italic_d ∈ caligraphic_D</annotation></semantics></math> to the comma category <math alttext="d\backslash\varphi" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.4.4.m4.1"><semantics id="S5.Thmtheorem3.p1.4.4.m4.1a"><mrow id="S5.Thmtheorem3.p1.4.4.m4.1.1" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml"><mi id="S5.Thmtheorem3.p1.4.4.m4.1.1.2" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.2.cmml">d</mi><mo id="S5.Thmtheorem3.p1.4.4.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.1.cmml">\</mo><mi id="S5.Thmtheorem3.p1.4.4.m4.1.1.3" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.4.4.m4.1b"><apply id="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1"><ci id="S5.Thmtheorem3.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.1">\</ci><ci id="S5.Thmtheorem3.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.2">𝑑</ci><ci id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.4.4.m4.1c">d\backslash\varphi</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.4.4.m4.1d">italic_d \ italic_φ</annotation></semantics></math>. Then for every coefficient system <math alttext="\mathcal{M}:\Delta(N\mathcal{C})\rightarrow R" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.5.5.m5.1"><semantics id="S5.Thmtheorem3.p1.5.5.m5.1a"><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.5.5.m5.1.1.3" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.3.cmml">ℳ</mi><mo id="S5.Thmtheorem3.p1.5.5.m5.1.1.2" lspace="0.278em" rspace="0.278em" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.cmml">:</mo><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.cmml"><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.3" mathvariant="normal" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.3.cmml">Δ</mi><mo id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.3.cmml">𝒞</mi></mrow><mo id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.2.cmml">→</mo><mi id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.3" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.5.5.m5.1b"><apply id="S5.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1"><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2">:</ci><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.3">ℳ</ci><apply id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1"><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.2">→</ci><apply id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1"><times id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.2"></times><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.3">Δ</ci><apply id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1"><times id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.1"></times><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.2">𝑁</ci><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.1.1.1.1.3">𝒞</ci></apply></apply><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.1.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.5.5.m5.1c">\mathcal{M}:\Delta(N\mathcal{C})\rightarrow R</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.5.5.m5.1d">caligraphic_M : roman_Δ ( italic_N caligraphic_C ) → italic_R</annotation></semantics></math>-Mod, the homotopy equivalence <math alttext="\upsilon:Y=\operatorname*{hocolim}_{\mathcal{D}^{op}}N(-\backslash\varphi)\to N% \mathcal{C}" class="ltx_math_unparsed" display="inline" id="S5.Thmtheorem3.p1.6.6.m6.1"><semantics id="S5.Thmtheorem3.p1.6.6.m6.1a"><mrow id="S5.Thmtheorem3.p1.6.6.m6.1b"><mi id="S5.Thmtheorem3.p1.6.6.m6.1.1">υ</mi><mo id="S5.Thmtheorem3.p1.6.6.m6.1.2" lspace="0.278em" rspace="0.278em">:</mo><mi id="S5.Thmtheorem3.p1.6.6.m6.1.3">Y</mi><mo id="S5.Thmtheorem3.p1.6.6.m6.1.4" rspace="0.1389em">=</mo><msub id="S5.Thmtheorem3.p1.6.6.m6.1.5"><mo id="S5.Thmtheorem3.p1.6.6.m6.1.5.2" lspace="0.1389em" rspace="0.167em">hocolim</mo><msup id="S5.Thmtheorem3.p1.6.6.m6.1.5.3"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.6.6.m6.1.5.3.2">𝒟</mi><mrow id="S5.Thmtheorem3.p1.6.6.m6.1.5.3.3"><mi id="S5.Thmtheorem3.p1.6.6.m6.1.5.3.3.2">o</mi><mo id="S5.Thmtheorem3.p1.6.6.m6.1.5.3.3.1">⁢</mo><mi id="S5.Thmtheorem3.p1.6.6.m6.1.5.3.3.3">p</mi></mrow></msup></msub><mi id="S5.Thmtheorem3.p1.6.6.m6.1.6">N</mi><mrow id="S5.Thmtheorem3.p1.6.6.m6.1.7"><mo id="S5.Thmtheorem3.p1.6.6.m6.1.7.1" stretchy="false">(</mo><mo id="S5.Thmtheorem3.p1.6.6.m6.1.7.2" lspace="0em" rspace="0em">−</mo><mo id="S5.Thmtheorem3.p1.6.6.m6.1.7.3" rspace="0.222em">\</mo><mi id="S5.Thmtheorem3.p1.6.6.m6.1.7.4">φ</mi><mo id="S5.Thmtheorem3.p1.6.6.m6.1.7.5" stretchy="false">)</mo></mrow><mo id="S5.Thmtheorem3.p1.6.6.m6.1.8" stretchy="false">→</mo><mi id="S5.Thmtheorem3.p1.6.6.m6.1.9">N</mi><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.6.6.m6.1.10">𝒞</mi></mrow><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.6.6.m6.1c">\upsilon:Y=\operatorname*{hocolim}_{\mathcal{D}^{op}}N(-\backslash\varphi)\to N% \mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.6.6.m6.1d">italic_υ : italic_Y = roman_hocolim start_POSTSUBSCRIPT caligraphic_D start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_N ( - \ italic_φ ) → italic_N caligraphic_C</annotation></semantics></math> induces an isomorphism</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex67"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{*}_{Th}(\mathcal{C};\mathcal{M})\cong H^{*}(Y;\upsilon^{*}\mathcal{M})" class="ltx_Math" display="block" id="S5.Ex67.m1.4"><semantics id="S5.Ex67.m1.4a"><mrow id="S5.Ex67.m1.4.4" xref="S5.Ex67.m1.4.4.cmml"><mrow id="S5.Ex67.m1.4.4.3" xref="S5.Ex67.m1.4.4.3.cmml"><msubsup id="S5.Ex67.m1.4.4.3.2" xref="S5.Ex67.m1.4.4.3.2.cmml"><mi id="S5.Ex67.m1.4.4.3.2.2.2" xref="S5.Ex67.m1.4.4.3.2.2.2.cmml">H</mi><mrow id="S5.Ex67.m1.4.4.3.2.3" xref="S5.Ex67.m1.4.4.3.2.3.cmml"><mi id="S5.Ex67.m1.4.4.3.2.3.2" xref="S5.Ex67.m1.4.4.3.2.3.2.cmml">T</mi><mo id="S5.Ex67.m1.4.4.3.2.3.1" xref="S5.Ex67.m1.4.4.3.2.3.1.cmml">⁢</mo><mi id="S5.Ex67.m1.4.4.3.2.3.3" xref="S5.Ex67.m1.4.4.3.2.3.3.cmml">h</mi></mrow><mo id="S5.Ex67.m1.4.4.3.2.2.3" xref="S5.Ex67.m1.4.4.3.2.2.3.cmml">∗</mo></msubsup><mo id="S5.Ex67.m1.4.4.3.1" xref="S5.Ex67.m1.4.4.3.1.cmml">⁢</mo><mrow id="S5.Ex67.m1.4.4.3.3.2" xref="S5.Ex67.m1.4.4.3.3.1.cmml"><mo id="S5.Ex67.m1.4.4.3.3.2.1" stretchy="false" xref="S5.Ex67.m1.4.4.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex67.m1.1.1" xref="S5.Ex67.m1.1.1.cmml">𝒞</mi><mo id="S5.Ex67.m1.4.4.3.3.2.2" xref="S5.Ex67.m1.4.4.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex67.m1.2.2" xref="S5.Ex67.m1.2.2.cmml">ℳ</mi><mo id="S5.Ex67.m1.4.4.3.3.2.3" stretchy="false" xref="S5.Ex67.m1.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex67.m1.4.4.2" xref="S5.Ex67.m1.4.4.2.cmml">≅</mo><mrow id="S5.Ex67.m1.4.4.1" xref="S5.Ex67.m1.4.4.1.cmml"><msup id="S5.Ex67.m1.4.4.1.3" xref="S5.Ex67.m1.4.4.1.3.cmml"><mi id="S5.Ex67.m1.4.4.1.3.2" xref="S5.Ex67.m1.4.4.1.3.2.cmml">H</mi><mo id="S5.Ex67.m1.4.4.1.3.3" xref="S5.Ex67.m1.4.4.1.3.3.cmml">∗</mo></msup><mo id="S5.Ex67.m1.4.4.1.2" xref="S5.Ex67.m1.4.4.1.2.cmml">⁢</mo><mrow id="S5.Ex67.m1.4.4.1.1.1" xref="S5.Ex67.m1.4.4.1.1.2.cmml"><mo id="S5.Ex67.m1.4.4.1.1.1.2" stretchy="false" xref="S5.Ex67.m1.4.4.1.1.2.cmml">(</mo><mi id="S5.Ex67.m1.3.3" xref="S5.Ex67.m1.3.3.cmml">Y</mi><mo id="S5.Ex67.m1.4.4.1.1.1.3" xref="S5.Ex67.m1.4.4.1.1.2.cmml">;</mo><mrow id="S5.Ex67.m1.4.4.1.1.1.1" xref="S5.Ex67.m1.4.4.1.1.1.1.cmml"><msup id="S5.Ex67.m1.4.4.1.1.1.1.2" xref="S5.Ex67.m1.4.4.1.1.1.1.2.cmml"><mi id="S5.Ex67.m1.4.4.1.1.1.1.2.2" xref="S5.Ex67.m1.4.4.1.1.1.1.2.2.cmml">υ</mi><mo id="S5.Ex67.m1.4.4.1.1.1.1.2.3" xref="S5.Ex67.m1.4.4.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S5.Ex67.m1.4.4.1.1.1.1.1" xref="S5.Ex67.m1.4.4.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex67.m1.4.4.1.1.1.1.3" xref="S5.Ex67.m1.4.4.1.1.1.1.3.cmml">ℳ</mi></mrow><mo id="S5.Ex67.m1.4.4.1.1.1.4" stretchy="false" xref="S5.Ex67.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex67.m1.4b"><apply id="S5.Ex67.m1.4.4.cmml" xref="S5.Ex67.m1.4.4"><approx id="S5.Ex67.m1.4.4.2.cmml" xref="S5.Ex67.m1.4.4.2"></approx><apply id="S5.Ex67.m1.4.4.3.cmml" xref="S5.Ex67.m1.4.4.3"><times id="S5.Ex67.m1.4.4.3.1.cmml" xref="S5.Ex67.m1.4.4.3.1"></times><apply id="S5.Ex67.m1.4.4.3.2.cmml" xref="S5.Ex67.m1.4.4.3.2"><csymbol cd="ambiguous" id="S5.Ex67.m1.4.4.3.2.1.cmml" xref="S5.Ex67.m1.4.4.3.2">subscript</csymbol><apply id="S5.Ex67.m1.4.4.3.2.2.cmml" xref="S5.Ex67.m1.4.4.3.2"><csymbol cd="ambiguous" id="S5.Ex67.m1.4.4.3.2.2.1.cmml" xref="S5.Ex67.m1.4.4.3.2">superscript</csymbol><ci id="S5.Ex67.m1.4.4.3.2.2.2.cmml" xref="S5.Ex67.m1.4.4.3.2.2.2">𝐻</ci><times id="S5.Ex67.m1.4.4.3.2.2.3.cmml" xref="S5.Ex67.m1.4.4.3.2.2.3"></times></apply><apply id="S5.Ex67.m1.4.4.3.2.3.cmml" xref="S5.Ex67.m1.4.4.3.2.3"><times id="S5.Ex67.m1.4.4.3.2.3.1.cmml" xref="S5.Ex67.m1.4.4.3.2.3.1"></times><ci id="S5.Ex67.m1.4.4.3.2.3.2.cmml" xref="S5.Ex67.m1.4.4.3.2.3.2">𝑇</ci><ci id="S5.Ex67.m1.4.4.3.2.3.3.cmml" xref="S5.Ex67.m1.4.4.3.2.3.3">ℎ</ci></apply></apply><list id="S5.Ex67.m1.4.4.3.3.1.cmml" xref="S5.Ex67.m1.4.4.3.3.2"><ci id="S5.Ex67.m1.1.1.cmml" xref="S5.Ex67.m1.1.1">𝒞</ci><ci id="S5.Ex67.m1.2.2.cmml" xref="S5.Ex67.m1.2.2">ℳ</ci></list></apply><apply id="S5.Ex67.m1.4.4.1.cmml" xref="S5.Ex67.m1.4.4.1"><times id="S5.Ex67.m1.4.4.1.2.cmml" xref="S5.Ex67.m1.4.4.1.2"></times><apply id="S5.Ex67.m1.4.4.1.3.cmml" xref="S5.Ex67.m1.4.4.1.3"><csymbol cd="ambiguous" id="S5.Ex67.m1.4.4.1.3.1.cmml" xref="S5.Ex67.m1.4.4.1.3">superscript</csymbol><ci id="S5.Ex67.m1.4.4.1.3.2.cmml" xref="S5.Ex67.m1.4.4.1.3.2">𝐻</ci><times id="S5.Ex67.m1.4.4.1.3.3.cmml" xref="S5.Ex67.m1.4.4.1.3.3"></times></apply><list id="S5.Ex67.m1.4.4.1.1.2.cmml" xref="S5.Ex67.m1.4.4.1.1.1"><ci id="S5.Ex67.m1.3.3.cmml" xref="S5.Ex67.m1.3.3">𝑌</ci><apply id="S5.Ex67.m1.4.4.1.1.1.1.cmml" xref="S5.Ex67.m1.4.4.1.1.1.1"><times id="S5.Ex67.m1.4.4.1.1.1.1.1.cmml" xref="S5.Ex67.m1.4.4.1.1.1.1.1"></times><apply id="S5.Ex67.m1.4.4.1.1.1.1.2.cmml" xref="S5.Ex67.m1.4.4.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex67.m1.4.4.1.1.1.1.2.1.cmml" xref="S5.Ex67.m1.4.4.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex67.m1.4.4.1.1.1.1.2.2.cmml" xref="S5.Ex67.m1.4.4.1.1.1.1.2.2">𝜐</ci><times id="S5.Ex67.m1.4.4.1.1.1.1.2.3.cmml" xref="S5.Ex67.m1.4.4.1.1.1.1.2.3"></times></apply><ci id="S5.Ex67.m1.4.4.1.1.1.1.3.cmml" xref="S5.Ex67.m1.4.4.1.1.1.1.3">ℳ</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex67.m1.4c">H^{*}_{Th}(\mathcal{C};\mathcal{M})\cong H^{*}(Y;\upsilon^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex67.m1.4d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_h end_POSTSUBSCRIPT ( caligraphic_C ; caligraphic_M ) ≅ italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_Y ; italic_υ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmtheorem3.p1.9"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p1.9.3">where <math alttext="\upsilon^{*}M" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.7.1.m1.1"><semantics id="S5.Thmtheorem3.p1.7.1.m1.1a"><mrow id="S5.Thmtheorem3.p1.7.1.m1.1.1" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.cmml"><msup id="S5.Thmtheorem3.p1.7.1.m1.1.1.2" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.2.cmml"><mi id="S5.Thmtheorem3.p1.7.1.m1.1.1.2.2" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.2.2.cmml">υ</mi><mo id="S5.Thmtheorem3.p1.7.1.m1.1.1.2.3" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p1.7.1.m1.1.1.1" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.7.1.m1.1.1.3" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.7.1.m1.1b"><apply id="S5.Thmtheorem3.p1.7.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.7.1.m1.1.1"><times id="S5.Thmtheorem3.p1.7.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.1"></times><apply id="S5.Thmtheorem3.p1.7.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.7.1.m1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.2">superscript</csymbol><ci id="S5.Thmtheorem3.p1.7.1.m1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.2.2">𝜐</ci><times id="S5.Thmtheorem3.p1.7.1.m1.1.1.2.3.cmml" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.2.3"></times></apply><ci id="S5.Thmtheorem3.p1.7.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.7.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.7.1.m1.1c">\upsilon^{*}M</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.7.1.m1.1d">italic_υ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_M</annotation></semantics></math> denotes the coefficient system for <math alttext="Y" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.8.2.m2.1"><semantics id="S5.Thmtheorem3.p1.8.2.m2.1a"><mi id="S5.Thmtheorem3.p1.8.2.m2.1.1" xref="S5.Thmtheorem3.p1.8.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.8.2.m2.1b"><ci id="S5.Thmtheorem3.p1.8.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.8.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.8.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.8.2.m2.1d">italic_Y</annotation></semantics></math> induced by <math alttext="\upsilon" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.9.3.m3.1"><semantics id="S5.Thmtheorem3.p1.9.3.m3.1a"><mi id="S5.Thmtheorem3.p1.9.3.m3.1.1" xref="S5.Thmtheorem3.p1.9.3.m3.1.1.cmml">υ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.9.3.m3.1b"><ci id="S5.Thmtheorem3.p1.9.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p1.9.3.m3.1.1">𝜐</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.9.3.m3.1c">\upsilon</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.9.3.m3.1d">italic_υ</annotation></semantics></math>.</span></p> </div> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6. </span>Cohomology of bisimplicial sets with nontrivial coefficients</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.13">Let <math alttext="X:\Delta^{op}\times\Delta^{op}\to Sets" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><mrow id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml"><mi id="S6.p1.1.m1.1.1.2" xref="S6.p1.1.m1.1.1.2.cmml">X</mi><mo id="S6.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S6.p1.1.m1.1.1.3" xref="S6.p1.1.m1.1.1.3.cmml"><mrow id="S6.p1.1.m1.1.1.3.2" xref="S6.p1.1.m1.1.1.3.2.cmml"><msup id="S6.p1.1.m1.1.1.3.2.2" xref="S6.p1.1.m1.1.1.3.2.2.cmml"><mi id="S6.p1.1.m1.1.1.3.2.2.2" mathvariant="normal" xref="S6.p1.1.m1.1.1.3.2.2.2.cmml">Δ</mi><mrow id="S6.p1.1.m1.1.1.3.2.2.3" xref="S6.p1.1.m1.1.1.3.2.2.3.cmml"><mi id="S6.p1.1.m1.1.1.3.2.2.3.2" xref="S6.p1.1.m1.1.1.3.2.2.3.2.cmml">o</mi><mo id="S6.p1.1.m1.1.1.3.2.2.3.1" xref="S6.p1.1.m1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S6.p1.1.m1.1.1.3.2.2.3.3" xref="S6.p1.1.m1.1.1.3.2.2.3.3.cmml">p</mi></mrow></msup><mo id="S6.p1.1.m1.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S6.p1.1.m1.1.1.3.2.1.cmml">×</mo><msup id="S6.p1.1.m1.1.1.3.2.3" xref="S6.p1.1.m1.1.1.3.2.3.cmml"><mi id="S6.p1.1.m1.1.1.3.2.3.2" mathvariant="normal" xref="S6.p1.1.m1.1.1.3.2.3.2.cmml">Δ</mi><mrow id="S6.p1.1.m1.1.1.3.2.3.3" xref="S6.p1.1.m1.1.1.3.2.3.3.cmml"><mi id="S6.p1.1.m1.1.1.3.2.3.3.2" xref="S6.p1.1.m1.1.1.3.2.3.3.2.cmml">o</mi><mo id="S6.p1.1.m1.1.1.3.2.3.3.1" xref="S6.p1.1.m1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S6.p1.1.m1.1.1.3.2.3.3.3" xref="S6.p1.1.m1.1.1.3.2.3.3.3.cmml">p</mi></mrow></msup></mrow><mo id="S6.p1.1.m1.1.1.3.1" stretchy="false" xref="S6.p1.1.m1.1.1.3.1.cmml">→</mo><mrow id="S6.p1.1.m1.1.1.3.3" xref="S6.p1.1.m1.1.1.3.3.cmml"><mi id="S6.p1.1.m1.1.1.3.3.2" xref="S6.p1.1.m1.1.1.3.3.2.cmml">S</mi><mo id="S6.p1.1.m1.1.1.3.3.1" xref="S6.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S6.p1.1.m1.1.1.3.3.3" xref="S6.p1.1.m1.1.1.3.3.3.cmml">e</mi><mo id="S6.p1.1.m1.1.1.3.3.1a" xref="S6.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S6.p1.1.m1.1.1.3.3.4" xref="S6.p1.1.m1.1.1.3.3.4.cmml">t</mi><mo id="S6.p1.1.m1.1.1.3.3.1b" xref="S6.p1.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S6.p1.1.m1.1.1.3.3.5" xref="S6.p1.1.m1.1.1.3.3.5.cmml">s</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><apply id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1"><ci id="S6.p1.1.m1.1.1.1.cmml" xref="S6.p1.1.m1.1.1.1">:</ci><ci id="S6.p1.1.m1.1.1.2.cmml" xref="S6.p1.1.m1.1.1.2">𝑋</ci><apply id="S6.p1.1.m1.1.1.3.cmml" xref="S6.p1.1.m1.1.1.3"><ci id="S6.p1.1.m1.1.1.3.1.cmml" xref="S6.p1.1.m1.1.1.3.1">→</ci><apply id="S6.p1.1.m1.1.1.3.2.cmml" xref="S6.p1.1.m1.1.1.3.2"><times id="S6.p1.1.m1.1.1.3.2.1.cmml" xref="S6.p1.1.m1.1.1.3.2.1"></times><apply id="S6.p1.1.m1.1.1.3.2.2.cmml" xref="S6.p1.1.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="S6.p1.1.m1.1.1.3.2.2.1.cmml" xref="S6.p1.1.m1.1.1.3.2.2">superscript</csymbol><ci id="S6.p1.1.m1.1.1.3.2.2.2.cmml" xref="S6.p1.1.m1.1.1.3.2.2.2">Δ</ci><apply id="S6.p1.1.m1.1.1.3.2.2.3.cmml" xref="S6.p1.1.m1.1.1.3.2.2.3"><times id="S6.p1.1.m1.1.1.3.2.2.3.1.cmml" xref="S6.p1.1.m1.1.1.3.2.2.3.1"></times><ci id="S6.p1.1.m1.1.1.3.2.2.3.2.cmml" xref="S6.p1.1.m1.1.1.3.2.2.3.2">𝑜</ci><ci id="S6.p1.1.m1.1.1.3.2.2.3.3.cmml" xref="S6.p1.1.m1.1.1.3.2.2.3.3">𝑝</ci></apply></apply><apply id="S6.p1.1.m1.1.1.3.2.3.cmml" xref="S6.p1.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S6.p1.1.m1.1.1.3.2.3.1.cmml" xref="S6.p1.1.m1.1.1.3.2.3">superscript</csymbol><ci id="S6.p1.1.m1.1.1.3.2.3.2.cmml" xref="S6.p1.1.m1.1.1.3.2.3.2">Δ</ci><apply id="S6.p1.1.m1.1.1.3.2.3.3.cmml" xref="S6.p1.1.m1.1.1.3.2.3.3"><times id="S6.p1.1.m1.1.1.3.2.3.3.1.cmml" xref="S6.p1.1.m1.1.1.3.2.3.3.1"></times><ci id="S6.p1.1.m1.1.1.3.2.3.3.2.cmml" xref="S6.p1.1.m1.1.1.3.2.3.3.2">𝑜</ci><ci id="S6.p1.1.m1.1.1.3.2.3.3.3.cmml" xref="S6.p1.1.m1.1.1.3.2.3.3.3">𝑝</ci></apply></apply></apply><apply id="S6.p1.1.m1.1.1.3.3.cmml" xref="S6.p1.1.m1.1.1.3.3"><times id="S6.p1.1.m1.1.1.3.3.1.cmml" xref="S6.p1.1.m1.1.1.3.3.1"></times><ci id="S6.p1.1.m1.1.1.3.3.2.cmml" xref="S6.p1.1.m1.1.1.3.3.2">𝑆</ci><ci id="S6.p1.1.m1.1.1.3.3.3.cmml" xref="S6.p1.1.m1.1.1.3.3.3">𝑒</ci><ci id="S6.p1.1.m1.1.1.3.3.4.cmml" xref="S6.p1.1.m1.1.1.3.3.4">𝑡</ci><ci id="S6.p1.1.m1.1.1.3.3.5.cmml" xref="S6.p1.1.m1.1.1.3.3.5">𝑠</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">X:\Delta^{op}\times\Delta^{op}\to Sets</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">italic_X : roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT × roman_Δ start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → italic_S italic_e italic_t italic_s</annotation></semantics></math> be a bisimplicial set and <math alttext="R" class="ltx_Math" display="inline" id="S6.p1.2.m2.1"><semantics id="S6.p1.2.m2.1a"><mi id="S6.p1.2.m2.1.1" xref="S6.p1.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S6.p1.2.m2.1b"><ci id="S6.p1.2.m2.1.1.cmml" xref="S6.p1.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S6.p1.2.m2.1d">italic_R</annotation></semantics></math> a commutative ring. The <em class="ltx_emph ltx_font_italic" id="S6.p1.3.1">simplex category of <math alttext="X" class="ltx_Math" display="inline" id="S6.p1.3.1.m1.1"><semantics id="S6.p1.3.1.m1.1a"><mi id="S6.p1.3.1.m1.1.1" xref="S6.p1.3.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.p1.3.1.m1.1b"><ci id="S6.p1.3.1.m1.1.1.cmml" xref="S6.p1.3.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.3.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.p1.3.1.m1.1d">italic_X</annotation></semantics></math></em> is the category <math alttext="\Delta(X)" class="ltx_Math" display="inline" id="S6.p1.4.m3.1"><semantics id="S6.p1.4.m3.1a"><mrow id="S6.p1.4.m3.1.2" xref="S6.p1.4.m3.1.2.cmml"><mi id="S6.p1.4.m3.1.2.2" mathvariant="normal" xref="S6.p1.4.m3.1.2.2.cmml">Δ</mi><mo id="S6.p1.4.m3.1.2.1" xref="S6.p1.4.m3.1.2.1.cmml">⁢</mo><mrow id="S6.p1.4.m3.1.2.3.2" xref="S6.p1.4.m3.1.2.cmml"><mo id="S6.p1.4.m3.1.2.3.2.1" stretchy="false" xref="S6.p1.4.m3.1.2.cmml">(</mo><mi id="S6.p1.4.m3.1.1" xref="S6.p1.4.m3.1.1.cmml">X</mi><mo id="S6.p1.4.m3.1.2.3.2.2" stretchy="false" xref="S6.p1.4.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.4.m3.1b"><apply id="S6.p1.4.m3.1.2.cmml" xref="S6.p1.4.m3.1.2"><times id="S6.p1.4.m3.1.2.1.cmml" xref="S6.p1.4.m3.1.2.1"></times><ci id="S6.p1.4.m3.1.2.2.cmml" xref="S6.p1.4.m3.1.2.2">Δ</ci><ci id="S6.p1.4.m3.1.1.cmml" xref="S6.p1.4.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.4.m3.1c">\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p1.4.m3.1d">roman_Δ ( italic_X )</annotation></semantics></math> whose objects are the simplices <math alttext="\sigma\in X_{p,q}" class="ltx_Math" display="inline" id="S6.p1.5.m4.2"><semantics id="S6.p1.5.m4.2a"><mrow id="S6.p1.5.m4.2.3" xref="S6.p1.5.m4.2.3.cmml"><mi id="S6.p1.5.m4.2.3.2" xref="S6.p1.5.m4.2.3.2.cmml">σ</mi><mo id="S6.p1.5.m4.2.3.1" xref="S6.p1.5.m4.2.3.1.cmml">∈</mo><msub id="S6.p1.5.m4.2.3.3" xref="S6.p1.5.m4.2.3.3.cmml"><mi id="S6.p1.5.m4.2.3.3.2" xref="S6.p1.5.m4.2.3.3.2.cmml">X</mi><mrow id="S6.p1.5.m4.2.2.2.4" xref="S6.p1.5.m4.2.2.2.3.cmml"><mi id="S6.p1.5.m4.1.1.1.1" xref="S6.p1.5.m4.1.1.1.1.cmml">p</mi><mo id="S6.p1.5.m4.2.2.2.4.1" xref="S6.p1.5.m4.2.2.2.3.cmml">,</mo><mi id="S6.p1.5.m4.2.2.2.2" xref="S6.p1.5.m4.2.2.2.2.cmml">q</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.5.m4.2b"><apply id="S6.p1.5.m4.2.3.cmml" xref="S6.p1.5.m4.2.3"><in id="S6.p1.5.m4.2.3.1.cmml" xref="S6.p1.5.m4.2.3.1"></in><ci id="S6.p1.5.m4.2.3.2.cmml" xref="S6.p1.5.m4.2.3.2">𝜎</ci><apply id="S6.p1.5.m4.2.3.3.cmml" xref="S6.p1.5.m4.2.3.3"><csymbol cd="ambiguous" id="S6.p1.5.m4.2.3.3.1.cmml" xref="S6.p1.5.m4.2.3.3">subscript</csymbol><ci id="S6.p1.5.m4.2.3.3.2.cmml" xref="S6.p1.5.m4.2.3.3.2">𝑋</ci><list id="S6.p1.5.m4.2.2.2.3.cmml" xref="S6.p1.5.m4.2.2.2.4"><ci id="S6.p1.5.m4.1.1.1.1.cmml" xref="S6.p1.5.m4.1.1.1.1">𝑝</ci><ci id="S6.p1.5.m4.2.2.2.2.cmml" xref="S6.p1.5.m4.2.2.2.2">𝑞</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.5.m4.2c">\sigma\in X_{p,q}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.5.m4.2d">italic_σ ∈ italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT</annotation></semantics></math> and where a morphism <math alttext="g\in\mathrm{Mor}_{\Delta(X)}(\tau,\sigma)" class="ltx_Math" display="inline" id="S6.p1.6.m5.3"><semantics id="S6.p1.6.m5.3a"><mrow id="S6.p1.6.m5.3.4" xref="S6.p1.6.m5.3.4.cmml"><mi id="S6.p1.6.m5.3.4.2" xref="S6.p1.6.m5.3.4.2.cmml">g</mi><mo id="S6.p1.6.m5.3.4.1" xref="S6.p1.6.m5.3.4.1.cmml">∈</mo><mrow id="S6.p1.6.m5.3.4.3" xref="S6.p1.6.m5.3.4.3.cmml"><msub id="S6.p1.6.m5.3.4.3.2" xref="S6.p1.6.m5.3.4.3.2.cmml"><mi id="S6.p1.6.m5.3.4.3.2.2" xref="S6.p1.6.m5.3.4.3.2.2.cmml">Mor</mi><mrow id="S6.p1.6.m5.1.1.1" xref="S6.p1.6.m5.1.1.1.cmml"><mi id="S6.p1.6.m5.1.1.1.3" mathvariant="normal" xref="S6.p1.6.m5.1.1.1.3.cmml">Δ</mi><mo id="S6.p1.6.m5.1.1.1.2" xref="S6.p1.6.m5.1.1.1.2.cmml">⁢</mo><mrow id="S6.p1.6.m5.1.1.1.4.2" xref="S6.p1.6.m5.1.1.1.cmml"><mo id="S6.p1.6.m5.1.1.1.4.2.1" stretchy="false" xref="S6.p1.6.m5.1.1.1.cmml">(</mo><mi id="S6.p1.6.m5.1.1.1.1" xref="S6.p1.6.m5.1.1.1.1.cmml">X</mi><mo id="S6.p1.6.m5.1.1.1.4.2.2" stretchy="false" xref="S6.p1.6.m5.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.p1.6.m5.3.4.3.1" xref="S6.p1.6.m5.3.4.3.1.cmml">⁢</mo><mrow id="S6.p1.6.m5.3.4.3.3.2" xref="S6.p1.6.m5.3.4.3.3.1.cmml"><mo id="S6.p1.6.m5.3.4.3.3.2.1" stretchy="false" xref="S6.p1.6.m5.3.4.3.3.1.cmml">(</mo><mi id="S6.p1.6.m5.2.2" xref="S6.p1.6.m5.2.2.cmml">τ</mi><mo id="S6.p1.6.m5.3.4.3.3.2.2" xref="S6.p1.6.m5.3.4.3.3.1.cmml">,</mo><mi id="S6.p1.6.m5.3.3" xref="S6.p1.6.m5.3.3.cmml">σ</mi><mo id="S6.p1.6.m5.3.4.3.3.2.3" stretchy="false" xref="S6.p1.6.m5.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.6.m5.3b"><apply id="S6.p1.6.m5.3.4.cmml" xref="S6.p1.6.m5.3.4"><in id="S6.p1.6.m5.3.4.1.cmml" xref="S6.p1.6.m5.3.4.1"></in><ci id="S6.p1.6.m5.3.4.2.cmml" xref="S6.p1.6.m5.3.4.2">𝑔</ci><apply id="S6.p1.6.m5.3.4.3.cmml" xref="S6.p1.6.m5.3.4.3"><times id="S6.p1.6.m5.3.4.3.1.cmml" xref="S6.p1.6.m5.3.4.3.1"></times><apply id="S6.p1.6.m5.3.4.3.2.cmml" xref="S6.p1.6.m5.3.4.3.2"><csymbol cd="ambiguous" id="S6.p1.6.m5.3.4.3.2.1.cmml" xref="S6.p1.6.m5.3.4.3.2">subscript</csymbol><ci id="S6.p1.6.m5.3.4.3.2.2.cmml" xref="S6.p1.6.m5.3.4.3.2.2">Mor</ci><apply id="S6.p1.6.m5.1.1.1.cmml" xref="S6.p1.6.m5.1.1.1"><times id="S6.p1.6.m5.1.1.1.2.cmml" xref="S6.p1.6.m5.1.1.1.2"></times><ci id="S6.p1.6.m5.1.1.1.3.cmml" xref="S6.p1.6.m5.1.1.1.3">Δ</ci><ci id="S6.p1.6.m5.1.1.1.1.cmml" xref="S6.p1.6.m5.1.1.1.1">𝑋</ci></apply></apply><interval closure="open" id="S6.p1.6.m5.3.4.3.3.1.cmml" xref="S6.p1.6.m5.3.4.3.3.2"><ci id="S6.p1.6.m5.2.2.cmml" xref="S6.p1.6.m5.2.2">𝜏</ci><ci id="S6.p1.6.m5.3.3.cmml" xref="S6.p1.6.m5.3.3">𝜎</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.6.m5.3c">g\in\mathrm{Mor}_{\Delta(X)}(\tau,\sigma)</annotation><annotation encoding="application/x-llamapun" id="S6.p1.6.m5.3d">italic_g ∈ roman_Mor start_POSTSUBSCRIPT roman_Δ ( italic_X ) end_POSTSUBSCRIPT ( italic_τ , italic_σ )</annotation></semantics></math> is given by a morphism <math alttext="g^{\prime}:[r]\times[s]\to[p]\times[q]" class="ltx_Math" display="inline" id="S6.p1.7.m6.4"><semantics id="S6.p1.7.m6.4a"><mrow id="S6.p1.7.m6.4.5" xref="S6.p1.7.m6.4.5.cmml"><msup id="S6.p1.7.m6.4.5.2" xref="S6.p1.7.m6.4.5.2.cmml"><mi id="S6.p1.7.m6.4.5.2.2" xref="S6.p1.7.m6.4.5.2.2.cmml">g</mi><mo id="S6.p1.7.m6.4.5.2.3" xref="S6.p1.7.m6.4.5.2.3.cmml">′</mo></msup><mo id="S6.p1.7.m6.4.5.1" lspace="0.278em" rspace="0.278em" xref="S6.p1.7.m6.4.5.1.cmml">:</mo><mrow id="S6.p1.7.m6.4.5.3" xref="S6.p1.7.m6.4.5.3.cmml"><mrow id="S6.p1.7.m6.4.5.3.2" xref="S6.p1.7.m6.4.5.3.2.cmml"><mrow id="S6.p1.7.m6.4.5.3.2.2.2" xref="S6.p1.7.m6.4.5.3.2.2.1.cmml"><mo id="S6.p1.7.m6.4.5.3.2.2.2.1" stretchy="false" xref="S6.p1.7.m6.4.5.3.2.2.1.1.cmml">[</mo><mi id="S6.p1.7.m6.1.1" xref="S6.p1.7.m6.1.1.cmml">r</mi><mo id="S6.p1.7.m6.4.5.3.2.2.2.2" rspace="0.055em" stretchy="false" xref="S6.p1.7.m6.4.5.3.2.2.1.1.cmml">]</mo></mrow><mo id="S6.p1.7.m6.4.5.3.2.1" rspace="0.222em" xref="S6.p1.7.m6.4.5.3.2.1.cmml">×</mo><mrow id="S6.p1.7.m6.4.5.3.2.3.2" xref="S6.p1.7.m6.4.5.3.2.3.1.cmml"><mo id="S6.p1.7.m6.4.5.3.2.3.2.1" stretchy="false" xref="S6.p1.7.m6.4.5.3.2.3.1.1.cmml">[</mo><mi id="S6.p1.7.m6.2.2" xref="S6.p1.7.m6.2.2.cmml">s</mi><mo id="S6.p1.7.m6.4.5.3.2.3.2.2" stretchy="false" xref="S6.p1.7.m6.4.5.3.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S6.p1.7.m6.4.5.3.1" stretchy="false" xref="S6.p1.7.m6.4.5.3.1.cmml">→</mo><mrow id="S6.p1.7.m6.4.5.3.3" xref="S6.p1.7.m6.4.5.3.3.cmml"><mrow id="S6.p1.7.m6.4.5.3.3.2.2" xref="S6.p1.7.m6.4.5.3.3.2.1.cmml"><mo id="S6.p1.7.m6.4.5.3.3.2.2.1" stretchy="false" xref="S6.p1.7.m6.4.5.3.3.2.1.1.cmml">[</mo><mi id="S6.p1.7.m6.3.3" xref="S6.p1.7.m6.3.3.cmml">p</mi><mo id="S6.p1.7.m6.4.5.3.3.2.2.2" rspace="0.055em" stretchy="false" xref="S6.p1.7.m6.4.5.3.3.2.1.1.cmml">]</mo></mrow><mo id="S6.p1.7.m6.4.5.3.3.1" rspace="0.222em" xref="S6.p1.7.m6.4.5.3.3.1.cmml">×</mo><mrow id="S6.p1.7.m6.4.5.3.3.3.2" xref="S6.p1.7.m6.4.5.3.3.3.1.cmml"><mo id="S6.p1.7.m6.4.5.3.3.3.2.1" stretchy="false" xref="S6.p1.7.m6.4.5.3.3.3.1.1.cmml">[</mo><mi id="S6.p1.7.m6.4.4" xref="S6.p1.7.m6.4.4.cmml">q</mi><mo id="S6.p1.7.m6.4.5.3.3.3.2.2" stretchy="false" xref="S6.p1.7.m6.4.5.3.3.3.1.1.cmml">]</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.7.m6.4b"><apply id="S6.p1.7.m6.4.5.cmml" xref="S6.p1.7.m6.4.5"><ci id="S6.p1.7.m6.4.5.1.cmml" xref="S6.p1.7.m6.4.5.1">:</ci><apply id="S6.p1.7.m6.4.5.2.cmml" xref="S6.p1.7.m6.4.5.2"><csymbol cd="ambiguous" id="S6.p1.7.m6.4.5.2.1.cmml" xref="S6.p1.7.m6.4.5.2">superscript</csymbol><ci id="S6.p1.7.m6.4.5.2.2.cmml" xref="S6.p1.7.m6.4.5.2.2">𝑔</ci><ci id="S6.p1.7.m6.4.5.2.3.cmml" xref="S6.p1.7.m6.4.5.2.3">′</ci></apply><apply id="S6.p1.7.m6.4.5.3.cmml" xref="S6.p1.7.m6.4.5.3"><ci id="S6.p1.7.m6.4.5.3.1.cmml" xref="S6.p1.7.m6.4.5.3.1">→</ci><apply id="S6.p1.7.m6.4.5.3.2.cmml" xref="S6.p1.7.m6.4.5.3.2"><times id="S6.p1.7.m6.4.5.3.2.1.cmml" xref="S6.p1.7.m6.4.5.3.2.1"></times><apply id="S6.p1.7.m6.4.5.3.2.2.1.cmml" xref="S6.p1.7.m6.4.5.3.2.2.2"><csymbol cd="latexml" id="S6.p1.7.m6.4.5.3.2.2.1.1.cmml" xref="S6.p1.7.m6.4.5.3.2.2.2.1">delimited-[]</csymbol><ci id="S6.p1.7.m6.1.1.cmml" xref="S6.p1.7.m6.1.1">𝑟</ci></apply><apply id="S6.p1.7.m6.4.5.3.2.3.1.cmml" xref="S6.p1.7.m6.4.5.3.2.3.2"><csymbol cd="latexml" id="S6.p1.7.m6.4.5.3.2.3.1.1.cmml" xref="S6.p1.7.m6.4.5.3.2.3.2.1">delimited-[]</csymbol><ci id="S6.p1.7.m6.2.2.cmml" xref="S6.p1.7.m6.2.2">𝑠</ci></apply></apply><apply id="S6.p1.7.m6.4.5.3.3.cmml" xref="S6.p1.7.m6.4.5.3.3"><times id="S6.p1.7.m6.4.5.3.3.1.cmml" xref="S6.p1.7.m6.4.5.3.3.1"></times><apply id="S6.p1.7.m6.4.5.3.3.2.1.cmml" xref="S6.p1.7.m6.4.5.3.3.2.2"><csymbol cd="latexml" id="S6.p1.7.m6.4.5.3.3.2.1.1.cmml" xref="S6.p1.7.m6.4.5.3.3.2.2.1">delimited-[]</csymbol><ci id="S6.p1.7.m6.3.3.cmml" xref="S6.p1.7.m6.3.3">𝑝</ci></apply><apply id="S6.p1.7.m6.4.5.3.3.3.1.cmml" xref="S6.p1.7.m6.4.5.3.3.3.2"><csymbol cd="latexml" id="S6.p1.7.m6.4.5.3.3.3.1.1.cmml" xref="S6.p1.7.m6.4.5.3.3.3.2.1">delimited-[]</csymbol><ci id="S6.p1.7.m6.4.4.cmml" xref="S6.p1.7.m6.4.4">𝑞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.7.m6.4c">g^{\prime}:[r]\times[s]\to[p]\times[q]</annotation><annotation encoding="application/x-llamapun" id="S6.p1.7.m6.4d">italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : [ italic_r ] × [ italic_s ] → [ italic_p ] × [ italic_q ]</annotation></semantics></math> in <math alttext="\Delta\times\Delta" class="ltx_Math" display="inline" id="S6.p1.8.m7.1"><semantics id="S6.p1.8.m7.1a"><mrow id="S6.p1.8.m7.1.1" xref="S6.p1.8.m7.1.1.cmml"><mi id="S6.p1.8.m7.1.1.2" mathvariant="normal" xref="S6.p1.8.m7.1.1.2.cmml">Δ</mi><mo id="S6.p1.8.m7.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.p1.8.m7.1.1.1.cmml">×</mo><mi id="S6.p1.8.m7.1.1.3" mathvariant="normal" xref="S6.p1.8.m7.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.8.m7.1b"><apply id="S6.p1.8.m7.1.1.cmml" xref="S6.p1.8.m7.1.1"><times id="S6.p1.8.m7.1.1.1.cmml" xref="S6.p1.8.m7.1.1.1"></times><ci id="S6.p1.8.m7.1.1.2.cmml" xref="S6.p1.8.m7.1.1.2">Δ</ci><ci id="S6.p1.8.m7.1.1.3.cmml" xref="S6.p1.8.m7.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.8.m7.1c">\Delta\times\Delta</annotation><annotation encoding="application/x-llamapun" id="S6.p1.8.m7.1d">roman_Δ × roman_Δ</annotation></semantics></math> such that <math alttext="(g^{\prime})^{*}(\sigma)=\tau" class="ltx_Math" display="inline" id="S6.p1.9.m8.2"><semantics id="S6.p1.9.m8.2a"><mrow id="S6.p1.9.m8.2.2" xref="S6.p1.9.m8.2.2.cmml"><mrow id="S6.p1.9.m8.2.2.1" xref="S6.p1.9.m8.2.2.1.cmml"><msup id="S6.p1.9.m8.2.2.1.1" xref="S6.p1.9.m8.2.2.1.1.cmml"><mrow id="S6.p1.9.m8.2.2.1.1.1.1" xref="S6.p1.9.m8.2.2.1.1.1.1.1.cmml"><mo id="S6.p1.9.m8.2.2.1.1.1.1.2" stretchy="false" xref="S6.p1.9.m8.2.2.1.1.1.1.1.cmml">(</mo><msup id="S6.p1.9.m8.2.2.1.1.1.1.1" xref="S6.p1.9.m8.2.2.1.1.1.1.1.cmml"><mi id="S6.p1.9.m8.2.2.1.1.1.1.1.2" xref="S6.p1.9.m8.2.2.1.1.1.1.1.2.cmml">g</mi><mo id="S6.p1.9.m8.2.2.1.1.1.1.1.3" xref="S6.p1.9.m8.2.2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.p1.9.m8.2.2.1.1.1.1.3" stretchy="false" xref="S6.p1.9.m8.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.p1.9.m8.2.2.1.1.3" xref="S6.p1.9.m8.2.2.1.1.3.cmml">∗</mo></msup><mo id="S6.p1.9.m8.2.2.1.2" xref="S6.p1.9.m8.2.2.1.2.cmml">⁢</mo><mrow id="S6.p1.9.m8.2.2.1.3.2" xref="S6.p1.9.m8.2.2.1.cmml"><mo id="S6.p1.9.m8.2.2.1.3.2.1" stretchy="false" xref="S6.p1.9.m8.2.2.1.cmml">(</mo><mi id="S6.p1.9.m8.1.1" xref="S6.p1.9.m8.1.1.cmml">σ</mi><mo id="S6.p1.9.m8.2.2.1.3.2.2" stretchy="false" xref="S6.p1.9.m8.2.2.1.cmml">)</mo></mrow></mrow><mo id="S6.p1.9.m8.2.2.2" xref="S6.p1.9.m8.2.2.2.cmml">=</mo><mi id="S6.p1.9.m8.2.2.3" xref="S6.p1.9.m8.2.2.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.9.m8.2b"><apply id="S6.p1.9.m8.2.2.cmml" xref="S6.p1.9.m8.2.2"><eq id="S6.p1.9.m8.2.2.2.cmml" xref="S6.p1.9.m8.2.2.2"></eq><apply id="S6.p1.9.m8.2.2.1.cmml" xref="S6.p1.9.m8.2.2.1"><times id="S6.p1.9.m8.2.2.1.2.cmml" xref="S6.p1.9.m8.2.2.1.2"></times><apply id="S6.p1.9.m8.2.2.1.1.cmml" xref="S6.p1.9.m8.2.2.1.1"><csymbol cd="ambiguous" id="S6.p1.9.m8.2.2.1.1.2.cmml" xref="S6.p1.9.m8.2.2.1.1">superscript</csymbol><apply id="S6.p1.9.m8.2.2.1.1.1.1.1.cmml" xref="S6.p1.9.m8.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.p1.9.m8.2.2.1.1.1.1.1.1.cmml" xref="S6.p1.9.m8.2.2.1.1.1.1">superscript</csymbol><ci id="S6.p1.9.m8.2.2.1.1.1.1.1.2.cmml" xref="S6.p1.9.m8.2.2.1.1.1.1.1.2">𝑔</ci><ci id="S6.p1.9.m8.2.2.1.1.1.1.1.3.cmml" xref="S6.p1.9.m8.2.2.1.1.1.1.1.3">′</ci></apply><times id="S6.p1.9.m8.2.2.1.1.3.cmml" xref="S6.p1.9.m8.2.2.1.1.3"></times></apply><ci id="S6.p1.9.m8.1.1.cmml" xref="S6.p1.9.m8.1.1">𝜎</ci></apply><ci id="S6.p1.9.m8.2.2.3.cmml" xref="S6.p1.9.m8.2.2.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.9.m8.2c">(g^{\prime})^{*}(\sigma)=\tau</annotation><annotation encoding="application/x-llamapun" id="S6.p1.9.m8.2d">( italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ ) = italic_τ</annotation></semantics></math>. For a bisimplicial set <math alttext="X" class="ltx_Math" display="inline" id="S6.p1.10.m9.1"><semantics id="S6.p1.10.m9.1a"><mi id="S6.p1.10.m9.1.1" xref="S6.p1.10.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.p1.10.m9.1b"><ci id="S6.p1.10.m9.1.1.cmml" xref="S6.p1.10.m9.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.10.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.p1.10.m9.1d">italic_X</annotation></semantics></math>, a functor <math alttext="\mathcal{M}:\Delta(X)\to R" class="ltx_Math" display="inline" id="S6.p1.11.m10.1"><semantics id="S6.p1.11.m10.1a"><mrow id="S6.p1.11.m10.1.2" xref="S6.p1.11.m10.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p1.11.m10.1.2.2" xref="S6.p1.11.m10.1.2.2.cmml">ℳ</mi><mo id="S6.p1.11.m10.1.2.1" lspace="0.278em" rspace="0.278em" xref="S6.p1.11.m10.1.2.1.cmml">:</mo><mrow id="S6.p1.11.m10.1.2.3" xref="S6.p1.11.m10.1.2.3.cmml"><mrow id="S6.p1.11.m10.1.2.3.2" xref="S6.p1.11.m10.1.2.3.2.cmml"><mi id="S6.p1.11.m10.1.2.3.2.2" mathvariant="normal" xref="S6.p1.11.m10.1.2.3.2.2.cmml">Δ</mi><mo id="S6.p1.11.m10.1.2.3.2.1" xref="S6.p1.11.m10.1.2.3.2.1.cmml">⁢</mo><mrow id="S6.p1.11.m10.1.2.3.2.3.2" xref="S6.p1.11.m10.1.2.3.2.cmml"><mo id="S6.p1.11.m10.1.2.3.2.3.2.1" stretchy="false" xref="S6.p1.11.m10.1.2.3.2.cmml">(</mo><mi id="S6.p1.11.m10.1.1" xref="S6.p1.11.m10.1.1.cmml">X</mi><mo id="S6.p1.11.m10.1.2.3.2.3.2.2" stretchy="false" xref="S6.p1.11.m10.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.p1.11.m10.1.2.3.1" stretchy="false" xref="S6.p1.11.m10.1.2.3.1.cmml">→</mo><mi id="S6.p1.11.m10.1.2.3.3" xref="S6.p1.11.m10.1.2.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.11.m10.1b"><apply id="S6.p1.11.m10.1.2.cmml" xref="S6.p1.11.m10.1.2"><ci id="S6.p1.11.m10.1.2.1.cmml" xref="S6.p1.11.m10.1.2.1">:</ci><ci id="S6.p1.11.m10.1.2.2.cmml" xref="S6.p1.11.m10.1.2.2">ℳ</ci><apply id="S6.p1.11.m10.1.2.3.cmml" xref="S6.p1.11.m10.1.2.3"><ci id="S6.p1.11.m10.1.2.3.1.cmml" xref="S6.p1.11.m10.1.2.3.1">→</ci><apply id="S6.p1.11.m10.1.2.3.2.cmml" xref="S6.p1.11.m10.1.2.3.2"><times id="S6.p1.11.m10.1.2.3.2.1.cmml" xref="S6.p1.11.m10.1.2.3.2.1"></times><ci id="S6.p1.11.m10.1.2.3.2.2.cmml" xref="S6.p1.11.m10.1.2.3.2.2">Δ</ci><ci id="S6.p1.11.m10.1.1.cmml" xref="S6.p1.11.m10.1.1">𝑋</ci></apply><ci id="S6.p1.11.m10.1.2.3.3.cmml" xref="S6.p1.11.m10.1.2.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.11.m10.1c">\mathcal{M}:\Delta(X)\to R</annotation><annotation encoding="application/x-llamapun" id="S6.p1.11.m10.1d">caligraphic_M : roman_Δ ( italic_X ) → italic_R</annotation></semantics></math>-Mod is called a <em class="ltx_emph ltx_font_italic" id="S6.p1.13.2">coefficient system</em> for <math alttext="X" class="ltx_Math" display="inline" id="S6.p1.12.m11.1"><semantics id="S6.p1.12.m11.1a"><mi id="S6.p1.12.m11.1.1" xref="S6.p1.12.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.p1.12.m11.1b"><ci id="S6.p1.12.m11.1.1.cmml" xref="S6.p1.12.m11.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.12.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.p1.12.m11.1d">italic_X</annotation></semantics></math> over <math alttext="R" class="ltx_Math" display="inline" id="S6.p1.13.m12.1"><semantics id="S6.p1.13.m12.1a"><mi id="S6.p1.13.m12.1.1" xref="S6.p1.13.m12.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S6.p1.13.m12.1b"><ci id="S6.p1.13.m12.1.1.cmml" xref="S6.p1.13.m12.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.13.m12.1c">R</annotation><annotation encoding="application/x-llamapun" id="S6.p1.13.m12.1d">italic_R</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.7">The cohomology of a bisimplicial set <math alttext="X" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><mi id="S6.p2.1.m1.1.1" xref="S6.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><ci id="S6.p2.1.m1.1.1.cmml" xref="S6.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">italic_X</annotation></semantics></math> with coefficients in a coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S6.p2.2.m2.1"><semantics id="S6.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S6.p2.2.m2.1.1" xref="S6.p2.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S6.p2.2.m2.1b"><ci id="S6.p2.2.m2.1.1.cmml" xref="S6.p2.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.2.m2.1d">caligraphic_M</annotation></semantics></math> is defined as the cohomology of the simplex category <math alttext="\Delta(X)" class="ltx_Math" display="inline" id="S6.p2.3.m3.1"><semantics id="S6.p2.3.m3.1a"><mrow id="S6.p2.3.m3.1.2" xref="S6.p2.3.m3.1.2.cmml"><mi id="S6.p2.3.m3.1.2.2" mathvariant="normal" xref="S6.p2.3.m3.1.2.2.cmml">Δ</mi><mo id="S6.p2.3.m3.1.2.1" xref="S6.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.p2.3.m3.1.2.3.2" xref="S6.p2.3.m3.1.2.cmml"><mo id="S6.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S6.p2.3.m3.1.2.cmml">(</mo><mi id="S6.p2.3.m3.1.1" xref="S6.p2.3.m3.1.1.cmml">X</mi><mo id="S6.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S6.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.3.m3.1b"><apply id="S6.p2.3.m3.1.2.cmml" xref="S6.p2.3.m3.1.2"><times id="S6.p2.3.m3.1.2.1.cmml" xref="S6.p2.3.m3.1.2.1"></times><ci id="S6.p2.3.m3.1.2.2.cmml" xref="S6.p2.3.m3.1.2.2">Δ</ci><ci id="S6.p2.3.m3.1.1.cmml" xref="S6.p2.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.3.m3.1c">\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.3.m3.1d">roman_Δ ( italic_X )</annotation></semantics></math> with coefficients in <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S6.p2.4.m4.1"><semantics id="S6.p2.4.m4.1a"><mrow id="S6.p2.4.m4.1.2" xref="S6.p2.4.m4.1.2.cmml"><mi id="S6.p2.4.m4.1.2.2" xref="S6.p2.4.m4.1.2.2.cmml">R</mi><mo id="S6.p2.4.m4.1.2.1" xref="S6.p2.4.m4.1.2.1.cmml">⁢</mo><mi id="S6.p2.4.m4.1.2.3" mathvariant="normal" xref="S6.p2.4.m4.1.2.3.cmml">Δ</mi><mo id="S6.p2.4.m4.1.2.1a" xref="S6.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S6.p2.4.m4.1.2.4.2" xref="S6.p2.4.m4.1.2.cmml"><mo id="S6.p2.4.m4.1.2.4.2.1" stretchy="false" xref="S6.p2.4.m4.1.2.cmml">(</mo><mi id="S6.p2.4.m4.1.1" xref="S6.p2.4.m4.1.1.cmml">X</mi><mo id="S6.p2.4.m4.1.2.4.2.2" stretchy="false" xref="S6.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.4.m4.1b"><apply id="S6.p2.4.m4.1.2.cmml" xref="S6.p2.4.m4.1.2"><times id="S6.p2.4.m4.1.2.1.cmml" xref="S6.p2.4.m4.1.2.1"></times><ci id="S6.p2.4.m4.1.2.2.cmml" xref="S6.p2.4.m4.1.2.2">𝑅</ci><ci id="S6.p2.4.m4.1.2.3.cmml" xref="S6.p2.4.m4.1.2.3">Δ</ci><ci id="S6.p2.4.m4.1.1.cmml" xref="S6.p2.4.m4.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.4.m4.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.4.m4.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S6.p2.5.m5.1"><semantics id="S6.p2.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S6.p2.5.m5.1.1" xref="S6.p2.5.m5.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S6.p2.5.m5.1b"><ci id="S6.p2.5.m5.1.1.cmml" xref="S6.p2.5.m5.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.5.m5.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.5.m5.1d">caligraphic_M</annotation></semantics></math>. As we did for the cohomology of simplicial sets, we can define a projective resolution for <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S6.p2.6.m6.1"><semantics id="S6.p2.6.m6.1a"><munder accentunder="true" id="S6.p2.6.m6.1.1" xref="S6.p2.6.m6.1.1.cmml"><mi id="S6.p2.6.m6.1.1.2" xref="S6.p2.6.m6.1.1.2.cmml">R</mi><mo id="S6.p2.6.m6.1.1.1" xref="S6.p2.6.m6.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S6.p2.6.m6.1b"><apply id="S6.p2.6.m6.1.1.cmml" xref="S6.p2.6.m6.1.1"><ci id="S6.p2.6.m6.1.1.1.cmml" xref="S6.p2.6.m6.1.1.1">¯</ci><ci id="S6.p2.6.m6.1.1.2.cmml" xref="S6.p2.6.m6.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.6.m6.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.6.m6.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S6.p2.7.m7.1"><semantics id="S6.p2.7.m7.1a"><mrow id="S6.p2.7.m7.1.2" xref="S6.p2.7.m7.1.2.cmml"><mi id="S6.p2.7.m7.1.2.2" xref="S6.p2.7.m7.1.2.2.cmml">R</mi><mo id="S6.p2.7.m7.1.2.1" xref="S6.p2.7.m7.1.2.1.cmml">⁢</mo><mi id="S6.p2.7.m7.1.2.3" mathvariant="normal" xref="S6.p2.7.m7.1.2.3.cmml">Δ</mi><mo id="S6.p2.7.m7.1.2.1a" xref="S6.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S6.p2.7.m7.1.2.4.2" xref="S6.p2.7.m7.1.2.cmml"><mo id="S6.p2.7.m7.1.2.4.2.1" stretchy="false" xref="S6.p2.7.m7.1.2.cmml">(</mo><mi id="S6.p2.7.m7.1.1" xref="S6.p2.7.m7.1.1.cmml">X</mi><mo id="S6.p2.7.m7.1.2.4.2.2" stretchy="false" xref="S6.p2.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.7.m7.1b"><apply id="S6.p2.7.m7.1.2.cmml" xref="S6.p2.7.m7.1.2"><times id="S6.p2.7.m7.1.2.1.cmml" xref="S6.p2.7.m7.1.2.1"></times><ci id="S6.p2.7.m7.1.2.2.cmml" xref="S6.p2.7.m7.1.2.2">𝑅</ci><ci id="S6.p2.7.m7.1.2.3.cmml" xref="S6.p2.7.m7.1.2.3">Δ</ci><ci id="S6.p2.7.m7.1.1.cmml" xref="S6.p2.7.m7.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.7.m7.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.7.m7.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module as follows:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S6.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.1.1.1">Definition 6.1</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem1.p1"> <p class="ltx_p" id="S6.Thmtheorem1.p1.3">Let <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.1.m1.1"><semantics id="S6.Thmtheorem1.p1.1.m1.1a"><mi id="S6.Thmtheorem1.p1.1.m1.1.1" xref="S6.Thmtheorem1.p1.1.m1.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.1.m1.1b"><ci 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xref="S6.Ex68.m1.7.7">?</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex68.m1.8c">\mathbb{P}_{r,s}=\bigoplus_{\tau\in X_{r,s}}R\mathrm{Mor}_{\Delta(X)}(\tau,?).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex68.m1.8d">blackboard_P start_POSTSUBSCRIPT italic_r , italic_s end_POSTSUBSCRIPT = ⨁ start_POSTSUBSCRIPT italic_τ ∈ italic_X start_POSTSUBSCRIPT italic_r , italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_R roman_Mor start_POSTSUBSCRIPT roman_Δ ( italic_X ) end_POSTSUBSCRIPT ( italic_τ , ? ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmtheorem1.p1.12">For every <math alttext="\theta\in X_{p,q}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.4.m1.2"><semantics id="S6.Thmtheorem1.p1.4.m1.2a"><mrow id="S6.Thmtheorem1.p1.4.m1.2.3" xref="S6.Thmtheorem1.p1.4.m1.2.3.cmml"><mi id="S6.Thmtheorem1.p1.4.m1.2.3.2" xref="S6.Thmtheorem1.p1.4.m1.2.3.2.cmml">θ</mi><mo id="S6.Thmtheorem1.p1.4.m1.2.3.1" xref="S6.Thmtheorem1.p1.4.m1.2.3.1.cmml">∈</mo><msub id="S6.Thmtheorem1.p1.4.m1.2.3.3" xref="S6.Thmtheorem1.p1.4.m1.2.3.3.cmml"><mi id="S6.Thmtheorem1.p1.4.m1.2.3.3.2" xref="S6.Thmtheorem1.p1.4.m1.2.3.3.2.cmml">X</mi><mrow id="S6.Thmtheorem1.p1.4.m1.2.2.2.4" xref="S6.Thmtheorem1.p1.4.m1.2.2.2.3.cmml"><mi id="S6.Thmtheorem1.p1.4.m1.1.1.1.1" xref="S6.Thmtheorem1.p1.4.m1.1.1.1.1.cmml">p</mi><mo id="S6.Thmtheorem1.p1.4.m1.2.2.2.4.1" xref="S6.Thmtheorem1.p1.4.m1.2.2.2.3.cmml">,</mo><mi id="S6.Thmtheorem1.p1.4.m1.2.2.2.2" xref="S6.Thmtheorem1.p1.4.m1.2.2.2.2.cmml">q</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.4.m1.2b"><apply id="S6.Thmtheorem1.p1.4.m1.2.3.cmml" xref="S6.Thmtheorem1.p1.4.m1.2.3"><in id="S6.Thmtheorem1.p1.4.m1.2.3.1.cmml" xref="S6.Thmtheorem1.p1.4.m1.2.3.1"></in><ci id="S6.Thmtheorem1.p1.4.m1.2.3.2.cmml" xref="S6.Thmtheorem1.p1.4.m1.2.3.2">𝜃</ci><apply id="S6.Thmtheorem1.p1.4.m1.2.3.3.cmml" xref="S6.Thmtheorem1.p1.4.m1.2.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.4.m1.2.3.3.1.cmml" xref="S6.Thmtheorem1.p1.4.m1.2.3.3">subscript</csymbol><ci id="S6.Thmtheorem1.p1.4.m1.2.3.3.2.cmml" xref="S6.Thmtheorem1.p1.4.m1.2.3.3.2">𝑋</ci><list id="S6.Thmtheorem1.p1.4.m1.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.4.m1.2.2.2.4"><ci id="S6.Thmtheorem1.p1.4.m1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.4.m1.1.1.1.1">𝑝</ci><ci id="S6.Thmtheorem1.p1.4.m1.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.4.m1.2.2.2.2">𝑞</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.4.m1.2c">\theta\in X_{p,q}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.4.m1.2d">italic_θ ∈ italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\mathbb{P}_{r,s}(\theta)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.5.m2.3"><semantics id="S6.Thmtheorem1.p1.5.m2.3a"><mrow id="S6.Thmtheorem1.p1.5.m2.3.4" xref="S6.Thmtheorem1.p1.5.m2.3.4.cmml"><msub id="S6.Thmtheorem1.p1.5.m2.3.4.2" xref="S6.Thmtheorem1.p1.5.m2.3.4.2.cmml"><mi id="S6.Thmtheorem1.p1.5.m2.3.4.2.2" xref="S6.Thmtheorem1.p1.5.m2.3.4.2.2.cmml">ℙ</mi><mrow id="S6.Thmtheorem1.p1.5.m2.2.2.2.4" xref="S6.Thmtheorem1.p1.5.m2.2.2.2.3.cmml"><mi id="S6.Thmtheorem1.p1.5.m2.1.1.1.1" xref="S6.Thmtheorem1.p1.5.m2.1.1.1.1.cmml">r</mi><mo id="S6.Thmtheorem1.p1.5.m2.2.2.2.4.1" xref="S6.Thmtheorem1.p1.5.m2.2.2.2.3.cmml">,</mo><mi id="S6.Thmtheorem1.p1.5.m2.2.2.2.2" xref="S6.Thmtheorem1.p1.5.m2.2.2.2.2.cmml">s</mi></mrow></msub><mo id="S6.Thmtheorem1.p1.5.m2.3.4.1" xref="S6.Thmtheorem1.p1.5.m2.3.4.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.5.m2.3.4.3.2" xref="S6.Thmtheorem1.p1.5.m2.3.4.cmml"><mo id="S6.Thmtheorem1.p1.5.m2.3.4.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.5.m2.3.4.cmml">(</mo><mi id="S6.Thmtheorem1.p1.5.m2.3.3" xref="S6.Thmtheorem1.p1.5.m2.3.3.cmml">θ</mi><mo id="S6.Thmtheorem1.p1.5.m2.3.4.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p1.5.m2.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.5.m2.3b"><apply id="S6.Thmtheorem1.p1.5.m2.3.4.cmml" xref="S6.Thmtheorem1.p1.5.m2.3.4"><times id="S6.Thmtheorem1.p1.5.m2.3.4.1.cmml" xref="S6.Thmtheorem1.p1.5.m2.3.4.1"></times><apply id="S6.Thmtheorem1.p1.5.m2.3.4.2.cmml" xref="S6.Thmtheorem1.p1.5.m2.3.4.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.5.m2.3.4.2.1.cmml" xref="S6.Thmtheorem1.p1.5.m2.3.4.2">subscript</csymbol><ci id="S6.Thmtheorem1.p1.5.m2.3.4.2.2.cmml" xref="S6.Thmtheorem1.p1.5.m2.3.4.2.2">ℙ</ci><list id="S6.Thmtheorem1.p1.5.m2.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.5.m2.2.2.2.4"><ci id="S6.Thmtheorem1.p1.5.m2.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.5.m2.1.1.1.1">𝑟</ci><ci id="S6.Thmtheorem1.p1.5.m2.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.5.m2.2.2.2.2">𝑠</ci></list></apply><ci id="S6.Thmtheorem1.p1.5.m2.3.3.cmml" xref="S6.Thmtheorem1.p1.5.m2.3.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.5.m2.3c">\mathbb{P}_{r,s}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.5.m2.3d">blackboard_P start_POSTSUBSCRIPT italic_r , italic_s end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> is a free <math alttext="R" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.6.m3.1"><semantics id="S6.Thmtheorem1.p1.6.m3.1a"><mi id="S6.Thmtheorem1.p1.6.m3.1.1" xref="S6.Thmtheorem1.p1.6.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.6.m3.1b"><ci id="S6.Thmtheorem1.p1.6.m3.1.1.cmml" xref="S6.Thmtheorem1.p1.6.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.6.m3.1c">R</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.6.m3.1d">italic_R</annotation></semantics></math>-module with basis given by the pairs <math alttext="(\tau,g)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.7.m4.2"><semantics id="S6.Thmtheorem1.p1.7.m4.2a"><mrow id="S6.Thmtheorem1.p1.7.m4.2.3.2" xref="S6.Thmtheorem1.p1.7.m4.2.3.1.cmml"><mo id="S6.Thmtheorem1.p1.7.m4.2.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.7.m4.2.3.1.cmml">(</mo><mi id="S6.Thmtheorem1.p1.7.m4.1.1" xref="S6.Thmtheorem1.p1.7.m4.1.1.cmml">τ</mi><mo id="S6.Thmtheorem1.p1.7.m4.2.3.2.2" xref="S6.Thmtheorem1.p1.7.m4.2.3.1.cmml">,</mo><mi id="S6.Thmtheorem1.p1.7.m4.2.2" xref="S6.Thmtheorem1.p1.7.m4.2.2.cmml">g</mi><mo id="S6.Thmtheorem1.p1.7.m4.2.3.2.3" stretchy="false" xref="S6.Thmtheorem1.p1.7.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.7.m4.2b"><interval closure="open" id="S6.Thmtheorem1.p1.7.m4.2.3.1.cmml" xref="S6.Thmtheorem1.p1.7.m4.2.3.2"><ci id="S6.Thmtheorem1.p1.7.m4.1.1.cmml" xref="S6.Thmtheorem1.p1.7.m4.1.1">𝜏</ci><ci id="S6.Thmtheorem1.p1.7.m4.2.2.cmml" xref="S6.Thmtheorem1.p1.7.m4.2.2">𝑔</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.7.m4.2c">(\tau,g)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.7.m4.2d">( italic_τ , italic_g )</annotation></semantics></math> where <math alttext="\tau\in X_{r,s}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.8.m5.2"><semantics id="S6.Thmtheorem1.p1.8.m5.2a"><mrow id="S6.Thmtheorem1.p1.8.m5.2.3" xref="S6.Thmtheorem1.p1.8.m5.2.3.cmml"><mi id="S6.Thmtheorem1.p1.8.m5.2.3.2" xref="S6.Thmtheorem1.p1.8.m5.2.3.2.cmml">τ</mi><mo id="S6.Thmtheorem1.p1.8.m5.2.3.1" xref="S6.Thmtheorem1.p1.8.m5.2.3.1.cmml">∈</mo><msub id="S6.Thmtheorem1.p1.8.m5.2.3.3" xref="S6.Thmtheorem1.p1.8.m5.2.3.3.cmml"><mi id="S6.Thmtheorem1.p1.8.m5.2.3.3.2" xref="S6.Thmtheorem1.p1.8.m5.2.3.3.2.cmml">X</mi><mrow id="S6.Thmtheorem1.p1.8.m5.2.2.2.4" xref="S6.Thmtheorem1.p1.8.m5.2.2.2.3.cmml"><mi id="S6.Thmtheorem1.p1.8.m5.1.1.1.1" xref="S6.Thmtheorem1.p1.8.m5.1.1.1.1.cmml">r</mi><mo id="S6.Thmtheorem1.p1.8.m5.2.2.2.4.1" xref="S6.Thmtheorem1.p1.8.m5.2.2.2.3.cmml">,</mo><mi id="S6.Thmtheorem1.p1.8.m5.2.2.2.2" xref="S6.Thmtheorem1.p1.8.m5.2.2.2.2.cmml">s</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.8.m5.2b"><apply id="S6.Thmtheorem1.p1.8.m5.2.3.cmml" xref="S6.Thmtheorem1.p1.8.m5.2.3"><in id="S6.Thmtheorem1.p1.8.m5.2.3.1.cmml" xref="S6.Thmtheorem1.p1.8.m5.2.3.1"></in><ci id="S6.Thmtheorem1.p1.8.m5.2.3.2.cmml" xref="S6.Thmtheorem1.p1.8.m5.2.3.2">𝜏</ci><apply id="S6.Thmtheorem1.p1.8.m5.2.3.3.cmml" xref="S6.Thmtheorem1.p1.8.m5.2.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.8.m5.2.3.3.1.cmml" xref="S6.Thmtheorem1.p1.8.m5.2.3.3">subscript</csymbol><ci id="S6.Thmtheorem1.p1.8.m5.2.3.3.2.cmml" xref="S6.Thmtheorem1.p1.8.m5.2.3.3.2">𝑋</ci><list id="S6.Thmtheorem1.p1.8.m5.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.8.m5.2.2.2.4"><ci id="S6.Thmtheorem1.p1.8.m5.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.8.m5.1.1.1.1">𝑟</ci><ci id="S6.Thmtheorem1.p1.8.m5.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.8.m5.2.2.2.2">𝑠</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.8.m5.2c">\tau\in X_{r,s}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.8.m5.2d">italic_τ ∈ italic_X start_POSTSUBSCRIPT italic_r , italic_s end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="g\in\mathrm{Mor}_{\Delta(X)}(\tau,\theta)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.9.m6.3"><semantics id="S6.Thmtheorem1.p1.9.m6.3a"><mrow id="S6.Thmtheorem1.p1.9.m6.3.4" xref="S6.Thmtheorem1.p1.9.m6.3.4.cmml"><mi id="S6.Thmtheorem1.p1.9.m6.3.4.2" xref="S6.Thmtheorem1.p1.9.m6.3.4.2.cmml">g</mi><mo id="S6.Thmtheorem1.p1.9.m6.3.4.1" xref="S6.Thmtheorem1.p1.9.m6.3.4.1.cmml">∈</mo><mrow id="S6.Thmtheorem1.p1.9.m6.3.4.3" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.cmml"><msub id="S6.Thmtheorem1.p1.9.m6.3.4.3.2" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.2.cmml"><mi id="S6.Thmtheorem1.p1.9.m6.3.4.3.2.2" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.2.2.cmml">Mor</mi><mrow id="S6.Thmtheorem1.p1.9.m6.1.1.1" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.9.m6.1.1.1.3" mathvariant="normal" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.3.cmml">Δ</mi><mo id="S6.Thmtheorem1.p1.9.m6.1.1.1.2" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.9.m6.1.1.1.4.2" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.cmml"><mo id="S6.Thmtheorem1.p1.9.m6.1.1.1.4.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.cmml">(</mo><mi id="S6.Thmtheorem1.p1.9.m6.1.1.1.1" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.1.cmml">X</mi><mo id="S6.Thmtheorem1.p1.9.m6.1.1.1.4.2.2" stretchy="false" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Thmtheorem1.p1.9.m6.3.4.3.1" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.9.m6.3.4.3.3.2" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.3.1.cmml"><mo id="S6.Thmtheorem1.p1.9.m6.3.4.3.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.3.1.cmml">(</mo><mi id="S6.Thmtheorem1.p1.9.m6.2.2" xref="S6.Thmtheorem1.p1.9.m6.2.2.cmml">τ</mi><mo id="S6.Thmtheorem1.p1.9.m6.3.4.3.3.2.2" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem1.p1.9.m6.3.3" xref="S6.Thmtheorem1.p1.9.m6.3.3.cmml">θ</mi><mo id="S6.Thmtheorem1.p1.9.m6.3.4.3.3.2.3" stretchy="false" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.9.m6.3b"><apply id="S6.Thmtheorem1.p1.9.m6.3.4.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4"><in id="S6.Thmtheorem1.p1.9.m6.3.4.1.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4.1"></in><ci id="S6.Thmtheorem1.p1.9.m6.3.4.2.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4.2">𝑔</ci><apply id="S6.Thmtheorem1.p1.9.m6.3.4.3.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4.3"><times id="S6.Thmtheorem1.p1.9.m6.3.4.3.1.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.1"></times><apply id="S6.Thmtheorem1.p1.9.m6.3.4.3.2.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.9.m6.3.4.3.2.1.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.2">subscript</csymbol><ci id="S6.Thmtheorem1.p1.9.m6.3.4.3.2.2.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.2.2">Mor</ci><apply id="S6.Thmtheorem1.p1.9.m6.1.1.1.cmml" xref="S6.Thmtheorem1.p1.9.m6.1.1.1"><times id="S6.Thmtheorem1.p1.9.m6.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.2"></times><ci id="S6.Thmtheorem1.p1.9.m6.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.3">Δ</ci><ci id="S6.Thmtheorem1.p1.9.m6.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.9.m6.1.1.1.1">𝑋</ci></apply></apply><interval closure="open" id="S6.Thmtheorem1.p1.9.m6.3.4.3.3.1.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.4.3.3.2"><ci id="S6.Thmtheorem1.p1.9.m6.2.2.cmml" xref="S6.Thmtheorem1.p1.9.m6.2.2">𝜏</ci><ci id="S6.Thmtheorem1.p1.9.m6.3.3.cmml" xref="S6.Thmtheorem1.p1.9.m6.3.3">𝜃</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.9.m6.3c">g\in\mathrm{Mor}_{\Delta(X)}(\tau,\theta)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.9.m6.3d">italic_g ∈ roman_Mor start_POSTSUBSCRIPT roman_Δ ( italic_X ) end_POSTSUBSCRIPT ( italic_τ , italic_θ )</annotation></semantics></math>. For every morphism <math alttext="f:[r^{\prime}]\times[s^{\prime}]\to[r]\times[s]" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.10.m7.4"><semantics id="S6.Thmtheorem1.p1.10.m7.4a"><mrow id="S6.Thmtheorem1.p1.10.m7.4.4" xref="S6.Thmtheorem1.p1.10.m7.4.4.cmml"><mi id="S6.Thmtheorem1.p1.10.m7.4.4.4" xref="S6.Thmtheorem1.p1.10.m7.4.4.4.cmml">f</mi><mo id="S6.Thmtheorem1.p1.10.m7.4.4.3" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem1.p1.10.m7.4.4.3.cmml">:</mo><mrow id="S6.Thmtheorem1.p1.10.m7.4.4.2" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.cmml"><mrow id="S6.Thmtheorem1.p1.10.m7.4.4.2.2" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.cmml"><mrow id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.2.cmml"><mo id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.2.1.cmml">[</mo><msup id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.2" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.2.cmml">r</mi><mo id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.3" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.2.1.cmml">]</mo></mrow><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.3" rspace="0.222em" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.3.cmml">×</mo><mrow id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.2.cmml"><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.2" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.2.1.cmml">[</mo><msup id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.cmml"><mi id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.2" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.2.cmml">s</mi><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.3" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.3.cmml">′</mo></msup><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.3" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.2.1.cmml">]</mo></mrow></mrow><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.3" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.3.cmml">→</mo><mrow id="S6.Thmtheorem1.p1.10.m7.4.4.2.4" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.cmml"><mrow id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.2" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.1.cmml"><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.1.1.cmml">[</mo><mi id="S6.Thmtheorem1.p1.10.m7.1.1" xref="S6.Thmtheorem1.p1.10.m7.1.1.cmml">r</mi><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.2.2" rspace="0.055em" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.1.1.cmml">]</mo></mrow><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.1" rspace="0.222em" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.1.cmml">×</mo><mrow id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.2" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.1.cmml"><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.1.1.cmml">[</mo><mi id="S6.Thmtheorem1.p1.10.m7.2.2" xref="S6.Thmtheorem1.p1.10.m7.2.2.cmml">s</mi><mo id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.1.1.cmml">]</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.10.m7.4b"><apply id="S6.Thmtheorem1.p1.10.m7.4.4.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4"><ci id="S6.Thmtheorem1.p1.10.m7.4.4.3.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.3">:</ci><ci id="S6.Thmtheorem1.p1.10.m7.4.4.4.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.4">𝑓</ci><apply id="S6.Thmtheorem1.p1.10.m7.4.4.2.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2"><ci id="S6.Thmtheorem1.p1.10.m7.4.4.2.3.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.3">→</ci><apply id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2"><times id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.3.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.3"></times><apply id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1"><csymbol cd="latexml" id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.2.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.2">delimited-[]</csymbol><apply id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.2">𝑟</ci><ci id="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.10.m7.3.3.1.1.1.1.1.3">′</ci></apply></apply><apply id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1"><csymbol cd="latexml" id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.2.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.2">delimited-[]</csymbol><apply id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1">superscript</csymbol><ci id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.2.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.2">𝑠</ci><ci id="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.3.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.2.2.1.1.3">′</ci></apply></apply></apply><apply id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4"><times id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.1"></times><apply id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.2"><csymbol cd="latexml" id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.1.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.2.2.1">delimited-[]</csymbol><ci id="S6.Thmtheorem1.p1.10.m7.1.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.1.1">𝑟</ci></apply><apply id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.2"><csymbol cd="latexml" id="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.1.1.cmml" xref="S6.Thmtheorem1.p1.10.m7.4.4.2.4.3.2.1">delimited-[]</csymbol><ci id="S6.Thmtheorem1.p1.10.m7.2.2.cmml" xref="S6.Thmtheorem1.p1.10.m7.2.2">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.10.m7.4c">f:[r^{\prime}]\times[s^{\prime}]\to[r]\times[s]</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.10.m7.4d">italic_f : [ italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] × [ italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] → [ italic_r ] × [ italic_s ]</annotation></semantics></math> in <math alttext="\Delta\times\Delta" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.11.m8.1"><semantics id="S6.Thmtheorem1.p1.11.m8.1a"><mrow id="S6.Thmtheorem1.p1.11.m8.1.1" xref="S6.Thmtheorem1.p1.11.m8.1.1.cmml"><mi id="S6.Thmtheorem1.p1.11.m8.1.1.2" mathvariant="normal" xref="S6.Thmtheorem1.p1.11.m8.1.1.2.cmml">Δ</mi><mo id="S6.Thmtheorem1.p1.11.m8.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.Thmtheorem1.p1.11.m8.1.1.1.cmml">×</mo><mi id="S6.Thmtheorem1.p1.11.m8.1.1.3" mathvariant="normal" xref="S6.Thmtheorem1.p1.11.m8.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.11.m8.1b"><apply id="S6.Thmtheorem1.p1.11.m8.1.1.cmml" xref="S6.Thmtheorem1.p1.11.m8.1.1"><times id="S6.Thmtheorem1.p1.11.m8.1.1.1.cmml" xref="S6.Thmtheorem1.p1.11.m8.1.1.1"></times><ci id="S6.Thmtheorem1.p1.11.m8.1.1.2.cmml" xref="S6.Thmtheorem1.p1.11.m8.1.1.2">Δ</ci><ci id="S6.Thmtheorem1.p1.11.m8.1.1.3.cmml" xref="S6.Thmtheorem1.p1.11.m8.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.11.m8.1c">\Delta\times\Delta</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.11.m8.1d">roman_Δ × roman_Δ</annotation></semantics></math>, <math alttext="f^{*}:\mathbb{P}_{r,s}(\theta)\to\mathbb{P}_{r^{\prime},s^{\prime}}(\theta)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.12.m9.6"><semantics id="S6.Thmtheorem1.p1.12.m9.6a"><mrow id="S6.Thmtheorem1.p1.12.m9.6.7" xref="S6.Thmtheorem1.p1.12.m9.6.7.cmml"><msup id="S6.Thmtheorem1.p1.12.m9.6.7.2" xref="S6.Thmtheorem1.p1.12.m9.6.7.2.cmml"><mi id="S6.Thmtheorem1.p1.12.m9.6.7.2.2" xref="S6.Thmtheorem1.p1.12.m9.6.7.2.2.cmml">f</mi><mo id="S6.Thmtheorem1.p1.12.m9.6.7.2.3" xref="S6.Thmtheorem1.p1.12.m9.6.7.2.3.cmml">∗</mo></msup><mo id="S6.Thmtheorem1.p1.12.m9.6.7.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem1.p1.12.m9.6.7.1.cmml">:</mo><mrow id="S6.Thmtheorem1.p1.12.m9.6.7.3" xref="S6.Thmtheorem1.p1.12.m9.6.7.3.cmml"><mrow id="S6.Thmtheorem1.p1.12.m9.6.7.3.2" 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id="S6.Thmtheorem1.p1.12.m9.6.7.3.3.2.2.cmml" xref="S6.Thmtheorem1.p1.12.m9.6.7.3.3.2.2">ℙ</ci><list id="S6.Thmtheorem1.p1.12.m9.4.4.2.3.cmml" xref="S6.Thmtheorem1.p1.12.m9.4.4.2.2"><apply id="S6.Thmtheorem1.p1.12.m9.3.3.1.1.1.cmml" xref="S6.Thmtheorem1.p1.12.m9.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.12.m9.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.12.m9.3.3.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem1.p1.12.m9.3.3.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.12.m9.3.3.1.1.1.2">𝑟</ci><ci id="S6.Thmtheorem1.p1.12.m9.3.3.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.12.m9.3.3.1.1.1.3">′</ci></apply><apply id="S6.Thmtheorem1.p1.12.m9.4.4.2.2.2.cmml" xref="S6.Thmtheorem1.p1.12.m9.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.12.m9.4.4.2.2.2.1.cmml" xref="S6.Thmtheorem1.p1.12.m9.4.4.2.2.2">superscript</csymbol><ci id="S6.Thmtheorem1.p1.12.m9.4.4.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.12.m9.4.4.2.2.2.2">𝑠</ci><ci id="S6.Thmtheorem1.p1.12.m9.4.4.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.12.m9.4.4.2.2.2.3">′</ci></apply></list></apply><ci id="S6.Thmtheorem1.p1.12.m9.6.6.cmml" xref="S6.Thmtheorem1.p1.12.m9.6.6">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.12.m9.6c">f^{*}:\mathbb{P}_{r,s}(\theta)\to\mathbb{P}_{r^{\prime},s^{\prime}}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.12.m9.6d">italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : blackboard_P start_POSTSUBSCRIPT italic_r , italic_s end_POSTSUBSCRIPT ( italic_θ ) → blackboard_P start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex69"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell 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id="S6.Ex69.m1.5.5.2.2.2.1.cmml" xref="S6.Ex69.m1.5.5.2.2.2.1"></compose><ci id="S6.Ex69.m1.5.5.2.2.2.2.cmml" xref="S6.Ex69.m1.5.5.2.2.2.2">𝑔</ci><ci id="S6.Ex69.m1.5.5.2.2.2.3.cmml" xref="S6.Ex69.m1.5.5.2.2.2.3">𝑓</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex69.m1.5c">f^{*}(\tau,g)=(f^{*}(\tau),g\circ f)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex69.m1.5d">italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ , italic_g ) = ( italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ ) , italic_g ∘ italic_f )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmtheorem1.p1.14">for every <math alttext="\tau\in X_{r,s}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.13.m1.2"><semantics id="S6.Thmtheorem1.p1.13.m1.2a"><mrow id="S6.Thmtheorem1.p1.13.m1.2.3" xref="S6.Thmtheorem1.p1.13.m1.2.3.cmml"><mi id="S6.Thmtheorem1.p1.13.m1.2.3.2" xref="S6.Thmtheorem1.p1.13.m1.2.3.2.cmml">τ</mi><mo id="S6.Thmtheorem1.p1.13.m1.2.3.1" xref="S6.Thmtheorem1.p1.13.m1.2.3.1.cmml">∈</mo><msub id="S6.Thmtheorem1.p1.13.m1.2.3.3" xref="S6.Thmtheorem1.p1.13.m1.2.3.3.cmml"><mi id="S6.Thmtheorem1.p1.13.m1.2.3.3.2" xref="S6.Thmtheorem1.p1.13.m1.2.3.3.2.cmml">X</mi><mrow id="S6.Thmtheorem1.p1.13.m1.2.2.2.4" xref="S6.Thmtheorem1.p1.13.m1.2.2.2.3.cmml"><mi id="S6.Thmtheorem1.p1.13.m1.1.1.1.1" xref="S6.Thmtheorem1.p1.13.m1.1.1.1.1.cmml">r</mi><mo id="S6.Thmtheorem1.p1.13.m1.2.2.2.4.1" xref="S6.Thmtheorem1.p1.13.m1.2.2.2.3.cmml">,</mo><mi id="S6.Thmtheorem1.p1.13.m1.2.2.2.2" xref="S6.Thmtheorem1.p1.13.m1.2.2.2.2.cmml">s</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.13.m1.2b"><apply id="S6.Thmtheorem1.p1.13.m1.2.3.cmml" xref="S6.Thmtheorem1.p1.13.m1.2.3"><in id="S6.Thmtheorem1.p1.13.m1.2.3.1.cmml" xref="S6.Thmtheorem1.p1.13.m1.2.3.1"></in><ci id="S6.Thmtheorem1.p1.13.m1.2.3.2.cmml" xref="S6.Thmtheorem1.p1.13.m1.2.3.2">𝜏</ci><apply id="S6.Thmtheorem1.p1.13.m1.2.3.3.cmml" xref="S6.Thmtheorem1.p1.13.m1.2.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.13.m1.2.3.3.1.cmml" xref="S6.Thmtheorem1.p1.13.m1.2.3.3">subscript</csymbol><ci id="S6.Thmtheorem1.p1.13.m1.2.3.3.2.cmml" xref="S6.Thmtheorem1.p1.13.m1.2.3.3.2">𝑋</ci><list id="S6.Thmtheorem1.p1.13.m1.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.13.m1.2.2.2.4"><ci id="S6.Thmtheorem1.p1.13.m1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.13.m1.1.1.1.1">𝑟</ci><ci id="S6.Thmtheorem1.p1.13.m1.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.13.m1.2.2.2.2">𝑠</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.13.m1.2c">\tau\in X_{r,s}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.13.m1.2d">italic_τ ∈ italic_X start_POSTSUBSCRIPT italic_r , italic_s end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="g\in\mathrm{Mor}_{\Delta(X)}(\tau,\sigma)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.14.m2.3"><semantics id="S6.Thmtheorem1.p1.14.m2.3a"><mrow id="S6.Thmtheorem1.p1.14.m2.3.4" xref="S6.Thmtheorem1.p1.14.m2.3.4.cmml"><mi id="S6.Thmtheorem1.p1.14.m2.3.4.2" xref="S6.Thmtheorem1.p1.14.m2.3.4.2.cmml">g</mi><mo id="S6.Thmtheorem1.p1.14.m2.3.4.1" xref="S6.Thmtheorem1.p1.14.m2.3.4.1.cmml">∈</mo><mrow id="S6.Thmtheorem1.p1.14.m2.3.4.3" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.cmml"><msub id="S6.Thmtheorem1.p1.14.m2.3.4.3.2" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.2.cmml"><mi id="S6.Thmtheorem1.p1.14.m2.3.4.3.2.2" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.2.2.cmml">Mor</mi><mrow id="S6.Thmtheorem1.p1.14.m2.1.1.1" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.14.m2.1.1.1.3" mathvariant="normal" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.3.cmml">Δ</mi><mo id="S6.Thmtheorem1.p1.14.m2.1.1.1.2" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.14.m2.1.1.1.4.2" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.cmml"><mo id="S6.Thmtheorem1.p1.14.m2.1.1.1.4.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.cmml">(</mo><mi id="S6.Thmtheorem1.p1.14.m2.1.1.1.1" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.1.cmml">X</mi><mo id="S6.Thmtheorem1.p1.14.m2.1.1.1.4.2.2" stretchy="false" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Thmtheorem1.p1.14.m2.3.4.3.1" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.14.m2.3.4.3.3.2" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.3.1.cmml"><mo id="S6.Thmtheorem1.p1.14.m2.3.4.3.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.3.1.cmml">(</mo><mi id="S6.Thmtheorem1.p1.14.m2.2.2" xref="S6.Thmtheorem1.p1.14.m2.2.2.cmml">τ</mi><mo id="S6.Thmtheorem1.p1.14.m2.3.4.3.3.2.2" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem1.p1.14.m2.3.3" xref="S6.Thmtheorem1.p1.14.m2.3.3.cmml">σ</mi><mo id="S6.Thmtheorem1.p1.14.m2.3.4.3.3.2.3" stretchy="false" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.14.m2.3b"><apply id="S6.Thmtheorem1.p1.14.m2.3.4.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4"><in id="S6.Thmtheorem1.p1.14.m2.3.4.1.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4.1"></in><ci id="S6.Thmtheorem1.p1.14.m2.3.4.2.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4.2">𝑔</ci><apply id="S6.Thmtheorem1.p1.14.m2.3.4.3.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4.3"><times id="S6.Thmtheorem1.p1.14.m2.3.4.3.1.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.1"></times><apply id="S6.Thmtheorem1.p1.14.m2.3.4.3.2.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.14.m2.3.4.3.2.1.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.2">subscript</csymbol><ci id="S6.Thmtheorem1.p1.14.m2.3.4.3.2.2.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.2.2">Mor</ci><apply id="S6.Thmtheorem1.p1.14.m2.1.1.1.cmml" xref="S6.Thmtheorem1.p1.14.m2.1.1.1"><times id="S6.Thmtheorem1.p1.14.m2.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.2"></times><ci id="S6.Thmtheorem1.p1.14.m2.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.3">Δ</ci><ci id="S6.Thmtheorem1.p1.14.m2.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.14.m2.1.1.1.1">𝑋</ci></apply></apply><interval closure="open" id="S6.Thmtheorem1.p1.14.m2.3.4.3.3.1.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.4.3.3.2"><ci id="S6.Thmtheorem1.p1.14.m2.2.2.cmml" xref="S6.Thmtheorem1.p1.14.m2.2.2">𝜏</ci><ci id="S6.Thmtheorem1.p1.14.m2.3.3.cmml" xref="S6.Thmtheorem1.p1.14.m2.3.3">𝜎</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.14.m2.3c">g\in\mathrm{Mor}_{\Delta(X)}(\tau,\sigma)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.14.m2.3d">italic_g ∈ roman_Mor start_POSTSUBSCRIPT roman_Δ ( italic_X ) end_POSTSUBSCRIPT ( italic_τ , italic_σ )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.4">Let <math alttext="P_{*,*}" class="ltx_Math" display="inline" id="S6.p3.1.m1.2"><semantics id="S6.p3.1.m1.2a"><msub id="S6.p3.1.m1.2.3" xref="S6.p3.1.m1.2.3.cmml"><mi id="S6.p3.1.m1.2.3.2" xref="S6.p3.1.m1.2.3.2.cmml">P</mi><mrow id="S6.p3.1.m1.2.2.2.4" xref="S6.p3.1.m1.2.2.2.3.cmml"><mo id="S6.p3.1.m1.1.1.1.1" rspace="0em" xref="S6.p3.1.m1.1.1.1.1.cmml">∗</mo><mo id="S6.p3.1.m1.2.2.2.4.1" rspace="0em" xref="S6.p3.1.m1.2.2.2.3.cmml">,</mo><mo id="S6.p3.1.m1.2.2.2.2" lspace="0em" xref="S6.p3.1.m1.2.2.2.2.cmml">∗</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.p3.1.m1.2b"><apply id="S6.p3.1.m1.2.3.cmml" xref="S6.p3.1.m1.2.3"><csymbol cd="ambiguous" id="S6.p3.1.m1.2.3.1.cmml" xref="S6.p3.1.m1.2.3">subscript</csymbol><ci id="S6.p3.1.m1.2.3.2.cmml" xref="S6.p3.1.m1.2.3.2">𝑃</ci><list id="S6.p3.1.m1.2.2.2.3.cmml" xref="S6.p3.1.m1.2.2.2.4"><times id="S6.p3.1.m1.1.1.1.1.cmml" xref="S6.p3.1.m1.1.1.1.1"></times><times id="S6.p3.1.m1.2.2.2.2.cmml" xref="S6.p3.1.m1.2.2.2.2"></times></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.1.m1.2c">P_{*,*}</annotation><annotation encoding="application/x-llamapun" id="S6.p3.1.m1.2d">italic_P start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT</annotation></semantics></math> denote the double complex of <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S6.p3.2.m2.1"><semantics id="S6.p3.2.m2.1a"><mrow id="S6.p3.2.m2.1.2" xref="S6.p3.2.m2.1.2.cmml"><mi id="S6.p3.2.m2.1.2.2" xref="S6.p3.2.m2.1.2.2.cmml">R</mi><mo id="S6.p3.2.m2.1.2.1" xref="S6.p3.2.m2.1.2.1.cmml">⁢</mo><mi id="S6.p3.2.m2.1.2.3" mathvariant="normal" xref="S6.p3.2.m2.1.2.3.cmml">Δ</mi><mo id="S6.p3.2.m2.1.2.1a" xref="S6.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S6.p3.2.m2.1.2.4.2" xref="S6.p3.2.m2.1.2.cmml"><mo id="S6.p3.2.m2.1.2.4.2.1" stretchy="false" xref="S6.p3.2.m2.1.2.cmml">(</mo><mi id="S6.p3.2.m2.1.1" xref="S6.p3.2.m2.1.1.cmml">X</mi><mo id="S6.p3.2.m2.1.2.4.2.2" stretchy="false" xref="S6.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.2.m2.1b"><apply id="S6.p3.2.m2.1.2.cmml" xref="S6.p3.2.m2.1.2"><times id="S6.p3.2.m2.1.2.1.cmml" xref="S6.p3.2.m2.1.2.1"></times><ci id="S6.p3.2.m2.1.2.2.cmml" xref="S6.p3.2.m2.1.2.2">𝑅</ci><ci id="S6.p3.2.m2.1.2.3.cmml" xref="S6.p3.2.m2.1.2.3">Δ</ci><ci id="S6.p3.2.m2.1.1.cmml" xref="S6.p3.2.m2.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.2.m2.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p3.2.m2.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-modules corresponding to the bisimplicial <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S6.p3.3.m3.1"><semantics id="S6.p3.3.m3.1a"><mrow id="S6.p3.3.m3.1.2" xref="S6.p3.3.m3.1.2.cmml"><mi id="S6.p3.3.m3.1.2.2" xref="S6.p3.3.m3.1.2.2.cmml">R</mi><mo id="S6.p3.3.m3.1.2.1" xref="S6.p3.3.m3.1.2.1.cmml">⁢</mo><mi id="S6.p3.3.m3.1.2.3" mathvariant="normal" xref="S6.p3.3.m3.1.2.3.cmml">Δ</mi><mo id="S6.p3.3.m3.1.2.1a" xref="S6.p3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.p3.3.m3.1.2.4.2" xref="S6.p3.3.m3.1.2.cmml"><mo id="S6.p3.3.m3.1.2.4.2.1" stretchy="false" xref="S6.p3.3.m3.1.2.cmml">(</mo><mi id="S6.p3.3.m3.1.1" xref="S6.p3.3.m3.1.1.cmml">X</mi><mo id="S6.p3.3.m3.1.2.4.2.2" stretchy="false" xref="S6.p3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.3.m3.1b"><apply id="S6.p3.3.m3.1.2.cmml" xref="S6.p3.3.m3.1.2"><times id="S6.p3.3.m3.1.2.1.cmml" xref="S6.p3.3.m3.1.2.1"></times><ci id="S6.p3.3.m3.1.2.2.cmml" xref="S6.p3.3.m3.1.2.2">𝑅</ci><ci id="S6.p3.3.m3.1.2.3.cmml" xref="S6.p3.3.m3.1.2.3">Δ</ci><ci id="S6.p3.3.m3.1.1.cmml" xref="S6.p3.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.3.m3.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p3.3.m3.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S6.p3.4.m4.1"><semantics id="S6.p3.4.m4.1a"><mi id="S6.p3.4.m4.1.1" xref="S6.p3.4.m4.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S6.p3.4.m4.1b"><ci id="S6.p3.4.m4.1.1.cmml" xref="S6.p3.4.m4.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.4.m4.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S6.p3.4.m4.1d">blackboard_P</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem2.1.1.1">Lemma 6.2</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem2.p1"> <p class="ltx_p" id="S6.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem2.p1.3.3">The chain complex <math alttext="\mathrm{Tot}(P_{*,*})\to\underline{R}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.1.1.m1.3"><semantics id="S6.Thmtheorem2.p1.1.1.m1.3a"><mrow id="S6.Thmtheorem2.p1.1.1.m1.3.3" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.cmml"><mrow id="S6.Thmtheorem2.p1.1.1.m1.3.3.1" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.cmml"><mi id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.3" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.3.cmml">Tot</mi><mo id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.2" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml"><mo id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml">(</mo><msub id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml"><mi id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.2" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.2.cmml">P</mi><mrow id="S6.Thmtheorem2.p1.1.1.m1.2.2.2.4" xref="S6.Thmtheorem2.p1.1.1.m1.2.2.2.3.cmml"><mo id="S6.Thmtheorem2.p1.1.1.m1.1.1.1.1" rspace="0em" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.1.1.cmml">∗</mo><mo id="S6.Thmtheorem2.p1.1.1.m1.2.2.2.4.1" rspace="0em" xref="S6.Thmtheorem2.p1.1.1.m1.2.2.2.3.cmml">,</mo><mo id="S6.Thmtheorem2.p1.1.1.m1.2.2.2.2" lspace="0em" xref="S6.Thmtheorem2.p1.1.1.m1.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3" stretchy="false" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem2.p1.1.1.m1.3.3.2" stretchy="false" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.2.cmml">→</mo><munder accentunder="true" id="S6.Thmtheorem2.p1.1.1.m1.3.3.3" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.3.cmml"><mi id="S6.Thmtheorem2.p1.1.1.m1.3.3.3.2" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.3.2.cmml">R</mi><mo id="S6.Thmtheorem2.p1.1.1.m1.3.3.3.1" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.3.1.cmml">¯</mo></munder></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.1.1.m1.3b"><apply id="S6.Thmtheorem2.p1.1.1.m1.3.3.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3"><ci id="S6.Thmtheorem2.p1.1.1.m1.3.3.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.2">→</ci><apply id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1"><times id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.2"></times><ci id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.3.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.3">Tot</ci><apply id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.2">𝑃</ci><list id="S6.Thmtheorem2.p1.1.1.m1.2.2.2.3.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.2.2.2.4"><times id="S6.Thmtheorem2.p1.1.1.m1.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.1.1"></times><times id="S6.Thmtheorem2.p1.1.1.m1.2.2.2.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.2.2.2.2"></times></list></apply></apply><apply id="S6.Thmtheorem2.p1.1.1.m1.3.3.3.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.3"><ci id="S6.Thmtheorem2.p1.1.1.m1.3.3.3.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.3.1">¯</ci><ci id="S6.Thmtheorem2.p1.1.1.m1.3.3.3.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.3.3.3.2">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.1.1.m1.3c">\mathrm{Tot}(P_{*,*})\to\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.1.1.m1.3d">roman_Tot ( italic_P start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ) → under¯ start_ARG italic_R end_ARG</annotation></semantics></math> is a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.2.2.m2.1"><semantics id="S6.Thmtheorem2.p1.2.2.m2.1a"><munder accentunder="true" id="S6.Thmtheorem2.p1.2.2.m2.1.1" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S6.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">R</mi><mo id="S6.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.2.2.m2.1b"><apply id="S6.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1"><ci id="S6.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.1">¯</ci><ci id="S6.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.2.2.m2.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.2.2.m2.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.3.3.m3.1"><semantics id="S6.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S6.Thmtheorem2.p1.3.3.m3.1.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.cmml"><mi id="S6.Thmtheorem2.p1.3.3.m3.1.2.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2.cmml">R</mi><mo id="S6.Thmtheorem2.p1.3.3.m3.1.2.1" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.1.cmml">⁢</mo><mi id="S6.Thmtheorem2.p1.3.3.m3.1.2.3" mathvariant="normal" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.cmml">Δ</mi><mo id="S6.Thmtheorem2.p1.3.3.m3.1.2.1a" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem2.p1.3.3.m3.1.2.4.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.cmml"><mo id="S6.Thmtheorem2.p1.3.3.m3.1.2.4.2.1" stretchy="false" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.cmml">(</mo><mi id="S6.Thmtheorem2.p1.3.3.m3.1.1" xref="S6.Thmtheorem2.p1.3.3.m3.1.1.cmml">X</mi><mo id="S6.Thmtheorem2.p1.3.3.m3.1.2.4.2.2" stretchy="false" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.3.3.m3.1b"><apply id="S6.Thmtheorem2.p1.3.3.m3.1.2.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2"><times id="S6.Thmtheorem2.p1.3.3.m3.1.2.1.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.1"></times><ci id="S6.Thmtheorem2.p1.3.3.m3.1.2.2.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2">𝑅</ci><ci id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3">Δ</ci><ci id="S6.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.3.3.m3.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.3.3.m3.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module.</span></p> </div> </div> <div class="ltx_proof" id="S6.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.1.p1"> <p class="ltx_p" id="S6.1.p1.14">Let <math alttext="\theta\in X_{p,q}" class="ltx_Math" display="inline" id="S6.1.p1.1.m1.2"><semantics id="S6.1.p1.1.m1.2a"><mrow id="S6.1.p1.1.m1.2.3" xref="S6.1.p1.1.m1.2.3.cmml"><mi id="S6.1.p1.1.m1.2.3.2" xref="S6.1.p1.1.m1.2.3.2.cmml">θ</mi><mo id="S6.1.p1.1.m1.2.3.1" xref="S6.1.p1.1.m1.2.3.1.cmml">∈</mo><msub id="S6.1.p1.1.m1.2.3.3" xref="S6.1.p1.1.m1.2.3.3.cmml"><mi id="S6.1.p1.1.m1.2.3.3.2" xref="S6.1.p1.1.m1.2.3.3.2.cmml">X</mi><mrow id="S6.1.p1.1.m1.2.2.2.4" xref="S6.1.p1.1.m1.2.2.2.3.cmml"><mi id="S6.1.p1.1.m1.1.1.1.1" xref="S6.1.p1.1.m1.1.1.1.1.cmml">p</mi><mo id="S6.1.p1.1.m1.2.2.2.4.1" xref="S6.1.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S6.1.p1.1.m1.2.2.2.2" xref="S6.1.p1.1.m1.2.2.2.2.cmml">q</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.1.m1.2b"><apply id="S6.1.p1.1.m1.2.3.cmml" xref="S6.1.p1.1.m1.2.3"><in id="S6.1.p1.1.m1.2.3.1.cmml" xref="S6.1.p1.1.m1.2.3.1"></in><ci id="S6.1.p1.1.m1.2.3.2.cmml" xref="S6.1.p1.1.m1.2.3.2">𝜃</ci><apply id="S6.1.p1.1.m1.2.3.3.cmml" xref="S6.1.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="S6.1.p1.1.m1.2.3.3.1.cmml" xref="S6.1.p1.1.m1.2.3.3">subscript</csymbol><ci id="S6.1.p1.1.m1.2.3.3.2.cmml" xref="S6.1.p1.1.m1.2.3.3.2">𝑋</ci><list id="S6.1.p1.1.m1.2.2.2.3.cmml" xref="S6.1.p1.1.m1.2.2.2.4"><ci id="S6.1.p1.1.m1.1.1.1.1.cmml" xref="S6.1.p1.1.m1.1.1.1.1">𝑝</ci><ci id="S6.1.p1.1.m1.2.2.2.2.cmml" xref="S6.1.p1.1.m1.2.2.2.2">𝑞</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.1.m1.2c">\theta\in X_{p,q}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.1.m1.2d">italic_θ ∈ italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT</annotation></semantics></math>. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S4.Thmtheorem3" title="Theorem 4.3 (Dold-Puppe Theorem [4]). ‣ 4.2. The Dold-Puppe Theorem ‣ 4. Bisimplicial objects and the Dold-Puppe theorem ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">4.3</span></a>, the inclusion map <math alttext="(\mathrm{diag}\mathbb{P}(\theta))_{*}\to\mathrm{Tot}(P_{*,*}(\theta))" class="ltx_Math" display="inline" id="S6.1.p1.2.m2.6"><semantics id="S6.1.p1.2.m2.6a"><mrow id="S6.1.p1.2.m2.6.6" xref="S6.1.p1.2.m2.6.6.cmml"><msub id="S6.1.p1.2.m2.5.5.1" xref="S6.1.p1.2.m2.5.5.1.cmml"><mrow id="S6.1.p1.2.m2.5.5.1.1.1" xref="S6.1.p1.2.m2.5.5.1.1.1.1.cmml"><mo id="S6.1.p1.2.m2.5.5.1.1.1.2" stretchy="false" xref="S6.1.p1.2.m2.5.5.1.1.1.1.cmml">(</mo><mrow id="S6.1.p1.2.m2.5.5.1.1.1.1" xref="S6.1.p1.2.m2.5.5.1.1.1.1.cmml"><mi id="S6.1.p1.2.m2.5.5.1.1.1.1.2" xref="S6.1.p1.2.m2.5.5.1.1.1.1.2.cmml">diag</mi><mo id="S6.1.p1.2.m2.5.5.1.1.1.1.1" xref="S6.1.p1.2.m2.5.5.1.1.1.1.1.cmml">⁢</mo><mi id="S6.1.p1.2.m2.5.5.1.1.1.1.3" xref="S6.1.p1.2.m2.5.5.1.1.1.1.3.cmml">ℙ</mi><mo id="S6.1.p1.2.m2.5.5.1.1.1.1.1a" xref="S6.1.p1.2.m2.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.1.p1.2.m2.5.5.1.1.1.1.4.2" xref="S6.1.p1.2.m2.5.5.1.1.1.1.cmml"><mo id="S6.1.p1.2.m2.5.5.1.1.1.1.4.2.1" stretchy="false" xref="S6.1.p1.2.m2.5.5.1.1.1.1.cmml">(</mo><mi id="S6.1.p1.2.m2.3.3" xref="S6.1.p1.2.m2.3.3.cmml">θ</mi><mo id="S6.1.p1.2.m2.5.5.1.1.1.1.4.2.2" stretchy="false" xref="S6.1.p1.2.m2.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.2.m2.5.5.1.1.1.3" stretchy="false" xref="S6.1.p1.2.m2.5.5.1.1.1.1.cmml">)</mo></mrow><mo id="S6.1.p1.2.m2.5.5.1.3" xref="S6.1.p1.2.m2.5.5.1.3.cmml">∗</mo></msub><mo id="S6.1.p1.2.m2.6.6.3" stretchy="false" xref="S6.1.p1.2.m2.6.6.3.cmml">→</mo><mrow id="S6.1.p1.2.m2.6.6.2" xref="S6.1.p1.2.m2.6.6.2.cmml"><mi id="S6.1.p1.2.m2.6.6.2.3" xref="S6.1.p1.2.m2.6.6.2.3.cmml">Tot</mi><mo id="S6.1.p1.2.m2.6.6.2.2" xref="S6.1.p1.2.m2.6.6.2.2.cmml">⁢</mo><mrow id="S6.1.p1.2.m2.6.6.2.1.1" xref="S6.1.p1.2.m2.6.6.2.1.1.1.cmml"><mo id="S6.1.p1.2.m2.6.6.2.1.1.2" stretchy="false" xref="S6.1.p1.2.m2.6.6.2.1.1.1.cmml">(</mo><mrow id="S6.1.p1.2.m2.6.6.2.1.1.1" xref="S6.1.p1.2.m2.6.6.2.1.1.1.cmml"><msub id="S6.1.p1.2.m2.6.6.2.1.1.1.2" xref="S6.1.p1.2.m2.6.6.2.1.1.1.2.cmml"><mi id="S6.1.p1.2.m2.6.6.2.1.1.1.2.2" xref="S6.1.p1.2.m2.6.6.2.1.1.1.2.2.cmml">P</mi><mrow id="S6.1.p1.2.m2.2.2.2.4" xref="S6.1.p1.2.m2.2.2.2.3.cmml"><mo id="S6.1.p1.2.m2.1.1.1.1" rspace="0em" xref="S6.1.p1.2.m2.1.1.1.1.cmml">∗</mo><mo id="S6.1.p1.2.m2.2.2.2.4.1" rspace="0em" xref="S6.1.p1.2.m2.2.2.2.3.cmml">,</mo><mo id="S6.1.p1.2.m2.2.2.2.2" lspace="0em" xref="S6.1.p1.2.m2.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S6.1.p1.2.m2.6.6.2.1.1.1.1" xref="S6.1.p1.2.m2.6.6.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.1.p1.2.m2.6.6.2.1.1.1.3.2" xref="S6.1.p1.2.m2.6.6.2.1.1.1.cmml"><mo id="S6.1.p1.2.m2.6.6.2.1.1.1.3.2.1" stretchy="false" xref="S6.1.p1.2.m2.6.6.2.1.1.1.cmml">(</mo><mi id="S6.1.p1.2.m2.4.4" xref="S6.1.p1.2.m2.4.4.cmml">θ</mi><mo id="S6.1.p1.2.m2.6.6.2.1.1.1.3.2.2" stretchy="false" xref="S6.1.p1.2.m2.6.6.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.2.m2.6.6.2.1.1.3" stretchy="false" xref="S6.1.p1.2.m2.6.6.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.2.m2.6b"><apply id="S6.1.p1.2.m2.6.6.cmml" xref="S6.1.p1.2.m2.6.6"><ci id="S6.1.p1.2.m2.6.6.3.cmml" xref="S6.1.p1.2.m2.6.6.3">→</ci><apply id="S6.1.p1.2.m2.5.5.1.cmml" xref="S6.1.p1.2.m2.5.5.1"><csymbol cd="ambiguous" id="S6.1.p1.2.m2.5.5.1.2.cmml" xref="S6.1.p1.2.m2.5.5.1">subscript</csymbol><apply id="S6.1.p1.2.m2.5.5.1.1.1.1.cmml" xref="S6.1.p1.2.m2.5.5.1.1.1"><times id="S6.1.p1.2.m2.5.5.1.1.1.1.1.cmml" xref="S6.1.p1.2.m2.5.5.1.1.1.1.1"></times><ci id="S6.1.p1.2.m2.5.5.1.1.1.1.2.cmml" xref="S6.1.p1.2.m2.5.5.1.1.1.1.2">diag</ci><ci id="S6.1.p1.2.m2.5.5.1.1.1.1.3.cmml" xref="S6.1.p1.2.m2.5.5.1.1.1.1.3">ℙ</ci><ci id="S6.1.p1.2.m2.3.3.cmml" xref="S6.1.p1.2.m2.3.3">𝜃</ci></apply><times id="S6.1.p1.2.m2.5.5.1.3.cmml" xref="S6.1.p1.2.m2.5.5.1.3"></times></apply><apply id="S6.1.p1.2.m2.6.6.2.cmml" xref="S6.1.p1.2.m2.6.6.2"><times id="S6.1.p1.2.m2.6.6.2.2.cmml" xref="S6.1.p1.2.m2.6.6.2.2"></times><ci id="S6.1.p1.2.m2.6.6.2.3.cmml" xref="S6.1.p1.2.m2.6.6.2.3">Tot</ci><apply id="S6.1.p1.2.m2.6.6.2.1.1.1.cmml" xref="S6.1.p1.2.m2.6.6.2.1.1"><times id="S6.1.p1.2.m2.6.6.2.1.1.1.1.cmml" xref="S6.1.p1.2.m2.6.6.2.1.1.1.1"></times><apply id="S6.1.p1.2.m2.6.6.2.1.1.1.2.cmml" xref="S6.1.p1.2.m2.6.6.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.1.p1.2.m2.6.6.2.1.1.1.2.1.cmml" xref="S6.1.p1.2.m2.6.6.2.1.1.1.2">subscript</csymbol><ci id="S6.1.p1.2.m2.6.6.2.1.1.1.2.2.cmml" xref="S6.1.p1.2.m2.6.6.2.1.1.1.2.2">𝑃</ci><list id="S6.1.p1.2.m2.2.2.2.3.cmml" xref="S6.1.p1.2.m2.2.2.2.4"><times id="S6.1.p1.2.m2.1.1.1.1.cmml" xref="S6.1.p1.2.m2.1.1.1.1"></times><times id="S6.1.p1.2.m2.2.2.2.2.cmml" xref="S6.1.p1.2.m2.2.2.2.2"></times></list></apply><ci id="S6.1.p1.2.m2.4.4.cmml" xref="S6.1.p1.2.m2.4.4">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.2.m2.6c">(\mathrm{diag}\mathbb{P}(\theta))_{*}\to\mathrm{Tot}(P_{*,*}(\theta))</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.2.m2.6d">( roman_diag blackboard_P ( italic_θ ) ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → roman_Tot ( italic_P start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ( italic_θ ) )</annotation></semantics></math> is a chain homotopy equivalence where <math alttext="(\mathrm{diag}\mathbb{P}(\theta))_{*}" class="ltx_Math" display="inline" id="S6.1.p1.3.m3.2"><semantics id="S6.1.p1.3.m3.2a"><msub id="S6.1.p1.3.m3.2.2" xref="S6.1.p1.3.m3.2.2.cmml"><mrow id="S6.1.p1.3.m3.2.2.1.1" xref="S6.1.p1.3.m3.2.2.1.1.1.cmml"><mo id="S6.1.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S6.1.p1.3.m3.2.2.1.1.1.cmml">(</mo><mrow id="S6.1.p1.3.m3.2.2.1.1.1" xref="S6.1.p1.3.m3.2.2.1.1.1.cmml"><mi id="S6.1.p1.3.m3.2.2.1.1.1.2" xref="S6.1.p1.3.m3.2.2.1.1.1.2.cmml">diag</mi><mo id="S6.1.p1.3.m3.2.2.1.1.1.1" xref="S6.1.p1.3.m3.2.2.1.1.1.1.cmml">⁢</mo><mi id="S6.1.p1.3.m3.2.2.1.1.1.3" xref="S6.1.p1.3.m3.2.2.1.1.1.3.cmml">ℙ</mi><mo id="S6.1.p1.3.m3.2.2.1.1.1.1a" xref="S6.1.p1.3.m3.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.1.p1.3.m3.2.2.1.1.1.4.2" xref="S6.1.p1.3.m3.2.2.1.1.1.cmml"><mo id="S6.1.p1.3.m3.2.2.1.1.1.4.2.1" stretchy="false" xref="S6.1.p1.3.m3.2.2.1.1.1.cmml">(</mo><mi id="S6.1.p1.3.m3.1.1" xref="S6.1.p1.3.m3.1.1.cmml">θ</mi><mo id="S6.1.p1.3.m3.2.2.1.1.1.4.2.2" stretchy="false" xref="S6.1.p1.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.3.m3.2.2.1.1.3" stretchy="false" xref="S6.1.p1.3.m3.2.2.1.1.1.cmml">)</mo></mrow><mo id="S6.1.p1.3.m3.2.2.3" xref="S6.1.p1.3.m3.2.2.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S6.1.p1.3.m3.2b"><apply id="S6.1.p1.3.m3.2.2.cmml" xref="S6.1.p1.3.m3.2.2"><csymbol cd="ambiguous" id="S6.1.p1.3.m3.2.2.2.cmml" xref="S6.1.p1.3.m3.2.2">subscript</csymbol><apply id="S6.1.p1.3.m3.2.2.1.1.1.cmml" xref="S6.1.p1.3.m3.2.2.1.1"><times id="S6.1.p1.3.m3.2.2.1.1.1.1.cmml" xref="S6.1.p1.3.m3.2.2.1.1.1.1"></times><ci id="S6.1.p1.3.m3.2.2.1.1.1.2.cmml" xref="S6.1.p1.3.m3.2.2.1.1.1.2">diag</ci><ci id="S6.1.p1.3.m3.2.2.1.1.1.3.cmml" xref="S6.1.p1.3.m3.2.2.1.1.1.3">ℙ</ci><ci id="S6.1.p1.3.m3.1.1.cmml" xref="S6.1.p1.3.m3.1.1">𝜃</ci></apply><times id="S6.1.p1.3.m3.2.2.3.cmml" xref="S6.1.p1.3.m3.2.2.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.3.m3.2c">(\mathrm{diag}\mathbb{P}(\theta))_{*}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.3.m3.2d">( roman_diag blackboard_P ( italic_θ ) ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> is the chain complex associated to the simplicial <math alttext="R" class="ltx_Math" display="inline" id="S6.1.p1.4.m4.1"><semantics id="S6.1.p1.4.m4.1a"><mi id="S6.1.p1.4.m4.1.1" xref="S6.1.p1.4.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S6.1.p1.4.m4.1b"><ci id="S6.1.p1.4.m4.1.1.cmml" xref="S6.1.p1.4.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.4.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.4.m4.1d">italic_R</annotation></semantics></math>-module <math alttext="\mathrm{diag}\mathbb{P}(\theta)" class="ltx_Math" display="inline" id="S6.1.p1.5.m5.1"><semantics id="S6.1.p1.5.m5.1a"><mrow id="S6.1.p1.5.m5.1.2" xref="S6.1.p1.5.m5.1.2.cmml"><mi id="S6.1.p1.5.m5.1.2.2" xref="S6.1.p1.5.m5.1.2.2.cmml">diag</mi><mo id="S6.1.p1.5.m5.1.2.1" xref="S6.1.p1.5.m5.1.2.1.cmml">⁢</mo><mi id="S6.1.p1.5.m5.1.2.3" xref="S6.1.p1.5.m5.1.2.3.cmml">ℙ</mi><mo id="S6.1.p1.5.m5.1.2.1a" xref="S6.1.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S6.1.p1.5.m5.1.2.4.2" xref="S6.1.p1.5.m5.1.2.cmml"><mo id="S6.1.p1.5.m5.1.2.4.2.1" stretchy="false" xref="S6.1.p1.5.m5.1.2.cmml">(</mo><mi id="S6.1.p1.5.m5.1.1" xref="S6.1.p1.5.m5.1.1.cmml">θ</mi><mo id="S6.1.p1.5.m5.1.2.4.2.2" stretchy="false" xref="S6.1.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.5.m5.1b"><apply id="S6.1.p1.5.m5.1.2.cmml" xref="S6.1.p1.5.m5.1.2"><times id="S6.1.p1.5.m5.1.2.1.cmml" xref="S6.1.p1.5.m5.1.2.1"></times><ci id="S6.1.p1.5.m5.1.2.2.cmml" xref="S6.1.p1.5.m5.1.2.2">diag</ci><ci id="S6.1.p1.5.m5.1.2.3.cmml" xref="S6.1.p1.5.m5.1.2.3">ℙ</ci><ci id="S6.1.p1.5.m5.1.1.cmml" xref="S6.1.p1.5.m5.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.5.m5.1c">\mathrm{diag}\mathbb{P}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.5.m5.1d">roman_diag blackboard_P ( italic_θ )</annotation></semantics></math>. The chain complex <math alttext="(\mathrm{diag}P(\theta))_{*}" class="ltx_Math" display="inline" id="S6.1.p1.6.m6.2"><semantics id="S6.1.p1.6.m6.2a"><msub id="S6.1.p1.6.m6.2.2" xref="S6.1.p1.6.m6.2.2.cmml"><mrow id="S6.1.p1.6.m6.2.2.1.1" xref="S6.1.p1.6.m6.2.2.1.1.1.cmml"><mo id="S6.1.p1.6.m6.2.2.1.1.2" stretchy="false" xref="S6.1.p1.6.m6.2.2.1.1.1.cmml">(</mo><mrow id="S6.1.p1.6.m6.2.2.1.1.1" xref="S6.1.p1.6.m6.2.2.1.1.1.cmml"><mi id="S6.1.p1.6.m6.2.2.1.1.1.2" xref="S6.1.p1.6.m6.2.2.1.1.1.2.cmml">diag</mi><mo id="S6.1.p1.6.m6.2.2.1.1.1.1" xref="S6.1.p1.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mi id="S6.1.p1.6.m6.2.2.1.1.1.3" xref="S6.1.p1.6.m6.2.2.1.1.1.3.cmml">P</mi><mo id="S6.1.p1.6.m6.2.2.1.1.1.1a" xref="S6.1.p1.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.1.p1.6.m6.2.2.1.1.1.4.2" xref="S6.1.p1.6.m6.2.2.1.1.1.cmml"><mo id="S6.1.p1.6.m6.2.2.1.1.1.4.2.1" stretchy="false" xref="S6.1.p1.6.m6.2.2.1.1.1.cmml">(</mo><mi id="S6.1.p1.6.m6.1.1" xref="S6.1.p1.6.m6.1.1.cmml">θ</mi><mo id="S6.1.p1.6.m6.2.2.1.1.1.4.2.2" stretchy="false" xref="S6.1.p1.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.6.m6.2.2.1.1.3" stretchy="false" xref="S6.1.p1.6.m6.2.2.1.1.1.cmml">)</mo></mrow><mo id="S6.1.p1.6.m6.2.2.3" xref="S6.1.p1.6.m6.2.2.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S6.1.p1.6.m6.2b"><apply id="S6.1.p1.6.m6.2.2.cmml" xref="S6.1.p1.6.m6.2.2"><csymbol cd="ambiguous" id="S6.1.p1.6.m6.2.2.2.cmml" xref="S6.1.p1.6.m6.2.2">subscript</csymbol><apply id="S6.1.p1.6.m6.2.2.1.1.1.cmml" xref="S6.1.p1.6.m6.2.2.1.1"><times id="S6.1.p1.6.m6.2.2.1.1.1.1.cmml" xref="S6.1.p1.6.m6.2.2.1.1.1.1"></times><ci id="S6.1.p1.6.m6.2.2.1.1.1.2.cmml" xref="S6.1.p1.6.m6.2.2.1.1.1.2">diag</ci><ci id="S6.1.p1.6.m6.2.2.1.1.1.3.cmml" xref="S6.1.p1.6.m6.2.2.1.1.1.3">𝑃</ci><ci id="S6.1.p1.6.m6.1.1.cmml" xref="S6.1.p1.6.m6.1.1">𝜃</ci></apply><times id="S6.1.p1.6.m6.2.2.3.cmml" xref="S6.1.p1.6.m6.2.2.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.6.m6.2c">(\mathrm{diag}P(\theta))_{*}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.6.m6.2d">( roman_diag italic_P ( italic_θ ) ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> is isomorphic to the simplicial chain complex of the simplicial set <math alttext="\Delta[p]\times\Delta[q]" class="ltx_Math" display="inline" id="S6.1.p1.7.m7.2"><semantics id="S6.1.p1.7.m7.2a"><mrow id="S6.1.p1.7.m7.2.3" xref="S6.1.p1.7.m7.2.3.cmml"><mrow id="S6.1.p1.7.m7.2.3.2" xref="S6.1.p1.7.m7.2.3.2.cmml"><mrow id="S6.1.p1.7.m7.2.3.2.2" xref="S6.1.p1.7.m7.2.3.2.2.cmml"><mi id="S6.1.p1.7.m7.2.3.2.2.2" mathvariant="normal" xref="S6.1.p1.7.m7.2.3.2.2.2.cmml">Δ</mi><mo id="S6.1.p1.7.m7.2.3.2.2.1" xref="S6.1.p1.7.m7.2.3.2.2.1.cmml">⁢</mo><mrow id="S6.1.p1.7.m7.2.3.2.2.3.2" xref="S6.1.p1.7.m7.2.3.2.2.3.1.cmml"><mo id="S6.1.p1.7.m7.2.3.2.2.3.2.1" stretchy="false" xref="S6.1.p1.7.m7.2.3.2.2.3.1.1.cmml">[</mo><mi id="S6.1.p1.7.m7.1.1" xref="S6.1.p1.7.m7.1.1.cmml">p</mi><mo id="S6.1.p1.7.m7.2.3.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S6.1.p1.7.m7.2.3.2.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S6.1.p1.7.m7.2.3.2.1" rspace="0.222em" xref="S6.1.p1.7.m7.2.3.2.1.cmml">×</mo><mi id="S6.1.p1.7.m7.2.3.2.3" mathvariant="normal" xref="S6.1.p1.7.m7.2.3.2.3.cmml">Δ</mi></mrow><mo id="S6.1.p1.7.m7.2.3.1" xref="S6.1.p1.7.m7.2.3.1.cmml">⁢</mo><mrow id="S6.1.p1.7.m7.2.3.3.2" xref="S6.1.p1.7.m7.2.3.3.1.cmml"><mo id="S6.1.p1.7.m7.2.3.3.2.1" stretchy="false" xref="S6.1.p1.7.m7.2.3.3.1.1.cmml">[</mo><mi id="S6.1.p1.7.m7.2.2" xref="S6.1.p1.7.m7.2.2.cmml">q</mi><mo id="S6.1.p1.7.m7.2.3.3.2.2" stretchy="false" xref="S6.1.p1.7.m7.2.3.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.7.m7.2b"><apply id="S6.1.p1.7.m7.2.3.cmml" xref="S6.1.p1.7.m7.2.3"><times id="S6.1.p1.7.m7.2.3.1.cmml" xref="S6.1.p1.7.m7.2.3.1"></times><apply id="S6.1.p1.7.m7.2.3.2.cmml" xref="S6.1.p1.7.m7.2.3.2"><times id="S6.1.p1.7.m7.2.3.2.1.cmml" xref="S6.1.p1.7.m7.2.3.2.1"></times><apply id="S6.1.p1.7.m7.2.3.2.2.cmml" xref="S6.1.p1.7.m7.2.3.2.2"><times id="S6.1.p1.7.m7.2.3.2.2.1.cmml" xref="S6.1.p1.7.m7.2.3.2.2.1"></times><ci id="S6.1.p1.7.m7.2.3.2.2.2.cmml" xref="S6.1.p1.7.m7.2.3.2.2.2">Δ</ci><apply id="S6.1.p1.7.m7.2.3.2.2.3.1.cmml" xref="S6.1.p1.7.m7.2.3.2.2.3.2"><csymbol cd="latexml" id="S6.1.p1.7.m7.2.3.2.2.3.1.1.cmml" xref="S6.1.p1.7.m7.2.3.2.2.3.2.1">delimited-[]</csymbol><ci id="S6.1.p1.7.m7.1.1.cmml" xref="S6.1.p1.7.m7.1.1">𝑝</ci></apply></apply><ci id="S6.1.p1.7.m7.2.3.2.3.cmml" xref="S6.1.p1.7.m7.2.3.2.3">Δ</ci></apply><apply id="S6.1.p1.7.m7.2.3.3.1.cmml" xref="S6.1.p1.7.m7.2.3.3.2"><csymbol cd="latexml" id="S6.1.p1.7.m7.2.3.3.1.1.cmml" xref="S6.1.p1.7.m7.2.3.3.2.1">delimited-[]</csymbol><ci id="S6.1.p1.7.m7.2.2.cmml" xref="S6.1.p1.7.m7.2.2">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.7.m7.2c">\Delta[p]\times\Delta[q]</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.7.m7.2d">roman_Δ [ italic_p ] × roman_Δ [ italic_q ]</annotation></semantics></math>. Since the realization of this simplicial set is homeomorphic to <math alttext="|\Delta[p]|\times|\Delta[q]|" class="ltx_Math" display="inline" id="S6.1.p1.8.m8.4"><semantics id="S6.1.p1.8.m8.4a"><mrow id="S6.1.p1.8.m8.4.4" xref="S6.1.p1.8.m8.4.4.cmml"><mrow id="S6.1.p1.8.m8.3.3.1.1" xref="S6.1.p1.8.m8.3.3.1.2.cmml"><mo id="S6.1.p1.8.m8.3.3.1.1.2" stretchy="false" xref="S6.1.p1.8.m8.3.3.1.2.1.cmml">|</mo><mrow id="S6.1.p1.8.m8.3.3.1.1.1" xref="S6.1.p1.8.m8.3.3.1.1.1.cmml"><mi id="S6.1.p1.8.m8.3.3.1.1.1.2" mathvariant="normal" xref="S6.1.p1.8.m8.3.3.1.1.1.2.cmml">Δ</mi><mo id="S6.1.p1.8.m8.3.3.1.1.1.1" xref="S6.1.p1.8.m8.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.1.p1.8.m8.3.3.1.1.1.3.2" xref="S6.1.p1.8.m8.3.3.1.1.1.3.1.cmml"><mo id="S6.1.p1.8.m8.3.3.1.1.1.3.2.1" stretchy="false" xref="S6.1.p1.8.m8.3.3.1.1.1.3.1.1.cmml">[</mo><mi id="S6.1.p1.8.m8.1.1" xref="S6.1.p1.8.m8.1.1.cmml">p</mi><mo id="S6.1.p1.8.m8.3.3.1.1.1.3.2.2" stretchy="false" xref="S6.1.p1.8.m8.3.3.1.1.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="S6.1.p1.8.m8.3.3.1.1.3" rspace="0.055em" stretchy="false" xref="S6.1.p1.8.m8.3.3.1.2.1.cmml">|</mo></mrow><mo id="S6.1.p1.8.m8.4.4.3" rspace="0.222em" xref="S6.1.p1.8.m8.4.4.3.cmml">×</mo><mrow id="S6.1.p1.8.m8.4.4.2.1" xref="S6.1.p1.8.m8.4.4.2.2.cmml"><mo id="S6.1.p1.8.m8.4.4.2.1.2" stretchy="false" xref="S6.1.p1.8.m8.4.4.2.2.1.cmml">|</mo><mrow id="S6.1.p1.8.m8.4.4.2.1.1" xref="S6.1.p1.8.m8.4.4.2.1.1.cmml"><mi id="S6.1.p1.8.m8.4.4.2.1.1.2" mathvariant="normal" xref="S6.1.p1.8.m8.4.4.2.1.1.2.cmml">Δ</mi><mo id="S6.1.p1.8.m8.4.4.2.1.1.1" xref="S6.1.p1.8.m8.4.4.2.1.1.1.cmml">⁢</mo><mrow id="S6.1.p1.8.m8.4.4.2.1.1.3.2" xref="S6.1.p1.8.m8.4.4.2.1.1.3.1.cmml"><mo id="S6.1.p1.8.m8.4.4.2.1.1.3.2.1" stretchy="false" xref="S6.1.p1.8.m8.4.4.2.1.1.3.1.1.cmml">[</mo><mi id="S6.1.p1.8.m8.2.2" xref="S6.1.p1.8.m8.2.2.cmml">q</mi><mo id="S6.1.p1.8.m8.4.4.2.1.1.3.2.2" stretchy="false" xref="S6.1.p1.8.m8.4.4.2.1.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="S6.1.p1.8.m8.4.4.2.1.3" stretchy="false" xref="S6.1.p1.8.m8.4.4.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.8.m8.4b"><apply id="S6.1.p1.8.m8.4.4.cmml" xref="S6.1.p1.8.m8.4.4"><times id="S6.1.p1.8.m8.4.4.3.cmml" xref="S6.1.p1.8.m8.4.4.3"></times><apply id="S6.1.p1.8.m8.3.3.1.2.cmml" xref="S6.1.p1.8.m8.3.3.1.1"><abs id="S6.1.p1.8.m8.3.3.1.2.1.cmml" xref="S6.1.p1.8.m8.3.3.1.1.2"></abs><apply id="S6.1.p1.8.m8.3.3.1.1.1.cmml" xref="S6.1.p1.8.m8.3.3.1.1.1"><times id="S6.1.p1.8.m8.3.3.1.1.1.1.cmml" xref="S6.1.p1.8.m8.3.3.1.1.1.1"></times><ci id="S6.1.p1.8.m8.3.3.1.1.1.2.cmml" xref="S6.1.p1.8.m8.3.3.1.1.1.2">Δ</ci><apply id="S6.1.p1.8.m8.3.3.1.1.1.3.1.cmml" xref="S6.1.p1.8.m8.3.3.1.1.1.3.2"><csymbol cd="latexml" id="S6.1.p1.8.m8.3.3.1.1.1.3.1.1.cmml" xref="S6.1.p1.8.m8.3.3.1.1.1.3.2.1">delimited-[]</csymbol><ci id="S6.1.p1.8.m8.1.1.cmml" xref="S6.1.p1.8.m8.1.1">𝑝</ci></apply></apply></apply><apply id="S6.1.p1.8.m8.4.4.2.2.cmml" xref="S6.1.p1.8.m8.4.4.2.1"><abs id="S6.1.p1.8.m8.4.4.2.2.1.cmml" xref="S6.1.p1.8.m8.4.4.2.1.2"></abs><apply id="S6.1.p1.8.m8.4.4.2.1.1.cmml" xref="S6.1.p1.8.m8.4.4.2.1.1"><times id="S6.1.p1.8.m8.4.4.2.1.1.1.cmml" xref="S6.1.p1.8.m8.4.4.2.1.1.1"></times><ci id="S6.1.p1.8.m8.4.4.2.1.1.2.cmml" xref="S6.1.p1.8.m8.4.4.2.1.1.2">Δ</ci><apply id="S6.1.p1.8.m8.4.4.2.1.1.3.1.cmml" xref="S6.1.p1.8.m8.4.4.2.1.1.3.2"><csymbol cd="latexml" id="S6.1.p1.8.m8.4.4.2.1.1.3.1.1.cmml" xref="S6.1.p1.8.m8.4.4.2.1.1.3.2.1">delimited-[]</csymbol><ci id="S6.1.p1.8.m8.2.2.cmml" xref="S6.1.p1.8.m8.2.2">𝑞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.8.m8.4c">|\Delta[p]|\times|\Delta[q]|</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.8.m8.4d">| roman_Δ [ italic_p ] | × | roman_Δ [ italic_q ] |</annotation></semantics></math> which is contractible, we can conclude that <math alttext="(\mathrm{diag}P(\theta))_{*}\to R" class="ltx_Math" display="inline" id="S6.1.p1.9.m9.2"><semantics id="S6.1.p1.9.m9.2a"><mrow id="S6.1.p1.9.m9.2.2" xref="S6.1.p1.9.m9.2.2.cmml"><msub id="S6.1.p1.9.m9.2.2.1" xref="S6.1.p1.9.m9.2.2.1.cmml"><mrow id="S6.1.p1.9.m9.2.2.1.1.1" xref="S6.1.p1.9.m9.2.2.1.1.1.1.cmml"><mo id="S6.1.p1.9.m9.2.2.1.1.1.2" stretchy="false" xref="S6.1.p1.9.m9.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.1.p1.9.m9.2.2.1.1.1.1" xref="S6.1.p1.9.m9.2.2.1.1.1.1.cmml"><mi id="S6.1.p1.9.m9.2.2.1.1.1.1.2" xref="S6.1.p1.9.m9.2.2.1.1.1.1.2.cmml">diag</mi><mo id="S6.1.p1.9.m9.2.2.1.1.1.1.1" xref="S6.1.p1.9.m9.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S6.1.p1.9.m9.2.2.1.1.1.1.3" xref="S6.1.p1.9.m9.2.2.1.1.1.1.3.cmml">P</mi><mo id="S6.1.p1.9.m9.2.2.1.1.1.1.1a" xref="S6.1.p1.9.m9.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.1.p1.9.m9.2.2.1.1.1.1.4.2" xref="S6.1.p1.9.m9.2.2.1.1.1.1.cmml"><mo id="S6.1.p1.9.m9.2.2.1.1.1.1.4.2.1" stretchy="false" xref="S6.1.p1.9.m9.2.2.1.1.1.1.cmml">(</mo><mi id="S6.1.p1.9.m9.1.1" xref="S6.1.p1.9.m9.1.1.cmml">θ</mi><mo id="S6.1.p1.9.m9.2.2.1.1.1.1.4.2.2" stretchy="false" xref="S6.1.p1.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.9.m9.2.2.1.1.1.3" stretchy="false" xref="S6.1.p1.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S6.1.p1.9.m9.2.2.1.3" xref="S6.1.p1.9.m9.2.2.1.3.cmml">∗</mo></msub><mo id="S6.1.p1.9.m9.2.2.2" stretchy="false" xref="S6.1.p1.9.m9.2.2.2.cmml">→</mo><mi id="S6.1.p1.9.m9.2.2.3" xref="S6.1.p1.9.m9.2.2.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.9.m9.2b"><apply id="S6.1.p1.9.m9.2.2.cmml" xref="S6.1.p1.9.m9.2.2"><ci id="S6.1.p1.9.m9.2.2.2.cmml" xref="S6.1.p1.9.m9.2.2.2">→</ci><apply id="S6.1.p1.9.m9.2.2.1.cmml" xref="S6.1.p1.9.m9.2.2.1"><csymbol cd="ambiguous" id="S6.1.p1.9.m9.2.2.1.2.cmml" xref="S6.1.p1.9.m9.2.2.1">subscript</csymbol><apply id="S6.1.p1.9.m9.2.2.1.1.1.1.cmml" xref="S6.1.p1.9.m9.2.2.1.1.1"><times id="S6.1.p1.9.m9.2.2.1.1.1.1.1.cmml" xref="S6.1.p1.9.m9.2.2.1.1.1.1.1"></times><ci id="S6.1.p1.9.m9.2.2.1.1.1.1.2.cmml" xref="S6.1.p1.9.m9.2.2.1.1.1.1.2">diag</ci><ci id="S6.1.p1.9.m9.2.2.1.1.1.1.3.cmml" xref="S6.1.p1.9.m9.2.2.1.1.1.1.3">𝑃</ci><ci id="S6.1.p1.9.m9.1.1.cmml" xref="S6.1.p1.9.m9.1.1">𝜃</ci></apply><times id="S6.1.p1.9.m9.2.2.1.3.cmml" xref="S6.1.p1.9.m9.2.2.1.3"></times></apply><ci id="S6.1.p1.9.m9.2.2.3.cmml" xref="S6.1.p1.9.m9.2.2.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.9.m9.2c">(\mathrm{diag}P(\theta))_{*}\to R</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.9.m9.2d">( roman_diag italic_P ( italic_θ ) ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → italic_R</annotation></semantics></math> is exact. This gives that the chain complex <math alttext="\mathrm{Tot}(P_{*,*}(\theta))\to R" class="ltx_Math" display="inline" id="S6.1.p1.10.m10.4"><semantics id="S6.1.p1.10.m10.4a"><mrow id="S6.1.p1.10.m10.4.4" xref="S6.1.p1.10.m10.4.4.cmml"><mrow id="S6.1.p1.10.m10.4.4.1" xref="S6.1.p1.10.m10.4.4.1.cmml"><mi id="S6.1.p1.10.m10.4.4.1.3" xref="S6.1.p1.10.m10.4.4.1.3.cmml">Tot</mi><mo id="S6.1.p1.10.m10.4.4.1.2" xref="S6.1.p1.10.m10.4.4.1.2.cmml">⁢</mo><mrow id="S6.1.p1.10.m10.4.4.1.1.1" xref="S6.1.p1.10.m10.4.4.1.1.1.1.cmml"><mo id="S6.1.p1.10.m10.4.4.1.1.1.2" stretchy="false" xref="S6.1.p1.10.m10.4.4.1.1.1.1.cmml">(</mo><mrow id="S6.1.p1.10.m10.4.4.1.1.1.1" xref="S6.1.p1.10.m10.4.4.1.1.1.1.cmml"><msub id="S6.1.p1.10.m10.4.4.1.1.1.1.2" xref="S6.1.p1.10.m10.4.4.1.1.1.1.2.cmml"><mi id="S6.1.p1.10.m10.4.4.1.1.1.1.2.2" xref="S6.1.p1.10.m10.4.4.1.1.1.1.2.2.cmml">P</mi><mrow id="S6.1.p1.10.m10.2.2.2.4" xref="S6.1.p1.10.m10.2.2.2.3.cmml"><mo id="S6.1.p1.10.m10.1.1.1.1" rspace="0em" xref="S6.1.p1.10.m10.1.1.1.1.cmml">∗</mo><mo id="S6.1.p1.10.m10.2.2.2.4.1" rspace="0em" xref="S6.1.p1.10.m10.2.2.2.3.cmml">,</mo><mo id="S6.1.p1.10.m10.2.2.2.2" lspace="0em" xref="S6.1.p1.10.m10.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S6.1.p1.10.m10.4.4.1.1.1.1.1" xref="S6.1.p1.10.m10.4.4.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.1.p1.10.m10.4.4.1.1.1.1.3.2" xref="S6.1.p1.10.m10.4.4.1.1.1.1.cmml"><mo id="S6.1.p1.10.m10.4.4.1.1.1.1.3.2.1" stretchy="false" xref="S6.1.p1.10.m10.4.4.1.1.1.1.cmml">(</mo><mi id="S6.1.p1.10.m10.3.3" xref="S6.1.p1.10.m10.3.3.cmml">θ</mi><mo id="S6.1.p1.10.m10.4.4.1.1.1.1.3.2.2" stretchy="false" xref="S6.1.p1.10.m10.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.10.m10.4.4.1.1.1.3" stretchy="false" xref="S6.1.p1.10.m10.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.10.m10.4.4.2" stretchy="false" xref="S6.1.p1.10.m10.4.4.2.cmml">→</mo><mi id="S6.1.p1.10.m10.4.4.3" xref="S6.1.p1.10.m10.4.4.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.10.m10.4b"><apply id="S6.1.p1.10.m10.4.4.cmml" xref="S6.1.p1.10.m10.4.4"><ci id="S6.1.p1.10.m10.4.4.2.cmml" xref="S6.1.p1.10.m10.4.4.2">→</ci><apply id="S6.1.p1.10.m10.4.4.1.cmml" xref="S6.1.p1.10.m10.4.4.1"><times id="S6.1.p1.10.m10.4.4.1.2.cmml" xref="S6.1.p1.10.m10.4.4.1.2"></times><ci id="S6.1.p1.10.m10.4.4.1.3.cmml" xref="S6.1.p1.10.m10.4.4.1.3">Tot</ci><apply id="S6.1.p1.10.m10.4.4.1.1.1.1.cmml" xref="S6.1.p1.10.m10.4.4.1.1.1"><times id="S6.1.p1.10.m10.4.4.1.1.1.1.1.cmml" xref="S6.1.p1.10.m10.4.4.1.1.1.1.1"></times><apply id="S6.1.p1.10.m10.4.4.1.1.1.1.2.cmml" xref="S6.1.p1.10.m10.4.4.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.1.p1.10.m10.4.4.1.1.1.1.2.1.cmml" xref="S6.1.p1.10.m10.4.4.1.1.1.1.2">subscript</csymbol><ci id="S6.1.p1.10.m10.4.4.1.1.1.1.2.2.cmml" xref="S6.1.p1.10.m10.4.4.1.1.1.1.2.2">𝑃</ci><list id="S6.1.p1.10.m10.2.2.2.3.cmml" xref="S6.1.p1.10.m10.2.2.2.4"><times id="S6.1.p1.10.m10.1.1.1.1.cmml" xref="S6.1.p1.10.m10.1.1.1.1"></times><times id="S6.1.p1.10.m10.2.2.2.2.cmml" xref="S6.1.p1.10.m10.2.2.2.2"></times></list></apply><ci id="S6.1.p1.10.m10.3.3.cmml" xref="S6.1.p1.10.m10.3.3">𝜃</ci></apply></apply><ci id="S6.1.p1.10.m10.4.4.3.cmml" xref="S6.1.p1.10.m10.4.4.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.10.m10.4c">\mathrm{Tot}(P_{*,*}(\theta))\to R</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.10.m10.4d">roman_Tot ( italic_P start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ( italic_θ ) ) → italic_R</annotation></semantics></math> is exact. Since this is true for all <math alttext="\theta\in X_{p,q}" class="ltx_Math" display="inline" id="S6.1.p1.11.m11.2"><semantics id="S6.1.p1.11.m11.2a"><mrow id="S6.1.p1.11.m11.2.3" xref="S6.1.p1.11.m11.2.3.cmml"><mi id="S6.1.p1.11.m11.2.3.2" xref="S6.1.p1.11.m11.2.3.2.cmml">θ</mi><mo id="S6.1.p1.11.m11.2.3.1" xref="S6.1.p1.11.m11.2.3.1.cmml">∈</mo><msub id="S6.1.p1.11.m11.2.3.3" xref="S6.1.p1.11.m11.2.3.3.cmml"><mi id="S6.1.p1.11.m11.2.3.3.2" xref="S6.1.p1.11.m11.2.3.3.2.cmml">X</mi><mrow id="S6.1.p1.11.m11.2.2.2.4" xref="S6.1.p1.11.m11.2.2.2.3.cmml"><mi id="S6.1.p1.11.m11.1.1.1.1" xref="S6.1.p1.11.m11.1.1.1.1.cmml">p</mi><mo id="S6.1.p1.11.m11.2.2.2.4.1" xref="S6.1.p1.11.m11.2.2.2.3.cmml">,</mo><mi id="S6.1.p1.11.m11.2.2.2.2" xref="S6.1.p1.11.m11.2.2.2.2.cmml">q</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.11.m11.2b"><apply id="S6.1.p1.11.m11.2.3.cmml" xref="S6.1.p1.11.m11.2.3"><in id="S6.1.p1.11.m11.2.3.1.cmml" xref="S6.1.p1.11.m11.2.3.1"></in><ci id="S6.1.p1.11.m11.2.3.2.cmml" xref="S6.1.p1.11.m11.2.3.2">𝜃</ci><apply id="S6.1.p1.11.m11.2.3.3.cmml" xref="S6.1.p1.11.m11.2.3.3"><csymbol cd="ambiguous" id="S6.1.p1.11.m11.2.3.3.1.cmml" xref="S6.1.p1.11.m11.2.3.3">subscript</csymbol><ci id="S6.1.p1.11.m11.2.3.3.2.cmml" xref="S6.1.p1.11.m11.2.3.3.2">𝑋</ci><list id="S6.1.p1.11.m11.2.2.2.3.cmml" xref="S6.1.p1.11.m11.2.2.2.4"><ci id="S6.1.p1.11.m11.1.1.1.1.cmml" xref="S6.1.p1.11.m11.1.1.1.1">𝑝</ci><ci id="S6.1.p1.11.m11.2.2.2.2.cmml" xref="S6.1.p1.11.m11.2.2.2.2">𝑞</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.11.m11.2c">\theta\in X_{p,q}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.11.m11.2d">italic_θ ∈ italic_X start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT</annotation></semantics></math>, the chain complex <math alttext="\mathrm{Tot}(P_{*,*})\to\underline{R}" class="ltx_Math" display="inline" id="S6.1.p1.12.m12.3"><semantics id="S6.1.p1.12.m12.3a"><mrow id="S6.1.p1.12.m12.3.3" xref="S6.1.p1.12.m12.3.3.cmml"><mrow id="S6.1.p1.12.m12.3.3.1" xref="S6.1.p1.12.m12.3.3.1.cmml"><mi id="S6.1.p1.12.m12.3.3.1.3" xref="S6.1.p1.12.m12.3.3.1.3.cmml">Tot</mi><mo id="S6.1.p1.12.m12.3.3.1.2" xref="S6.1.p1.12.m12.3.3.1.2.cmml">⁢</mo><mrow id="S6.1.p1.12.m12.3.3.1.1.1" xref="S6.1.p1.12.m12.3.3.1.1.1.1.cmml"><mo id="S6.1.p1.12.m12.3.3.1.1.1.2" stretchy="false" xref="S6.1.p1.12.m12.3.3.1.1.1.1.cmml">(</mo><msub id="S6.1.p1.12.m12.3.3.1.1.1.1" xref="S6.1.p1.12.m12.3.3.1.1.1.1.cmml"><mi id="S6.1.p1.12.m12.3.3.1.1.1.1.2" xref="S6.1.p1.12.m12.3.3.1.1.1.1.2.cmml">P</mi><mrow id="S6.1.p1.12.m12.2.2.2.4" xref="S6.1.p1.12.m12.2.2.2.3.cmml"><mo id="S6.1.p1.12.m12.1.1.1.1" rspace="0em" xref="S6.1.p1.12.m12.1.1.1.1.cmml">∗</mo><mo id="S6.1.p1.12.m12.2.2.2.4.1" rspace="0em" xref="S6.1.p1.12.m12.2.2.2.3.cmml">,</mo><mo id="S6.1.p1.12.m12.2.2.2.2" lspace="0em" xref="S6.1.p1.12.m12.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S6.1.p1.12.m12.3.3.1.1.1.3" stretchy="false" xref="S6.1.p1.12.m12.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.12.m12.3.3.2" stretchy="false" xref="S6.1.p1.12.m12.3.3.2.cmml">→</mo><munder accentunder="true" id="S6.1.p1.12.m12.3.3.3" xref="S6.1.p1.12.m12.3.3.3.cmml"><mi id="S6.1.p1.12.m12.3.3.3.2" xref="S6.1.p1.12.m12.3.3.3.2.cmml">R</mi><mo id="S6.1.p1.12.m12.3.3.3.1" xref="S6.1.p1.12.m12.3.3.3.1.cmml">¯</mo></munder></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.12.m12.3b"><apply id="S6.1.p1.12.m12.3.3.cmml" xref="S6.1.p1.12.m12.3.3"><ci id="S6.1.p1.12.m12.3.3.2.cmml" xref="S6.1.p1.12.m12.3.3.2">→</ci><apply id="S6.1.p1.12.m12.3.3.1.cmml" xref="S6.1.p1.12.m12.3.3.1"><times id="S6.1.p1.12.m12.3.3.1.2.cmml" xref="S6.1.p1.12.m12.3.3.1.2"></times><ci id="S6.1.p1.12.m12.3.3.1.3.cmml" xref="S6.1.p1.12.m12.3.3.1.3">Tot</ci><apply id="S6.1.p1.12.m12.3.3.1.1.1.1.cmml" xref="S6.1.p1.12.m12.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.1.p1.12.m12.3.3.1.1.1.1.1.cmml" xref="S6.1.p1.12.m12.3.3.1.1.1">subscript</csymbol><ci id="S6.1.p1.12.m12.3.3.1.1.1.1.2.cmml" xref="S6.1.p1.12.m12.3.3.1.1.1.1.2">𝑃</ci><list id="S6.1.p1.12.m12.2.2.2.3.cmml" xref="S6.1.p1.12.m12.2.2.2.4"><times id="S6.1.p1.12.m12.1.1.1.1.cmml" xref="S6.1.p1.12.m12.1.1.1.1"></times><times id="S6.1.p1.12.m12.2.2.2.2.cmml" xref="S6.1.p1.12.m12.2.2.2.2"></times></list></apply></apply><apply id="S6.1.p1.12.m12.3.3.3.cmml" xref="S6.1.p1.12.m12.3.3.3"><ci id="S6.1.p1.12.m12.3.3.3.1.cmml" xref="S6.1.p1.12.m12.3.3.3.1">¯</ci><ci id="S6.1.p1.12.m12.3.3.3.2.cmml" xref="S6.1.p1.12.m12.3.3.3.2">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.12.m12.3c">\mathrm{Tot}(P_{*,*})\to\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.12.m12.3d">roman_Tot ( italic_P start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ) → under¯ start_ARG italic_R end_ARG</annotation></semantics></math> of <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S6.1.p1.13.m13.1"><semantics id="S6.1.p1.13.m13.1a"><mrow id="S6.1.p1.13.m13.1.2" xref="S6.1.p1.13.m13.1.2.cmml"><mi id="S6.1.p1.13.m13.1.2.2" xref="S6.1.p1.13.m13.1.2.2.cmml">R</mi><mo id="S6.1.p1.13.m13.1.2.1" xref="S6.1.p1.13.m13.1.2.1.cmml">⁢</mo><mi id="S6.1.p1.13.m13.1.2.3" mathvariant="normal" xref="S6.1.p1.13.m13.1.2.3.cmml">Δ</mi><mo id="S6.1.p1.13.m13.1.2.1a" xref="S6.1.p1.13.m13.1.2.1.cmml">⁢</mo><mrow id="S6.1.p1.13.m13.1.2.4.2" xref="S6.1.p1.13.m13.1.2.cmml"><mo id="S6.1.p1.13.m13.1.2.4.2.1" stretchy="false" xref="S6.1.p1.13.m13.1.2.cmml">(</mo><mi id="S6.1.p1.13.m13.1.1" xref="S6.1.p1.13.m13.1.1.cmml">X</mi><mo id="S6.1.p1.13.m13.1.2.4.2.2" stretchy="false" xref="S6.1.p1.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.13.m13.1b"><apply id="S6.1.p1.13.m13.1.2.cmml" xref="S6.1.p1.13.m13.1.2"><times id="S6.1.p1.13.m13.1.2.1.cmml" xref="S6.1.p1.13.m13.1.2.1"></times><ci id="S6.1.p1.13.m13.1.2.2.cmml" xref="S6.1.p1.13.m13.1.2.2">𝑅</ci><ci id="S6.1.p1.13.m13.1.2.3.cmml" xref="S6.1.p1.13.m13.1.2.3">Δ</ci><ci id="S6.1.p1.13.m13.1.1.cmml" xref="S6.1.p1.13.m13.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.13.m13.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.13.m13.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-modules is a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S6.1.p1.14.m14.1"><semantics id="S6.1.p1.14.m14.1a"><munder accentunder="true" id="S6.1.p1.14.m14.1.1" xref="S6.1.p1.14.m14.1.1.cmml"><mi id="S6.1.p1.14.m14.1.1.2" xref="S6.1.p1.14.m14.1.1.2.cmml">R</mi><mo id="S6.1.p1.14.m14.1.1.1" xref="S6.1.p1.14.m14.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S6.1.p1.14.m14.1b"><apply id="S6.1.p1.14.m14.1.1.cmml" xref="S6.1.p1.14.m14.1.1"><ci id="S6.1.p1.14.m14.1.1.1.cmml" xref="S6.1.p1.14.m14.1.1.1">¯</ci><ci id="S6.1.p1.14.m14.1.1.2.cmml" xref="S6.1.p1.14.m14.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.14.m14.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.14.m14.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S6.p4"> <p class="ltx_p" id="S6.p4.5">Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S6.Thmtheorem2" title="Lemma 6.2. ‣ 6. Cohomology of bisimplicial sets with nontrivial coefficients ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">6.2</span></a> shows that there is a standard cochain complex to calculate the cohomology groups <math alttext="H^{*}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S6.p4.1.m1.2"><semantics id="S6.p4.1.m1.2a"><mrow id="S6.p4.1.m1.2.3" xref="S6.p4.1.m1.2.3.cmml"><msup id="S6.p4.1.m1.2.3.2" xref="S6.p4.1.m1.2.3.2.cmml"><mi id="S6.p4.1.m1.2.3.2.2" xref="S6.p4.1.m1.2.3.2.2.cmml">H</mi><mo id="S6.p4.1.m1.2.3.2.3" xref="S6.p4.1.m1.2.3.2.3.cmml">∗</mo></msup><mo id="S6.p4.1.m1.2.3.1" xref="S6.p4.1.m1.2.3.1.cmml">⁢</mo><mrow id="S6.p4.1.m1.2.3.3.2" xref="S6.p4.1.m1.2.3.3.1.cmml"><mo id="S6.p4.1.m1.2.3.3.2.1" stretchy="false" xref="S6.p4.1.m1.2.3.3.1.cmml">(</mo><mi id="S6.p4.1.m1.1.1" xref="S6.p4.1.m1.1.1.cmml">X</mi><mo id="S6.p4.1.m1.2.3.3.2.2" xref="S6.p4.1.m1.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S6.p4.1.m1.2.2" xref="S6.p4.1.m1.2.2.cmml">ℳ</mi><mo id="S6.p4.1.m1.2.3.3.2.3" stretchy="false" xref="S6.p4.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.1.m1.2b"><apply id="S6.p4.1.m1.2.3.cmml" xref="S6.p4.1.m1.2.3"><times id="S6.p4.1.m1.2.3.1.cmml" xref="S6.p4.1.m1.2.3.1"></times><apply id="S6.p4.1.m1.2.3.2.cmml" xref="S6.p4.1.m1.2.3.2"><csymbol cd="ambiguous" id="S6.p4.1.m1.2.3.2.1.cmml" xref="S6.p4.1.m1.2.3.2">superscript</csymbol><ci id="S6.p4.1.m1.2.3.2.2.cmml" xref="S6.p4.1.m1.2.3.2.2">𝐻</ci><times id="S6.p4.1.m1.2.3.2.3.cmml" xref="S6.p4.1.m1.2.3.2.3"></times></apply><list id="S6.p4.1.m1.2.3.3.1.cmml" xref="S6.p4.1.m1.2.3.3.2"><ci id="S6.p4.1.m1.1.1.cmml" xref="S6.p4.1.m1.1.1">𝑋</ci><ci id="S6.p4.1.m1.2.2.cmml" xref="S6.p4.1.m1.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.1.m1.2c">H^{*}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S6.p4.1.m1.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math> for a bisimplicial set <math alttext="X" class="ltx_Math" display="inline" id="S6.p4.2.m2.1"><semantics id="S6.p4.2.m2.1a"><mi id="S6.p4.2.m2.1.1" xref="S6.p4.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.p4.2.m2.1b"><ci id="S6.p4.2.m2.1.1.cmml" xref="S6.p4.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.p4.2.m2.1d">italic_X</annotation></semantics></math> with coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S6.p4.3.m3.1"><semantics id="S6.p4.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S6.p4.3.m3.1.1" xref="S6.p4.3.m3.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S6.p4.3.m3.1b"><ci id="S6.p4.3.m3.1.1.cmml" xref="S6.p4.3.m3.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.3.m3.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.3.m3.1d">caligraphic_M</annotation></semantics></math>. For each <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S6.p4.4.m4.1"><semantics id="S6.p4.4.m4.1a"><mrow id="S6.p4.4.m4.1.1" xref="S6.p4.4.m4.1.1.cmml"><mi id="S6.p4.4.m4.1.1.2" xref="S6.p4.4.m4.1.1.2.cmml">n</mi><mo id="S6.p4.4.m4.1.1.1" xref="S6.p4.4.m4.1.1.1.cmml">≥</mo><mn id="S6.p4.4.m4.1.1.3" xref="S6.p4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.4.m4.1b"><apply id="S6.p4.4.m4.1.1.cmml" xref="S6.p4.4.m4.1.1"><geq id="S6.p4.4.m4.1.1.1.cmml" xref="S6.p4.4.m4.1.1.1"></geq><ci id="S6.p4.4.m4.1.1.2.cmml" xref="S6.p4.4.m4.1.1.2">𝑛</ci><cn id="S6.p4.4.m4.1.1.3.cmml" type="integer" xref="S6.p4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.4.m4.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.p4.4.m4.1d">italic_n ≥ 0</annotation></semantics></math>, this cochain complex has <math alttext="n" class="ltx_Math" display="inline" id="S6.p4.5.m5.1"><semantics id="S6.p4.5.m5.1a"><mi id="S6.p4.5.m5.1.1" xref="S6.p4.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.p4.5.m5.1b"><ci id="S6.p4.5.m5.1.1.cmml" xref="S6.p4.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.p4.5.m5.1d">italic_n</annotation></semantics></math>-cochains</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex70"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C^{n}(X;\mathcal{M})=\mathrm{Hom}_{R\Delta(X)}(\mathrm{Tot}(P_{*,*})_{n};% \mathcal{M})\cong\prod_{r+s=n}\prod_{\tau\in X_{r,s}}\mathcal{M}(\tau)" class="ltx_Math" display="block" id="S6.Ex70.m1.10"><semantics id="S6.Ex70.m1.10a"><mrow id="S6.Ex70.m1.10.10" xref="S6.Ex70.m1.10.10.cmml"><mrow id="S6.Ex70.m1.10.10.3" xref="S6.Ex70.m1.10.10.3.cmml"><msup id="S6.Ex70.m1.10.10.3.2" xref="S6.Ex70.m1.10.10.3.2.cmml"><mi id="S6.Ex70.m1.10.10.3.2.2" xref="S6.Ex70.m1.10.10.3.2.2.cmml">C</mi><mi id="S6.Ex70.m1.10.10.3.2.3" xref="S6.Ex70.m1.10.10.3.2.3.cmml">n</mi></msup><mo id="S6.Ex70.m1.10.10.3.1" xref="S6.Ex70.m1.10.10.3.1.cmml">⁢</mo><mrow id="S6.Ex70.m1.10.10.3.3.2" xref="S6.Ex70.m1.10.10.3.3.1.cmml"><mo id="S6.Ex70.m1.10.10.3.3.2.1" stretchy="false" xref="S6.Ex70.m1.10.10.3.3.1.cmml">(</mo><mi id="S6.Ex70.m1.6.6" xref="S6.Ex70.m1.6.6.cmml">X</mi><mo id="S6.Ex70.m1.10.10.3.3.2.2" xref="S6.Ex70.m1.10.10.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex70.m1.7.7" xref="S6.Ex70.m1.7.7.cmml">ℳ</mi><mo id="S6.Ex70.m1.10.10.3.3.2.3" stretchy="false" xref="S6.Ex70.m1.10.10.3.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex70.m1.10.10.4" xref="S6.Ex70.m1.10.10.4.cmml">=</mo><mrow id="S6.Ex70.m1.10.10.1" xref="S6.Ex70.m1.10.10.1.cmml"><msub id="S6.Ex70.m1.10.10.1.3" xref="S6.Ex70.m1.10.10.1.3.cmml"><mi id="S6.Ex70.m1.10.10.1.3.2" xref="S6.Ex70.m1.10.10.1.3.2.cmml">Hom</mi><mrow id="S6.Ex70.m1.1.1.1" xref="S6.Ex70.m1.1.1.1.cmml"><mi id="S6.Ex70.m1.1.1.1.3" xref="S6.Ex70.m1.1.1.1.3.cmml">R</mi><mo id="S6.Ex70.m1.1.1.1.2" xref="S6.Ex70.m1.1.1.1.2.cmml">⁢</mo><mi id="S6.Ex70.m1.1.1.1.4" mathvariant="normal" xref="S6.Ex70.m1.1.1.1.4.cmml">Δ</mi><mo id="S6.Ex70.m1.1.1.1.2a" xref="S6.Ex70.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex70.m1.1.1.1.5.2" xref="S6.Ex70.m1.1.1.1.cmml"><mo id="S6.Ex70.m1.1.1.1.5.2.1" stretchy="false" xref="S6.Ex70.m1.1.1.1.cmml">(</mo><mi id="S6.Ex70.m1.1.1.1.1" xref="S6.Ex70.m1.1.1.1.1.cmml">X</mi><mo id="S6.Ex70.m1.1.1.1.5.2.2" stretchy="false" xref="S6.Ex70.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Ex70.m1.10.10.1.2" xref="S6.Ex70.m1.10.10.1.2.cmml">⁢</mo><mrow id="S6.Ex70.m1.10.10.1.1.1" xref="S6.Ex70.m1.10.10.1.1.2.cmml"><mo id="S6.Ex70.m1.10.10.1.1.1.2" stretchy="false" xref="S6.Ex70.m1.10.10.1.1.2.cmml">(</mo><mrow id="S6.Ex70.m1.10.10.1.1.1.1" xref="S6.Ex70.m1.10.10.1.1.1.1.cmml"><mi id="S6.Ex70.m1.10.10.1.1.1.1.3" xref="S6.Ex70.m1.10.10.1.1.1.1.3.cmml">Tot</mi><mo id="S6.Ex70.m1.10.10.1.1.1.1.2" xref="S6.Ex70.m1.10.10.1.1.1.1.2.cmml">⁢</mo><msub id="S6.Ex70.m1.10.10.1.1.1.1.1" xref="S6.Ex70.m1.10.10.1.1.1.1.1.cmml"><mrow id="S6.Ex70.m1.10.10.1.1.1.1.1.1.1" xref="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.1" xref="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.1.2" xref="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.1.2.cmml">P</mi><mrow id="S6.Ex70.m1.3.3.2.4" xref="S6.Ex70.m1.3.3.2.3.cmml"><mo id="S6.Ex70.m1.2.2.1.1" rspace="0em" xref="S6.Ex70.m1.2.2.1.1.cmml">∗</mo><mo id="S6.Ex70.m1.3.3.2.4.1" rspace="0em" xref="S6.Ex70.m1.3.3.2.3.cmml">,</mo><mo id="S6.Ex70.m1.3.3.2.2" lspace="0em" xref="S6.Ex70.m1.3.3.2.2.cmml">∗</mo></mrow></msub><mo id="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex70.m1.10.10.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S6.Ex70.m1.10.10.1.1.1.1.1.3" xref="S6.Ex70.m1.10.10.1.1.1.1.1.3.cmml">n</mi></msub></mrow><mo id="S6.Ex70.m1.10.10.1.1.1.3" xref="S6.Ex70.m1.10.10.1.1.2.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex70.m1.8.8" xref="S6.Ex70.m1.8.8.cmml">ℳ</mi><mo id="S6.Ex70.m1.10.10.1.1.1.4" stretchy="false" xref="S6.Ex70.m1.10.10.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Ex70.m1.10.10.5" rspace="0.111em" xref="S6.Ex70.m1.10.10.5.cmml">≅</mo><mrow id="S6.Ex70.m1.10.10.6" xref="S6.Ex70.m1.10.10.6.cmml"><munder id="S6.Ex70.m1.10.10.6.1" xref="S6.Ex70.m1.10.10.6.1.cmml"><mo id="S6.Ex70.m1.10.10.6.1.2" movablelimits="false" rspace="0em" xref="S6.Ex70.m1.10.10.6.1.2.cmml">∏</mo><mrow id="S6.Ex70.m1.10.10.6.1.3" xref="S6.Ex70.m1.10.10.6.1.3.cmml"><mrow id="S6.Ex70.m1.10.10.6.1.3.2" xref="S6.Ex70.m1.10.10.6.1.3.2.cmml"><mi id="S6.Ex70.m1.10.10.6.1.3.2.2" xref="S6.Ex70.m1.10.10.6.1.3.2.2.cmml">r</mi><mo id="S6.Ex70.m1.10.10.6.1.3.2.1" xref="S6.Ex70.m1.10.10.6.1.3.2.1.cmml">+</mo><mi id="S6.Ex70.m1.10.10.6.1.3.2.3" xref="S6.Ex70.m1.10.10.6.1.3.2.3.cmml">s</mi></mrow><mo id="S6.Ex70.m1.10.10.6.1.3.1" xref="S6.Ex70.m1.10.10.6.1.3.1.cmml">=</mo><mi id="S6.Ex70.m1.10.10.6.1.3.3" xref="S6.Ex70.m1.10.10.6.1.3.3.cmml">n</mi></mrow></munder><mrow id="S6.Ex70.m1.10.10.6.2" xref="S6.Ex70.m1.10.10.6.2.cmml"><munder id="S6.Ex70.m1.10.10.6.2.1" xref="S6.Ex70.m1.10.10.6.2.1.cmml"><mo id="S6.Ex70.m1.10.10.6.2.1.2" movablelimits="false" xref="S6.Ex70.m1.10.10.6.2.1.2.cmml">∏</mo><mrow id="S6.Ex70.m1.5.5.2" xref="S6.Ex70.m1.5.5.2.cmml"><mi id="S6.Ex70.m1.5.5.2.4" xref="S6.Ex70.m1.5.5.2.4.cmml">τ</mi><mo id="S6.Ex70.m1.5.5.2.3" xref="S6.Ex70.m1.5.5.2.3.cmml">∈</mo><msub id="S6.Ex70.m1.5.5.2.5" xref="S6.Ex70.m1.5.5.2.5.cmml"><mi id="S6.Ex70.m1.5.5.2.5.2" xref="S6.Ex70.m1.5.5.2.5.2.cmml">X</mi><mrow id="S6.Ex70.m1.5.5.2.2.2.4" 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class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem3.1.1.1">Proposition 6.3</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem3.p1"> <p class="ltx_p" id="S6.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem3.p1.5.5">Let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.1.1.m1.1"><semantics id="S6.Thmtheorem3.p1.1.1.m1.1a"><mi id="S6.Thmtheorem3.p1.1.1.m1.1.1" xref="S6.Thmtheorem3.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.1.1.m1.1b"><ci id="S6.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem3.p1.1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a bisimplicial set and <math alttext="J:\Delta(\mathrm{diag}X)\to\Delta(X)" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.2.2.m2.2"><semantics id="S6.Thmtheorem3.p1.2.2.m2.2a"><mrow id="S6.Thmtheorem3.p1.2.2.m2.2.2" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.cmml"><mi id="S6.Thmtheorem3.p1.2.2.m2.2.2.3" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.3.cmml">J</mi><mo id="S6.Thmtheorem3.p1.2.2.m2.2.2.2" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.2.cmml">:</mo><mrow id="S6.Thmtheorem3.p1.2.2.m2.2.2.1" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.cmml"><mrow id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.cmml"><mi id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.3" mathvariant="normal" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.3.cmml">Δ</mi><mo id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.2" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1" 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mathvariant="normal" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.2.cmml">Δ</mi><mo id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.1" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.3.2" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.cmml"><mo id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.3.2.1" stretchy="false" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.cmml">(</mo><mi id="S6.Thmtheorem3.p1.2.2.m2.1.1" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.cmml">X</mi><mo id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.3.2.2" stretchy="false" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.2.2.m2.2b"><apply id="S6.Thmtheorem3.p1.2.2.m2.2.2.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2"><ci id="S6.Thmtheorem3.p1.2.2.m2.2.2.2.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.2">:</ci><ci id="S6.Thmtheorem3.p1.2.2.m2.2.2.3.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.3">𝐽</ci><apply id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1"><ci id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.2.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.2">→</ci><apply id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1"><times id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.2.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.2"></times><ci id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.3.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.3">Δ</ci><apply id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1.1.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1"><times id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1.1.1"></times><ci id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1.1.2">diag</ci><ci id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.1.1.1.1.3">𝑋</ci></apply></apply><apply id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3"><times id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.1.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.1"></times><ci id="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.2.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.2.2.1.3.2">Δ</ci><ci id="S6.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.1.1">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.2.2.m2.2c">J:\Delta(\mathrm{diag}X)\to\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.2.2.m2.2d">italic_J : roman_Δ ( roman_diag italic_X ) → roman_Δ ( italic_X )</annotation></semantics></math> be the functor defined by the inclusion of simplices in the diagonal <math alttext="\mathrm{diag}X" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.3.3.m3.1"><semantics id="S6.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S6.Thmtheorem3.p1.3.3.m3.1.1" xref="S6.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi id="S6.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S6.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">diag</mi><mo id="S6.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S6.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">⁢</mo><mi id="S6.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S6.Thmtheorem3.p1.3.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.3.3.m3.1b"><apply id="S6.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem3.p1.3.3.m3.1.1"><times id="S6.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S6.Thmtheorem3.p1.3.3.m3.1.1.1"></times><ci id="S6.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S6.Thmtheorem3.p1.3.3.m3.1.1.2">diag</ci><ci id="S6.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S6.Thmtheorem3.p1.3.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.3.3.m3.1c">\mathrm{diag}X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.3.3.m3.1d">roman_diag italic_X</annotation></semantics></math>. For every coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.4.4.m4.1"><semantics id="S6.Thmtheorem3.p1.4.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem3.p1.4.4.m4.1.1" xref="S6.Thmtheorem3.p1.4.4.m4.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.4.4.m4.1b"><ci id="S6.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.4.4.m4.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.4.4.m4.1d">caligraphic_M</annotation></semantics></math> for <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.5.5.m5.1"><semantics id="S6.Thmtheorem3.p1.5.5.m5.1a"><mi id="S6.Thmtheorem3.p1.5.5.m5.1.1" xref="S6.Thmtheorem3.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.5.5.m5.1b"><ci id="S6.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.5.5.m5.1d">italic_X</annotation></semantics></math>, the induced homomorphism</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex71"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="J^{*}:H^{*}(X;\mathcal{M})\to H^{*}(\mathrm{diag}X;J^{*}\mathcal{M})" class="ltx_Math" display="block" id="S6.Ex71.m1.4"><semantics id="S6.Ex71.m1.4a"><mrow id="S6.Ex71.m1.4.4" xref="S6.Ex71.m1.4.4.cmml"><msup id="S6.Ex71.m1.4.4.4" xref="S6.Ex71.m1.4.4.4.cmml"><mi id="S6.Ex71.m1.4.4.4.2" xref="S6.Ex71.m1.4.4.4.2.cmml">J</mi><mo id="S6.Ex71.m1.4.4.4.3" xref="S6.Ex71.m1.4.4.4.3.cmml">∗</mo></msup><mo id="S6.Ex71.m1.4.4.3" lspace="0.278em" rspace="0.278em" xref="S6.Ex71.m1.4.4.3.cmml">:</mo><mrow id="S6.Ex71.m1.4.4.2" xref="S6.Ex71.m1.4.4.2.cmml"><mrow id="S6.Ex71.m1.4.4.2.4" xref="S6.Ex71.m1.4.4.2.4.cmml"><msup id="S6.Ex71.m1.4.4.2.4.2" xref="S6.Ex71.m1.4.4.2.4.2.cmml"><mi id="S6.Ex71.m1.4.4.2.4.2.2" xref="S6.Ex71.m1.4.4.2.4.2.2.cmml">H</mi><mo id="S6.Ex71.m1.4.4.2.4.2.3" xref="S6.Ex71.m1.4.4.2.4.2.3.cmml">∗</mo></msup><mo id="S6.Ex71.m1.4.4.2.4.1" xref="S6.Ex71.m1.4.4.2.4.1.cmml">⁢</mo><mrow id="S6.Ex71.m1.4.4.2.4.3.2" xref="S6.Ex71.m1.4.4.2.4.3.1.cmml"><mo id="S6.Ex71.m1.4.4.2.4.3.2.1" stretchy="false" xref="S6.Ex71.m1.4.4.2.4.3.1.cmml">(</mo><mi id="S6.Ex71.m1.1.1" xref="S6.Ex71.m1.1.1.cmml">X</mi><mo id="S6.Ex71.m1.4.4.2.4.3.2.2" xref="S6.Ex71.m1.4.4.2.4.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex71.m1.2.2" xref="S6.Ex71.m1.2.2.cmml">ℳ</mi><mo id="S6.Ex71.m1.4.4.2.4.3.2.3" stretchy="false" xref="S6.Ex71.m1.4.4.2.4.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex71.m1.4.4.2.3" stretchy="false" xref="S6.Ex71.m1.4.4.2.3.cmml">→</mo><mrow id="S6.Ex71.m1.4.4.2.2" xref="S6.Ex71.m1.4.4.2.2.cmml"><msup id="S6.Ex71.m1.4.4.2.2.4" xref="S6.Ex71.m1.4.4.2.2.4.cmml"><mi id="S6.Ex71.m1.4.4.2.2.4.2" xref="S6.Ex71.m1.4.4.2.2.4.2.cmml">H</mi><mo id="S6.Ex71.m1.4.4.2.2.4.3" xref="S6.Ex71.m1.4.4.2.2.4.3.cmml">∗</mo></msup><mo id="S6.Ex71.m1.4.4.2.2.3" xref="S6.Ex71.m1.4.4.2.2.3.cmml">⁢</mo><mrow id="S6.Ex71.m1.4.4.2.2.2.2" xref="S6.Ex71.m1.4.4.2.2.2.3.cmml"><mo id="S6.Ex71.m1.4.4.2.2.2.2.3" stretchy="false" xref="S6.Ex71.m1.4.4.2.2.2.3.cmml">(</mo><mrow id="S6.Ex71.m1.3.3.1.1.1.1.1" xref="S6.Ex71.m1.3.3.1.1.1.1.1.cmml"><mi id="S6.Ex71.m1.3.3.1.1.1.1.1.2" xref="S6.Ex71.m1.3.3.1.1.1.1.1.2.cmml">diag</mi><mo id="S6.Ex71.m1.3.3.1.1.1.1.1.1" xref="S6.Ex71.m1.3.3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S6.Ex71.m1.3.3.1.1.1.1.1.3" xref="S6.Ex71.m1.3.3.1.1.1.1.1.3.cmml">X</mi></mrow><mo id="S6.Ex71.m1.4.4.2.2.2.2.4" xref="S6.Ex71.m1.4.4.2.2.2.3.cmml">;</mo><mrow id="S6.Ex71.m1.4.4.2.2.2.2.2" xref="S6.Ex71.m1.4.4.2.2.2.2.2.cmml"><msup id="S6.Ex71.m1.4.4.2.2.2.2.2.2" xref="S6.Ex71.m1.4.4.2.2.2.2.2.2.cmml"><mi id="S6.Ex71.m1.4.4.2.2.2.2.2.2.2" xref="S6.Ex71.m1.4.4.2.2.2.2.2.2.2.cmml">J</mi><mo id="S6.Ex71.m1.4.4.2.2.2.2.2.2.3" xref="S6.Ex71.m1.4.4.2.2.2.2.2.2.3.cmml">∗</mo></msup><mo id="S6.Ex71.m1.4.4.2.2.2.2.2.1" xref="S6.Ex71.m1.4.4.2.2.2.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex71.m1.4.4.2.2.2.2.2.3" xref="S6.Ex71.m1.4.4.2.2.2.2.2.3.cmml">ℳ</mi></mrow><mo id="S6.Ex71.m1.4.4.2.2.2.2.5" stretchy="false" xref="S6.Ex71.m1.4.4.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex71.m1.4b"><apply id="S6.Ex71.m1.4.4.cmml" xref="S6.Ex71.m1.4.4"><ci id="S6.Ex71.m1.4.4.3.cmml" xref="S6.Ex71.m1.4.4.3">:</ci><apply id="S6.Ex71.m1.4.4.4.cmml" xref="S6.Ex71.m1.4.4.4"><csymbol cd="ambiguous" id="S6.Ex71.m1.4.4.4.1.cmml" xref="S6.Ex71.m1.4.4.4">superscript</csymbol><ci id="S6.Ex71.m1.4.4.4.2.cmml" xref="S6.Ex71.m1.4.4.4.2">𝐽</ci><times id="S6.Ex71.m1.4.4.4.3.cmml" xref="S6.Ex71.m1.4.4.4.3"></times></apply><apply id="S6.Ex71.m1.4.4.2.cmml" xref="S6.Ex71.m1.4.4.2"><ci id="S6.Ex71.m1.4.4.2.3.cmml" xref="S6.Ex71.m1.4.4.2.3">→</ci><apply id="S6.Ex71.m1.4.4.2.4.cmml" xref="S6.Ex71.m1.4.4.2.4"><times id="S6.Ex71.m1.4.4.2.4.1.cmml" xref="S6.Ex71.m1.4.4.2.4.1"></times><apply id="S6.Ex71.m1.4.4.2.4.2.cmml" xref="S6.Ex71.m1.4.4.2.4.2"><csymbol cd="ambiguous" id="S6.Ex71.m1.4.4.2.4.2.1.cmml" xref="S6.Ex71.m1.4.4.2.4.2">superscript</csymbol><ci id="S6.Ex71.m1.4.4.2.4.2.2.cmml" xref="S6.Ex71.m1.4.4.2.4.2.2">𝐻</ci><times id="S6.Ex71.m1.4.4.2.4.2.3.cmml" xref="S6.Ex71.m1.4.4.2.4.2.3"></times></apply><list id="S6.Ex71.m1.4.4.2.4.3.1.cmml" xref="S6.Ex71.m1.4.4.2.4.3.2"><ci id="S6.Ex71.m1.1.1.cmml" xref="S6.Ex71.m1.1.1">𝑋</ci><ci id="S6.Ex71.m1.2.2.cmml" xref="S6.Ex71.m1.2.2">ℳ</ci></list></apply><apply id="S6.Ex71.m1.4.4.2.2.cmml" xref="S6.Ex71.m1.4.4.2.2"><times id="S6.Ex71.m1.4.4.2.2.3.cmml" xref="S6.Ex71.m1.4.4.2.2.3"></times><apply id="S6.Ex71.m1.4.4.2.2.4.cmml" xref="S6.Ex71.m1.4.4.2.2.4"><csymbol cd="ambiguous" id="S6.Ex71.m1.4.4.2.2.4.1.cmml" xref="S6.Ex71.m1.4.4.2.2.4">superscript</csymbol><ci id="S6.Ex71.m1.4.4.2.2.4.2.cmml" xref="S6.Ex71.m1.4.4.2.2.4.2">𝐻</ci><times id="S6.Ex71.m1.4.4.2.2.4.3.cmml" xref="S6.Ex71.m1.4.4.2.2.4.3"></times></apply><list id="S6.Ex71.m1.4.4.2.2.2.3.cmml" xref="S6.Ex71.m1.4.4.2.2.2.2"><apply id="S6.Ex71.m1.3.3.1.1.1.1.1.cmml" xref="S6.Ex71.m1.3.3.1.1.1.1.1"><times id="S6.Ex71.m1.3.3.1.1.1.1.1.1.cmml" xref="S6.Ex71.m1.3.3.1.1.1.1.1.1"></times><ci id="S6.Ex71.m1.3.3.1.1.1.1.1.2.cmml" xref="S6.Ex71.m1.3.3.1.1.1.1.1.2">diag</ci><ci id="S6.Ex71.m1.3.3.1.1.1.1.1.3.cmml" xref="S6.Ex71.m1.3.3.1.1.1.1.1.3">𝑋</ci></apply><apply id="S6.Ex71.m1.4.4.2.2.2.2.2.cmml" xref="S6.Ex71.m1.4.4.2.2.2.2.2"><times id="S6.Ex71.m1.4.4.2.2.2.2.2.1.cmml" xref="S6.Ex71.m1.4.4.2.2.2.2.2.1"></times><apply id="S6.Ex71.m1.4.4.2.2.2.2.2.2.cmml" xref="S6.Ex71.m1.4.4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex71.m1.4.4.2.2.2.2.2.2.1.cmml" xref="S6.Ex71.m1.4.4.2.2.2.2.2.2">superscript</csymbol><ci id="S6.Ex71.m1.4.4.2.2.2.2.2.2.2.cmml" xref="S6.Ex71.m1.4.4.2.2.2.2.2.2.2">𝐽</ci><times id="S6.Ex71.m1.4.4.2.2.2.2.2.2.3.cmml" xref="S6.Ex71.m1.4.4.2.2.2.2.2.2.3"></times></apply><ci id="S6.Ex71.m1.4.4.2.2.2.2.2.3.cmml" xref="S6.Ex71.m1.4.4.2.2.2.2.2.3">ℳ</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex71.m1.4c">J^{*}:H^{*}(X;\mathcal{M})\to H^{*}(\mathrm{diag}X;J^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S6.Ex71.m1.4d">italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M ) → italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_diag italic_X ; italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmtheorem3.p1.6"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem3.p1.6.1">is an isomorphism.</span></p> </div> </div> <div class="ltx_proof" id="S6.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.2.p1"> <p class="ltx_p" id="S6.2.p1.7">By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S6.Thmtheorem2" title="Lemma 6.2. ‣ 6. Cohomology of bisimplicial sets with nontrivial coefficients ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">6.2</span></a>, <math alttext="\mathrm{Tot}(P_{*,*})\to\underline{R}" class="ltx_Math" display="inline" id="S6.2.p1.1.m1.3"><semantics id="S6.2.p1.1.m1.3a"><mrow id="S6.2.p1.1.m1.3.3" xref="S6.2.p1.1.m1.3.3.cmml"><mrow id="S6.2.p1.1.m1.3.3.1" xref="S6.2.p1.1.m1.3.3.1.cmml"><mi id="S6.2.p1.1.m1.3.3.1.3" xref="S6.2.p1.1.m1.3.3.1.3.cmml">Tot</mi><mo id="S6.2.p1.1.m1.3.3.1.2" xref="S6.2.p1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S6.2.p1.1.m1.3.3.1.1.1" xref="S6.2.p1.1.m1.3.3.1.1.1.1.cmml"><mo id="S6.2.p1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S6.2.p1.1.m1.3.3.1.1.1.1.cmml">(</mo><msub id="S6.2.p1.1.m1.3.3.1.1.1.1" xref="S6.2.p1.1.m1.3.3.1.1.1.1.cmml"><mi id="S6.2.p1.1.m1.3.3.1.1.1.1.2" xref="S6.2.p1.1.m1.3.3.1.1.1.1.2.cmml">P</mi><mrow id="S6.2.p1.1.m1.2.2.2.4" xref="S6.2.p1.1.m1.2.2.2.3.cmml"><mo id="S6.2.p1.1.m1.1.1.1.1" rspace="0em" xref="S6.2.p1.1.m1.1.1.1.1.cmml">∗</mo><mo id="S6.2.p1.1.m1.2.2.2.4.1" rspace="0em" xref="S6.2.p1.1.m1.2.2.2.3.cmml">,</mo><mo id="S6.2.p1.1.m1.2.2.2.2" lspace="0em" xref="S6.2.p1.1.m1.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S6.2.p1.1.m1.3.3.1.1.1.3" stretchy="false" xref="S6.2.p1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.2.p1.1.m1.3.3.2" stretchy="false" xref="S6.2.p1.1.m1.3.3.2.cmml">→</mo><munder accentunder="true" id="S6.2.p1.1.m1.3.3.3" xref="S6.2.p1.1.m1.3.3.3.cmml"><mi id="S6.2.p1.1.m1.3.3.3.2" xref="S6.2.p1.1.m1.3.3.3.2.cmml">R</mi><mo id="S6.2.p1.1.m1.3.3.3.1" xref="S6.2.p1.1.m1.3.3.3.1.cmml">¯</mo></munder></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.1.m1.3b"><apply id="S6.2.p1.1.m1.3.3.cmml" xref="S6.2.p1.1.m1.3.3"><ci id="S6.2.p1.1.m1.3.3.2.cmml" xref="S6.2.p1.1.m1.3.3.2">→</ci><apply id="S6.2.p1.1.m1.3.3.1.cmml" xref="S6.2.p1.1.m1.3.3.1"><times id="S6.2.p1.1.m1.3.3.1.2.cmml" xref="S6.2.p1.1.m1.3.3.1.2"></times><ci id="S6.2.p1.1.m1.3.3.1.3.cmml" xref="S6.2.p1.1.m1.3.3.1.3">Tot</ci><apply id="S6.2.p1.1.m1.3.3.1.1.1.1.cmml" xref="S6.2.p1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.2.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="S6.2.p1.1.m1.3.3.1.1.1">subscript</csymbol><ci id="S6.2.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S6.2.p1.1.m1.3.3.1.1.1.1.2">𝑃</ci><list id="S6.2.p1.1.m1.2.2.2.3.cmml" xref="S6.2.p1.1.m1.2.2.2.4"><times id="S6.2.p1.1.m1.1.1.1.1.cmml" xref="S6.2.p1.1.m1.1.1.1.1"></times><times id="S6.2.p1.1.m1.2.2.2.2.cmml" xref="S6.2.p1.1.m1.2.2.2.2"></times></list></apply></apply><apply id="S6.2.p1.1.m1.3.3.3.cmml" xref="S6.2.p1.1.m1.3.3.3"><ci id="S6.2.p1.1.m1.3.3.3.1.cmml" xref="S6.2.p1.1.m1.3.3.3.1">¯</ci><ci id="S6.2.p1.1.m1.3.3.3.2.cmml" xref="S6.2.p1.1.m1.3.3.3.2">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.1.m1.3c">\mathrm{Tot}(P_{*,*})\to\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.1.m1.3d">roman_Tot ( italic_P start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT ) → under¯ start_ARG italic_R end_ARG</annotation></semantics></math> is a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S6.2.p1.2.m2.1"><semantics id="S6.2.p1.2.m2.1a"><munder accentunder="true" id="S6.2.p1.2.m2.1.1" xref="S6.2.p1.2.m2.1.1.cmml"><mi id="S6.2.p1.2.m2.1.1.2" xref="S6.2.p1.2.m2.1.1.2.cmml">R</mi><mo id="S6.2.p1.2.m2.1.1.1" xref="S6.2.p1.2.m2.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S6.2.p1.2.m2.1b"><apply id="S6.2.p1.2.m2.1.1.cmml" xref="S6.2.p1.2.m2.1.1"><ci id="S6.2.p1.2.m2.1.1.1.cmml" xref="S6.2.p1.2.m2.1.1.1">¯</ci><ci id="S6.2.p1.2.m2.1.1.2.cmml" xref="S6.2.p1.2.m2.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.2.m2.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.2.m2.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S6.2.p1.3.m3.1"><semantics id="S6.2.p1.3.m3.1a"><mrow id="S6.2.p1.3.m3.1.2" xref="S6.2.p1.3.m3.1.2.cmml"><mi id="S6.2.p1.3.m3.1.2.2" xref="S6.2.p1.3.m3.1.2.2.cmml">R</mi><mo id="S6.2.p1.3.m3.1.2.1" xref="S6.2.p1.3.m3.1.2.1.cmml">⁢</mo><mi id="S6.2.p1.3.m3.1.2.3" mathvariant="normal" xref="S6.2.p1.3.m3.1.2.3.cmml">Δ</mi><mo id="S6.2.p1.3.m3.1.2.1a" xref="S6.2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.2.p1.3.m3.1.2.4.2" xref="S6.2.p1.3.m3.1.2.cmml"><mo id="S6.2.p1.3.m3.1.2.4.2.1" stretchy="false" xref="S6.2.p1.3.m3.1.2.cmml">(</mo><mi id="S6.2.p1.3.m3.1.1" xref="S6.2.p1.3.m3.1.1.cmml">X</mi><mo id="S6.2.p1.3.m3.1.2.4.2.2" stretchy="false" xref="S6.2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.3.m3.1b"><apply id="S6.2.p1.3.m3.1.2.cmml" xref="S6.2.p1.3.m3.1.2"><times id="S6.2.p1.3.m3.1.2.1.cmml" xref="S6.2.p1.3.m3.1.2.1"></times><ci id="S6.2.p1.3.m3.1.2.2.cmml" xref="S6.2.p1.3.m3.1.2.2">𝑅</ci><ci id="S6.2.p1.3.m3.1.2.3.cmml" xref="S6.2.p1.3.m3.1.2.3">Δ</ci><ci id="S6.2.p1.3.m3.1.1.cmml" xref="S6.2.p1.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.3.m3.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.3.m3.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S4.Thmtheorem3" title="Theorem 4.3 (Dold-Puppe Theorem [4]). ‣ 4.2. The Dold-Puppe Theorem ‣ 4. Bisimplicial objects and the Dold-Puppe theorem ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">4.3</span></a>, we see that <math alttext="(\mathrm{diag}\mathbb{P})_{*}\to\underline{R}" class="ltx_Math" display="inline" id="S6.2.p1.4.m4.1"><semantics id="S6.2.p1.4.m4.1a"><mrow id="S6.2.p1.4.m4.1.1" xref="S6.2.p1.4.m4.1.1.cmml"><msub id="S6.2.p1.4.m4.1.1.1" xref="S6.2.p1.4.m4.1.1.1.cmml"><mrow id="S6.2.p1.4.m4.1.1.1.1.1" xref="S6.2.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S6.2.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S6.2.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.2.p1.4.m4.1.1.1.1.1.1" xref="S6.2.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S6.2.p1.4.m4.1.1.1.1.1.1.2" xref="S6.2.p1.4.m4.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S6.2.p1.4.m4.1.1.1.1.1.1.1" xref="S6.2.p1.4.m4.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S6.2.p1.4.m4.1.1.1.1.1.1.3" xref="S6.2.p1.4.m4.1.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S6.2.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S6.2.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.2.p1.4.m4.1.1.1.3" xref="S6.2.p1.4.m4.1.1.1.3.cmml">∗</mo></msub><mo id="S6.2.p1.4.m4.1.1.2" stretchy="false" xref="S6.2.p1.4.m4.1.1.2.cmml">→</mo><munder accentunder="true" id="S6.2.p1.4.m4.1.1.3" xref="S6.2.p1.4.m4.1.1.3.cmml"><mi id="S6.2.p1.4.m4.1.1.3.2" xref="S6.2.p1.4.m4.1.1.3.2.cmml">R</mi><mo id="S6.2.p1.4.m4.1.1.3.1" xref="S6.2.p1.4.m4.1.1.3.1.cmml">¯</mo></munder></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.4.m4.1b"><apply id="S6.2.p1.4.m4.1.1.cmml" xref="S6.2.p1.4.m4.1.1"><ci id="S6.2.p1.4.m4.1.1.2.cmml" xref="S6.2.p1.4.m4.1.1.2">→</ci><apply id="S6.2.p1.4.m4.1.1.1.cmml" xref="S6.2.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="S6.2.p1.4.m4.1.1.1.2.cmml" xref="S6.2.p1.4.m4.1.1.1">subscript</csymbol><apply id="S6.2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S6.2.p1.4.m4.1.1.1.1.1"><times id="S6.2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.2.p1.4.m4.1.1.1.1.1.1.1"></times><ci id="S6.2.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S6.2.p1.4.m4.1.1.1.1.1.1.2">diag</ci><ci id="S6.2.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S6.2.p1.4.m4.1.1.1.1.1.1.3">ℙ</ci></apply><times id="S6.2.p1.4.m4.1.1.1.3.cmml" xref="S6.2.p1.4.m4.1.1.1.3"></times></apply><apply id="S6.2.p1.4.m4.1.1.3.cmml" xref="S6.2.p1.4.m4.1.1.3"><ci id="S6.2.p1.4.m4.1.1.3.1.cmml" xref="S6.2.p1.4.m4.1.1.3.1">¯</ci><ci id="S6.2.p1.4.m4.1.1.3.2.cmml" xref="S6.2.p1.4.m4.1.1.3.2">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.4.m4.1c">(\mathrm{diag}\mathbb{P})_{*}\to\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.4.m4.1d">( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → under¯ start_ARG italic_R end_ARG</annotation></semantics></math> is a also a projective resolution of <math alttext="\underline{R}" class="ltx_Math" display="inline" id="S6.2.p1.5.m5.1"><semantics id="S6.2.p1.5.m5.1a"><munder accentunder="true" id="S6.2.p1.5.m5.1.1" xref="S6.2.p1.5.m5.1.1.cmml"><mi id="S6.2.p1.5.m5.1.1.2" xref="S6.2.p1.5.m5.1.1.2.cmml">R</mi><mo id="S6.2.p1.5.m5.1.1.1" xref="S6.2.p1.5.m5.1.1.1.cmml">¯</mo></munder><annotation-xml encoding="MathML-Content" id="S6.2.p1.5.m5.1b"><apply id="S6.2.p1.5.m5.1.1.cmml" xref="S6.2.p1.5.m5.1.1"><ci id="S6.2.p1.5.m5.1.1.1.cmml" xref="S6.2.p1.5.m5.1.1.1">¯</ci><ci id="S6.2.p1.5.m5.1.1.2.cmml" xref="S6.2.p1.5.m5.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.5.m5.1c">\underline{R}</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.5.m5.1d">under¯ start_ARG italic_R end_ARG</annotation></semantics></math> as an <math alttext="R\Delta(X)" class="ltx_Math" display="inline" id="S6.2.p1.6.m6.1"><semantics id="S6.2.p1.6.m6.1a"><mrow id="S6.2.p1.6.m6.1.2" xref="S6.2.p1.6.m6.1.2.cmml"><mi id="S6.2.p1.6.m6.1.2.2" xref="S6.2.p1.6.m6.1.2.2.cmml">R</mi><mo id="S6.2.p1.6.m6.1.2.1" xref="S6.2.p1.6.m6.1.2.1.cmml">⁢</mo><mi id="S6.2.p1.6.m6.1.2.3" mathvariant="normal" xref="S6.2.p1.6.m6.1.2.3.cmml">Δ</mi><mo id="S6.2.p1.6.m6.1.2.1a" xref="S6.2.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S6.2.p1.6.m6.1.2.4.2" xref="S6.2.p1.6.m6.1.2.cmml"><mo id="S6.2.p1.6.m6.1.2.4.2.1" stretchy="false" xref="S6.2.p1.6.m6.1.2.cmml">(</mo><mi id="S6.2.p1.6.m6.1.1" xref="S6.2.p1.6.m6.1.1.cmml">X</mi><mo id="S6.2.p1.6.m6.1.2.4.2.2" stretchy="false" xref="S6.2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.6.m6.1b"><apply id="S6.2.p1.6.m6.1.2.cmml" xref="S6.2.p1.6.m6.1.2"><times id="S6.2.p1.6.m6.1.2.1.cmml" xref="S6.2.p1.6.m6.1.2.1"></times><ci id="S6.2.p1.6.m6.1.2.2.cmml" xref="S6.2.p1.6.m6.1.2.2">𝑅</ci><ci id="S6.2.p1.6.m6.1.2.3.cmml" xref="S6.2.p1.6.m6.1.2.3">Δ</ci><ci id="S6.2.p1.6.m6.1.1.cmml" xref="S6.2.p1.6.m6.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.6.m6.1c">R\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.6.m6.1d">italic_R roman_Δ ( italic_X )</annotation></semantics></math>-module, hence the chain map defined by the inclusion of diagonal modules induces a chain homotopy equivalence <math alttext="j_{*}:(\mathrm{diag}\mathbb{P})_{*}\to\mathrm{Tot}(\mathbb{P}_{*,*})" class="ltx_Math" display="inline" id="S6.2.p1.7.m7.4"><semantics id="S6.2.p1.7.m7.4a"><mrow id="S6.2.p1.7.m7.4.4" xref="S6.2.p1.7.m7.4.4.cmml"><msub id="S6.2.p1.7.m7.4.4.4" xref="S6.2.p1.7.m7.4.4.4.cmml"><mi id="S6.2.p1.7.m7.4.4.4.2" xref="S6.2.p1.7.m7.4.4.4.2.cmml">j</mi><mo id="S6.2.p1.7.m7.4.4.4.3" xref="S6.2.p1.7.m7.4.4.4.3.cmml">∗</mo></msub><mo id="S6.2.p1.7.m7.4.4.3" lspace="0.278em" rspace="0.278em" xref="S6.2.p1.7.m7.4.4.3.cmml">:</mo><mrow id="S6.2.p1.7.m7.4.4.2" xref="S6.2.p1.7.m7.4.4.2.cmml"><msub id="S6.2.p1.7.m7.3.3.1.1" xref="S6.2.p1.7.m7.3.3.1.1.cmml"><mrow id="S6.2.p1.7.m7.3.3.1.1.1.1" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.cmml"><mo id="S6.2.p1.7.m7.3.3.1.1.1.1.2" stretchy="false" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S6.2.p1.7.m7.3.3.1.1.1.1.1" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.cmml"><mi id="S6.2.p1.7.m7.3.3.1.1.1.1.1.2" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.2.cmml">diag</mi><mo id="S6.2.p1.7.m7.3.3.1.1.1.1.1.1" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S6.2.p1.7.m7.3.3.1.1.1.1.1.3" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S6.2.p1.7.m7.3.3.1.1.1.1.3" stretchy="false" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.2.p1.7.m7.3.3.1.1.3" xref="S6.2.p1.7.m7.3.3.1.1.3.cmml">∗</mo></msub><mo id="S6.2.p1.7.m7.4.4.2.3" stretchy="false" xref="S6.2.p1.7.m7.4.4.2.3.cmml">→</mo><mrow id="S6.2.p1.7.m7.4.4.2.2" xref="S6.2.p1.7.m7.4.4.2.2.cmml"><mi id="S6.2.p1.7.m7.4.4.2.2.3" xref="S6.2.p1.7.m7.4.4.2.2.3.cmml">Tot</mi><mo id="S6.2.p1.7.m7.4.4.2.2.2" xref="S6.2.p1.7.m7.4.4.2.2.2.cmml">⁢</mo><mrow id="S6.2.p1.7.m7.4.4.2.2.1.1" xref="S6.2.p1.7.m7.4.4.2.2.1.1.1.cmml"><mo id="S6.2.p1.7.m7.4.4.2.2.1.1.2" stretchy="false" xref="S6.2.p1.7.m7.4.4.2.2.1.1.1.cmml">(</mo><msub id="S6.2.p1.7.m7.4.4.2.2.1.1.1" xref="S6.2.p1.7.m7.4.4.2.2.1.1.1.cmml"><mi id="S6.2.p1.7.m7.4.4.2.2.1.1.1.2" xref="S6.2.p1.7.m7.4.4.2.2.1.1.1.2.cmml">ℙ</mi><mrow id="S6.2.p1.7.m7.2.2.2.4" xref="S6.2.p1.7.m7.2.2.2.3.cmml"><mo id="S6.2.p1.7.m7.1.1.1.1" rspace="0em" xref="S6.2.p1.7.m7.1.1.1.1.cmml">∗</mo><mo id="S6.2.p1.7.m7.2.2.2.4.1" rspace="0em" xref="S6.2.p1.7.m7.2.2.2.3.cmml">,</mo><mo id="S6.2.p1.7.m7.2.2.2.2" lspace="0em" xref="S6.2.p1.7.m7.2.2.2.2.cmml">∗</mo></mrow></msub><mo id="S6.2.p1.7.m7.4.4.2.2.1.1.3" stretchy="false" xref="S6.2.p1.7.m7.4.4.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.7.m7.4b"><apply id="S6.2.p1.7.m7.4.4.cmml" xref="S6.2.p1.7.m7.4.4"><ci id="S6.2.p1.7.m7.4.4.3.cmml" xref="S6.2.p1.7.m7.4.4.3">:</ci><apply id="S6.2.p1.7.m7.4.4.4.cmml" xref="S6.2.p1.7.m7.4.4.4"><csymbol cd="ambiguous" id="S6.2.p1.7.m7.4.4.4.1.cmml" xref="S6.2.p1.7.m7.4.4.4">subscript</csymbol><ci id="S6.2.p1.7.m7.4.4.4.2.cmml" xref="S6.2.p1.7.m7.4.4.4.2">𝑗</ci><times id="S6.2.p1.7.m7.4.4.4.3.cmml" xref="S6.2.p1.7.m7.4.4.4.3"></times></apply><apply id="S6.2.p1.7.m7.4.4.2.cmml" xref="S6.2.p1.7.m7.4.4.2"><ci id="S6.2.p1.7.m7.4.4.2.3.cmml" xref="S6.2.p1.7.m7.4.4.2.3">→</ci><apply id="S6.2.p1.7.m7.3.3.1.1.cmml" xref="S6.2.p1.7.m7.3.3.1.1"><csymbol cd="ambiguous" id="S6.2.p1.7.m7.3.3.1.1.2.cmml" xref="S6.2.p1.7.m7.3.3.1.1">subscript</csymbol><apply id="S6.2.p1.7.m7.3.3.1.1.1.1.1.cmml" xref="S6.2.p1.7.m7.3.3.1.1.1.1"><times id="S6.2.p1.7.m7.3.3.1.1.1.1.1.1.cmml" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.1"></times><ci id="S6.2.p1.7.m7.3.3.1.1.1.1.1.2.cmml" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.2">diag</ci><ci id="S6.2.p1.7.m7.3.3.1.1.1.1.1.3.cmml" xref="S6.2.p1.7.m7.3.3.1.1.1.1.1.3">ℙ</ci></apply><times id="S6.2.p1.7.m7.3.3.1.1.3.cmml" xref="S6.2.p1.7.m7.3.3.1.1.3"></times></apply><apply id="S6.2.p1.7.m7.4.4.2.2.cmml" xref="S6.2.p1.7.m7.4.4.2.2"><times id="S6.2.p1.7.m7.4.4.2.2.2.cmml" xref="S6.2.p1.7.m7.4.4.2.2.2"></times><ci id="S6.2.p1.7.m7.4.4.2.2.3.cmml" xref="S6.2.p1.7.m7.4.4.2.2.3">Tot</ci><apply id="S6.2.p1.7.m7.4.4.2.2.1.1.1.cmml" xref="S6.2.p1.7.m7.4.4.2.2.1.1"><csymbol cd="ambiguous" id="S6.2.p1.7.m7.4.4.2.2.1.1.1.1.cmml" xref="S6.2.p1.7.m7.4.4.2.2.1.1">subscript</csymbol><ci id="S6.2.p1.7.m7.4.4.2.2.1.1.1.2.cmml" xref="S6.2.p1.7.m7.4.4.2.2.1.1.1.2">ℙ</ci><list id="S6.2.p1.7.m7.2.2.2.3.cmml" xref="S6.2.p1.7.m7.2.2.2.4"><times id="S6.2.p1.7.m7.1.1.1.1.cmml" xref="S6.2.p1.7.m7.1.1.1.1"></times><times id="S6.2.p1.7.m7.2.2.2.2.cmml" xref="S6.2.p1.7.m7.2.2.2.2"></times></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.7.m7.4c">j_{*}:(\mathrm{diag}\mathbb{P})_{*}\to\mathrm{Tot}(\mathbb{P}_{*,*})</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.7.m7.4d">italic_j start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : ( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT → roman_Tot ( blackboard_P start_POSTSUBSCRIPT ∗ , ∗ end_POSTSUBSCRIPT )</annotation></semantics></math>. We have</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex72"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathrm{Hom}_{\Delta(X)}((\mathrm{diag}\mathbb{P})_{n},\mathcal{M})\cong\prod_% {x\in X_{n,n}}\mathcal{M}(x)\cong C^{n}(\mathrm{diag}X;J^{*}\mathcal{M})" class="ltx_Math" display="block" id="S6.Ex72.m1.8"><semantics id="S6.Ex72.m1.8a"><mrow id="S6.Ex72.m1.8.8" xref="S6.Ex72.m1.8.8.cmml"><mrow id="S6.Ex72.m1.6.6.1" xref="S6.Ex72.m1.6.6.1.cmml"><msub id="S6.Ex72.m1.6.6.1.3" xref="S6.Ex72.m1.6.6.1.3.cmml"><mi id="S6.Ex72.m1.6.6.1.3.2" xref="S6.Ex72.m1.6.6.1.3.2.cmml">Hom</mi><mrow id="S6.Ex72.m1.1.1.1" xref="S6.Ex72.m1.1.1.1.cmml"><mi id="S6.Ex72.m1.1.1.1.3" mathvariant="normal" xref="S6.Ex72.m1.1.1.1.3.cmml">Δ</mi><mo id="S6.Ex72.m1.1.1.1.2" xref="S6.Ex72.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex72.m1.1.1.1.4.2" xref="S6.Ex72.m1.1.1.1.cmml"><mo id="S6.Ex72.m1.1.1.1.4.2.1" stretchy="false" xref="S6.Ex72.m1.1.1.1.cmml">(</mo><mi id="S6.Ex72.m1.1.1.1.1" xref="S6.Ex72.m1.1.1.1.1.cmml">X</mi><mo id="S6.Ex72.m1.1.1.1.4.2.2" stretchy="false" xref="S6.Ex72.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Ex72.m1.6.6.1.2" xref="S6.Ex72.m1.6.6.1.2.cmml">⁢</mo><mrow id="S6.Ex72.m1.6.6.1.1.1" xref="S6.Ex72.m1.6.6.1.1.2.cmml"><mo id="S6.Ex72.m1.6.6.1.1.1.2" stretchy="false" xref="S6.Ex72.m1.6.6.1.1.2.cmml">(</mo><msub id="S6.Ex72.m1.6.6.1.1.1.1" xref="S6.Ex72.m1.6.6.1.1.1.1.cmml"><mrow id="S6.Ex72.m1.6.6.1.1.1.1.1.1" xref="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex72.m1.6.6.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex72.m1.6.6.1.1.1.1.1.1.1" xref="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.2" xref="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.2.cmml">diag</mi><mo id="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.1" xref="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.3" xref="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.3.cmml">ℙ</mi></mrow><mo id="S6.Ex72.m1.6.6.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex72.m1.6.6.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S6.Ex72.m1.6.6.1.1.1.1.3" xref="S6.Ex72.m1.6.6.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.Ex72.m1.6.6.1.1.1.3" xref="S6.Ex72.m1.6.6.1.1.2.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex72.m1.4.4" xref="S6.Ex72.m1.4.4.cmml">ℳ</mi><mo id="S6.Ex72.m1.6.6.1.1.1.4" stretchy="false" xref="S6.Ex72.m1.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Ex72.m1.8.8.5" rspace="0.111em" xref="S6.Ex72.m1.8.8.5.cmml">≅</mo><mrow id="S6.Ex72.m1.8.8.6" xref="S6.Ex72.m1.8.8.6.cmml"><munder id="S6.Ex72.m1.8.8.6.1" xref="S6.Ex72.m1.8.8.6.1.cmml"><mo id="S6.Ex72.m1.8.8.6.1.2" movablelimits="false" xref="S6.Ex72.m1.8.8.6.1.2.cmml">∏</mo><mrow id="S6.Ex72.m1.3.3.2" xref="S6.Ex72.m1.3.3.2.cmml"><mi id="S6.Ex72.m1.3.3.2.4" xref="S6.Ex72.m1.3.3.2.4.cmml">x</mi><mo id="S6.Ex72.m1.3.3.2.3" xref="S6.Ex72.m1.3.3.2.3.cmml">∈</mo><msub id="S6.Ex72.m1.3.3.2.5" xref="S6.Ex72.m1.3.3.2.5.cmml"><mi id="S6.Ex72.m1.3.3.2.5.2" xref="S6.Ex72.m1.3.3.2.5.2.cmml">X</mi><mrow id="S6.Ex72.m1.3.3.2.2.2.4" xref="S6.Ex72.m1.3.3.2.2.2.3.cmml"><mi id="S6.Ex72.m1.2.2.1.1.1.1" xref="S6.Ex72.m1.2.2.1.1.1.1.cmml">n</mi><mo id="S6.Ex72.m1.3.3.2.2.2.4.1" xref="S6.Ex72.m1.3.3.2.2.2.3.cmml">,</mo><mi id="S6.Ex72.m1.3.3.2.2.2.2" xref="S6.Ex72.m1.3.3.2.2.2.2.cmml">n</mi></mrow></msub></mrow></munder><mrow id="S6.Ex72.m1.8.8.6.2" xref="S6.Ex72.m1.8.8.6.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex72.m1.8.8.6.2.2" xref="S6.Ex72.m1.8.8.6.2.2.cmml">ℳ</mi><mo id="S6.Ex72.m1.8.8.6.2.1" xref="S6.Ex72.m1.8.8.6.2.1.cmml">⁢</mo><mrow id="S6.Ex72.m1.8.8.6.2.3.2" xref="S6.Ex72.m1.8.8.6.2.cmml"><mo id="S6.Ex72.m1.8.8.6.2.3.2.1" stretchy="false" xref="S6.Ex72.m1.8.8.6.2.cmml">(</mo><mi id="S6.Ex72.m1.5.5" xref="S6.Ex72.m1.5.5.cmml">x</mi><mo id="S6.Ex72.m1.8.8.6.2.3.2.2" stretchy="false" 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id="S6.Ex72.m1.8.8.3.2.2.2" xref="S6.Ex72.m1.8.8.3.2.2.2.cmml"><msup id="S6.Ex72.m1.8.8.3.2.2.2.2" xref="S6.Ex72.m1.8.8.3.2.2.2.2.cmml"><mi id="S6.Ex72.m1.8.8.3.2.2.2.2.2" xref="S6.Ex72.m1.8.8.3.2.2.2.2.2.cmml">J</mi><mo id="S6.Ex72.m1.8.8.3.2.2.2.2.3" xref="S6.Ex72.m1.8.8.3.2.2.2.2.3.cmml">∗</mo></msup><mo id="S6.Ex72.m1.8.8.3.2.2.2.1" xref="S6.Ex72.m1.8.8.3.2.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex72.m1.8.8.3.2.2.2.3" xref="S6.Ex72.m1.8.8.3.2.2.2.3.cmml">ℳ</mi></mrow><mo id="S6.Ex72.m1.8.8.3.2.2.5" stretchy="false" xref="S6.Ex72.m1.8.8.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex72.m1.8b"><apply id="S6.Ex72.m1.8.8.cmml" xref="S6.Ex72.m1.8.8"><and id="S6.Ex72.m1.8.8a.cmml" xref="S6.Ex72.m1.8.8"></and><apply id="S6.Ex72.m1.8.8b.cmml" xref="S6.Ex72.m1.8.8"><approx id="S6.Ex72.m1.8.8.5.cmml" xref="S6.Ex72.m1.8.8.5"></approx><apply id="S6.Ex72.m1.6.6.1.cmml" xref="S6.Ex72.m1.6.6.1"><times id="S6.Ex72.m1.6.6.1.2.cmml" 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xref="S6.Ex72.m1.8.8.3.3"></times><apply id="S6.Ex72.m1.8.8.3.4.cmml" xref="S6.Ex72.m1.8.8.3.4"><csymbol cd="ambiguous" id="S6.Ex72.m1.8.8.3.4.1.cmml" xref="S6.Ex72.m1.8.8.3.4">superscript</csymbol><ci id="S6.Ex72.m1.8.8.3.4.2.cmml" xref="S6.Ex72.m1.8.8.3.4.2">𝐶</ci><ci id="S6.Ex72.m1.8.8.3.4.3.cmml" xref="S6.Ex72.m1.8.8.3.4.3">𝑛</ci></apply><list id="S6.Ex72.m1.8.8.3.2.3.cmml" xref="S6.Ex72.m1.8.8.3.2.2"><apply id="S6.Ex72.m1.7.7.2.1.1.1.cmml" xref="S6.Ex72.m1.7.7.2.1.1.1"><times id="S6.Ex72.m1.7.7.2.1.1.1.1.cmml" xref="S6.Ex72.m1.7.7.2.1.1.1.1"></times><ci id="S6.Ex72.m1.7.7.2.1.1.1.2.cmml" xref="S6.Ex72.m1.7.7.2.1.1.1.2">diag</ci><ci id="S6.Ex72.m1.7.7.2.1.1.1.3.cmml" xref="S6.Ex72.m1.7.7.2.1.1.1.3">𝑋</ci></apply><apply id="S6.Ex72.m1.8.8.3.2.2.2.cmml" xref="S6.Ex72.m1.8.8.3.2.2.2"><times id="S6.Ex72.m1.8.8.3.2.2.2.1.cmml" xref="S6.Ex72.m1.8.8.3.2.2.2.1"></times><apply id="S6.Ex72.m1.8.8.3.2.2.2.2.cmml" xref="S6.Ex72.m1.8.8.3.2.2.2.2"><csymbol cd="ambiguous" 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italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( roman_diag italic_X ; italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.2.p1.9">where <math alttext="C^{n}(\mathrm{diag}X;J^{*}\mathcal{M})" class="ltx_Math" display="inline" id="S6.2.p1.8.m1.2"><semantics id="S6.2.p1.8.m1.2a"><mrow id="S6.2.p1.8.m1.2.2" xref="S6.2.p1.8.m1.2.2.cmml"><msup id="S6.2.p1.8.m1.2.2.4" xref="S6.2.p1.8.m1.2.2.4.cmml"><mi id="S6.2.p1.8.m1.2.2.4.2" xref="S6.2.p1.8.m1.2.2.4.2.cmml">C</mi><mi id="S6.2.p1.8.m1.2.2.4.3" xref="S6.2.p1.8.m1.2.2.4.3.cmml">n</mi></msup><mo id="S6.2.p1.8.m1.2.2.3" xref="S6.2.p1.8.m1.2.2.3.cmml">⁢</mo><mrow id="S6.2.p1.8.m1.2.2.2.2" xref="S6.2.p1.8.m1.2.2.2.3.cmml"><mo id="S6.2.p1.8.m1.2.2.2.2.3" stretchy="false" xref="S6.2.p1.8.m1.2.2.2.3.cmml">(</mo><mrow id="S6.2.p1.8.m1.1.1.1.1.1" xref="S6.2.p1.8.m1.1.1.1.1.1.cmml"><mi 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id="S6.2.p1.8.m1.2.2.2.2.2.2.cmml" xref="S6.2.p1.8.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.2.p1.8.m1.2.2.2.2.2.2.1.cmml" xref="S6.2.p1.8.m1.2.2.2.2.2.2">superscript</csymbol><ci id="S6.2.p1.8.m1.2.2.2.2.2.2.2.cmml" xref="S6.2.p1.8.m1.2.2.2.2.2.2.2">𝐽</ci><times id="S6.2.p1.8.m1.2.2.2.2.2.2.3.cmml" xref="S6.2.p1.8.m1.2.2.2.2.2.2.3"></times></apply><ci id="S6.2.p1.8.m1.2.2.2.2.2.3.cmml" xref="S6.2.p1.8.m1.2.2.2.2.2.3">ℳ</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.8.m1.2c">C^{n}(\mathrm{diag}X;J^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.8.m1.2d">italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( roman_diag italic_X ; italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math> is the cochain module of the simplicial set <math alttext="\mathrm{diag}X" class="ltx_Math" display="inline" id="S6.2.p1.9.m2.1"><semantics id="S6.2.p1.9.m2.1a"><mrow id="S6.2.p1.9.m2.1.1" xref="S6.2.p1.9.m2.1.1.cmml"><mi id="S6.2.p1.9.m2.1.1.2" xref="S6.2.p1.9.m2.1.1.2.cmml">diag</mi><mo id="S6.2.p1.9.m2.1.1.1" xref="S6.2.p1.9.m2.1.1.1.cmml">⁢</mo><mi id="S6.2.p1.9.m2.1.1.3" xref="S6.2.p1.9.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.9.m2.1b"><apply id="S6.2.p1.9.m2.1.1.cmml" xref="S6.2.p1.9.m2.1.1"><times id="S6.2.p1.9.m2.1.1.1.cmml" xref="S6.2.p1.9.m2.1.1.1"></times><ci id="S6.2.p1.9.m2.1.1.2.cmml" xref="S6.2.p1.9.m2.1.1.2">diag</ci><ci id="S6.2.p1.9.m2.1.1.3.cmml" xref="S6.2.p1.9.m2.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.9.m2.1c">\mathrm{diag}X</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.9.m2.1d">roman_diag italic_X</annotation></semantics></math>. The coboundary maps are compatible, so we have an isomorphism of cochain complexes</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex73"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C^{*}(\mathrm{diag}X;J^{*}\mathcal{M})\cong\mathrm{Hom}_{\Delta(X)}((\mathrm{% diag}\mathbb{P})_{*},\mathcal{M}))." class="ltx_math_unparsed" display="block" id="S6.Ex73.m1.2"><semantics id="S6.Ex73.m1.2a"><mrow id="S6.Ex73.m1.2b"><msup id="S6.Ex73.m1.2.3"><mi id="S6.Ex73.m1.2.3.2">C</mi><mo id="S6.Ex73.m1.2.3.3">∗</mo></msup><mrow id="S6.Ex73.m1.2.4"><mo id="S6.Ex73.m1.2.4.1" stretchy="false">(</mo><mi id="S6.Ex73.m1.2.4.2">diag</mi><mi id="S6.Ex73.m1.2.4.3">X</mi><mo id="S6.Ex73.m1.2.4.4">;</mo><msup id="S6.Ex73.m1.2.4.5"><mi id="S6.Ex73.m1.2.4.5.2">J</mi><mo id="S6.Ex73.m1.2.4.5.3">∗</mo></msup><mi class="ltx_font_mathcaligraphic" id="S6.Ex73.m1.2.4.6">ℳ</mi><mo id="S6.Ex73.m1.2.4.7" stretchy="false">)</mo></mrow><mo id="S6.Ex73.m1.2.5">≅</mo><msub id="S6.Ex73.m1.2.6"><mi id="S6.Ex73.m1.2.6.2">Hom</mi><mrow id="S6.Ex73.m1.1.1.1"><mi id="S6.Ex73.m1.1.1.1.3" mathvariant="normal">Δ</mi><mo id="S6.Ex73.m1.1.1.1.2">⁢</mo><mrow id="S6.Ex73.m1.1.1.1.4.2"><mo id="S6.Ex73.m1.1.1.1.4.2.1" stretchy="false">(</mo><mi id="S6.Ex73.m1.1.1.1.1">X</mi><mo id="S6.Ex73.m1.1.1.1.4.2.2" stretchy="false">)</mo></mrow></mrow></msub><mrow id="S6.Ex73.m1.2.7"><mo id="S6.Ex73.m1.2.7.1" stretchy="false">(</mo><msub id="S6.Ex73.m1.2.7.2"><mrow id="S6.Ex73.m1.2.7.2.2"><mo id="S6.Ex73.m1.2.7.2.2.1" stretchy="false">(</mo><mi id="S6.Ex73.m1.2.7.2.2.2">diag</mi><mi id="S6.Ex73.m1.2.7.2.2.3">ℙ</mi><mo id="S6.Ex73.m1.2.7.2.2.4" stretchy="false">)</mo></mrow><mo id="S6.Ex73.m1.2.7.2.3">∗</mo></msub><mo id="S6.Ex73.m1.2.7.3">,</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex73.m1.2.2">ℳ</mi><mo id="S6.Ex73.m1.2.7.4" stretchy="false">)</mo></mrow><mo id="S6.Ex73.m1.2.8" stretchy="false">)</mo><mo id="S6.Ex73.m1.2.9" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S6.Ex73.m1.2c">C^{*}(\mathrm{diag}X;J^{*}\mathcal{M})\cong\mathrm{Hom}_{\Delta(X)}((\mathrm{% diag}\mathbb{P})_{*},\mathcal{M})).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex73.m1.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_diag italic_X ; italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M ) ≅ roman_Hom start_POSTSUBSCRIPT roman_Δ ( italic_X ) end_POSTSUBSCRIPT ( ( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , caligraphic_M ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S6.3.p2"> <p class="ltx_p" id="S6.3.p2.1">Applying the hom-functor to <math alttext="j^{*}" class="ltx_Math" display="inline" id="S6.3.p2.1.m1.1"><semantics id="S6.3.p2.1.m1.1a"><msup id="S6.3.p2.1.m1.1.1" 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xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.3">subscript</csymbol><ci id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.3.2.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.3.2">Hom</ci><apply id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1"><times id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.2.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.2"></times><ci id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.3.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.3">𝑅</ci><ci id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.4.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.4">Δ</ci><ci id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.1.1.1.1">𝑋</ci></apply></apply><interval closure="open" id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.2.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1"><apply id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.2.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1">subscript</csymbol><apply id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.1.1.1.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.1.1"><times id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.1.1.1.1"></times><ci id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.1.1.1.2">diag</ci><ci id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.1.1.1.3">ℙ</ci></apply><times id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.3.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3.3.1.1.1.3"></times></apply><ci id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.2.2.cmml" xref="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.2.2">ℳ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3c">\textstyle{\mathrm{Hom}_{R\Delta(X)}((\mathrm{diag}\mathbb{P})_{*},\mathcal{M})}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex74.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.3d">roman_Hom start_POSTSUBSCRIPT italic_R roman_Δ ( italic_X ) end_POSTSUBSCRIPT ( ( roman_diag blackboard_P ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT , caligraphic_M )</annotation></semantics></math></foreignobject></g></g></svg></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.3.p2.3">where the horizontal maps are the chain isomorphisms described above, and the second vertical map is a chain homotopy equivalence. This means <math alttext="J^{*}" class="ltx_Math" display="inline" id="S6.3.p2.2.m1.1"><semantics id="S6.3.p2.2.m1.1a"><msup id="S6.3.p2.2.m1.1.1" xref="S6.3.p2.2.m1.1.1.cmml"><mi id="S6.3.p2.2.m1.1.1.2" xref="S6.3.p2.2.m1.1.1.2.cmml">J</mi><mo id="S6.3.p2.2.m1.1.1.3" xref="S6.3.p2.2.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S6.3.p2.2.m1.1b"><apply id="S6.3.p2.2.m1.1.1.cmml" xref="S6.3.p2.2.m1.1.1"><csymbol cd="ambiguous" id="S6.3.p2.2.m1.1.1.1.cmml" xref="S6.3.p2.2.m1.1.1">superscript</csymbol><ci id="S6.3.p2.2.m1.1.1.2.cmml" xref="S6.3.p2.2.m1.1.1.2">𝐽</ci><times id="S6.3.p2.2.m1.1.1.3.cmml" xref="S6.3.p2.2.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.2.m1.1c">J^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.2.m1.1d">italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is also a chain homotopy equivalence, hence <math alttext="J^{*}" class="ltx_Math" display="inline" id="S6.3.p2.3.m2.1"><semantics id="S6.3.p2.3.m2.1a"><msup id="S6.3.p2.3.m2.1.1" xref="S6.3.p2.3.m2.1.1.cmml"><mi id="S6.3.p2.3.m2.1.1.2" xref="S6.3.p2.3.m2.1.1.2.cmml">J</mi><mo id="S6.3.p2.3.m2.1.1.3" xref="S6.3.p2.3.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S6.3.p2.3.m2.1b"><apply id="S6.3.p2.3.m2.1.1.cmml" xref="S6.3.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S6.3.p2.3.m2.1.1.1.cmml" xref="S6.3.p2.3.m2.1.1">superscript</csymbol><ci id="S6.3.p2.3.m2.1.1.2.cmml" xref="S6.3.p2.3.m2.1.1.2">𝐽</ci><times id="S6.3.p2.3.m2.1.1.3.cmml" xref="S6.3.p2.3.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.3.m2.1c">J^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.3.m2.1d">italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> induces isomorphism on cohomology. ∎</p> </div> </div> <div class="ltx_para" id="S6.p5"> <p class="ltx_p" id="S6.p5.1">The following definition will be important for our applications with bisimplicial sets.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S6.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem4.1.1.1">Definition 6.4</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem4.p1"> <p class="ltx_p" id="S6.Thmtheorem4.p1.6">Let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.1.m1.1"><semantics id="S6.Thmtheorem4.p1.1.m1.1a"><mi id="S6.Thmtheorem4.p1.1.m1.1.1" xref="S6.Thmtheorem4.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.1.m1.1b"><ci id="S6.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem4.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.1.m1.1d">italic_X</annotation></semantics></math> be a bisimplicial set. A coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.2.m2.1"><semantics id="S6.Thmtheorem4.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem4.p1.2.m2.1.1" xref="S6.Thmtheorem4.p1.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.2.m2.1b"><ci id="S6.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem4.p1.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.2.m2.1d">caligraphic_M</annotation></semantics></math> on <math alttext="\mathrm{diag}X" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.3.m3.1"><semantics id="S6.Thmtheorem4.p1.3.m3.1a"><mrow id="S6.Thmtheorem4.p1.3.m3.1.1" xref="S6.Thmtheorem4.p1.3.m3.1.1.cmml"><mi id="S6.Thmtheorem4.p1.3.m3.1.1.2" xref="S6.Thmtheorem4.p1.3.m3.1.1.2.cmml">diag</mi><mo id="S6.Thmtheorem4.p1.3.m3.1.1.1" xref="S6.Thmtheorem4.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S6.Thmtheorem4.p1.3.m3.1.1.3" xref="S6.Thmtheorem4.p1.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.3.m3.1b"><apply id="S6.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem4.p1.3.m3.1.1"><times id="S6.Thmtheorem4.p1.3.m3.1.1.1.cmml" xref="S6.Thmtheorem4.p1.3.m3.1.1.1"></times><ci id="S6.Thmtheorem4.p1.3.m3.1.1.2.cmml" xref="S6.Thmtheorem4.p1.3.m3.1.1.2">diag</ci><ci id="S6.Thmtheorem4.p1.3.m3.1.1.3.cmml" xref="S6.Thmtheorem4.p1.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.3.m3.1c">\mathrm{diag}X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.3.m3.1d">roman_diag italic_X</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S6.Thmtheorem4.p1.6.1">extendable</em> if there is a coefficient system <math alttext="\mathcal{M}^{\prime}" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.4.m4.1"><semantics id="S6.Thmtheorem4.p1.4.m4.1a"><msup id="S6.Thmtheorem4.p1.4.m4.1.1" xref="S6.Thmtheorem4.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem4.p1.4.m4.1.1.2" xref="S6.Thmtheorem4.p1.4.m4.1.1.2.cmml">ℳ</mi><mo id="S6.Thmtheorem4.p1.4.m4.1.1.3" xref="S6.Thmtheorem4.p1.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.4.m4.1b"><apply id="S6.Thmtheorem4.p1.4.m4.1.1.cmml" xref="S6.Thmtheorem4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.4.m4.1.1.1.cmml" xref="S6.Thmtheorem4.p1.4.m4.1.1">superscript</csymbol><ci id="S6.Thmtheorem4.p1.4.m4.1.1.2.cmml" xref="S6.Thmtheorem4.p1.4.m4.1.1.2">ℳ</ci><ci id="S6.Thmtheorem4.p1.4.m4.1.1.3.cmml" xref="S6.Thmtheorem4.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.4.m4.1c">\mathcal{M}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.4.m4.1d">caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> on <math alttext="\Delta(X)" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.5.m5.1"><semantics id="S6.Thmtheorem4.p1.5.m5.1a"><mrow id="S6.Thmtheorem4.p1.5.m5.1.2" xref="S6.Thmtheorem4.p1.5.m5.1.2.cmml"><mi id="S6.Thmtheorem4.p1.5.m5.1.2.2" mathvariant="normal" xref="S6.Thmtheorem4.p1.5.m5.1.2.2.cmml">Δ</mi><mo id="S6.Thmtheorem4.p1.5.m5.1.2.1" xref="S6.Thmtheorem4.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem4.p1.5.m5.1.2.3.2" xref="S6.Thmtheorem4.p1.5.m5.1.2.cmml"><mo id="S6.Thmtheorem4.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S6.Thmtheorem4.p1.5.m5.1.2.cmml">(</mo><mi id="S6.Thmtheorem4.p1.5.m5.1.1" xref="S6.Thmtheorem4.p1.5.m5.1.1.cmml">X</mi><mo id="S6.Thmtheorem4.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S6.Thmtheorem4.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.5.m5.1b"><apply id="S6.Thmtheorem4.p1.5.m5.1.2.cmml" xref="S6.Thmtheorem4.p1.5.m5.1.2"><times id="S6.Thmtheorem4.p1.5.m5.1.2.1.cmml" xref="S6.Thmtheorem4.p1.5.m5.1.2.1"></times><ci id="S6.Thmtheorem4.p1.5.m5.1.2.2.cmml" xref="S6.Thmtheorem4.p1.5.m5.1.2.2">Δ</ci><ci id="S6.Thmtheorem4.p1.5.m5.1.1.cmml" xref="S6.Thmtheorem4.p1.5.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.5.m5.1c">\Delta(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.5.m5.1d">roman_Δ ( italic_X )</annotation></semantics></math> such that <math alttext="\mathcal{M}=J^{*}(\mathcal{M}^{\prime})" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.6.m6.1"><semantics id="S6.Thmtheorem4.p1.6.m6.1a"><mrow id="S6.Thmtheorem4.p1.6.m6.1.1" xref="S6.Thmtheorem4.p1.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem4.p1.6.m6.1.1.3" xref="S6.Thmtheorem4.p1.6.m6.1.1.3.cmml">ℳ</mi><mo id="S6.Thmtheorem4.p1.6.m6.1.1.2" xref="S6.Thmtheorem4.p1.6.m6.1.1.2.cmml">=</mo><mrow id="S6.Thmtheorem4.p1.6.m6.1.1.1" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.cmml"><msup id="S6.Thmtheorem4.p1.6.m6.1.1.1.3" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.3.cmml"><mi id="S6.Thmtheorem4.p1.6.m6.1.1.1.3.2" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.3.2.cmml">J</mi><mo id="S6.Thmtheorem4.p1.6.m6.1.1.1.3.3" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.3.3.cmml">∗</mo></msup><mo id="S6.Thmtheorem4.p1.6.m6.1.1.1.2" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.2" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.2.cmml">ℳ</mi><mo id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.3" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.6.m6.1b"><apply id="S6.Thmtheorem4.p1.6.m6.1.1.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1"><eq id="S6.Thmtheorem4.p1.6.m6.1.1.2.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.2"></eq><ci id="S6.Thmtheorem4.p1.6.m6.1.1.3.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.3">ℳ</ci><apply id="S6.Thmtheorem4.p1.6.m6.1.1.1.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1"><times id="S6.Thmtheorem4.p1.6.m6.1.1.1.2.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.2"></times><apply id="S6.Thmtheorem4.p1.6.m6.1.1.1.3.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.6.m6.1.1.1.3.1.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.3">superscript</csymbol><ci id="S6.Thmtheorem4.p1.6.m6.1.1.1.3.2.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.3.2">𝐽</ci><times id="S6.Thmtheorem4.p1.6.m6.1.1.1.3.3.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.3.3"></times></apply><apply id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.2">ℳ</ci><ci id="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.6.m6.1c">\mathcal{M}=J^{*}(\mathcal{M}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.6.m6.1d">caligraphic_M = italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S6.p6"> <p class="ltx_p" id="S6.p6.11">Let <math alttext="F:\mathcal{D}\to sSet" class="ltx_Math" display="inline" id="S6.p6.1.m1.1"><semantics id="S6.p6.1.m1.1a"><mrow id="S6.p6.1.m1.1.1" xref="S6.p6.1.m1.1.1.cmml"><mi id="S6.p6.1.m1.1.1.2" xref="S6.p6.1.m1.1.1.2.cmml">F</mi><mo id="S6.p6.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.p6.1.m1.1.1.1.cmml">:</mo><mrow id="S6.p6.1.m1.1.1.3" xref="S6.p6.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p6.1.m1.1.1.3.2" xref="S6.p6.1.m1.1.1.3.2.cmml">𝒟</mi><mo id="S6.p6.1.m1.1.1.3.1" stretchy="false" xref="S6.p6.1.m1.1.1.3.1.cmml">→</mo><mrow id="S6.p6.1.m1.1.1.3.3" xref="S6.p6.1.m1.1.1.3.3.cmml"><mi id="S6.p6.1.m1.1.1.3.3.2" xref="S6.p6.1.m1.1.1.3.3.2.cmml">s</mi><mo id="S6.p6.1.m1.1.1.3.3.1" xref="S6.p6.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S6.p6.1.m1.1.1.3.3.3" xref="S6.p6.1.m1.1.1.3.3.3.cmml">S</mi><mo id="S6.p6.1.m1.1.1.3.3.1a" xref="S6.p6.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S6.p6.1.m1.1.1.3.3.4" xref="S6.p6.1.m1.1.1.3.3.4.cmml">e</mi><mo id="S6.p6.1.m1.1.1.3.3.1b" xref="S6.p6.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S6.p6.1.m1.1.1.3.3.5" xref="S6.p6.1.m1.1.1.3.3.5.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.1.m1.1b"><apply id="S6.p6.1.m1.1.1.cmml" xref="S6.p6.1.m1.1.1"><ci id="S6.p6.1.m1.1.1.1.cmml" xref="S6.p6.1.m1.1.1.1">:</ci><ci id="S6.p6.1.m1.1.1.2.cmml" xref="S6.p6.1.m1.1.1.2">𝐹</ci><apply id="S6.p6.1.m1.1.1.3.cmml" xref="S6.p6.1.m1.1.1.3"><ci id="S6.p6.1.m1.1.1.3.1.cmml" xref="S6.p6.1.m1.1.1.3.1">→</ci><ci id="S6.p6.1.m1.1.1.3.2.cmml" xref="S6.p6.1.m1.1.1.3.2">𝒟</ci><apply id="S6.p6.1.m1.1.1.3.3.cmml" xref="S6.p6.1.m1.1.1.3.3"><times id="S6.p6.1.m1.1.1.3.3.1.cmml" xref="S6.p6.1.m1.1.1.3.3.1"></times><ci id="S6.p6.1.m1.1.1.3.3.2.cmml" xref="S6.p6.1.m1.1.1.3.3.2">𝑠</ci><ci id="S6.p6.1.m1.1.1.3.3.3.cmml" xref="S6.p6.1.m1.1.1.3.3.3">𝑆</ci><ci id="S6.p6.1.m1.1.1.3.3.4.cmml" xref="S6.p6.1.m1.1.1.3.3.4">𝑒</ci><ci id="S6.p6.1.m1.1.1.3.3.5.cmml" xref="S6.p6.1.m1.1.1.3.3.5">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.1.m1.1c">F:\mathcal{D}\to sSet</annotation><annotation encoding="application/x-llamapun" id="S6.p6.1.m1.1d">italic_F : caligraphic_D → italic_s italic_S italic_e italic_t</annotation></semantics></math> be a functor. The homotopy colimit <math alttext="\operatorname*{hocolim}_{\mathcal{D}}F" class="ltx_Math" display="inline" id="S6.p6.2.m2.1"><semantics id="S6.p6.2.m2.1a"><mrow id="S6.p6.2.m2.1.1" xref="S6.p6.2.m2.1.1.cmml"><msub id="S6.p6.2.m2.1.1.1" xref="S6.p6.2.m2.1.1.1.cmml"><mo id="S6.p6.2.m2.1.1.1.2" xref="S6.p6.2.m2.1.1.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S6.p6.2.m2.1.1.1.3" xref="S6.p6.2.m2.1.1.1.3.cmml">𝒟</mi></msub><mi id="S6.p6.2.m2.1.1.2" xref="S6.p6.2.m2.1.1.2.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.2.m2.1b"><apply id="S6.p6.2.m2.1.1.cmml" xref="S6.p6.2.m2.1.1"><apply id="S6.p6.2.m2.1.1.1.cmml" xref="S6.p6.2.m2.1.1.1"><csymbol cd="ambiguous" id="S6.p6.2.m2.1.1.1.1.cmml" xref="S6.p6.2.m2.1.1.1">subscript</csymbol><ci id="S6.p6.2.m2.1.1.1.2.cmml" xref="S6.p6.2.m2.1.1.1.2">hocolim</ci><ci id="S6.p6.2.m2.1.1.1.3.cmml" xref="S6.p6.2.m2.1.1.1.3">𝒟</ci></apply><ci id="S6.p6.2.m2.1.1.2.cmml" xref="S6.p6.2.m2.1.1.2">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.2.m2.1c">\operatorname*{hocolim}_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="S6.p6.2.m2.1d">roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> is the diagonal of the bisimplicial set <math alttext="N(\mathcal{D};F)" class="ltx_Math" display="inline" id="S6.p6.3.m3.2"><semantics id="S6.p6.3.m3.2a"><mrow id="S6.p6.3.m3.2.3" xref="S6.p6.3.m3.2.3.cmml"><mi id="S6.p6.3.m3.2.3.2" xref="S6.p6.3.m3.2.3.2.cmml">N</mi><mo id="S6.p6.3.m3.2.3.1" xref="S6.p6.3.m3.2.3.1.cmml">⁢</mo><mrow id="S6.p6.3.m3.2.3.3.2" xref="S6.p6.3.m3.2.3.3.1.cmml"><mo id="S6.p6.3.m3.2.3.3.2.1" stretchy="false" xref="S6.p6.3.m3.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p6.3.m3.1.1" xref="S6.p6.3.m3.1.1.cmml">𝒟</mi><mo id="S6.p6.3.m3.2.3.3.2.2" xref="S6.p6.3.m3.2.3.3.1.cmml">;</mo><mi id="S6.p6.3.m3.2.2" xref="S6.p6.3.m3.2.2.cmml">F</mi><mo id="S6.p6.3.m3.2.3.3.2.3" stretchy="false" xref="S6.p6.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.3.m3.2b"><apply id="S6.p6.3.m3.2.3.cmml" xref="S6.p6.3.m3.2.3"><times id="S6.p6.3.m3.2.3.1.cmml" xref="S6.p6.3.m3.2.3.1"></times><ci id="S6.p6.3.m3.2.3.2.cmml" xref="S6.p6.3.m3.2.3.2">𝑁</ci><list id="S6.p6.3.m3.2.3.3.1.cmml" xref="S6.p6.3.m3.2.3.3.2"><ci id="S6.p6.3.m3.1.1.cmml" xref="S6.p6.3.m3.1.1">𝒟</ci><ci id="S6.p6.3.m3.2.2.cmml" xref="S6.p6.3.m3.2.2">𝐹</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.3.m3.2c">N(\mathcal{D};F)</annotation><annotation encoding="application/x-llamapun" id="S6.p6.3.m3.2d">italic_N ( caligraphic_D ; italic_F )</annotation></semantics></math> whose <math alttext="(p,q)" class="ltx_Math" display="inline" id="S6.p6.4.m4.2"><semantics id="S6.p6.4.m4.2a"><mrow id="S6.p6.4.m4.2.3.2" xref="S6.p6.4.m4.2.3.1.cmml"><mo id="S6.p6.4.m4.2.3.2.1" stretchy="false" xref="S6.p6.4.m4.2.3.1.cmml">(</mo><mi id="S6.p6.4.m4.1.1" xref="S6.p6.4.m4.1.1.cmml">p</mi><mo id="S6.p6.4.m4.2.3.2.2" xref="S6.p6.4.m4.2.3.1.cmml">,</mo><mi id="S6.p6.4.m4.2.2" xref="S6.p6.4.m4.2.2.cmml">q</mi><mo id="S6.p6.4.m4.2.3.2.3" stretchy="false" xref="S6.p6.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.4.m4.2b"><interval closure="open" id="S6.p6.4.m4.2.3.1.cmml" xref="S6.p6.4.m4.2.3.2"><ci id="S6.p6.4.m4.1.1.cmml" xref="S6.p6.4.m4.1.1">𝑝</ci><ci id="S6.p6.4.m4.2.2.cmml" xref="S6.p6.4.m4.2.2">𝑞</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.4.m4.2c">(p,q)</annotation><annotation encoding="application/x-llamapun" id="S6.p6.4.m4.2d">( italic_p , italic_q )</annotation></semantics></math>-simplices are given by pairs <math alttext="(\sigma,\tau)" class="ltx_Math" display="inline" id="S6.p6.5.m5.2"><semantics id="S6.p6.5.m5.2a"><mrow id="S6.p6.5.m5.2.3.2" xref="S6.p6.5.m5.2.3.1.cmml"><mo id="S6.p6.5.m5.2.3.2.1" stretchy="false" xref="S6.p6.5.m5.2.3.1.cmml">(</mo><mi id="S6.p6.5.m5.1.1" xref="S6.p6.5.m5.1.1.cmml">σ</mi><mo id="S6.p6.5.m5.2.3.2.2" xref="S6.p6.5.m5.2.3.1.cmml">,</mo><mi id="S6.p6.5.m5.2.2" xref="S6.p6.5.m5.2.2.cmml">τ</mi><mo id="S6.p6.5.m5.2.3.2.3" stretchy="false" xref="S6.p6.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.5.m5.2b"><interval closure="open" id="S6.p6.5.m5.2.3.1.cmml" xref="S6.p6.5.m5.2.3.2"><ci id="S6.p6.5.m5.1.1.cmml" xref="S6.p6.5.m5.1.1">𝜎</ci><ci id="S6.p6.5.m5.2.2.cmml" xref="S6.p6.5.m5.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.5.m5.2c">(\sigma,\tau)</annotation><annotation encoding="application/x-llamapun" id="S6.p6.5.m5.2d">( italic_σ , italic_τ )</annotation></semantics></math> where <math alttext="\sigma=(d_{0}\to\cdots\to d_{p})\in N\mathcal{D}_{p}" class="ltx_Math" display="inline" id="S6.p6.6.m6.1"><semantics id="S6.p6.6.m6.1a"><mrow id="S6.p6.6.m6.1.1" xref="S6.p6.6.m6.1.1.cmml"><mi id="S6.p6.6.m6.1.1.3" xref="S6.p6.6.m6.1.1.3.cmml">σ</mi><mo id="S6.p6.6.m6.1.1.4" xref="S6.p6.6.m6.1.1.4.cmml">=</mo><mrow id="S6.p6.6.m6.1.1.1.1" xref="S6.p6.6.m6.1.1.1.1.1.cmml"><mo id="S6.p6.6.m6.1.1.1.1.2" stretchy="false" xref="S6.p6.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S6.p6.6.m6.1.1.1.1.1" xref="S6.p6.6.m6.1.1.1.1.1.cmml"><msub id="S6.p6.6.m6.1.1.1.1.1.2" xref="S6.p6.6.m6.1.1.1.1.1.2.cmml"><mi id="S6.p6.6.m6.1.1.1.1.1.2.2" xref="S6.p6.6.m6.1.1.1.1.1.2.2.cmml">d</mi><mn id="S6.p6.6.m6.1.1.1.1.1.2.3" xref="S6.p6.6.m6.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.p6.6.m6.1.1.1.1.1.3" stretchy="false" xref="S6.p6.6.m6.1.1.1.1.1.3.cmml">→</mo><mi id="S6.p6.6.m6.1.1.1.1.1.4" mathvariant="normal" xref="S6.p6.6.m6.1.1.1.1.1.4.cmml">⋯</mi><mo id="S6.p6.6.m6.1.1.1.1.1.5" stretchy="false" xref="S6.p6.6.m6.1.1.1.1.1.5.cmml">→</mo><msub id="S6.p6.6.m6.1.1.1.1.1.6" xref="S6.p6.6.m6.1.1.1.1.1.6.cmml"><mi id="S6.p6.6.m6.1.1.1.1.1.6.2" xref="S6.p6.6.m6.1.1.1.1.1.6.2.cmml">d</mi><mi id="S6.p6.6.m6.1.1.1.1.1.6.3" xref="S6.p6.6.m6.1.1.1.1.1.6.3.cmml">p</mi></msub></mrow><mo id="S6.p6.6.m6.1.1.1.1.3" stretchy="false" xref="S6.p6.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.p6.6.m6.1.1.5" xref="S6.p6.6.m6.1.1.5.cmml">∈</mo><mrow id="S6.p6.6.m6.1.1.6" xref="S6.p6.6.m6.1.1.6.cmml"><mi id="S6.p6.6.m6.1.1.6.2" xref="S6.p6.6.m6.1.1.6.2.cmml">N</mi><mo id="S6.p6.6.m6.1.1.6.1" xref="S6.p6.6.m6.1.1.6.1.cmml">⁢</mo><msub id="S6.p6.6.m6.1.1.6.3" xref="S6.p6.6.m6.1.1.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p6.6.m6.1.1.6.3.2" xref="S6.p6.6.m6.1.1.6.3.2.cmml">𝒟</mi><mi id="S6.p6.6.m6.1.1.6.3.3" xref="S6.p6.6.m6.1.1.6.3.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.6.m6.1b"><apply id="S6.p6.6.m6.1.1.cmml" xref="S6.p6.6.m6.1.1"><and id="S6.p6.6.m6.1.1a.cmml" xref="S6.p6.6.m6.1.1"></and><apply id="S6.p6.6.m6.1.1b.cmml" xref="S6.p6.6.m6.1.1"><eq id="S6.p6.6.m6.1.1.4.cmml" xref="S6.p6.6.m6.1.1.4"></eq><ci id="S6.p6.6.m6.1.1.3.cmml" xref="S6.p6.6.m6.1.1.3">𝜎</ci><apply id="S6.p6.6.m6.1.1.1.1.1.cmml" xref="S6.p6.6.m6.1.1.1.1"><and id="S6.p6.6.m6.1.1.1.1.1a.cmml" xref="S6.p6.6.m6.1.1.1.1"></and><apply id="S6.p6.6.m6.1.1.1.1.1b.cmml" xref="S6.p6.6.m6.1.1.1.1"><ci id="S6.p6.6.m6.1.1.1.1.1.3.cmml" xref="S6.p6.6.m6.1.1.1.1.1.3">→</ci><apply id="S6.p6.6.m6.1.1.1.1.1.2.cmml" xref="S6.p6.6.m6.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.p6.6.m6.1.1.1.1.1.2.1.cmml" xref="S6.p6.6.m6.1.1.1.1.1.2">subscript</csymbol><ci id="S6.p6.6.m6.1.1.1.1.1.2.2.cmml" xref="S6.p6.6.m6.1.1.1.1.1.2.2">𝑑</ci><cn id="S6.p6.6.m6.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.p6.6.m6.1.1.1.1.1.2.3">0</cn></apply><ci id="S6.p6.6.m6.1.1.1.1.1.4.cmml" xref="S6.p6.6.m6.1.1.1.1.1.4">⋯</ci></apply><apply id="S6.p6.6.m6.1.1.1.1.1c.cmml" xref="S6.p6.6.m6.1.1.1.1"><ci id="S6.p6.6.m6.1.1.1.1.1.5.cmml" xref="S6.p6.6.m6.1.1.1.1.1.5">→</ci><share href="https://arxiv.org/html/2503.14659v1#S6.p6.6.m6.1.1.1.1.1.4.cmml" id="S6.p6.6.m6.1.1.1.1.1d.cmml" xref="S6.p6.6.m6.1.1.1.1"></share><apply id="S6.p6.6.m6.1.1.1.1.1.6.cmml" xref="S6.p6.6.m6.1.1.1.1.1.6"><csymbol cd="ambiguous" id="S6.p6.6.m6.1.1.1.1.1.6.1.cmml" xref="S6.p6.6.m6.1.1.1.1.1.6">subscript</csymbol><ci id="S6.p6.6.m6.1.1.1.1.1.6.2.cmml" xref="S6.p6.6.m6.1.1.1.1.1.6.2">𝑑</ci><ci id="S6.p6.6.m6.1.1.1.1.1.6.3.cmml" xref="S6.p6.6.m6.1.1.1.1.1.6.3">𝑝</ci></apply></apply></apply></apply><apply id="S6.p6.6.m6.1.1c.cmml" xref="S6.p6.6.m6.1.1"><in id="S6.p6.6.m6.1.1.5.cmml" xref="S6.p6.6.m6.1.1.5"></in><share href="https://arxiv.org/html/2503.14659v1#S6.p6.6.m6.1.1.1.cmml" id="S6.p6.6.m6.1.1d.cmml" xref="S6.p6.6.m6.1.1"></share><apply id="S6.p6.6.m6.1.1.6.cmml" xref="S6.p6.6.m6.1.1.6"><times id="S6.p6.6.m6.1.1.6.1.cmml" xref="S6.p6.6.m6.1.1.6.1"></times><ci id="S6.p6.6.m6.1.1.6.2.cmml" xref="S6.p6.6.m6.1.1.6.2">𝑁</ci><apply id="S6.p6.6.m6.1.1.6.3.cmml" xref="S6.p6.6.m6.1.1.6.3"><csymbol cd="ambiguous" id="S6.p6.6.m6.1.1.6.3.1.cmml" xref="S6.p6.6.m6.1.1.6.3">subscript</csymbol><ci id="S6.p6.6.m6.1.1.6.3.2.cmml" xref="S6.p6.6.m6.1.1.6.3.2">𝒟</ci><ci id="S6.p6.6.m6.1.1.6.3.3.cmml" xref="S6.p6.6.m6.1.1.6.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.6.m6.1c">\sigma=(d_{0}\to\cdots\to d_{p})\in N\mathcal{D}_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.p6.6.m6.1d">italic_σ = ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ⋯ → italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ∈ italic_N caligraphic_D start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tau\in F(d_{0})_{q}" class="ltx_Math" display="inline" id="S6.p6.7.m7.1"><semantics id="S6.p6.7.m7.1a"><mrow id="S6.p6.7.m7.1.1" xref="S6.p6.7.m7.1.1.cmml"><mi id="S6.p6.7.m7.1.1.3" xref="S6.p6.7.m7.1.1.3.cmml">τ</mi><mo id="S6.p6.7.m7.1.1.2" xref="S6.p6.7.m7.1.1.2.cmml">∈</mo><mrow id="S6.p6.7.m7.1.1.1" xref="S6.p6.7.m7.1.1.1.cmml"><mi id="S6.p6.7.m7.1.1.1.3" xref="S6.p6.7.m7.1.1.1.3.cmml">F</mi><mo id="S6.p6.7.m7.1.1.1.2" xref="S6.p6.7.m7.1.1.1.2.cmml">⁢</mo><msub id="S6.p6.7.m7.1.1.1.1" xref="S6.p6.7.m7.1.1.1.1.cmml"><mrow id="S6.p6.7.m7.1.1.1.1.1.1" xref="S6.p6.7.m7.1.1.1.1.1.1.1.cmml"><mo id="S6.p6.7.m7.1.1.1.1.1.1.2" stretchy="false" xref="S6.p6.7.m7.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.p6.7.m7.1.1.1.1.1.1.1" xref="S6.p6.7.m7.1.1.1.1.1.1.1.cmml"><mi id="S6.p6.7.m7.1.1.1.1.1.1.1.2" xref="S6.p6.7.m7.1.1.1.1.1.1.1.2.cmml">d</mi><mn id="S6.p6.7.m7.1.1.1.1.1.1.1.3" xref="S6.p6.7.m7.1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.p6.7.m7.1.1.1.1.1.1.3" stretchy="false" xref="S6.p6.7.m7.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S6.p6.7.m7.1.1.1.1.3" xref="S6.p6.7.m7.1.1.1.1.3.cmml">q</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.7.m7.1b"><apply id="S6.p6.7.m7.1.1.cmml" xref="S6.p6.7.m7.1.1"><in id="S6.p6.7.m7.1.1.2.cmml" xref="S6.p6.7.m7.1.1.2"></in><ci id="S6.p6.7.m7.1.1.3.cmml" xref="S6.p6.7.m7.1.1.3">𝜏</ci><apply id="S6.p6.7.m7.1.1.1.cmml" xref="S6.p6.7.m7.1.1.1"><times id="S6.p6.7.m7.1.1.1.2.cmml" xref="S6.p6.7.m7.1.1.1.2"></times><ci id="S6.p6.7.m7.1.1.1.3.cmml" xref="S6.p6.7.m7.1.1.1.3">𝐹</ci><apply id="S6.p6.7.m7.1.1.1.1.cmml" xref="S6.p6.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S6.p6.7.m7.1.1.1.1.2.cmml" xref="S6.p6.7.m7.1.1.1.1">subscript</csymbol><apply id="S6.p6.7.m7.1.1.1.1.1.1.1.cmml" xref="S6.p6.7.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p6.7.m7.1.1.1.1.1.1.1.1.cmml" xref="S6.p6.7.m7.1.1.1.1.1.1">subscript</csymbol><ci id="S6.p6.7.m7.1.1.1.1.1.1.1.2.cmml" xref="S6.p6.7.m7.1.1.1.1.1.1.1.2">𝑑</ci><cn id="S6.p6.7.m7.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.p6.7.m7.1.1.1.1.1.1.1.3">0</cn></apply><ci id="S6.p6.7.m7.1.1.1.1.3.cmml" xref="S6.p6.7.m7.1.1.1.1.3">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.7.m7.1c">\tau\in F(d_{0})_{q}</annotation><annotation encoding="application/x-llamapun" id="S6.p6.7.m7.1d">italic_τ ∈ italic_F ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math>. A coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S6.p6.8.m8.1"><semantics id="S6.p6.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S6.p6.8.m8.1.1" xref="S6.p6.8.m8.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S6.p6.8.m8.1b"><ci id="S6.p6.8.m8.1.1.cmml" xref="S6.p6.8.m8.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.8.m8.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.p6.8.m8.1d">caligraphic_M</annotation></semantics></math> for <math alttext="\operatorname*{hocolim}_{\mathcal{D}}F" class="ltx_Math" display="inline" id="S6.p6.9.m9.1"><semantics id="S6.p6.9.m9.1a"><mrow id="S6.p6.9.m9.1.1" xref="S6.p6.9.m9.1.1.cmml"><msub id="S6.p6.9.m9.1.1.1" xref="S6.p6.9.m9.1.1.1.cmml"><mo id="S6.p6.9.m9.1.1.1.2" xref="S6.p6.9.m9.1.1.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S6.p6.9.m9.1.1.1.3" xref="S6.p6.9.m9.1.1.1.3.cmml">𝒟</mi></msub><mi id="S6.p6.9.m9.1.1.2" xref="S6.p6.9.m9.1.1.2.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.9.m9.1b"><apply id="S6.p6.9.m9.1.1.cmml" xref="S6.p6.9.m9.1.1"><apply id="S6.p6.9.m9.1.1.1.cmml" xref="S6.p6.9.m9.1.1.1"><csymbol cd="ambiguous" id="S6.p6.9.m9.1.1.1.1.cmml" xref="S6.p6.9.m9.1.1.1">subscript</csymbol><ci id="S6.p6.9.m9.1.1.1.2.cmml" xref="S6.p6.9.m9.1.1.1.2">hocolim</ci><ci id="S6.p6.9.m9.1.1.1.3.cmml" xref="S6.p6.9.m9.1.1.1.3">𝒟</ci></apply><ci id="S6.p6.9.m9.1.1.2.cmml" xref="S6.p6.9.m9.1.1.2">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.9.m9.1c">\operatorname*{hocolim}_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="S6.p6.9.m9.1d">roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> is a functor from the simplex category <math alttext="\Delta(\operatorname*{hocolim}_{\mathcal{D}}F)" class="ltx_Math" display="inline" id="S6.p6.10.m10.1"><semantics id="S6.p6.10.m10.1a"><mrow id="S6.p6.10.m10.1.1" xref="S6.p6.10.m10.1.1.cmml"><mi id="S6.p6.10.m10.1.1.3" mathvariant="normal" xref="S6.p6.10.m10.1.1.3.cmml">Δ</mi><mo id="S6.p6.10.m10.1.1.2" xref="S6.p6.10.m10.1.1.2.cmml">⁢</mo><mrow id="S6.p6.10.m10.1.1.1.1" xref="S6.p6.10.m10.1.1.1.1.1.cmml"><mo id="S6.p6.10.m10.1.1.1.1.2" stretchy="false" xref="S6.p6.10.m10.1.1.1.1.1.cmml">(</mo><mrow id="S6.p6.10.m10.1.1.1.1.1" xref="S6.p6.10.m10.1.1.1.1.1.cmml"><msub id="S6.p6.10.m10.1.1.1.1.1.1" xref="S6.p6.10.m10.1.1.1.1.1.1.cmml"><mo id="S6.p6.10.m10.1.1.1.1.1.1.2" lspace="0em" rspace="0.167em" xref="S6.p6.10.m10.1.1.1.1.1.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S6.p6.10.m10.1.1.1.1.1.1.3" xref="S6.p6.10.m10.1.1.1.1.1.1.3.cmml">𝒟</mi></msub><mi id="S6.p6.10.m10.1.1.1.1.1.2" xref="S6.p6.10.m10.1.1.1.1.1.2.cmml">F</mi></mrow><mo id="S6.p6.10.m10.1.1.1.1.3" stretchy="false" xref="S6.p6.10.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.10.m10.1b"><apply id="S6.p6.10.m10.1.1.cmml" xref="S6.p6.10.m10.1.1"><times id="S6.p6.10.m10.1.1.2.cmml" xref="S6.p6.10.m10.1.1.2"></times><ci id="S6.p6.10.m10.1.1.3.cmml" xref="S6.p6.10.m10.1.1.3">Δ</ci><apply id="S6.p6.10.m10.1.1.1.1.1.cmml" xref="S6.p6.10.m10.1.1.1.1"><apply id="S6.p6.10.m10.1.1.1.1.1.1.cmml" xref="S6.p6.10.m10.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p6.10.m10.1.1.1.1.1.1.1.cmml" xref="S6.p6.10.m10.1.1.1.1.1.1">subscript</csymbol><ci id="S6.p6.10.m10.1.1.1.1.1.1.2.cmml" xref="S6.p6.10.m10.1.1.1.1.1.1.2">hocolim</ci><ci id="S6.p6.10.m10.1.1.1.1.1.1.3.cmml" xref="S6.p6.10.m10.1.1.1.1.1.1.3">𝒟</ci></apply><ci id="S6.p6.10.m10.1.1.1.1.1.2.cmml" xref="S6.p6.10.m10.1.1.1.1.1.2">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.10.m10.1c">\Delta(\operatorname*{hocolim}_{\mathcal{D}}F)</annotation><annotation encoding="application/x-llamapun" id="S6.p6.10.m10.1d">roman_Δ ( roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F )</annotation></semantics></math> to the category of <math alttext="R" class="ltx_Math" display="inline" id="S6.p6.11.m11.1"><semantics id="S6.p6.11.m11.1a"><mi id="S6.p6.11.m11.1.1" xref="S6.p6.11.m11.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S6.p6.11.m11.1b"><ci id="S6.p6.11.m11.1.1.cmml" xref="S6.p6.11.m11.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.11.m11.1c">R</annotation><annotation encoding="application/x-llamapun" id="S6.p6.11.m11.1d">italic_R</annotation></semantics></math>-modules.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem5.1.1.1">Lemma 6.5</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem5.p1"> <p class="ltx_p" id="S6.Thmtheorem5.p1.3"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem5.p1.3.3">Let <math alttext="U:F_{1}\to F_{2}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.1.1.m1.1"><semantics id="S6.Thmtheorem5.p1.1.1.m1.1a"><mrow id="S6.Thmtheorem5.p1.1.1.m1.1.1" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem5.p1.1.1.m1.1.1.2" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.2.cmml">U</mi><mo id="S6.Thmtheorem5.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S6.Thmtheorem5.p1.1.1.m1.1.1.3" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.cmml"><msub id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.cmml"><mi id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.2" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.2.cmml">F</mi><mn id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.3" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.1.cmml">→</mo><msub id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.cmml"><mi id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.2" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.2.cmml">F</mi><mn id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.3" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.1.1.m1.1b"><apply id="S6.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1"><ci id="S6.Thmtheorem5.p1.1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.1">:</ci><ci id="S6.Thmtheorem5.p1.1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.2">𝑈</ci><apply id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3"><ci id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.1">→</ci><apply id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.1.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.2.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.2">𝐹</ci><cn id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.2.3">1</cn></apply><apply id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.1.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.2.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.2">𝐹</ci><cn id="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.1.1.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.1.1.m1.1c">U:F_{1}\to F_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.1.1.m1.1d">italic_U : italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be a natural transformation between functors <math alttext="F_{1},F_{2}:\mathcal{D}\to Cat" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.2.2.m2.2"><semantics id="S6.Thmtheorem5.p1.2.2.m2.2a"><mrow id="S6.Thmtheorem5.p1.2.2.m2.2.2" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.cmml"><mrow id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.3.cmml"><msub id="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1" xref="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.2" xref="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.2.cmml">F</mi><mn id="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.3" xref="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.3" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.3.cmml">,</mo><msub id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.cmml"><mi id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.2" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.2.cmml">F</mi><mn id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.3" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.3.cmml">:</mo><mrow id="S6.Thmtheorem5.p1.2.2.m2.2.2.4" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.2" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.2.cmml">𝒟</mi><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.1" stretchy="false" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.1.cmml">→</mo><mrow id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.cmml"><mi id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.2" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.2.cmml">C</mi><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.1" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.1.cmml">⁢</mo><mi id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.3" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.3.cmml">a</mi><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.1a" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.1.cmml">⁢</mo><mi id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.4" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.4.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.2.2.m2.2b"><apply id="S6.Thmtheorem5.p1.2.2.m2.2.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2"><ci id="S6.Thmtheorem5.p1.2.2.m2.2.2.3.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.3">:</ci><list id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.3.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2"><apply id="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.2">𝐹</ci><cn id="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2">subscript</csymbol><ci id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.2">𝐹</ci><cn id="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.2.2.2.3">2</cn></apply></list><apply id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4"><ci id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.1">→</ci><ci id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.2">𝒟</ci><apply id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3"><times id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.1"></times><ci id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.2">𝐶</ci><ci id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.3.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.3">𝑎</ci><ci id="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.4.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.4.3.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.2.2.m2.2c">F_{1},F_{2}:\mathcal{D}\to Cat</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.2.2.m2.2d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : caligraphic_D → italic_C italic_a italic_t</annotation></semantics></math>. Then <math alttext="U" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.3.3.m3.1"><semantics id="S6.Thmtheorem5.p1.3.3.m3.1a"><mi id="S6.Thmtheorem5.p1.3.3.m3.1.1" xref="S6.Thmtheorem5.p1.3.3.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.3.3.m3.1b"><ci id="S6.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem5.p1.3.3.m3.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.3.3.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.3.3.m3.1d">italic_U</annotation></semantics></math> induces a bisimplicial map</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex75"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathfrak{U}:N(\mathcal{D};NF_{1})\to N(\mathcal{D};NF_{2})," class="ltx_Math" display="block" id="S6.Ex75.m1.3"><semantics id="S6.Ex75.m1.3a"><mrow id="S6.Ex75.m1.3.3.1" xref="S6.Ex75.m1.3.3.1.1.cmml"><mrow id="S6.Ex75.m1.3.3.1.1" xref="S6.Ex75.m1.3.3.1.1.cmml"><mi id="S6.Ex75.m1.3.3.1.1.4" xref="S6.Ex75.m1.3.3.1.1.4.cmml">𝔘</mi><mo id="S6.Ex75.m1.3.3.1.1.3" lspace="0.278em" rspace="0.278em" xref="S6.Ex75.m1.3.3.1.1.3.cmml">:</mo><mrow id="S6.Ex75.m1.3.3.1.1.2" xref="S6.Ex75.m1.3.3.1.1.2.cmml"><mrow id="S6.Ex75.m1.3.3.1.1.1.1" xref="S6.Ex75.m1.3.3.1.1.1.1.cmml"><mi id="S6.Ex75.m1.3.3.1.1.1.1.3" xref="S6.Ex75.m1.3.3.1.1.1.1.3.cmml">N</mi><mo id="S6.Ex75.m1.3.3.1.1.1.1.2" xref="S6.Ex75.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex75.m1.3.3.1.1.1.1.1.1" xref="S6.Ex75.m1.3.3.1.1.1.1.1.2.cmml"><mo id="S6.Ex75.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex75.m1.3.3.1.1.1.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex75.m1.1.1" xref="S6.Ex75.m1.1.1.cmml">𝒟</mi><mo id="S6.Ex75.m1.3.3.1.1.1.1.1.1.3" xref="S6.Ex75.m1.3.3.1.1.1.1.1.2.cmml">;</mo><mrow id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.2" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.2.cmml">N</mi><mo id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.1" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.2" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.2.cmml">F</mi><mn id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.3" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.Ex75.m1.3.3.1.1.1.1.1.1.4" stretchy="false" xref="S6.Ex75.m1.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Ex75.m1.3.3.1.1.2.3" stretchy="false" xref="S6.Ex75.m1.3.3.1.1.2.3.cmml">→</mo><mrow id="S6.Ex75.m1.3.3.1.1.2.2" xref="S6.Ex75.m1.3.3.1.1.2.2.cmml"><mi id="S6.Ex75.m1.3.3.1.1.2.2.3" xref="S6.Ex75.m1.3.3.1.1.2.2.3.cmml">N</mi><mo id="S6.Ex75.m1.3.3.1.1.2.2.2" xref="S6.Ex75.m1.3.3.1.1.2.2.2.cmml">⁢</mo><mrow id="S6.Ex75.m1.3.3.1.1.2.2.1.1" xref="S6.Ex75.m1.3.3.1.1.2.2.1.2.cmml"><mo id="S6.Ex75.m1.3.3.1.1.2.2.1.1.2" stretchy="false" xref="S6.Ex75.m1.3.3.1.1.2.2.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex75.m1.2.2" xref="S6.Ex75.m1.2.2.cmml">𝒟</mi><mo id="S6.Ex75.m1.3.3.1.1.2.2.1.1.3" xref="S6.Ex75.m1.3.3.1.1.2.2.1.2.cmml">;</mo><mrow id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.cmml"><mi id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.2" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.2.cmml">N</mi><mo id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.1" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.1.cmml">⁢</mo><msub id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.cmml"><mi id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.2" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.2.cmml">F</mi><mn id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.3" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S6.Ex75.m1.3.3.1.1.2.2.1.1.4" stretchy="false" xref="S6.Ex75.m1.3.3.1.1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.Ex75.m1.3.3.1.2" xref="S6.Ex75.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex75.m1.3b"><apply id="S6.Ex75.m1.3.3.1.1.cmml" xref="S6.Ex75.m1.3.3.1"><ci id="S6.Ex75.m1.3.3.1.1.3.cmml" xref="S6.Ex75.m1.3.3.1.1.3">:</ci><ci id="S6.Ex75.m1.3.3.1.1.4.cmml" xref="S6.Ex75.m1.3.3.1.1.4">𝔘</ci><apply id="S6.Ex75.m1.3.3.1.1.2.cmml" xref="S6.Ex75.m1.3.3.1.1.2"><ci id="S6.Ex75.m1.3.3.1.1.2.3.cmml" xref="S6.Ex75.m1.3.3.1.1.2.3">→</ci><apply id="S6.Ex75.m1.3.3.1.1.1.1.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1"><times id="S6.Ex75.m1.3.3.1.1.1.1.2.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.2"></times><ci id="S6.Ex75.m1.3.3.1.1.1.1.3.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.3">𝑁</ci><list id="S6.Ex75.m1.3.3.1.1.1.1.1.2.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1"><ci id="S6.Ex75.m1.1.1.cmml" xref="S6.Ex75.m1.1.1">𝒟</ci><apply id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1"><times id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.1"></times><ci id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.2">𝑁</ci><apply id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.1.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.2.cmml" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.2">𝐹</ci><cn id="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Ex75.m1.3.3.1.1.1.1.1.1.1.3.3">1</cn></apply></apply></list></apply><apply id="S6.Ex75.m1.3.3.1.1.2.2.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2"><times id="S6.Ex75.m1.3.3.1.1.2.2.2.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.2"></times><ci id="S6.Ex75.m1.3.3.1.1.2.2.3.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.3">𝑁</ci><list id="S6.Ex75.m1.3.3.1.1.2.2.1.2.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1"><ci id="S6.Ex75.m1.2.2.cmml" xref="S6.Ex75.m1.2.2">𝒟</ci><apply id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1"><times id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.1.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.1"></times><ci id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.2.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.2">𝑁</ci><apply id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.1.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3">subscript</csymbol><ci id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.2.cmml" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.2">𝐹</ci><cn id="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.3.cmml" type="integer" xref="S6.Ex75.m1.3.3.1.1.2.2.1.1.1.3.3">2</cn></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex75.m1.3c">\mathfrak{U}:N(\mathcal{D};NF_{1})\to N(\mathcal{D};NF_{2}),</annotation><annotation encoding="application/x-llamapun" id="S6.Ex75.m1.3d">fraktur_U : italic_N ( caligraphic_D ; italic_N italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) → italic_N ( caligraphic_D ; italic_N italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmtheorem5.p1.4"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem5.p1.4.1">hence a simplicial map <math alttext="\mathfrak{u}:\operatorname*{hocolim}_{\mathcal{D}}NF_{1}\to\operatorname*{% hocolim}_{\mathcal{D}}NF_{2}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.4.1.m1.1"><semantics id="S6.Thmtheorem5.p1.4.1.m1.1a"><mrow id="S6.Thmtheorem5.p1.4.1.m1.1.1" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.cmml"><mi id="S6.Thmtheorem5.p1.4.1.m1.1.1.2" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.2.cmml">𝔲</mi><mo id="S6.Thmtheorem5.p1.4.1.m1.1.1.1" lspace="0.278em" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.1.cmml">:</mo><mrow id="S6.Thmtheorem5.p1.4.1.m1.1.1.3" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.cmml"><mrow id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.cmml"><msub id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.cmml"><mo id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.2" lspace="0.111em" rspace="0.167em" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.3" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.3.cmml">𝒟</mi></msub><mrow id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.cmml"><mi id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.2" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.2.cmml">N</mi><mo id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.1" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.1.cmml">⁢</mo><msub id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.cmml"><mi id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.2" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.2.cmml">F</mi><mn id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.3" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.3.cmml">1</mn></msub></mrow></mrow><mo id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.1" rspace="0.1389em" stretchy="false" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.1.cmml">→</mo><mrow id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.cmml"><msub id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.cmml"><mo id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.2" lspace="0.1389em" rspace="0.167em" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.3" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.3.cmml">𝒟</mi></msub><mrow id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.cmml"><mi id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.2" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.2.cmml">N</mi><mo id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.1" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.1.cmml">⁢</mo><msub id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.cmml"><mi id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.2" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.2.cmml">F</mi><mn id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.3" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.3.cmml">2</mn></msub></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.4.1.m1.1b"><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1"><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.1">:</ci><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.2">𝔲</ci><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3"><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.1">→</ci><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2"><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1">subscript</csymbol><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.2">hocolim</ci><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.3.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.1.3">𝒟</ci></apply><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2"><times id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.1"></times><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.2">𝑁</ci><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3">subscript</csymbol><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.2">𝐹</ci><cn id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.2.2.3.3">1</cn></apply></apply></apply><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3"><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1">subscript</csymbol><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.2">hocolim</ci><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.3.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.1.3">𝒟</ci></apply><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2"><times id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.1"></times><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.2">𝑁</ci><apply id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.1.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3">subscript</csymbol><ci id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.2.cmml" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.2">𝐹</ci><cn id="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.4.1.m1.1.1.3.3.2.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.4.1.m1.1c">\mathfrak{u}:\operatorname*{hocolim}_{\mathcal{D}}NF_{1}\to\operatorname*{% hocolim}_{\mathcal{D}}NF_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.4.1.m1.1d">fraktur_u : roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_N italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_N italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S6.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.4.p1"> <p class="ltx_p" id="S6.4.p1.2">Let <math alttext="\mathfrak{U}" class="ltx_Math" display="inline" id="S6.4.p1.1.m1.1"><semantics id="S6.4.p1.1.m1.1a"><mi id="S6.4.p1.1.m1.1.1" xref="S6.4.p1.1.m1.1.1.cmml">𝔘</mi><annotation-xml encoding="MathML-Content" id="S6.4.p1.1.m1.1b"><ci id="S6.4.p1.1.m1.1.1.cmml" xref="S6.4.p1.1.m1.1.1">𝔘</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.1.m1.1c">\mathfrak{U}</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.1.m1.1d">fraktur_U</annotation></semantics></math> be the map that sends a pair <math alttext="(\sigma,\tau)" class="ltx_Math" display="inline" id="S6.4.p1.2.m2.2"><semantics id="S6.4.p1.2.m2.2a"><mrow id="S6.4.p1.2.m2.2.3.2" xref="S6.4.p1.2.m2.2.3.1.cmml"><mo id="S6.4.p1.2.m2.2.3.2.1" stretchy="false" xref="S6.4.p1.2.m2.2.3.1.cmml">(</mo><mi id="S6.4.p1.2.m2.1.1" xref="S6.4.p1.2.m2.1.1.cmml">σ</mi><mo id="S6.4.p1.2.m2.2.3.2.2" xref="S6.4.p1.2.m2.2.3.1.cmml">,</mo><mi id="S6.4.p1.2.m2.2.2" xref="S6.4.p1.2.m2.2.2.cmml">τ</mi><mo id="S6.4.p1.2.m2.2.3.2.3" stretchy="false" xref="S6.4.p1.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.2.m2.2b"><interval closure="open" id="S6.4.p1.2.m2.2.3.1.cmml" xref="S6.4.p1.2.m2.2.3.2"><ci id="S6.4.p1.2.m2.1.1.cmml" xref="S6.4.p1.2.m2.1.1">𝜎</ci><ci id="S6.4.p1.2.m2.2.2.cmml" xref="S6.4.p1.2.m2.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.2.m2.2c">(\sigma,\tau)</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.2.m2.2d">( italic_σ , italic_τ )</annotation></semantics></math> where</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex76"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma=(d_{0}\to\cdots\to d_{p})\in N\mathcal{D}_{p}\text{ and }\tau=(x_{0}% \smash{\,\mathop{\longrightarrow}\limits^{\beta_{1}}\,}\cdots\smash{\,\mathop{% \longrightarrow}\limits^{\beta_{q}}\,}x_{q})\in NF_{1}(d_{0})_{q}," class="ltx_Math" display="block" id="S6.Ex76.m1.1"><semantics id="S6.Ex76.m1.1a"><mrow id="S6.Ex76.m1.1.1.1" xref="S6.Ex76.m1.1.1.1.1.cmml"><mrow id="S6.Ex76.m1.1.1.1.1" xref="S6.Ex76.m1.1.1.1.1.cmml"><mi id="S6.Ex76.m1.1.1.1.1.5" xref="S6.Ex76.m1.1.1.1.1.5.cmml">σ</mi><mo id="S6.Ex76.m1.1.1.1.1.6" xref="S6.Ex76.m1.1.1.1.1.6.cmml">=</mo><mrow id="S6.Ex76.m1.1.1.1.1.1.1" xref="S6.Ex76.m1.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex76.m1.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex76.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex76.m1.1.1.1.1.1.1.1" xref="S6.Ex76.m1.1.1.1.1.1.1.1.cmml"><msub id="S6.Ex76.m1.1.1.1.1.1.1.1.2" xref="S6.Ex76.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S6.Ex76.m1.1.1.1.1.1.1.1.2.2" xref="S6.Ex76.m1.1.1.1.1.1.1.1.2.2.cmml">d</mi><mn id="S6.Ex76.m1.1.1.1.1.1.1.1.2.3" xref="S6.Ex76.m1.1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.Ex76.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex76.m1.1.1.1.1.1.1.1.3.cmml">→</mo><mi id="S6.Ex76.m1.1.1.1.1.1.1.1.4" mathvariant="normal" xref="S6.Ex76.m1.1.1.1.1.1.1.1.4.cmml">⋯</mi><mo id="S6.Ex76.m1.1.1.1.1.1.1.1.5" stretchy="false" xref="S6.Ex76.m1.1.1.1.1.1.1.1.5.cmml">→</mo><msub id="S6.Ex76.m1.1.1.1.1.1.1.1.6" xref="S6.Ex76.m1.1.1.1.1.1.1.1.6.cmml"><mi id="S6.Ex76.m1.1.1.1.1.1.1.1.6.2" xref="S6.Ex76.m1.1.1.1.1.1.1.1.6.2.cmml">d</mi><mi id="S6.Ex76.m1.1.1.1.1.1.1.1.6.3" xref="S6.Ex76.m1.1.1.1.1.1.1.1.6.3.cmml">p</mi></msub></mrow><mo id="S6.Ex76.m1.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex76.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex76.m1.1.1.1.1.7" xref="S6.Ex76.m1.1.1.1.1.7.cmml">∈</mo><mrow id="S6.Ex76.m1.1.1.1.1.8" xref="S6.Ex76.m1.1.1.1.1.8.cmml"><mi id="S6.Ex76.m1.1.1.1.1.8.2" xref="S6.Ex76.m1.1.1.1.1.8.2.cmml">N</mi><mo id="S6.Ex76.m1.1.1.1.1.8.1" xref="S6.Ex76.m1.1.1.1.1.8.1.cmml">⁢</mo><msub id="S6.Ex76.m1.1.1.1.1.8.3" xref="S6.Ex76.m1.1.1.1.1.8.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex76.m1.1.1.1.1.8.3.2" xref="S6.Ex76.m1.1.1.1.1.8.3.2.cmml">𝒟</mi><mi id="S6.Ex76.m1.1.1.1.1.8.3.3" xref="S6.Ex76.m1.1.1.1.1.8.3.3.cmml">p</mi></msub><mo id="S6.Ex76.m1.1.1.1.1.8.1a" xref="S6.Ex76.m1.1.1.1.1.8.1.cmml">⁢</mo><mtext id="S6.Ex76.m1.1.1.1.1.8.4" xref="S6.Ex76.m1.1.1.1.1.8.4a.cmml"> and </mtext><mo id="S6.Ex76.m1.1.1.1.1.8.1b" xref="S6.Ex76.m1.1.1.1.1.8.1.cmml">⁢</mo><mi id="S6.Ex76.m1.1.1.1.1.8.5" xref="S6.Ex76.m1.1.1.1.1.8.5.cmml">τ</mi></mrow><mo id="S6.Ex76.m1.1.1.1.1.9" xref="S6.Ex76.m1.1.1.1.1.9.cmml">=</mo><mrow id="S6.Ex76.m1.1.1.1.1.2.1" xref="S6.Ex76.m1.1.1.1.1.2.1.1.cmml"><mo id="S6.Ex76.m1.1.1.1.1.2.1.2" stretchy="false" xref="S6.Ex76.m1.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S6.Ex76.m1.1.1.1.1.2.1.1" xref="S6.Ex76.m1.1.1.1.1.2.1.1.cmml"><msub id="S6.Ex76.m1.1.1.1.1.2.1.1.2" xref="S6.Ex76.m1.1.1.1.1.2.1.1.2.cmml"><mi id="S6.Ex76.m1.1.1.1.1.2.1.1.2.2" xref="S6.Ex76.m1.1.1.1.1.2.1.1.2.2.cmml">x</mi><mn id="S6.Ex76.m1.1.1.1.1.2.1.1.2.3" xref="S6.Ex76.m1.1.1.1.1.2.1.1.2.3.cmml">0</mn></msub><mo id="S6.Ex76.m1.1.1.1.1.2.1.1.1" lspace="0.167em" xref="S6.Ex76.m1.1.1.1.1.2.1.1.1.cmml">⁢</mo><mrow id="S6.Ex76.m1.1.1.1.1.2.1.1.3" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.cmml"><mover id="S6.Ex76.m1.1.1.1.1.2.1.1.3.1" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.cmml"><mo id="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.2.cmml">⟶</mo><msub id="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.3" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.3.cmml"><mi id="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.3.2" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.3.2.cmml">β</mi><mn id="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.3.3" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.1.3.3.cmml">1</mn></msub></mover><mrow id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.cmml"><mi id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.2" mathvariant="normal" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.2.cmml">⋯</mi><mo id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.1" lspace="0.337em" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.1.cmml">⁢</mo><mrow id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.cmml"><mover id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.cmml"><mo id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.2.cmml">⟶</mo><msub id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.3" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.3.cmml"><mi id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.3.2" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.3.2.cmml">β</mi><mi id="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.3.3" xref="S6.Ex76.m1.1.1.1.1.2.1.1.3.2.3.1.3.3.cmml">q</mi></msub></mover><msub 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NF_{1}(d_{0})_{q},</annotation><annotation encoding="application/x-llamapun" id="S6.Ex76.m1.1d">italic_σ = ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ⋯ → italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ∈ italic_N caligraphic_D start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and italic_τ = ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟶ start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ∈ italic_N italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.4.p1.3">to the pair <math alttext="(\sigma,\tau^{\prime})" class="ltx_Math" display="inline" id="S6.4.p1.3.m1.2"><semantics id="S6.4.p1.3.m1.2a"><mrow id="S6.4.p1.3.m1.2.2.1" xref="S6.4.p1.3.m1.2.2.2.cmml"><mo id="S6.4.p1.3.m1.2.2.1.2" stretchy="false" xref="S6.4.p1.3.m1.2.2.2.cmml">(</mo><mi id="S6.4.p1.3.m1.1.1" xref="S6.4.p1.3.m1.1.1.cmml">σ</mi><mo id="S6.4.p1.3.m1.2.2.1.3" xref="S6.4.p1.3.m1.2.2.2.cmml">,</mo><msup id="S6.4.p1.3.m1.2.2.1.1" xref="S6.4.p1.3.m1.2.2.1.1.cmml"><mi id="S6.4.p1.3.m1.2.2.1.1.2" xref="S6.4.p1.3.m1.2.2.1.1.2.cmml">τ</mi><mo id="S6.4.p1.3.m1.2.2.1.1.3" xref="S6.4.p1.3.m1.2.2.1.1.3.cmml">′</mo></msup><mo id="S6.4.p1.3.m1.2.2.1.4" stretchy="false" xref="S6.4.p1.3.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.3.m1.2b"><interval closure="open" id="S6.4.p1.3.m1.2.2.2.cmml" xref="S6.4.p1.3.m1.2.2.1"><ci id="S6.4.p1.3.m1.1.1.cmml" xref="S6.4.p1.3.m1.1.1">𝜎</ci><apply id="S6.4.p1.3.m1.2.2.1.1.cmml" xref="S6.4.p1.3.m1.2.2.1.1"><csymbol cd="ambiguous" id="S6.4.p1.3.m1.2.2.1.1.1.cmml" xref="S6.4.p1.3.m1.2.2.1.1">superscript</csymbol><ci id="S6.4.p1.3.m1.2.2.1.1.2.cmml" xref="S6.4.p1.3.m1.2.2.1.1.2">𝜏</ci><ci id="S6.4.p1.3.m1.2.2.1.1.3.cmml" xref="S6.4.p1.3.m1.2.2.1.1.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.3.m1.2c">(\sigma,\tau^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.3.m1.2d">( italic_σ , italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> where</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex77"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\tau^{\prime}=U(d_{0})(\tau):=(U(d_{0})(x_{0})\xrightarrow{U(d_{0})(\beta_{1})% }U(d_{0})(x_{1})\xrightarrow{}\cdots\xrightarrow{U(d_{0})(\beta_{q})}U(d_{0})(% x_{q}))" class="ltx_Math" display="block" id="S6.Ex77.m1.7"><semantics id="S6.Ex77.m1.7a"><mrow 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id="S6.Ex77.m1.6.6.1.2a" xref="S6.Ex77.m1.6.6.1.2.cmml">⁢</mo><mrow id="S6.Ex77.m1.6.6.1.4.2" xref="S6.Ex77.m1.6.6.1.cmml"><mo id="S6.Ex77.m1.6.6.1.4.2.1" stretchy="false" xref="S6.Ex77.m1.6.6.1.cmml">(</mo><mi id="S6.Ex77.m1.5.5" xref="S6.Ex77.m1.5.5.cmml">τ</mi><mo id="S6.Ex77.m1.6.6.1.4.2.2" rspace="0.278em" stretchy="false" xref="S6.Ex77.m1.6.6.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex77.m1.7.7.6" rspace="0.278em" xref="S6.Ex77.m1.7.7.6.cmml">:=</mo><mrow id="S6.Ex77.m1.7.7.2.1" xref="S6.Ex77.m1.7.7.2.1.1.cmml"><mo id="S6.Ex77.m1.7.7.2.1.2" stretchy="false" xref="S6.Ex77.m1.7.7.2.1.1.cmml">(</mo><mrow id="S6.Ex77.m1.7.7.2.1.1" xref="S6.Ex77.m1.7.7.2.1.1.cmml"><mrow id="S6.Ex77.m1.7.7.2.1.1.2" xref="S6.Ex77.m1.7.7.2.1.1.2.cmml"><mi id="S6.Ex77.m1.7.7.2.1.1.2.4" xref="S6.Ex77.m1.7.7.2.1.1.2.4.cmml">U</mi><mo id="S6.Ex77.m1.7.7.2.1.1.2.3" xref="S6.Ex77.m1.7.7.2.1.1.2.3.cmml">⁢</mo><mrow id="S6.Ex77.m1.7.7.2.1.1.1.1.1" xref="S6.Ex77.m1.7.7.2.1.1.1.1.1.1.cmml"><mo 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xref="S6.Ex77.m1.7.7.2.1.1.2.2.1.1.3.cmml">0</mn></msub><mo id="S6.Ex77.m1.7.7.2.1.1.2.2.1.3" stretchy="false" xref="S6.Ex77.m1.7.7.2.1.1.2.2.1.1.cmml">)</mo></mrow></mrow><mover accent="true" id="S6.Ex77.m1.2.2" xref="S6.Ex77.m1.2.2.cmml"><mo id="S6.Ex77.m1.2.2.3" stretchy="false" xref="S6.Ex77.m1.2.2.3.cmml">→</mo><mrow id="S6.Ex77.m1.2.2.2" xref="S6.Ex77.m1.2.2.2.cmml"><mi id="S6.Ex77.m1.2.2.2.4" xref="S6.Ex77.m1.2.2.2.4.cmml">U</mi><mo id="S6.Ex77.m1.2.2.2.3" xref="S6.Ex77.m1.2.2.2.3.cmml">⁢</mo><mrow id="S6.Ex77.m1.1.1.1.1.1" xref="S6.Ex77.m1.1.1.1.1.1.1.cmml"><mo id="S6.Ex77.m1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex77.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex77.m1.1.1.1.1.1.1" xref="S6.Ex77.m1.1.1.1.1.1.1.cmml"><mi id="S6.Ex77.m1.1.1.1.1.1.1.2" xref="S6.Ex77.m1.1.1.1.1.1.1.2.cmml">d</mi><mn id="S6.Ex77.m1.1.1.1.1.1.1.3" xref="S6.Ex77.m1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.Ex77.m1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex77.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo 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xref="S6.Ex77.m1.4.4.2.3"></times><ci id="S6.Ex77.m1.4.4.2.4.cmml" xref="S6.Ex77.m1.4.4.2.4">𝑈</ci><apply id="S6.Ex77.m1.3.3.1.1.1.1.cmml" xref="S6.Ex77.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.Ex77.m1.3.3.1.1.1.1.1.cmml" xref="S6.Ex77.m1.3.3.1.1.1">subscript</csymbol><ci id="S6.Ex77.m1.3.3.1.1.1.1.2.cmml" xref="S6.Ex77.m1.3.3.1.1.1.1.2">𝑑</ci><cn id="S6.Ex77.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="S6.Ex77.m1.3.3.1.1.1.1.3">0</cn></apply><apply id="S6.Ex77.m1.4.4.2.2.1.1.cmml" xref="S6.Ex77.m1.4.4.2.2.1"><csymbol cd="ambiguous" id="S6.Ex77.m1.4.4.2.2.1.1.1.cmml" xref="S6.Ex77.m1.4.4.2.2.1">subscript</csymbol><ci id="S6.Ex77.m1.4.4.2.2.1.1.2.cmml" xref="S6.Ex77.m1.4.4.2.2.1.1.2">𝛽</ci><ci id="S6.Ex77.m1.4.4.2.2.1.1.3.cmml" xref="S6.Ex77.m1.4.4.2.2.1.1.3">𝑞</ci></apply></apply><ci id="S6.Ex77.m1.4.4.3.cmml" xref="S6.Ex77.m1.4.4.3">→</ci></apply><share href="https://arxiv.org/html/2503.14659v1#S6.Ex77.m1.7.7.2.1.1.9.cmml" id="S6.Ex77.m1.7.7.2.1.1f.cmml" 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xref="S6.Ex77.m1.7.7.2.1.1.6.2.1.1.3">𝑞</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex77.m1.7c">\tau^{\prime}=U(d_{0})(\tau):=(U(d_{0})(x_{0})\xrightarrow{U(d_{0})(\beta_{1})% }U(d_{0})(x_{1})\xrightarrow{}\cdots\xrightarrow{U(d_{0})(\beta_{q})}U(d_{0})(% x_{q}))</annotation><annotation encoding="application/x-llamapun" id="S6.Ex77.m1.7d">italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_τ ) := ( italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_ARROW start_OVERACCENT italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_OVERACCENT → end_ARROW italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_ARROW start_OVERACCENT end_OVERACCENT → end_ARROW ⋯ start_ARROW start_OVERACCENT italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_β start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) end_OVERACCENT → end_ARROW italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_x start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.4.p1.7">in <math alttext="N(F_{2}(d_{0}))_{q}" class="ltx_Math" display="inline" id="S6.4.p1.4.m1.1"><semantics id="S6.4.p1.4.m1.1a"><mrow id="S6.4.p1.4.m1.1.1" xref="S6.4.p1.4.m1.1.1.cmml"><mi id="S6.4.p1.4.m1.1.1.3" xref="S6.4.p1.4.m1.1.1.3.cmml">N</mi><mo id="S6.4.p1.4.m1.1.1.2" xref="S6.4.p1.4.m1.1.1.2.cmml">⁢</mo><msub id="S6.4.p1.4.m1.1.1.1" xref="S6.4.p1.4.m1.1.1.1.cmml"><mrow id="S6.4.p1.4.m1.1.1.1.1.1" xref="S6.4.p1.4.m1.1.1.1.1.1.1.cmml"><mo id="S6.4.p1.4.m1.1.1.1.1.1.2" stretchy="false" xref="S6.4.p1.4.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.4.p1.4.m1.1.1.1.1.1.1" xref="S6.4.p1.4.m1.1.1.1.1.1.1.cmml"><msub id="S6.4.p1.4.m1.1.1.1.1.1.1.3" xref="S6.4.p1.4.m1.1.1.1.1.1.1.3.cmml"><mi id="S6.4.p1.4.m1.1.1.1.1.1.1.3.2" xref="S6.4.p1.4.m1.1.1.1.1.1.1.3.2.cmml">F</mi><mn id="S6.4.p1.4.m1.1.1.1.1.1.1.3.3" xref="S6.4.p1.4.m1.1.1.1.1.1.1.3.3.cmml">2</mn></msub><mo id="S6.4.p1.4.m1.1.1.1.1.1.1.2" xref="S6.4.p1.4.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.4.p1.4.m1.1.1.1.1.1.1.1.1" xref="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1" xref="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1.2" xref="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1.2.cmml">d</mi><mn id="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1.3" xref="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.4.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.4.p1.4.m1.1.1.1.1.1.3" stretchy="false" xref="S6.4.p1.4.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S6.4.p1.4.m1.1.1.1.3" xref="S6.4.p1.4.m1.1.1.1.3.cmml">q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.4.m1.1b"><apply id="S6.4.p1.4.m1.1.1.cmml" xref="S6.4.p1.4.m1.1.1"><times id="S6.4.p1.4.m1.1.1.2.cmml" xref="S6.4.p1.4.m1.1.1.2"></times><ci id="S6.4.p1.4.m1.1.1.3.cmml" xref="S6.4.p1.4.m1.1.1.3">𝑁</ci><apply id="S6.4.p1.4.m1.1.1.1.cmml" xref="S6.4.p1.4.m1.1.1.1"><csymbol cd="ambiguous" id="S6.4.p1.4.m1.1.1.1.2.cmml" xref="S6.4.p1.4.m1.1.1.1">subscript</csymbol><apply id="S6.4.p1.4.m1.1.1.1.1.1.1.cmml" xref="S6.4.p1.4.m1.1.1.1.1.1"><times id="S6.4.p1.4.m1.1.1.1.1.1.1.2.cmml" xref="S6.4.p1.4.m1.1.1.1.1.1.1.2"></times><apply id="S6.4.p1.4.m1.1.1.1.1.1.1.3.cmml" xref="S6.4.p1.4.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.4.p1.4.m1.1.1.1.1.1.1.3.1.cmml" 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italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math>. 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xref="S6.4.p1.5.m2.6.6.4.cmml"><mrow id="S6.4.p1.5.m2.4.4.2.2.2" xref="S6.4.p1.5.m2.4.4.2.2.3.cmml"><mo id="S6.4.p1.5.m2.4.4.2.2.2.3" stretchy="false" xref="S6.4.p1.5.m2.4.4.2.2.3.cmml">(</mo><mrow id="S6.4.p1.5.m2.3.3.1.1.1.1.1" xref="S6.4.p1.5.m2.3.3.1.1.1.1.2.cmml"><mo id="S6.4.p1.5.m2.3.3.1.1.1.1.1.2" stretchy="false" xref="S6.4.p1.5.m2.3.3.1.1.1.1.2.1.cmml">[</mo><msup id="S6.4.p1.5.m2.3.3.1.1.1.1.1.1" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.cmml"><mi id="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.2" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.2.cmml">p</mi><mo id="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.3" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.4.p1.5.m2.3.3.1.1.1.1.1.3" stretchy="false" xref="S6.4.p1.5.m2.3.3.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S6.4.p1.5.m2.4.4.2.2.2.4" xref="S6.4.p1.5.m2.4.4.2.2.3.cmml">,</mo><mrow id="S6.4.p1.5.m2.4.4.2.2.2.2.1" xref="S6.4.p1.5.m2.4.4.2.2.2.2.2.cmml"><mo id="S6.4.p1.5.m2.4.4.2.2.2.2.1.2" stretchy="false" xref="S6.4.p1.5.m2.4.4.2.2.2.2.2.1.cmml">[</mo><msup id="S6.4.p1.5.m2.4.4.2.2.2.2.1.1" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.cmml"><mi id="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.2" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.2.cmml">q</mi><mo id="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.3" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.3.cmml">′</mo></msup><mo id="S6.4.p1.5.m2.4.4.2.2.2.2.1.3" stretchy="false" xref="S6.4.p1.5.m2.4.4.2.2.2.2.2.1.cmml">]</mo></mrow><mo id="S6.4.p1.5.m2.4.4.2.2.2.5" stretchy="false" xref="S6.4.p1.5.m2.4.4.2.2.3.cmml">)</mo></mrow><mo id="S6.4.p1.5.m2.6.6.4.5" stretchy="false" xref="S6.4.p1.5.m2.6.6.4.5.cmml">→</mo><mrow id="S6.4.p1.5.m2.6.6.4.4.2" xref="S6.4.p1.5.m2.6.6.4.4.3.cmml"><mo id="S6.4.p1.5.m2.6.6.4.4.2.3" stretchy="false" xref="S6.4.p1.5.m2.6.6.4.4.3.cmml">(</mo><mrow id="S6.4.p1.5.m2.5.5.3.3.1.1.2" xref="S6.4.p1.5.m2.5.5.3.3.1.1.1.cmml"><mo id="S6.4.p1.5.m2.5.5.3.3.1.1.2.1" stretchy="false" xref="S6.4.p1.5.m2.5.5.3.3.1.1.1.1.cmml">[</mo><mi id="S6.4.p1.5.m2.1.1" xref="S6.4.p1.5.m2.1.1.cmml">p</mi><mo id="S6.4.p1.5.m2.5.5.3.3.1.1.2.2" stretchy="false" xref="S6.4.p1.5.m2.5.5.3.3.1.1.1.1.cmml">]</mo></mrow><mo id="S6.4.p1.5.m2.6.6.4.4.2.4" xref="S6.4.p1.5.m2.6.6.4.4.3.cmml">,</mo><mrow id="S6.4.p1.5.m2.6.6.4.4.2.2.2" xref="S6.4.p1.5.m2.6.6.4.4.2.2.1.cmml"><mo id="S6.4.p1.5.m2.6.6.4.4.2.2.2.1" stretchy="false" xref="S6.4.p1.5.m2.6.6.4.4.2.2.1.1.cmml">[</mo><mi id="S6.4.p1.5.m2.2.2" xref="S6.4.p1.5.m2.2.2.cmml">q</mi><mo id="S6.4.p1.5.m2.6.6.4.4.2.2.2.2" stretchy="false" xref="S6.4.p1.5.m2.6.6.4.4.2.2.1.1.cmml">]</mo></mrow><mo id="S6.4.p1.5.m2.6.6.4.4.2.5" stretchy="false" xref="S6.4.p1.5.m2.6.6.4.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.5.m2.6b"><apply id="S6.4.p1.5.m2.6.6.cmml" xref="S6.4.p1.5.m2.6.6"><ci id="S6.4.p1.5.m2.6.6.5.cmml" xref="S6.4.p1.5.m2.6.6.5">:</ci><apply id="S6.4.p1.5.m2.6.6.6.cmml" xref="S6.4.p1.5.m2.6.6.6"><times id="S6.4.p1.5.m2.6.6.6.1.cmml" xref="S6.4.p1.5.m2.6.6.6.1"></times><apply id="S6.4.p1.5.m2.6.6.6.2.cmml" xref="S6.4.p1.5.m2.6.6.6.2"><csymbol cd="ambiguous" id="S6.4.p1.5.m2.6.6.6.2.1.cmml" xref="S6.4.p1.5.m2.6.6.6.2">subscript</csymbol><ci id="S6.4.p1.5.m2.6.6.6.2.2.cmml" xref="S6.4.p1.5.m2.6.6.6.2.2">𝑓</ci><ci id="S6.4.p1.5.m2.6.6.6.2.3.cmml" xref="S6.4.p1.5.m2.6.6.6.2.3">ℎ</ci></apply><apply id="S6.4.p1.5.m2.6.6.6.3.cmml" xref="S6.4.p1.5.m2.6.6.6.3"><csymbol cd="ambiguous" id="S6.4.p1.5.m2.6.6.6.3.1.cmml" xref="S6.4.p1.5.m2.6.6.6.3">subscript</csymbol><ci id="S6.4.p1.5.m2.6.6.6.3.2.cmml" xref="S6.4.p1.5.m2.6.6.6.3.2">𝑓</ci><ci id="S6.4.p1.5.m2.6.6.6.3.3.cmml" xref="S6.4.p1.5.m2.6.6.6.3.3">𝑣</ci></apply></apply><apply id="S6.4.p1.5.m2.6.6.4.cmml" xref="S6.4.p1.5.m2.6.6.4"><ci id="S6.4.p1.5.m2.6.6.4.5.cmml" xref="S6.4.p1.5.m2.6.6.4.5">→</ci><interval closure="open" id="S6.4.p1.5.m2.4.4.2.2.3.cmml" xref="S6.4.p1.5.m2.4.4.2.2.2"><apply id="S6.4.p1.5.m2.3.3.1.1.1.1.2.cmml" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1"><csymbol cd="latexml" id="S6.4.p1.5.m2.3.3.1.1.1.1.2.1.cmml" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.cmml" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.1.cmml" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1.1">superscript</csymbol><ci id="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.2.cmml" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.2">𝑝</ci><ci id="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.3.cmml" xref="S6.4.p1.5.m2.3.3.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S6.4.p1.5.m2.4.4.2.2.2.2.2.cmml" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1"><csymbol cd="latexml" id="S6.4.p1.5.m2.4.4.2.2.2.2.2.1.cmml" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.cmml" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.1.cmml" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1.1">superscript</csymbol><ci id="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.2.cmml" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.2">𝑞</ci><ci id="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.3.cmml" xref="S6.4.p1.5.m2.4.4.2.2.2.2.1.1.3">′</ci></apply></apply></interval><interval closure="open" id="S6.4.p1.5.m2.6.6.4.4.3.cmml" xref="S6.4.p1.5.m2.6.6.4.4.2"><apply id="S6.4.p1.5.m2.5.5.3.3.1.1.1.cmml" xref="S6.4.p1.5.m2.5.5.3.3.1.1.2"><csymbol cd="latexml" id="S6.4.p1.5.m2.5.5.3.3.1.1.1.1.cmml" xref="S6.4.p1.5.m2.5.5.3.3.1.1.2.1">delimited-[]</csymbol><ci id="S6.4.p1.5.m2.1.1.cmml" xref="S6.4.p1.5.m2.1.1">𝑝</ci></apply><apply id="S6.4.p1.5.m2.6.6.4.4.2.2.1.cmml" xref="S6.4.p1.5.m2.6.6.4.4.2.2.2"><csymbol cd="latexml" id="S6.4.p1.5.m2.6.6.4.4.2.2.1.1.cmml" xref="S6.4.p1.5.m2.6.6.4.4.2.2.2.1">delimited-[]</csymbol><ci id="S6.4.p1.5.m2.2.2.cmml" xref="S6.4.p1.5.m2.2.2">𝑞</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.5.m2.6c">f_{h}\times f_{v}:([p^{\prime}],[q^{\prime}])\to([p],[q])</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.5.m2.6d">italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT : ( [ italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] , [ italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] ) → ( [ italic_p ] , [ italic_q ] )</annotation></semantics></math> in <math alttext="\Delta\times\Delta" class="ltx_Math" display="inline" id="S6.4.p1.6.m3.1"><semantics id="S6.4.p1.6.m3.1a"><mrow id="S6.4.p1.6.m3.1.1" xref="S6.4.p1.6.m3.1.1.cmml"><mi id="S6.4.p1.6.m3.1.1.2" mathvariant="normal" xref="S6.4.p1.6.m3.1.1.2.cmml">Δ</mi><mo id="S6.4.p1.6.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.4.p1.6.m3.1.1.1.cmml">×</mo><mi id="S6.4.p1.6.m3.1.1.3" mathvariant="normal" xref="S6.4.p1.6.m3.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.6.m3.1b"><apply id="S6.4.p1.6.m3.1.1.cmml" xref="S6.4.p1.6.m3.1.1"><times id="S6.4.p1.6.m3.1.1.1.cmml" xref="S6.4.p1.6.m3.1.1.1"></times><ci id="S6.4.p1.6.m3.1.1.2.cmml" xref="S6.4.p1.6.m3.1.1.2">Δ</ci><ci id="S6.4.p1.6.m3.1.1.3.cmml" xref="S6.4.p1.6.m3.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.6.m3.1c">\Delta\times\Delta</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.6.m3.1d">roman_Δ × roman_Δ</annotation></semantics></math>, and for <math alttext="(\sigma,\tau)\in N(\mathcal{D};NF_{1})" class="ltx_Math" display="inline" id="S6.4.p1.7.m4.4"><semantics id="S6.4.p1.7.m4.4a"><mrow id="S6.4.p1.7.m4.4.4" xref="S6.4.p1.7.m4.4.4.cmml"><mrow id="S6.4.p1.7.m4.4.4.3.2" xref="S6.4.p1.7.m4.4.4.3.1.cmml"><mo id="S6.4.p1.7.m4.4.4.3.2.1" stretchy="false" xref="S6.4.p1.7.m4.4.4.3.1.cmml">(</mo><mi id="S6.4.p1.7.m4.1.1" xref="S6.4.p1.7.m4.1.1.cmml">σ</mi><mo id="S6.4.p1.7.m4.4.4.3.2.2" xref="S6.4.p1.7.m4.4.4.3.1.cmml">,</mo><mi id="S6.4.p1.7.m4.2.2" xref="S6.4.p1.7.m4.2.2.cmml">τ</mi><mo id="S6.4.p1.7.m4.4.4.3.2.3" stretchy="false" xref="S6.4.p1.7.m4.4.4.3.1.cmml">)</mo></mrow><mo id="S6.4.p1.7.m4.4.4.2" xref="S6.4.p1.7.m4.4.4.2.cmml">∈</mo><mrow id="S6.4.p1.7.m4.4.4.1" xref="S6.4.p1.7.m4.4.4.1.cmml"><mi id="S6.4.p1.7.m4.4.4.1.3" xref="S6.4.p1.7.m4.4.4.1.3.cmml">N</mi><mo id="S6.4.p1.7.m4.4.4.1.2" xref="S6.4.p1.7.m4.4.4.1.2.cmml">⁢</mo><mrow id="S6.4.p1.7.m4.4.4.1.1.1" xref="S6.4.p1.7.m4.4.4.1.1.2.cmml"><mo id="S6.4.p1.7.m4.4.4.1.1.1.2" stretchy="false" xref="S6.4.p1.7.m4.4.4.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.4.p1.7.m4.3.3" xref="S6.4.p1.7.m4.3.3.cmml">𝒟</mi><mo id="S6.4.p1.7.m4.4.4.1.1.1.3" xref="S6.4.p1.7.m4.4.4.1.1.2.cmml">;</mo><mrow id="S6.4.p1.7.m4.4.4.1.1.1.1" xref="S6.4.p1.7.m4.4.4.1.1.1.1.cmml"><mi id="S6.4.p1.7.m4.4.4.1.1.1.1.2" xref="S6.4.p1.7.m4.4.4.1.1.1.1.2.cmml">N</mi><mo id="S6.4.p1.7.m4.4.4.1.1.1.1.1" xref="S6.4.p1.7.m4.4.4.1.1.1.1.1.cmml">⁢</mo><msub id="S6.4.p1.7.m4.4.4.1.1.1.1.3" xref="S6.4.p1.7.m4.4.4.1.1.1.1.3.cmml"><mi id="S6.4.p1.7.m4.4.4.1.1.1.1.3.2" xref="S6.4.p1.7.m4.4.4.1.1.1.1.3.2.cmml">F</mi><mn id="S6.4.p1.7.m4.4.4.1.1.1.1.3.3" xref="S6.4.p1.7.m4.4.4.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.4.p1.7.m4.4.4.1.1.1.4" stretchy="false" xref="S6.4.p1.7.m4.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.7.m4.4b"><apply id="S6.4.p1.7.m4.4.4.cmml" xref="S6.4.p1.7.m4.4.4"><in id="S6.4.p1.7.m4.4.4.2.cmml" xref="S6.4.p1.7.m4.4.4.2"></in><interval closure="open" id="S6.4.p1.7.m4.4.4.3.1.cmml" xref="S6.4.p1.7.m4.4.4.3.2"><ci id="S6.4.p1.7.m4.1.1.cmml" xref="S6.4.p1.7.m4.1.1">𝜎</ci><ci id="S6.4.p1.7.m4.2.2.cmml" xref="S6.4.p1.7.m4.2.2">𝜏</ci></interval><apply id="S6.4.p1.7.m4.4.4.1.cmml" xref="S6.4.p1.7.m4.4.4.1"><times id="S6.4.p1.7.m4.4.4.1.2.cmml" xref="S6.4.p1.7.m4.4.4.1.2"></times><ci id="S6.4.p1.7.m4.4.4.1.3.cmml" xref="S6.4.p1.7.m4.4.4.1.3">𝑁</ci><list id="S6.4.p1.7.m4.4.4.1.1.2.cmml" xref="S6.4.p1.7.m4.4.4.1.1.1"><ci id="S6.4.p1.7.m4.3.3.cmml" xref="S6.4.p1.7.m4.3.3">𝒟</ci><apply id="S6.4.p1.7.m4.4.4.1.1.1.1.cmml" xref="S6.4.p1.7.m4.4.4.1.1.1.1"><times id="S6.4.p1.7.m4.4.4.1.1.1.1.1.cmml" xref="S6.4.p1.7.m4.4.4.1.1.1.1.1"></times><ci id="S6.4.p1.7.m4.4.4.1.1.1.1.2.cmml" xref="S6.4.p1.7.m4.4.4.1.1.1.1.2">𝑁</ci><apply id="S6.4.p1.7.m4.4.4.1.1.1.1.3.cmml" xref="S6.4.p1.7.m4.4.4.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.4.p1.7.m4.4.4.1.1.1.1.3.1.cmml" xref="S6.4.p1.7.m4.4.4.1.1.1.1.3">subscript</csymbol><ci id="S6.4.p1.7.m4.4.4.1.1.1.1.3.2.cmml" xref="S6.4.p1.7.m4.4.4.1.1.1.1.3.2">𝐹</ci><cn id="S6.4.p1.7.m4.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S6.4.p1.7.m4.4.4.1.1.1.1.3.3">1</cn></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.7.m4.4c">(\sigma,\tau)\in N(\mathcal{D};NF_{1})</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.7.m4.4d">( italic_σ , italic_τ ) ∈ italic_N ( caligraphic_D ; italic_N italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex78"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\bigl{(}(f_{h}\times f_{v})^{*}\circ\mathfrak{U}\bigr{)}(\sigma,\tau)=(f_{h}^{% *}(\sigma),F_{2}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(f_{v}^{*}(\tau^{\prime}))" class="ltx_math_unparsed" display="block" id="S6.Ex78.m1.4"><semantics id="S6.Ex78.m1.4a"><mrow id="S6.Ex78.m1.4b"><mrow id="S6.Ex78.m1.4.5"><mo id="S6.Ex78.m1.4.5.1" maxsize="120%" minsize="120%">(</mo><msup id="S6.Ex78.m1.4.5.2"><mrow id="S6.Ex78.m1.4.5.2.2"><mo id="S6.Ex78.m1.4.5.2.2.1" stretchy="false">(</mo><msub id="S6.Ex78.m1.4.5.2.2.2"><mi id="S6.Ex78.m1.4.5.2.2.2.2">f</mi><mi id="S6.Ex78.m1.4.5.2.2.2.3">h</mi></msub><mo id="S6.Ex78.m1.4.5.2.2.3" lspace="0.222em" rspace="0.222em">×</mo><msub id="S6.Ex78.m1.4.5.2.2.4"><mi id="S6.Ex78.m1.4.5.2.2.4.2">f</mi><mi id="S6.Ex78.m1.4.5.2.2.4.3">v</mi></msub><mo id="S6.Ex78.m1.4.5.2.2.5" rspace="0.055em" stretchy="false">)</mo></mrow><mo id="S6.Ex78.m1.4.5.2.3">∗</mo></msup><mo id="S6.Ex78.m1.4.5.3" rspace="0.222em">∘</mo><mi id="S6.Ex78.m1.4.5.4">𝔘</mi><mo id="S6.Ex78.m1.4.5.5" maxsize="120%" minsize="120%">)</mo></mrow><mrow id="S6.Ex78.m1.4.6"><mo id="S6.Ex78.m1.4.6.1" stretchy="false">(</mo><mi id="S6.Ex78.m1.2.2">σ</mi><mo id="S6.Ex78.m1.4.6.2">,</mo><mi id="S6.Ex78.m1.3.3">τ</mi><mo id="S6.Ex78.m1.4.6.3" stretchy="false">)</mo></mrow><mo id="S6.Ex78.m1.4.7">=</mo><mrow id="S6.Ex78.m1.4.8"><mo id="S6.Ex78.m1.4.8.1" stretchy="false">(</mo><msubsup id="S6.Ex78.m1.4.8.2"><mi id="S6.Ex78.m1.4.8.2.2.2">f</mi><mi id="S6.Ex78.m1.4.8.2.2.3">h</mi><mo id="S6.Ex78.m1.4.8.2.3">∗</mo></msubsup><mrow id="S6.Ex78.m1.4.8.3"><mo id="S6.Ex78.m1.4.8.3.1" stretchy="false">(</mo><mi id="S6.Ex78.m1.4.4">σ</mi><mo id="S6.Ex78.m1.4.8.3.2" stretchy="false">)</mo></mrow><mo id="S6.Ex78.m1.4.8.4">,</mo><msub id="S6.Ex78.m1.4.8.5"><mi id="S6.Ex78.m1.4.8.5.2">F</mi><mn id="S6.Ex78.m1.4.8.5.3">2</mn></msub><mrow id="S6.Ex78.m1.4.8.6"><mo id="S6.Ex78.m1.4.8.6.1" stretchy="false">(</mo><msub id="S6.Ex78.m1.4.8.6.2"><mi id="S6.Ex78.m1.4.8.6.2.2">α</mi><mrow id="S6.Ex78.m1.1.1.1"><msub id="S6.Ex78.m1.1.1.1.3"><mi id="S6.Ex78.m1.1.1.1.3.2">f</mi><mi id="S6.Ex78.m1.1.1.1.3.3">h</mi></msub><mo id="S6.Ex78.m1.1.1.1.2">⁢</mo><mrow id="S6.Ex78.m1.1.1.1.4.2"><mo id="S6.Ex78.m1.1.1.1.4.2.1" stretchy="false">(</mo><mn id="S6.Ex78.m1.1.1.1.1">0</mn><mo id="S6.Ex78.m1.1.1.1.4.2.2" stretchy="false">)</mo></mrow></mrow></msub><mi id="S6.Ex78.m1.4.8.6.3" mathvariant="normal">⋯</mi><msub id="S6.Ex78.m1.4.8.6.4"><mi id="S6.Ex78.m1.4.8.6.4.2">α</mi><mn id="S6.Ex78.m1.4.8.6.4.3">1</mn></msub><mo id="S6.Ex78.m1.4.8.6.5" stretchy="false">)</mo></mrow><mrow id="S6.Ex78.m1.4.8.7"><mo id="S6.Ex78.m1.4.8.7.1" stretchy="false">(</mo><msubsup id="S6.Ex78.m1.4.8.7.2"><mi id="S6.Ex78.m1.4.8.7.2.2.2">f</mi><mi id="S6.Ex78.m1.4.8.7.2.2.3">v</mi><mo id="S6.Ex78.m1.4.8.7.2.3">∗</mo></msubsup><mrow id="S6.Ex78.m1.4.8.7.3"><mo id="S6.Ex78.m1.4.8.7.3.1" stretchy="false">(</mo><msup id="S6.Ex78.m1.4.8.7.3.2"><mi id="S6.Ex78.m1.4.8.7.3.2.2">τ</mi><mo id="S6.Ex78.m1.4.8.7.3.2.3">′</mo></msup><mo id="S6.Ex78.m1.4.8.7.3.3" stretchy="false">)</mo></mrow><mo id="S6.Ex78.m1.4.8.7.4" stretchy="false">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S6.Ex78.m1.4c">\bigl{(}(f_{h}\times f_{v})^{*}\circ\mathfrak{U}\bigr{)}(\sigma,\tau)=(f_{h}^{% *}(\sigma),F_{2}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(f_{v}^{*}(\tau^{\prime}))</annotation><annotation encoding="application/x-llamapun" id="S6.Ex78.m1.4d">( ( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∘ fraktur_U ) ( italic_σ , italic_τ ) = ( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ ) , italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.4.p1.10">and</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex79"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\bigl{(}\mathfrak{U}\circ(f_{h}\times f_{v})^{*}\bigr{)}(\sigma,\tau)=(f_{h}^{% *}(\sigma),\bigl{(}F_{1}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(f_{v}^{*}(\tau))% \bigr{)}^{\prime})." class="ltx_Math" display="block" id="S6.Ex79.m1.6"><semantics id="S6.Ex79.m1.6a"><mrow id="S6.Ex79.m1.6.6.1" xref="S6.Ex79.m1.6.6.1.1.cmml"><mrow id="S6.Ex79.m1.6.6.1.1" xref="S6.Ex79.m1.6.6.1.1.cmml"><mrow id="S6.Ex79.m1.6.6.1.1.1" xref="S6.Ex79.m1.6.6.1.1.1.cmml"><mrow id="S6.Ex79.m1.6.6.1.1.1.1.1" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.cmml"><mo id="S6.Ex79.m1.6.6.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex79.m1.6.6.1.1.1.1.1.1" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.cmml"><mi id="S6.Ex79.m1.6.6.1.1.1.1.1.1.3" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.3.cmml">𝔘</mi><mo id="S6.Ex79.m1.6.6.1.1.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.2.cmml">∘</mo><msup id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.cmml"><mrow id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.2" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.2" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.2.cmml">f</mi><mi id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.3" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.3.cmml">h</mi></msub><mo id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.cmml">×</mo><msub id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.3" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.2" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.2.cmml">f</mi><mi id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.3" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.3.cmml">v</mi></msub></mrow><mo id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.3" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.1.3.cmml">∗</mo></msup></mrow><mo id="S6.Ex79.m1.6.6.1.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S6.Ex79.m1.6.6.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex79.m1.6.6.1.1.1.2" xref="S6.Ex79.m1.6.6.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex79.m1.6.6.1.1.1.3.2" xref="S6.Ex79.m1.6.6.1.1.1.3.1.cmml"><mo id="S6.Ex79.m1.6.6.1.1.1.3.2.1" stretchy="false" xref="S6.Ex79.m1.6.6.1.1.1.3.1.cmml">(</mo><mi id="S6.Ex79.m1.2.2" xref="S6.Ex79.m1.2.2.cmml">σ</mi><mo id="S6.Ex79.m1.6.6.1.1.1.3.2.2" xref="S6.Ex79.m1.6.6.1.1.1.3.1.cmml">,</mo><mi id="S6.Ex79.m1.3.3" xref="S6.Ex79.m1.3.3.cmml">τ</mi><mo id="S6.Ex79.m1.6.6.1.1.1.3.2.3" stretchy="false" xref="S6.Ex79.m1.6.6.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex79.m1.6.6.1.1.4" xref="S6.Ex79.m1.6.6.1.1.4.cmml">=</mo><mrow id="S6.Ex79.m1.6.6.1.1.3.2" xref="S6.Ex79.m1.6.6.1.1.3.3.cmml"><mo id="S6.Ex79.m1.6.6.1.1.3.2.3" stretchy="false" xref="S6.Ex79.m1.6.6.1.1.3.3.cmml">(</mo><mrow id="S6.Ex79.m1.6.6.1.1.2.1.1" xref="S6.Ex79.m1.6.6.1.1.2.1.1.cmml"><msubsup id="S6.Ex79.m1.6.6.1.1.2.1.1.2" xref="S6.Ex79.m1.6.6.1.1.2.1.1.2.cmml"><mi id="S6.Ex79.m1.6.6.1.1.2.1.1.2.2.2" xref="S6.Ex79.m1.6.6.1.1.2.1.1.2.2.2.cmml">f</mi><mi id="S6.Ex79.m1.6.6.1.1.2.1.1.2.2.3" xref="S6.Ex79.m1.6.6.1.1.2.1.1.2.2.3.cmml">h</mi><mo id="S6.Ex79.m1.6.6.1.1.2.1.1.2.3" xref="S6.Ex79.m1.6.6.1.1.2.1.1.2.3.cmml">∗</mo></msubsup><mo id="S6.Ex79.m1.6.6.1.1.2.1.1.1" xref="S6.Ex79.m1.6.6.1.1.2.1.1.1.cmml">⁢</mo><mrow id="S6.Ex79.m1.6.6.1.1.2.1.1.3.2" xref="S6.Ex79.m1.6.6.1.1.2.1.1.cmml"><mo id="S6.Ex79.m1.6.6.1.1.2.1.1.3.2.1" stretchy="false" xref="S6.Ex79.m1.6.6.1.1.2.1.1.cmml">(</mo><mi id="S6.Ex79.m1.4.4" xref="S6.Ex79.m1.4.4.cmml">σ</mi><mo id="S6.Ex79.m1.6.6.1.1.2.1.1.3.2.2" stretchy="false" xref="S6.Ex79.m1.6.6.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex79.m1.6.6.1.1.3.2.4" xref="S6.Ex79.m1.6.6.1.1.3.3.cmml">,</mo><msup 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xref="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.1.1.1.cmml"><msub id="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.1.1.1.2" xref="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.1.1.1.2.cmml"><mi id="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.1.1.1.2.2" xref="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.1.1.1.2.2.cmml">α</mi><mrow id="S6.Ex79.m1.1.1.1" xref="S6.Ex79.m1.1.1.1.cmml"><msub id="S6.Ex79.m1.1.1.1.3" xref="S6.Ex79.m1.1.1.1.3.cmml"><mi id="S6.Ex79.m1.1.1.1.3.2" xref="S6.Ex79.m1.1.1.1.3.2.cmml">f</mi><mi id="S6.Ex79.m1.1.1.1.3.3" xref="S6.Ex79.m1.1.1.1.3.3.cmml">h</mi></msub><mo id="S6.Ex79.m1.1.1.1.2" xref="S6.Ex79.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex79.m1.1.1.1.4.2" xref="S6.Ex79.m1.1.1.1.cmml"><mo id="S6.Ex79.m1.1.1.1.4.2.1" stretchy="false" xref="S6.Ex79.m1.1.1.1.cmml">(</mo><mn id="S6.Ex79.m1.1.1.1.1" xref="S6.Ex79.m1.1.1.1.1.cmml">0</mn><mo id="S6.Ex79.m1.1.1.1.4.2.2" stretchy="false" xref="S6.Ex79.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.1.1.1.1" 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id="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2.2.cmml" xref="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2"><csymbol cd="ambiguous" id="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2.2.1.cmml" xref="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2">subscript</csymbol><ci id="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2.2.2.cmml" xref="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2.2.2">𝑓</ci><ci id="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2.2.3.cmml" xref="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2.2.3">𝑣</ci></apply><times id="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2.3.cmml" xref="S6.Ex79.m1.6.6.1.1.3.2.2.1.1.1.2.1.1.2.3"></times></apply><ci id="S6.Ex79.m1.5.5.cmml" xref="S6.Ex79.m1.5.5">𝜏</ci></apply></apply><ci id="S6.Ex79.m1.6.6.1.1.3.2.2.3.cmml" xref="S6.Ex79.m1.6.6.1.1.3.2.2.3">′</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex79.m1.6c">\bigl{(}\mathfrak{U}\circ(f_{h}\times f_{v})^{*}\bigr{)}(\sigma,\tau)=(f_{h}^{% *}(\sigma),\bigl{(}F_{1}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(f_{v}^{*}(\tau))% \bigr{)}^{\prime}).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex79.m1.6d">( fraktur_U ∘ ( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ( italic_σ , italic_τ ) = ( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_σ ) , ( italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ ) ) ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.4.p1.11">Note that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S7.EGx5"> <tbody id="S6.Ex80"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle F_{2}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(f_{v}^{*}(\tau^{\prime}))" class="ltx_Math" display="inline" id="S6.Ex80.m1.3"><semantics id="S6.Ex80.m1.3a"><mrow id="S6.Ex80.m1.3.3" xref="S6.Ex80.m1.3.3.cmml"><msub id="S6.Ex80.m1.3.3.4" xref="S6.Ex80.m1.3.3.4.cmml"><mi id="S6.Ex80.m1.3.3.4.2" xref="S6.Ex80.m1.3.3.4.2.cmml">F</mi><mn id="S6.Ex80.m1.3.3.4.3" xref="S6.Ex80.m1.3.3.4.3.cmml">2</mn></msub><mo id="S6.Ex80.m1.3.3.3" xref="S6.Ex80.m1.3.3.3.cmml">⁢</mo><mrow id="S6.Ex80.m1.2.2.1.1" xref="S6.Ex80.m1.2.2.1.1.1.cmml"><mo id="S6.Ex80.m1.2.2.1.1.2" stretchy="false" xref="S6.Ex80.m1.2.2.1.1.1.cmml">(</mo><mrow id="S6.Ex80.m1.2.2.1.1.1" xref="S6.Ex80.m1.2.2.1.1.1.cmml"><msub id="S6.Ex80.m1.2.2.1.1.1.2" xref="S6.Ex80.m1.2.2.1.1.1.2.cmml"><mi id="S6.Ex80.m1.2.2.1.1.1.2.2" xref="S6.Ex80.m1.2.2.1.1.1.2.2.cmml">α</mi><mrow id="S6.Ex80.m1.1.1.1" xref="S6.Ex80.m1.1.1.1.cmml"><msub id="S6.Ex80.m1.1.1.1.3" xref="S6.Ex80.m1.1.1.1.3.cmml"><mi id="S6.Ex80.m1.1.1.1.3.2" xref="S6.Ex80.m1.1.1.1.3.2.cmml">f</mi><mi id="S6.Ex80.m1.1.1.1.3.3" xref="S6.Ex80.m1.1.1.1.3.3.cmml">h</mi></msub><mo id="S6.Ex80.m1.1.1.1.2" xref="S6.Ex80.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex80.m1.1.1.1.4.2" xref="S6.Ex80.m1.1.1.1.cmml"><mo id="S6.Ex80.m1.1.1.1.4.2.1" stretchy="false" xref="S6.Ex80.m1.1.1.1.cmml">(</mo><mn id="S6.Ex80.m1.1.1.1.1" xref="S6.Ex80.m1.1.1.1.1.cmml">0</mn><mo id="S6.Ex80.m1.1.1.1.4.2.2" stretchy="false" xref="S6.Ex80.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Ex80.m1.2.2.1.1.1.1" xref="S6.Ex80.m1.2.2.1.1.1.1.cmml">⁢</mo><mi id="S6.Ex80.m1.2.2.1.1.1.3" mathvariant="normal" xref="S6.Ex80.m1.2.2.1.1.1.3.cmml">⋯</mi><mo id="S6.Ex80.m1.2.2.1.1.1.1a" xref="S6.Ex80.m1.2.2.1.1.1.1.cmml">⁢</mo><msub id="S6.Ex80.m1.2.2.1.1.1.4" xref="S6.Ex80.m1.2.2.1.1.1.4.cmml"><mi id="S6.Ex80.m1.2.2.1.1.1.4.2" xref="S6.Ex80.m1.2.2.1.1.1.4.2.cmml">α</mi><mn id="S6.Ex80.m1.2.2.1.1.1.4.3" xref="S6.Ex80.m1.2.2.1.1.1.4.3.cmml">1</mn></msub></mrow><mo id="S6.Ex80.m1.2.2.1.1.3" stretchy="false" xref="S6.Ex80.m1.2.2.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex80.m1.3.3.3a" xref="S6.Ex80.m1.3.3.3.cmml">⁢</mo><mrow id="S6.Ex80.m1.3.3.2.1" xref="S6.Ex80.m1.3.3.2.1.1.cmml"><mo id="S6.Ex80.m1.3.3.2.1.2" stretchy="false" xref="S6.Ex80.m1.3.3.2.1.1.cmml">(</mo><mrow id="S6.Ex80.m1.3.3.2.1.1" xref="S6.Ex80.m1.3.3.2.1.1.cmml"><msubsup id="S6.Ex80.m1.3.3.2.1.1.3" xref="S6.Ex80.m1.3.3.2.1.1.3.cmml"><mi id="S6.Ex80.m1.3.3.2.1.1.3.2.2" xref="S6.Ex80.m1.3.3.2.1.1.3.2.2.cmml">f</mi><mi id="S6.Ex80.m1.3.3.2.1.1.3.2.3" xref="S6.Ex80.m1.3.3.2.1.1.3.2.3.cmml">v</mi><mo id="S6.Ex80.m1.3.3.2.1.1.3.3" xref="S6.Ex80.m1.3.3.2.1.1.3.3.cmml">∗</mo></msubsup><mo 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xref="S6.Ex80.m1.3.3.4">subscript</csymbol><ci id="S6.Ex80.m1.3.3.4.2.cmml" xref="S6.Ex80.m1.3.3.4.2">𝐹</ci><cn id="S6.Ex80.m1.3.3.4.3.cmml" type="integer" xref="S6.Ex80.m1.3.3.4.3">2</cn></apply><apply id="S6.Ex80.m1.2.2.1.1.1.cmml" xref="S6.Ex80.m1.2.2.1.1"><times id="S6.Ex80.m1.2.2.1.1.1.1.cmml" xref="S6.Ex80.m1.2.2.1.1.1.1"></times><apply id="S6.Ex80.m1.2.2.1.1.1.2.cmml" xref="S6.Ex80.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex80.m1.2.2.1.1.1.2.1.cmml" xref="S6.Ex80.m1.2.2.1.1.1.2">subscript</csymbol><ci id="S6.Ex80.m1.2.2.1.1.1.2.2.cmml" xref="S6.Ex80.m1.2.2.1.1.1.2.2">𝛼</ci><apply id="S6.Ex80.m1.1.1.1.cmml" xref="S6.Ex80.m1.1.1.1"><times id="S6.Ex80.m1.1.1.1.2.cmml" xref="S6.Ex80.m1.1.1.1.2"></times><apply id="S6.Ex80.m1.1.1.1.3.cmml" xref="S6.Ex80.m1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex80.m1.1.1.1.3.1.cmml" xref="S6.Ex80.m1.1.1.1.3">subscript</csymbol><ci id="S6.Ex80.m1.1.1.1.3.2.cmml" xref="S6.Ex80.m1.1.1.1.3.2">𝑓</ci><ci id="S6.Ex80.m1.1.1.1.3.3.cmml" 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xref="S6.Ex80.m2.4.4.2.4.3.cmml">2</mn></msub><mo id="S6.Ex80.m2.4.4.2.3" xref="S6.Ex80.m2.4.4.2.3.cmml">⁢</mo><mrow id="S6.Ex80.m2.3.3.1.1.1" xref="S6.Ex80.m2.3.3.1.1.1.1.cmml"><mo id="S6.Ex80.m2.3.3.1.1.1.2" stretchy="false" xref="S6.Ex80.m2.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.Ex80.m2.3.3.1.1.1.1" xref="S6.Ex80.m2.3.3.1.1.1.1.cmml"><msub id="S6.Ex80.m2.3.3.1.1.1.1.2" xref="S6.Ex80.m2.3.3.1.1.1.1.2.cmml"><mi id="S6.Ex80.m2.3.3.1.1.1.1.2.2" xref="S6.Ex80.m2.3.3.1.1.1.1.2.2.cmml">α</mi><mrow id="S6.Ex80.m2.1.1.1" xref="S6.Ex80.m2.1.1.1.cmml"><msub id="S6.Ex80.m2.1.1.1.3" xref="S6.Ex80.m2.1.1.1.3.cmml"><mi id="S6.Ex80.m2.1.1.1.3.2" xref="S6.Ex80.m2.1.1.1.3.2.cmml">f</mi><mi id="S6.Ex80.m2.1.1.1.3.3" xref="S6.Ex80.m2.1.1.1.3.3.cmml">h</mi></msub><mo id="S6.Ex80.m2.1.1.1.2" xref="S6.Ex80.m2.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex80.m2.1.1.1.4.2" xref="S6.Ex80.m2.1.1.1.cmml"><mo id="S6.Ex80.m2.1.1.1.4.2.1" stretchy="false" xref="S6.Ex80.m2.1.1.1.cmml">(</mo><mn id="S6.Ex80.m2.1.1.1.1" 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xref="S6.Ex80.m2.2.2">𝜏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex80.m2.4c">\displaystyle=F_{2}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(f_{v}^{*}(U(d_{0})(\tau% )))</annotation><annotation encoding="application/x-llamapun" id="S6.Ex80.m2.4d">= italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_τ ) ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex81"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td 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xref="S6.Ex81.m1.3.3.1.1.1.1.2.cmml"><mi id="S6.Ex81.m1.3.3.1.1.1.1.2.2" xref="S6.Ex81.m1.3.3.1.1.1.1.2.2.cmml">α</mi><mrow id="S6.Ex81.m1.1.1.1" xref="S6.Ex81.m1.1.1.1.cmml"><msub id="S6.Ex81.m1.1.1.1.3" xref="S6.Ex81.m1.1.1.1.3.cmml"><mi id="S6.Ex81.m1.1.1.1.3.2" xref="S6.Ex81.m1.1.1.1.3.2.cmml">f</mi><mi id="S6.Ex81.m1.1.1.1.3.3" xref="S6.Ex81.m1.1.1.1.3.3.cmml">h</mi></msub><mo id="S6.Ex81.m1.1.1.1.2" xref="S6.Ex81.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex81.m1.1.1.1.4.2" xref="S6.Ex81.m1.1.1.1.cmml"><mo id="S6.Ex81.m1.1.1.1.4.2.1" stretchy="false" xref="S6.Ex81.m1.1.1.1.cmml">(</mo><mn id="S6.Ex81.m1.1.1.1.1" xref="S6.Ex81.m1.1.1.1.1.cmml">0</mn><mo id="S6.Ex81.m1.1.1.1.4.2.2" stretchy="false" xref="S6.Ex81.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Ex81.m1.3.3.1.1.1.1.1" xref="S6.Ex81.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mi id="S6.Ex81.m1.3.3.1.1.1.1.3" mathvariant="normal" xref="S6.Ex81.m1.3.3.1.1.1.1.3.cmml">⋯</mi><mo id="S6.Ex81.m1.3.3.1.1.1.1.1a" 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xref="S6.Ex81.m1.4.4.2.2.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex81.m1.4.4.2.2.1.1.1.1.1" xref="S6.Ex81.m1.4.4.2.2.1.1.1.1.1.cmml"><mi id="S6.Ex81.m1.4.4.2.2.1.1.1.1.1.2" xref="S6.Ex81.m1.4.4.2.2.1.1.1.1.1.2.cmml">d</mi><mn id="S6.Ex81.m1.4.4.2.2.1.1.1.1.1.3" xref="S6.Ex81.m1.4.4.2.2.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.Ex81.m1.4.4.2.2.1.1.1.1.3" stretchy="false" xref="S6.Ex81.m1.4.4.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex81.m1.4.4.2.2.1.1.3a" xref="S6.Ex81.m1.4.4.2.2.1.1.3.cmml">⁢</mo><mrow id="S6.Ex81.m1.4.4.2.2.1.1.2.1" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.cmml"><mo id="S6.Ex81.m1.4.4.2.2.1.1.2.1.2" stretchy="false" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.cmml">(</mo><mrow id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.cmml"><msubsup id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.cmml"><mi id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.2" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.2.cmml">f</mi><mi id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.3" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.3.cmml">v</mi><mo id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.3" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.3.cmml">∗</mo></msubsup><mo id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.1" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.1.cmml">⁢</mo><mrow id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.3.2" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.cmml"><mo id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.3.2.1" stretchy="false" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.cmml">(</mo><mi id="S6.Ex81.m1.2.2" xref="S6.Ex81.m1.2.2.cmml">τ</mi><mo id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.3.2.2" stretchy="false" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex81.m1.4.4.2.2.1.1.2.1.3" stretchy="false" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex81.m1.4.4.2.2.1.3" stretchy="false" xref="S6.Ex81.m1.4.4.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex81.m1.4b"><apply id="S6.Ex81.m1.4.4.cmml" xref="S6.Ex81.m1.4.4"><eq id="S6.Ex81.m1.4.4.3.cmml" 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xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2">superscript</csymbol><apply id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.cmml" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2"><csymbol cd="ambiguous" id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.1.cmml" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2">subscript</csymbol><ci id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.2.cmml" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.2">𝑓</ci><ci id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.3.cmml" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.2.3">𝑣</ci></apply><times id="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.3.cmml" xref="S6.Ex81.m1.4.4.2.2.1.1.2.1.1.2.3"></times></apply><ci id="S6.Ex81.m1.2.2.cmml" xref="S6.Ex81.m1.2.2">𝜏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex81.m1.4c">\displaystyle=F_{2}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(U(d_{0})(f_{v}^{*}(\tau% )))</annotation><annotation encoding="application/x-llamapun" id="S6.Ex81.m1.4d">= italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT italic_f 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start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ ) ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex83"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\bigl{(}F_{1}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(f_{v}^{*}(\tau)% )\bigr{)}^{\prime}." class="ltx_Math" display="inline" id="S6.Ex83.m1.3"><semantics id="S6.Ex83.m1.3a"><mrow id="S6.Ex83.m1.3.3.1" xref="S6.Ex83.m1.3.3.1.1.cmml"><mrow id="S6.Ex83.m1.3.3.1.1" xref="S6.Ex83.m1.3.3.1.1.cmml"><mi id="S6.Ex83.m1.3.3.1.1.3" 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xref="S6.Ex83.m1.3.3.1.1.1.1.1.1.2.1.1.2.3"></times></apply><ci id="S6.Ex83.m1.2.2.cmml" xref="S6.Ex83.m1.2.2">𝜏</ci></apply></apply><ci id="S6.Ex83.m1.3.3.1.1.1.3.cmml" xref="S6.Ex83.m1.3.3.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex83.m1.3c">\displaystyle=\bigl{(}F_{1}(\alpha_{f_{h}(0)}\cdots\alpha_{1})(f_{v}^{*}(\tau)% )\bigr{)}^{\prime}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex83.m1.3d">= ( italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT ⋯ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ ) ) ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.4.p1.9">The second equality holds because <math alttext="U(d_{0}):F_{1}(d_{0})\to F_{2}(d_{0})" class="ltx_Math" display="inline" id="S6.4.p1.8.m1.3"><semantics id="S6.4.p1.8.m1.3a"><mrow id="S6.4.p1.8.m1.3.3" xref="S6.4.p1.8.m1.3.3.cmml"><mrow id="S6.4.p1.8.m1.1.1.1" xref="S6.4.p1.8.m1.1.1.1.cmml"><mi id="S6.4.p1.8.m1.1.1.1.3" xref="S6.4.p1.8.m1.1.1.1.3.cmml">U</mi><mo id="S6.4.p1.8.m1.1.1.1.2" xref="S6.4.p1.8.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.4.p1.8.m1.1.1.1.1.1" xref="S6.4.p1.8.m1.1.1.1.1.1.1.cmml"><mo id="S6.4.p1.8.m1.1.1.1.1.1.2" stretchy="false" xref="S6.4.p1.8.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.4.p1.8.m1.1.1.1.1.1.1" xref="S6.4.p1.8.m1.1.1.1.1.1.1.cmml"><mi id="S6.4.p1.8.m1.1.1.1.1.1.1.2" xref="S6.4.p1.8.m1.1.1.1.1.1.1.2.cmml">d</mi><mn id="S6.4.p1.8.m1.1.1.1.1.1.1.3" xref="S6.4.p1.8.m1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.4.p1.8.m1.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S6.4.p1.8.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.4.p1.8.m1.3.3.4" rspace="0.278em" xref="S6.4.p1.8.m1.3.3.4.cmml">:</mo><mrow id="S6.4.p1.8.m1.3.3.3" xref="S6.4.p1.8.m1.3.3.3.cmml"><mrow id="S6.4.p1.8.m1.2.2.2.1" xref="S6.4.p1.8.m1.2.2.2.1.cmml"><msub id="S6.4.p1.8.m1.2.2.2.1.3" xref="S6.4.p1.8.m1.2.2.2.1.3.cmml"><mi id="S6.4.p1.8.m1.2.2.2.1.3.2" xref="S6.4.p1.8.m1.2.2.2.1.3.2.cmml">F</mi><mn id="S6.4.p1.8.m1.2.2.2.1.3.3" xref="S6.4.p1.8.m1.2.2.2.1.3.3.cmml">1</mn></msub><mo id="S6.4.p1.8.m1.2.2.2.1.2" xref="S6.4.p1.8.m1.2.2.2.1.2.cmml">⁢</mo><mrow id="S6.4.p1.8.m1.2.2.2.1.1.1" xref="S6.4.p1.8.m1.2.2.2.1.1.1.1.cmml"><mo id="S6.4.p1.8.m1.2.2.2.1.1.1.2" stretchy="false" xref="S6.4.p1.8.m1.2.2.2.1.1.1.1.cmml">(</mo><msub id="S6.4.p1.8.m1.2.2.2.1.1.1.1" xref="S6.4.p1.8.m1.2.2.2.1.1.1.1.cmml"><mi id="S6.4.p1.8.m1.2.2.2.1.1.1.1.2" xref="S6.4.p1.8.m1.2.2.2.1.1.1.1.2.cmml">d</mi><mn id="S6.4.p1.8.m1.2.2.2.1.1.1.1.3" xref="S6.4.p1.8.m1.2.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.4.p1.8.m1.2.2.2.1.1.1.3" stretchy="false" xref="S6.4.p1.8.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.4.p1.8.m1.3.3.3.3" stretchy="false" xref="S6.4.p1.8.m1.3.3.3.3.cmml">→</mo><mrow id="S6.4.p1.8.m1.3.3.3.2" xref="S6.4.p1.8.m1.3.3.3.2.cmml"><msub id="S6.4.p1.8.m1.3.3.3.2.3" xref="S6.4.p1.8.m1.3.3.3.2.3.cmml"><mi id="S6.4.p1.8.m1.3.3.3.2.3.2" xref="S6.4.p1.8.m1.3.3.3.2.3.2.cmml">F</mi><mn id="S6.4.p1.8.m1.3.3.3.2.3.3" xref="S6.4.p1.8.m1.3.3.3.2.3.3.cmml">2</mn></msub><mo id="S6.4.p1.8.m1.3.3.3.2.2" xref="S6.4.p1.8.m1.3.3.3.2.2.cmml">⁢</mo><mrow id="S6.4.p1.8.m1.3.3.3.2.1.1" xref="S6.4.p1.8.m1.3.3.3.2.1.1.1.cmml"><mo id="S6.4.p1.8.m1.3.3.3.2.1.1.2" stretchy="false" xref="S6.4.p1.8.m1.3.3.3.2.1.1.1.cmml">(</mo><msub id="S6.4.p1.8.m1.3.3.3.2.1.1.1" xref="S6.4.p1.8.m1.3.3.3.2.1.1.1.cmml"><mi id="S6.4.p1.8.m1.3.3.3.2.1.1.1.2" xref="S6.4.p1.8.m1.3.3.3.2.1.1.1.2.cmml">d</mi><mn id="S6.4.p1.8.m1.3.3.3.2.1.1.1.3" xref="S6.4.p1.8.m1.3.3.3.2.1.1.1.3.cmml">0</mn></msub><mo id="S6.4.p1.8.m1.3.3.3.2.1.1.3" stretchy="false" xref="S6.4.p1.8.m1.3.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.8.m1.3b"><apply id="S6.4.p1.8.m1.3.3.cmml" xref="S6.4.p1.8.m1.3.3"><ci id="S6.4.p1.8.m1.3.3.4.cmml" xref="S6.4.p1.8.m1.3.3.4">:</ci><apply id="S6.4.p1.8.m1.1.1.1.cmml" xref="S6.4.p1.8.m1.1.1.1"><times id="S6.4.p1.8.m1.1.1.1.2.cmml" xref="S6.4.p1.8.m1.1.1.1.2"></times><ci id="S6.4.p1.8.m1.1.1.1.3.cmml" xref="S6.4.p1.8.m1.1.1.1.3">𝑈</ci><apply id="S6.4.p1.8.m1.1.1.1.1.1.1.cmml" xref="S6.4.p1.8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.4.p1.8.m1.1.1.1.1.1.1.1.cmml" xref="S6.4.p1.8.m1.1.1.1.1.1">subscript</csymbol><ci id="S6.4.p1.8.m1.1.1.1.1.1.1.2.cmml" xref="S6.4.p1.8.m1.1.1.1.1.1.1.2">𝑑</ci><cn id="S6.4.p1.8.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.4.p1.8.m1.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S6.4.p1.8.m1.3.3.3.cmml" xref="S6.4.p1.8.m1.3.3.3"><ci id="S6.4.p1.8.m1.3.3.3.3.cmml" xref="S6.4.p1.8.m1.3.3.3.3">→</ci><apply id="S6.4.p1.8.m1.2.2.2.1.cmml" xref="S6.4.p1.8.m1.2.2.2.1"><times id="S6.4.p1.8.m1.2.2.2.1.2.cmml" xref="S6.4.p1.8.m1.2.2.2.1.2"></times><apply id="S6.4.p1.8.m1.2.2.2.1.3.cmml" xref="S6.4.p1.8.m1.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.4.p1.8.m1.2.2.2.1.3.1.cmml" xref="S6.4.p1.8.m1.2.2.2.1.3">subscript</csymbol><ci id="S6.4.p1.8.m1.2.2.2.1.3.2.cmml" xref="S6.4.p1.8.m1.2.2.2.1.3.2">𝐹</ci><cn id="S6.4.p1.8.m1.2.2.2.1.3.3.cmml" type="integer" xref="S6.4.p1.8.m1.2.2.2.1.3.3">1</cn></apply><apply id="S6.4.p1.8.m1.2.2.2.1.1.1.1.cmml" xref="S6.4.p1.8.m1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.4.p1.8.m1.2.2.2.1.1.1.1.1.cmml" xref="S6.4.p1.8.m1.2.2.2.1.1.1">subscript</csymbol><ci id="S6.4.p1.8.m1.2.2.2.1.1.1.1.2.cmml" xref="S6.4.p1.8.m1.2.2.2.1.1.1.1.2">𝑑</ci><cn id="S6.4.p1.8.m1.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.4.p1.8.m1.2.2.2.1.1.1.1.3">0</cn></apply></apply><apply id="S6.4.p1.8.m1.3.3.3.2.cmml" xref="S6.4.p1.8.m1.3.3.3.2"><times id="S6.4.p1.8.m1.3.3.3.2.2.cmml" xref="S6.4.p1.8.m1.3.3.3.2.2"></times><apply id="S6.4.p1.8.m1.3.3.3.2.3.cmml" xref="S6.4.p1.8.m1.3.3.3.2.3"><csymbol cd="ambiguous" id="S6.4.p1.8.m1.3.3.3.2.3.1.cmml" xref="S6.4.p1.8.m1.3.3.3.2.3">subscript</csymbol><ci id="S6.4.p1.8.m1.3.3.3.2.3.2.cmml" xref="S6.4.p1.8.m1.3.3.3.2.3.2">𝐹</ci><cn id="S6.4.p1.8.m1.3.3.3.2.3.3.cmml" type="integer" xref="S6.4.p1.8.m1.3.3.3.2.3.3">2</cn></apply><apply id="S6.4.p1.8.m1.3.3.3.2.1.1.1.cmml" xref="S6.4.p1.8.m1.3.3.3.2.1.1"><csymbol cd="ambiguous" id="S6.4.p1.8.m1.3.3.3.2.1.1.1.1.cmml" xref="S6.4.p1.8.m1.3.3.3.2.1.1">subscript</csymbol><ci id="S6.4.p1.8.m1.3.3.3.2.1.1.1.2.cmml" xref="S6.4.p1.8.m1.3.3.3.2.1.1.1.2">𝑑</ci><cn id="S6.4.p1.8.m1.3.3.3.2.1.1.1.3.cmml" type="integer" xref="S6.4.p1.8.m1.3.3.3.2.1.1.1.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.8.m1.3c">U(d_{0}):F_{1}(d_{0})\to F_{2}(d_{0})</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.8.m1.3d">italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) : italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) → italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math> is a functor, and the third equality holds because <math alttext="U" class="ltx_Math" display="inline" id="S6.4.p1.9.m2.1"><semantics id="S6.4.p1.9.m2.1a"><mi id="S6.4.p1.9.m2.1.1" xref="S6.4.p1.9.m2.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S6.4.p1.9.m2.1b"><ci id="S6.4.p1.9.m2.1.1.cmml" xref="S6.4.p1.9.m2.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.9.m2.1c">U</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.9.m2.1d">italic_U</annotation></semantics></math> is a natural transformation. So the equality</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex84"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(f_{h}\times f_{v})^{*}\circ\mathfrak{U}=\mathfrak{U}\circ(f_{h}\times f_{v})^% {*}" class="ltx_Math" display="block" id="S6.Ex84.m1.2"><semantics id="S6.Ex84.m1.2a"><mrow id="S6.Ex84.m1.2.2" xref="S6.Ex84.m1.2.2.cmml"><mrow id="S6.Ex84.m1.1.1.1" xref="S6.Ex84.m1.1.1.1.cmml"><msup id="S6.Ex84.m1.1.1.1.1" xref="S6.Ex84.m1.1.1.1.1.cmml"><mrow id="S6.Ex84.m1.1.1.1.1.1.1" xref="S6.Ex84.m1.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex84.m1.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex84.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex84.m1.1.1.1.1.1.1.1" xref="S6.Ex84.m1.1.1.1.1.1.1.1.cmml"><msub id="S6.Ex84.m1.1.1.1.1.1.1.1.2" xref="S6.Ex84.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S6.Ex84.m1.1.1.1.1.1.1.1.2.2" xref="S6.Ex84.m1.1.1.1.1.1.1.1.2.2.cmml">f</mi><mi id="S6.Ex84.m1.1.1.1.1.1.1.1.2.3" xref="S6.Ex84.m1.1.1.1.1.1.1.1.2.3.cmml">h</mi></msub><mo id="S6.Ex84.m1.1.1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.Ex84.m1.1.1.1.1.1.1.1.1.cmml">×</mo><msub id="S6.Ex84.m1.1.1.1.1.1.1.1.3" xref="S6.Ex84.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex84.m1.1.1.1.1.1.1.1.3.2" xref="S6.Ex84.m1.1.1.1.1.1.1.1.3.2.cmml">f</mi><mi id="S6.Ex84.m1.1.1.1.1.1.1.1.3.3" xref="S6.Ex84.m1.1.1.1.1.1.1.1.3.3.cmml">v</mi></msub></mrow><mo id="S6.Ex84.m1.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex84.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex84.m1.1.1.1.1.3" xref="S6.Ex84.m1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S6.Ex84.m1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S6.Ex84.m1.1.1.1.2.cmml">∘</mo><mi id="S6.Ex84.m1.1.1.1.3" xref="S6.Ex84.m1.1.1.1.3.cmml">𝔘</mi></mrow><mo id="S6.Ex84.m1.2.2.3" xref="S6.Ex84.m1.2.2.3.cmml">=</mo><mrow id="S6.Ex84.m1.2.2.2" xref="S6.Ex84.m1.2.2.2.cmml"><mi id="S6.Ex84.m1.2.2.2.3" xref="S6.Ex84.m1.2.2.2.3.cmml">𝔘</mi><mo id="S6.Ex84.m1.2.2.2.2" lspace="0.222em" rspace="0.222em" xref="S6.Ex84.m1.2.2.2.2.cmml">∘</mo><msup id="S6.Ex84.m1.2.2.2.1" xref="S6.Ex84.m1.2.2.2.1.cmml"><mrow id="S6.Ex84.m1.2.2.2.1.1.1" xref="S6.Ex84.m1.2.2.2.1.1.1.1.cmml"><mo id="S6.Ex84.m1.2.2.2.1.1.1.2" stretchy="false" xref="S6.Ex84.m1.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.Ex84.m1.2.2.2.1.1.1.1" xref="S6.Ex84.m1.2.2.2.1.1.1.1.cmml"><msub id="S6.Ex84.m1.2.2.2.1.1.1.1.2" xref="S6.Ex84.m1.2.2.2.1.1.1.1.2.cmml"><mi id="S6.Ex84.m1.2.2.2.1.1.1.1.2.2" xref="S6.Ex84.m1.2.2.2.1.1.1.1.2.2.cmml">f</mi><mi id="S6.Ex84.m1.2.2.2.1.1.1.1.2.3" xref="S6.Ex84.m1.2.2.2.1.1.1.1.2.3.cmml">h</mi></msub><mo id="S6.Ex84.m1.2.2.2.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.Ex84.m1.2.2.2.1.1.1.1.1.cmml">×</mo><msub id="S6.Ex84.m1.2.2.2.1.1.1.1.3" xref="S6.Ex84.m1.2.2.2.1.1.1.1.3.cmml"><mi id="S6.Ex84.m1.2.2.2.1.1.1.1.3.2" xref="S6.Ex84.m1.2.2.2.1.1.1.1.3.2.cmml">f</mi><mi id="S6.Ex84.m1.2.2.2.1.1.1.1.3.3" xref="S6.Ex84.m1.2.2.2.1.1.1.1.3.3.cmml">v</mi></msub></mrow><mo id="S6.Ex84.m1.2.2.2.1.1.1.3" stretchy="false" xref="S6.Ex84.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex84.m1.2.2.2.1.3" xref="S6.Ex84.m1.2.2.2.1.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex84.m1.2b"><apply id="S6.Ex84.m1.2.2.cmml" xref="S6.Ex84.m1.2.2"><eq id="S6.Ex84.m1.2.2.3.cmml" xref="S6.Ex84.m1.2.2.3"></eq><apply id="S6.Ex84.m1.1.1.1.cmml" xref="S6.Ex84.m1.1.1.1"><compose id="S6.Ex84.m1.1.1.1.2.cmml" xref="S6.Ex84.m1.1.1.1.2"></compose><apply id="S6.Ex84.m1.1.1.1.1.cmml" xref="S6.Ex84.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex84.m1.1.1.1.1.2.cmml" xref="S6.Ex84.m1.1.1.1.1">superscript</csymbol><apply id="S6.Ex84.m1.1.1.1.1.1.1.1.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1"><times id="S6.Ex84.m1.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.1"></times><apply id="S6.Ex84.m1.1.1.1.1.1.1.1.2.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex84.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.Ex84.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.2.2">𝑓</ci><ci id="S6.Ex84.m1.1.1.1.1.1.1.1.2.3.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.2.3">ℎ</ci></apply><apply id="S6.Ex84.m1.1.1.1.1.1.1.1.3.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex84.m1.1.1.1.1.1.1.1.3.1.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex84.m1.1.1.1.1.1.1.1.3.2.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.3.2">𝑓</ci><ci id="S6.Ex84.m1.1.1.1.1.1.1.1.3.3.cmml" xref="S6.Ex84.m1.1.1.1.1.1.1.1.3.3">𝑣</ci></apply></apply><times id="S6.Ex84.m1.1.1.1.1.3.cmml" xref="S6.Ex84.m1.1.1.1.1.3"></times></apply><ci id="S6.Ex84.m1.1.1.1.3.cmml" xref="S6.Ex84.m1.1.1.1.3">𝔘</ci></apply><apply id="S6.Ex84.m1.2.2.2.cmml" xref="S6.Ex84.m1.2.2.2"><compose id="S6.Ex84.m1.2.2.2.2.cmml" xref="S6.Ex84.m1.2.2.2.2"></compose><ci id="S6.Ex84.m1.2.2.2.3.cmml" xref="S6.Ex84.m1.2.2.2.3">𝔘</ci><apply id="S6.Ex84.m1.2.2.2.1.cmml" xref="S6.Ex84.m1.2.2.2.1"><csymbol cd="ambiguous" id="S6.Ex84.m1.2.2.2.1.2.cmml" xref="S6.Ex84.m1.2.2.2.1">superscript</csymbol><apply id="S6.Ex84.m1.2.2.2.1.1.1.1.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1"><times id="S6.Ex84.m1.2.2.2.1.1.1.1.1.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.1"></times><apply id="S6.Ex84.m1.2.2.2.1.1.1.1.2.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex84.m1.2.2.2.1.1.1.1.2.1.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.2">subscript</csymbol><ci id="S6.Ex84.m1.2.2.2.1.1.1.1.2.2.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.2.2">𝑓</ci><ci id="S6.Ex84.m1.2.2.2.1.1.1.1.2.3.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.2.3">ℎ</ci></apply><apply id="S6.Ex84.m1.2.2.2.1.1.1.1.3.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex84.m1.2.2.2.1.1.1.1.3.1.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex84.m1.2.2.2.1.1.1.1.3.2.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.3.2">𝑓</ci><ci id="S6.Ex84.m1.2.2.2.1.1.1.1.3.3.cmml" xref="S6.Ex84.m1.2.2.2.1.1.1.1.3.3">𝑣</ci></apply></apply><times id="S6.Ex84.m1.2.2.2.1.3.cmml" xref="S6.Ex84.m1.2.2.2.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex84.m1.2c">(f_{h}\times f_{v})^{*}\circ\mathfrak{U}=\mathfrak{U}\circ(f_{h}\times f_{v})^% {*}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex84.m1.2d">( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∘ fraktur_U = fraktur_U ∘ ( italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_f start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.4.p1.12">holds. ∎</p> </div> </div> <div class="ltx_para" id="S6.p7"> <p class="ltx_p" id="S6.p7.5">For every <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S6.p7.1.m1.1"><semantics id="S6.p7.1.m1.1a"><mrow id="S6.p7.1.m1.1.1" xref="S6.p7.1.m1.1.1.cmml"><mi id="S6.p7.1.m1.1.1.2" xref="S6.p7.1.m1.1.1.2.cmml">d</mi><mo id="S6.p7.1.m1.1.1.1" xref="S6.p7.1.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S6.p7.1.m1.1.1.3" xref="S6.p7.1.m1.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p7.1.m1.1b"><apply id="S6.p7.1.m1.1.1.cmml" xref="S6.p7.1.m1.1.1"><in id="S6.p7.1.m1.1.1.1.cmml" xref="S6.p7.1.m1.1.1.1"></in><ci id="S6.p7.1.m1.1.1.2.cmml" xref="S6.p7.1.m1.1.1.2">𝑑</ci><ci id="S6.p7.1.m1.1.1.3.cmml" xref="S6.p7.1.m1.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p7.1.m1.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S6.p7.1.m1.1d">italic_d ∈ caligraphic_D</annotation></semantics></math>, there is a simplicial map <math alttext="i_{d}:NF(d)\to\operatorname*{hocolim}_{\mathcal{D}}F" class="ltx_Math" display="inline" id="S6.p7.2.m2.1"><semantics id="S6.p7.2.m2.1a"><mrow id="S6.p7.2.m2.1.2" xref="S6.p7.2.m2.1.2.cmml"><msub id="S6.p7.2.m2.1.2.2" xref="S6.p7.2.m2.1.2.2.cmml"><mi id="S6.p7.2.m2.1.2.2.2" xref="S6.p7.2.m2.1.2.2.2.cmml">i</mi><mi id="S6.p7.2.m2.1.2.2.3" xref="S6.p7.2.m2.1.2.2.3.cmml">d</mi></msub><mo id="S6.p7.2.m2.1.2.1" lspace="0.278em" rspace="0.278em" xref="S6.p7.2.m2.1.2.1.cmml">:</mo><mrow id="S6.p7.2.m2.1.2.3" xref="S6.p7.2.m2.1.2.3.cmml"><mrow id="S6.p7.2.m2.1.2.3.2" xref="S6.p7.2.m2.1.2.3.2.cmml"><mi id="S6.p7.2.m2.1.2.3.2.2" xref="S6.p7.2.m2.1.2.3.2.2.cmml">N</mi><mo id="S6.p7.2.m2.1.2.3.2.1" xref="S6.p7.2.m2.1.2.3.2.1.cmml">⁢</mo><mi id="S6.p7.2.m2.1.2.3.2.3" xref="S6.p7.2.m2.1.2.3.2.3.cmml">F</mi><mo id="S6.p7.2.m2.1.2.3.2.1a" xref="S6.p7.2.m2.1.2.3.2.1.cmml">⁢</mo><mrow id="S6.p7.2.m2.1.2.3.2.4.2" xref="S6.p7.2.m2.1.2.3.2.cmml"><mo id="S6.p7.2.m2.1.2.3.2.4.2.1" stretchy="false" xref="S6.p7.2.m2.1.2.3.2.cmml">(</mo><mi id="S6.p7.2.m2.1.1" xref="S6.p7.2.m2.1.1.cmml">d</mi><mo id="S6.p7.2.m2.1.2.3.2.4.2.2" stretchy="false" xref="S6.p7.2.m2.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.p7.2.m2.1.2.3.1" rspace="0.1389em" stretchy="false" xref="S6.p7.2.m2.1.2.3.1.cmml">→</mo><mrow id="S6.p7.2.m2.1.2.3.3" xref="S6.p7.2.m2.1.2.3.3.cmml"><msub id="S6.p7.2.m2.1.2.3.3.1" xref="S6.p7.2.m2.1.2.3.3.1.cmml"><mo id="S6.p7.2.m2.1.2.3.3.1.2" lspace="0.1389em" rspace="0.167em" xref="S6.p7.2.m2.1.2.3.3.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S6.p7.2.m2.1.2.3.3.1.3" xref="S6.p7.2.m2.1.2.3.3.1.3.cmml">𝒟</mi></msub><mi id="S6.p7.2.m2.1.2.3.3.2" xref="S6.p7.2.m2.1.2.3.3.2.cmml">F</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p7.2.m2.1b"><apply id="S6.p7.2.m2.1.2.cmml" xref="S6.p7.2.m2.1.2"><ci id="S6.p7.2.m2.1.2.1.cmml" xref="S6.p7.2.m2.1.2.1">:</ci><apply id="S6.p7.2.m2.1.2.2.cmml" xref="S6.p7.2.m2.1.2.2"><csymbol cd="ambiguous" id="S6.p7.2.m2.1.2.2.1.cmml" xref="S6.p7.2.m2.1.2.2">subscript</csymbol><ci id="S6.p7.2.m2.1.2.2.2.cmml" xref="S6.p7.2.m2.1.2.2.2">𝑖</ci><ci id="S6.p7.2.m2.1.2.2.3.cmml" xref="S6.p7.2.m2.1.2.2.3">𝑑</ci></apply><apply id="S6.p7.2.m2.1.2.3.cmml" xref="S6.p7.2.m2.1.2.3"><ci id="S6.p7.2.m2.1.2.3.1.cmml" xref="S6.p7.2.m2.1.2.3.1">→</ci><apply id="S6.p7.2.m2.1.2.3.2.cmml" xref="S6.p7.2.m2.1.2.3.2"><times id="S6.p7.2.m2.1.2.3.2.1.cmml" xref="S6.p7.2.m2.1.2.3.2.1"></times><ci id="S6.p7.2.m2.1.2.3.2.2.cmml" xref="S6.p7.2.m2.1.2.3.2.2">𝑁</ci><ci id="S6.p7.2.m2.1.2.3.2.3.cmml" xref="S6.p7.2.m2.1.2.3.2.3">𝐹</ci><ci id="S6.p7.2.m2.1.1.cmml" xref="S6.p7.2.m2.1.1">𝑑</ci></apply><apply id="S6.p7.2.m2.1.2.3.3.cmml" xref="S6.p7.2.m2.1.2.3.3"><apply id="S6.p7.2.m2.1.2.3.3.1.cmml" xref="S6.p7.2.m2.1.2.3.3.1"><csymbol cd="ambiguous" id="S6.p7.2.m2.1.2.3.3.1.1.cmml" xref="S6.p7.2.m2.1.2.3.3.1">subscript</csymbol><ci id="S6.p7.2.m2.1.2.3.3.1.2.cmml" xref="S6.p7.2.m2.1.2.3.3.1.2">hocolim</ci><ci id="S6.p7.2.m2.1.2.3.3.1.3.cmml" xref="S6.p7.2.m2.1.2.3.3.1.3">𝒟</ci></apply><ci id="S6.p7.2.m2.1.2.3.3.2.cmml" xref="S6.p7.2.m2.1.2.3.3.2">𝐹</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p7.2.m2.1c">i_{d}:NF(d)\to\operatorname*{hocolim}_{\mathcal{D}}F</annotation><annotation encoding="application/x-llamapun" id="S6.p7.2.m2.1d">italic_i start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT : italic_N italic_F ( italic_d ) → roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F</annotation></semantics></math> that sends a simplex <math alttext="\tau\in NF(d)_{p}" class="ltx_Math" display="inline" id="S6.p7.3.m3.1"><semantics id="S6.p7.3.m3.1a"><mrow id="S6.p7.3.m3.1.2" xref="S6.p7.3.m3.1.2.cmml"><mi id="S6.p7.3.m3.1.2.2" xref="S6.p7.3.m3.1.2.2.cmml">τ</mi><mo id="S6.p7.3.m3.1.2.1" xref="S6.p7.3.m3.1.2.1.cmml">∈</mo><mrow id="S6.p7.3.m3.1.2.3" xref="S6.p7.3.m3.1.2.3.cmml"><mi id="S6.p7.3.m3.1.2.3.2" xref="S6.p7.3.m3.1.2.3.2.cmml">N</mi><mo id="S6.p7.3.m3.1.2.3.1" xref="S6.p7.3.m3.1.2.3.1.cmml">⁢</mo><mi id="S6.p7.3.m3.1.2.3.3" xref="S6.p7.3.m3.1.2.3.3.cmml">F</mi><mo id="S6.p7.3.m3.1.2.3.1a" xref="S6.p7.3.m3.1.2.3.1.cmml">⁢</mo><msub id="S6.p7.3.m3.1.2.3.4" xref="S6.p7.3.m3.1.2.3.4.cmml"><mrow id="S6.p7.3.m3.1.2.3.4.2.2" xref="S6.p7.3.m3.1.2.3.4.cmml"><mo id="S6.p7.3.m3.1.2.3.4.2.2.1" stretchy="false" xref="S6.p7.3.m3.1.2.3.4.cmml">(</mo><mi id="S6.p7.3.m3.1.1" xref="S6.p7.3.m3.1.1.cmml">d</mi><mo id="S6.p7.3.m3.1.2.3.4.2.2.2" stretchy="false" xref="S6.p7.3.m3.1.2.3.4.cmml">)</mo></mrow><mi id="S6.p7.3.m3.1.2.3.4.3" xref="S6.p7.3.m3.1.2.3.4.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p7.3.m3.1b"><apply id="S6.p7.3.m3.1.2.cmml" xref="S6.p7.3.m3.1.2"><in id="S6.p7.3.m3.1.2.1.cmml" xref="S6.p7.3.m3.1.2.1"></in><ci id="S6.p7.3.m3.1.2.2.cmml" xref="S6.p7.3.m3.1.2.2">𝜏</ci><apply id="S6.p7.3.m3.1.2.3.cmml" xref="S6.p7.3.m3.1.2.3"><times id="S6.p7.3.m3.1.2.3.1.cmml" xref="S6.p7.3.m3.1.2.3.1"></times><ci id="S6.p7.3.m3.1.2.3.2.cmml" xref="S6.p7.3.m3.1.2.3.2">𝑁</ci><ci id="S6.p7.3.m3.1.2.3.3.cmml" xref="S6.p7.3.m3.1.2.3.3">𝐹</ci><apply id="S6.p7.3.m3.1.2.3.4.cmml" xref="S6.p7.3.m3.1.2.3.4"><csymbol cd="ambiguous" id="S6.p7.3.m3.1.2.3.4.1.cmml" xref="S6.p7.3.m3.1.2.3.4">subscript</csymbol><ci id="S6.p7.3.m3.1.1.cmml" xref="S6.p7.3.m3.1.1">𝑑</ci><ci id="S6.p7.3.m3.1.2.3.4.3.cmml" xref="S6.p7.3.m3.1.2.3.4.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p7.3.m3.1c">\tau\in NF(d)_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.p7.3.m3.1d">italic_τ ∈ italic_N italic_F ( italic_d ) start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> to the pair <math alttext="((d\smash{\,\mathop{\longrightarrow}\limits^{\mathrm{id}}\,}d\smash{\,\mathop{% \longrightarrow}\limits^{\mathrm{id}}\,}\cdots\smash{\,\mathop{\longrightarrow% }\limits^{\mathrm{id}}\,}d),\tau)" class="ltx_Math" display="inline" id="S6.p7.4.m4.2"><semantics id="S6.p7.4.m4.2a"><mrow id="S6.p7.4.m4.2.2.1" xref="S6.p7.4.m4.2.2.2.cmml"><mo id="S6.p7.4.m4.2.2.1.2" stretchy="false" xref="S6.p7.4.m4.2.2.2.cmml">(</mo><mrow id="S6.p7.4.m4.2.2.1.1.1" xref="S6.p7.4.m4.2.2.1.1.1.1.cmml"><mo id="S6.p7.4.m4.2.2.1.1.1.2" stretchy="false" xref="S6.p7.4.m4.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.p7.4.m4.2.2.1.1.1.1" xref="S6.p7.4.m4.2.2.1.1.1.1.cmml"><mi id="S6.p7.4.m4.2.2.1.1.1.1.2" xref="S6.p7.4.m4.2.2.1.1.1.1.2.cmml">d</mi><mo id="S6.p7.4.m4.2.2.1.1.1.1.1" lspace="0.337em" xref="S6.p7.4.m4.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.p7.4.m4.2.2.1.1.1.1.3" xref="S6.p7.4.m4.2.2.1.1.1.1.3.cmml"><mover id="S6.p7.4.m4.2.2.1.1.1.1.3.1" xref="S6.p7.4.m4.2.2.1.1.1.1.3.1.cmml"><mo id="S6.p7.4.m4.2.2.1.1.1.1.3.1.2" movablelimits="false" rspace="0.167em" xref="S6.p7.4.m4.2.2.1.1.1.1.3.1.2.cmml">⟶</mo><mi id="S6.p7.4.m4.2.2.1.1.1.1.3.1.3" xref="S6.p7.4.m4.2.2.1.1.1.1.3.1.3.cmml">id</mi></mover><mrow id="S6.p7.4.m4.2.2.1.1.1.1.3.2" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.cmml"><mi id="S6.p7.4.m4.2.2.1.1.1.1.3.2.2" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.2.cmml">d</mi><mo id="S6.p7.4.m4.2.2.1.1.1.1.3.2.1" lspace="0.337em" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.cmml"><mover id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.cmml"><mo id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.2.cmml">⟶</mo><mi id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.3" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.3.cmml">id</mi></mover><mrow id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.cmml"><mi id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.2" mathvariant="normal" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.2.cmml">⋯</mi><mo id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.1" lspace="0.337em" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.1.cmml">⁢</mo><mrow id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.cmml"><mover id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.cmml"><mo id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.2" movablelimits="false" rspace="0.167em" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.2.cmml">⟶</mo><mi id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.3" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.3.cmml">id</mi></mover><mi id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.2" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.2.cmml">d</mi></mrow></mrow></mrow></mrow></mrow></mrow><mo id="S6.p7.4.m4.2.2.1.1.1.3" stretchy="false" xref="S6.p7.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S6.p7.4.m4.2.2.1.3" xref="S6.p7.4.m4.2.2.2.cmml">,</mo><mi id="S6.p7.4.m4.1.1" xref="S6.p7.4.m4.1.1.cmml">τ</mi><mo id="S6.p7.4.m4.2.2.1.4" stretchy="false" xref="S6.p7.4.m4.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p7.4.m4.2b"><interval closure="open" id="S6.p7.4.m4.2.2.2.cmml" xref="S6.p7.4.m4.2.2.1"><apply id="S6.p7.4.m4.2.2.1.1.1.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1"><times id="S6.p7.4.m4.2.2.1.1.1.1.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.1"></times><ci id="S6.p7.4.m4.2.2.1.1.1.1.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.2">𝑑</ci><apply id="S6.p7.4.m4.2.2.1.1.1.1.3.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3"><apply id="S6.p7.4.m4.2.2.1.1.1.1.3.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S6.p7.4.m4.2.2.1.1.1.1.3.1.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.1">superscript</csymbol><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.1.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.1.2">⟶</ci><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.1.3.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.1.3">id</ci></apply><apply id="S6.p7.4.m4.2.2.1.1.1.1.3.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2"><times id="S6.p7.4.m4.2.2.1.1.1.1.3.2.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.1"></times><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.2.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.2">𝑑</ci><apply id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3"><apply id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1">superscript</csymbol><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.2">⟶</ci><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.3.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.1.3">id</ci></apply><apply id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2"><times id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.1"></times><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.2">⋯</ci><apply id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3"><apply id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1"><csymbol cd="ambiguous" id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.1.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1">superscript</csymbol><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.2">⟶</ci><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.3.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.1.3">id</ci></apply><ci id="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.2.cmml" xref="S6.p7.4.m4.2.2.1.1.1.1.3.2.3.2.3.2">𝑑</ci></apply></apply></apply></apply></apply></apply><ci id="S6.p7.4.m4.1.1.cmml" xref="S6.p7.4.m4.1.1">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.p7.4.m4.2c">((d\smash{\,\mathop{\longrightarrow}\limits^{\mathrm{id}}\,}d\smash{\,\mathop{% \longrightarrow}\limits^{\mathrm{id}}\,}\cdots\smash{\,\mathop{\longrightarrow% }\limits^{\mathrm{id}}\,}d),\tau)</annotation><annotation encoding="application/x-llamapun" id="S6.p7.4.m4.2d">( ( italic_d ⟶ start_POSTSUPERSCRIPT roman_id end_POSTSUPERSCRIPT italic_d ⟶ start_POSTSUPERSCRIPT roman_id end_POSTSUPERSCRIPT ⋯ ⟶ start_POSTSUPERSCRIPT roman_id end_POSTSUPERSCRIPT italic_d ) , italic_τ )</annotation></semantics></math> in <math alttext="N(\mathcal{D};F)_{p,p}" class="ltx_Math" display="inline" id="S6.p7.5.m5.4"><semantics id="S6.p7.5.m5.4a"><mrow id="S6.p7.5.m5.4.5" xref="S6.p7.5.m5.4.5.cmml"><mi id="S6.p7.5.m5.4.5.2" xref="S6.p7.5.m5.4.5.2.cmml">N</mi><mo id="S6.p7.5.m5.4.5.1" xref="S6.p7.5.m5.4.5.1.cmml">⁢</mo><msub id="S6.p7.5.m5.4.5.3" xref="S6.p7.5.m5.4.5.3.cmml"><mrow id="S6.p7.5.m5.4.5.3.2.2" xref="S6.p7.5.m5.4.5.3.2.1.cmml"><mo id="S6.p7.5.m5.4.5.3.2.2.1" stretchy="false" xref="S6.p7.5.m5.4.5.3.2.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p7.5.m5.3.3" xref="S6.p7.5.m5.3.3.cmml">𝒟</mi><mo id="S6.p7.5.m5.4.5.3.2.2.2" xref="S6.p7.5.m5.4.5.3.2.1.cmml">;</mo><mi id="S6.p7.5.m5.4.4" xref="S6.p7.5.m5.4.4.cmml">F</mi><mo id="S6.p7.5.m5.4.5.3.2.2.3" stretchy="false" xref="S6.p7.5.m5.4.5.3.2.1.cmml">)</mo></mrow><mrow id="S6.p7.5.m5.2.2.2.4" xref="S6.p7.5.m5.2.2.2.3.cmml"><mi id="S6.p7.5.m5.1.1.1.1" xref="S6.p7.5.m5.1.1.1.1.cmml">p</mi><mo id="S6.p7.5.m5.2.2.2.4.1" xref="S6.p7.5.m5.2.2.2.3.cmml">,</mo><mi id="S6.p7.5.m5.2.2.2.2" xref="S6.p7.5.m5.2.2.2.2.cmml">p</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.p7.5.m5.4b"><apply id="S6.p7.5.m5.4.5.cmml" xref="S6.p7.5.m5.4.5"><times id="S6.p7.5.m5.4.5.1.cmml" xref="S6.p7.5.m5.4.5.1"></times><ci id="S6.p7.5.m5.4.5.2.cmml" xref="S6.p7.5.m5.4.5.2">𝑁</ci><apply id="S6.p7.5.m5.4.5.3.cmml" xref="S6.p7.5.m5.4.5.3"><csymbol cd="ambiguous" id="S6.p7.5.m5.4.5.3.1.cmml" xref="S6.p7.5.m5.4.5.3">subscript</csymbol><list id="S6.p7.5.m5.4.5.3.2.1.cmml" xref="S6.p7.5.m5.4.5.3.2.2"><ci id="S6.p7.5.m5.3.3.cmml" xref="S6.p7.5.m5.3.3">𝒟</ci><ci id="S6.p7.5.m5.4.4.cmml" xref="S6.p7.5.m5.4.4">𝐹</ci></list><list id="S6.p7.5.m5.2.2.2.3.cmml" xref="S6.p7.5.m5.2.2.2.4"><ci id="S6.p7.5.m5.1.1.1.1.cmml" xref="S6.p7.5.m5.1.1.1.1">𝑝</ci><ci id="S6.p7.5.m5.2.2.2.2.cmml" xref="S6.p7.5.m5.2.2.2.2">𝑝</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p7.5.m5.4c">N(\mathcal{D};F)_{p,p}</annotation><annotation encoding="application/x-llamapun" id="S6.p7.5.m5.4d">italic_N ( caligraphic_D ; italic_F ) start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem6.1.1.1">Proposition 6.6</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem6.p1"> <p class="ltx_p" id="S6.Thmtheorem6.p1.5"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem6.p1.5.5">Let <math alttext="U:F_{1}\to F_{2}" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.1.1.m1.1"><semantics id="S6.Thmtheorem6.p1.1.1.m1.1a"><mrow id="S6.Thmtheorem6.p1.1.1.m1.1.1" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem6.p1.1.1.m1.1.1.2" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.2.cmml">U</mi><mo id="S6.Thmtheorem6.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S6.Thmtheorem6.p1.1.1.m1.1.1.3" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.cmml"><msub id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.cmml"><mi id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.2" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.2.cmml">F</mi><mn id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.3" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.1.cmml">→</mo><msub id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.cmml"><mi id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.2" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.2.cmml">F</mi><mn id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.3" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.1.1.m1.1b"><apply id="S6.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1"><ci id="S6.Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.1">:</ci><ci id="S6.Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.2">𝑈</ci><apply id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3"><ci id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.1">→</ci><apply id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.1.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.2.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.2">𝐹</ci><cn id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.2.3">1</cn></apply><apply id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.1.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.2.cmml" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.2">𝐹</ci><cn id="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.1.1.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.1.1.m1.1c">U:F_{1}\to F_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.1.1.m1.1d">italic_U : italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be a natural transformation between functors <math alttext="F_{1},F_{2}:\mathcal{D}\to Cat" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.2.2.m2.2"><semantics id="S6.Thmtheorem6.p1.2.2.m2.2a"><mrow id="S6.Thmtheorem6.p1.2.2.m2.2.2" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.cmml"><mrow id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.3.cmml"><msub id="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1" xref="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.2" xref="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.2.cmml">F</mi><mn id="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.3" xref="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.3" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.3.cmml">,</mo><msub id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.cmml"><mi id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.2" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.2.cmml">F</mi><mn id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.3" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S6.Thmtheorem6.p1.2.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.3.cmml">:</mo><mrow id="S6.Thmtheorem6.p1.2.2.m2.2.2.4" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.2" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.2.cmml">𝒟</mi><mo id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.1" stretchy="false" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.1.cmml">→</mo><mrow id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.cmml"><mi id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.2" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.2.cmml">C</mi><mo id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.1" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.1.cmml">⁢</mo><mi id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.3" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.3.cmml">a</mi><mo id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.1a" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.1.cmml">⁢</mo><mi id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.4" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.4.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.2.2.m2.2b"><apply id="S6.Thmtheorem6.p1.2.2.m2.2.2.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2"><ci id="S6.Thmtheorem6.p1.2.2.m2.2.2.3.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.3">:</ci><list id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.3.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2"><apply id="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.2">𝐹</ci><cn id="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.1.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2">subscript</csymbol><ci id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.2">𝐹</ci><cn id="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.2.2.2.3">2</cn></apply></list><apply id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4"><ci id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.1.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.1">→</ci><ci id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.2.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.2">𝒟</ci><apply id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3"><times id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.1.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.1"></times><ci id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.2.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.2">𝐶</ci><ci id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.3.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.3">𝑎</ci><ci id="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.4.cmml" xref="S6.Thmtheorem6.p1.2.2.m2.2.2.4.3.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.2.2.m2.2c">F_{1},F_{2}:\mathcal{D}\to Cat</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.2.2.m2.2d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : caligraphic_D → italic_C italic_a italic_t</annotation></semantics></math>, and let <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.3.3.m3.1"><semantics id="S6.Thmtheorem6.p1.3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem6.p1.3.3.m3.1.1" xref="S6.Thmtheorem6.p1.3.3.m3.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.3.3.m3.1b"><ci id="S6.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem6.p1.3.3.m3.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.3.3.m3.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.3.3.m3.1d">caligraphic_M</annotation></semantics></math> be an extendable coefficient system defined on <math alttext="\operatorname*{hocolim}_{\mathcal{D}}F_{2}" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.4.4.m4.1"><semantics id="S6.Thmtheorem6.p1.4.4.m4.1a"><mrow id="S6.Thmtheorem6.p1.4.4.m4.1.1" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.cmml"><msub id="S6.Thmtheorem6.p1.4.4.m4.1.1.1" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.1.cmml"><mo id="S6.Thmtheorem6.p1.4.4.m4.1.1.1.2" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.1.2.cmml">hocolim</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem6.p1.4.4.m4.1.1.1.3" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.1.3.cmml">𝒟</mi></msub><msub id="S6.Thmtheorem6.p1.4.4.m4.1.1.2" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.2.cmml"><mi id="S6.Thmtheorem6.p1.4.4.m4.1.1.2.2" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.2.2.cmml">F</mi><mn id="S6.Thmtheorem6.p1.4.4.m4.1.1.2.3" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.4.4.m4.1b"><apply id="S6.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.1.1"><apply id="S6.Thmtheorem6.p1.4.4.m4.1.1.1.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.4.4.m4.1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem6.p1.4.4.m4.1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.1.2">hocolim</ci><ci id="S6.Thmtheorem6.p1.4.4.m4.1.1.1.3.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.1.3">𝒟</ci></apply><apply id="S6.Thmtheorem6.p1.4.4.m4.1.1.2.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.4.4.m4.1.1.2.1.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem6.p1.4.4.m4.1.1.2.2.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.2.2">𝐹</ci><cn id="S6.Thmtheorem6.p1.4.4.m4.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.4.4.m4.1.1.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.4.4.m4.1c">\operatorname*{hocolim}_{\mathcal{D}}F_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.4.4.m4.1d">roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. If for every <math alttext="d\in\mathcal{D}" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.5.5.m5.1"><semantics id="S6.Thmtheorem6.p1.5.5.m5.1a"><mrow id="S6.Thmtheorem6.p1.5.5.m5.1.1" xref="S6.Thmtheorem6.p1.5.5.m5.1.1.cmml"><mi id="S6.Thmtheorem6.p1.5.5.m5.1.1.2" xref="S6.Thmtheorem6.p1.5.5.m5.1.1.2.cmml">d</mi><mo id="S6.Thmtheorem6.p1.5.5.m5.1.1.1" xref="S6.Thmtheorem6.p1.5.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem6.p1.5.5.m5.1.1.3" xref="S6.Thmtheorem6.p1.5.5.m5.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.5.5.m5.1b"><apply id="S6.Thmtheorem6.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.1.1"><in id="S6.Thmtheorem6.p1.5.5.m5.1.1.1.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.1.1.1"></in><ci id="S6.Thmtheorem6.p1.5.5.m5.1.1.2.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.1.1.2">𝑑</ci><ci id="S6.Thmtheorem6.p1.5.5.m5.1.1.3.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.5.5.m5.1c">d\in\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.5.5.m5.1d">italic_d ∈ caligraphic_D</annotation></semantics></math>, the induced map</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex85"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="NU(d)^{*}:H^{*}(NF_{2}(d),i_{d}^{*}\mathcal{M})\to H^{*}(NF_{1}(d);NU(d)^{*}i_% {d}^{*}\mathcal{M})" class="ltx_Math" display="block" id="S6.Ex85.m1.8"><semantics id="S6.Ex85.m1.8a"><mrow id="S6.Ex85.m1.8.8" xref="S6.Ex85.m1.8.8.cmml"><mrow id="S6.Ex85.m1.8.8.6" xref="S6.Ex85.m1.8.8.6.cmml"><mi id="S6.Ex85.m1.8.8.6.2" xref="S6.Ex85.m1.8.8.6.2.cmml">N</mi><mo id="S6.Ex85.m1.8.8.6.1" xref="S6.Ex85.m1.8.8.6.1.cmml">⁢</mo><mi id="S6.Ex85.m1.8.8.6.3" xref="S6.Ex85.m1.8.8.6.3.cmml">U</mi><mo id="S6.Ex85.m1.8.8.6.1a" xref="S6.Ex85.m1.8.8.6.1.cmml">⁢</mo><msup id="S6.Ex85.m1.8.8.6.4" xref="S6.Ex85.m1.8.8.6.4.cmml"><mrow id="S6.Ex85.m1.8.8.6.4.2.2" xref="S6.Ex85.m1.8.8.6.4.cmml"><mo id="S6.Ex85.m1.8.8.6.4.2.2.1" stretchy="false" xref="S6.Ex85.m1.8.8.6.4.cmml">(</mo><mi id="S6.Ex85.m1.1.1" xref="S6.Ex85.m1.1.1.cmml">d</mi><mo id="S6.Ex85.m1.8.8.6.4.2.2.2" rspace="0.278em" stretchy="false" xref="S6.Ex85.m1.8.8.6.4.cmml">)</mo></mrow><mo id="S6.Ex85.m1.8.8.6.4.3" xref="S6.Ex85.m1.8.8.6.4.3.cmml">∗</mo></msup></mrow><mo id="S6.Ex85.m1.8.8.5" rspace="0.278em" xref="S6.Ex85.m1.8.8.5.cmml">:</mo><mrow id="S6.Ex85.m1.8.8.4" xref="S6.Ex85.m1.8.8.4.cmml"><mrow id="S6.Ex85.m1.6.6.2.2" xref="S6.Ex85.m1.6.6.2.2.cmml"><msup id="S6.Ex85.m1.6.6.2.2.4" xref="S6.Ex85.m1.6.6.2.2.4.cmml"><mi id="S6.Ex85.m1.6.6.2.2.4.2" xref="S6.Ex85.m1.6.6.2.2.4.2.cmml">H</mi><mo id="S6.Ex85.m1.6.6.2.2.4.3" xref="S6.Ex85.m1.6.6.2.2.4.3.cmml">∗</mo></msup><mo id="S6.Ex85.m1.6.6.2.2.3" xref="S6.Ex85.m1.6.6.2.2.3.cmml">⁢</mo><mrow id="S6.Ex85.m1.6.6.2.2.2.2" xref="S6.Ex85.m1.6.6.2.2.2.3.cmml"><mo id="S6.Ex85.m1.6.6.2.2.2.2.3" stretchy="false" xref="S6.Ex85.m1.6.6.2.2.2.3.cmml">(</mo><mrow id="S6.Ex85.m1.5.5.1.1.1.1.1" xref="S6.Ex85.m1.5.5.1.1.1.1.1.cmml"><mi id="S6.Ex85.m1.5.5.1.1.1.1.1.2" xref="S6.Ex85.m1.5.5.1.1.1.1.1.2.cmml">N</mi><mo id="S6.Ex85.m1.5.5.1.1.1.1.1.1" xref="S6.Ex85.m1.5.5.1.1.1.1.1.1.cmml">⁢</mo><msub id="S6.Ex85.m1.5.5.1.1.1.1.1.3" xref="S6.Ex85.m1.5.5.1.1.1.1.1.3.cmml"><mi id="S6.Ex85.m1.5.5.1.1.1.1.1.3.2" xref="S6.Ex85.m1.5.5.1.1.1.1.1.3.2.cmml">F</mi><mn id="S6.Ex85.m1.5.5.1.1.1.1.1.3.3" xref="S6.Ex85.m1.5.5.1.1.1.1.1.3.3.cmml">2</mn></msub><mo id="S6.Ex85.m1.5.5.1.1.1.1.1.1a" xref="S6.Ex85.m1.5.5.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Ex85.m1.5.5.1.1.1.1.1.4.2" xref="S6.Ex85.m1.5.5.1.1.1.1.1.cmml"><mo id="S6.Ex85.m1.5.5.1.1.1.1.1.4.2.1" stretchy="false" xref="S6.Ex85.m1.5.5.1.1.1.1.1.cmml">(</mo><mi id="S6.Ex85.m1.2.2" xref="S6.Ex85.m1.2.2.cmml">d</mi><mo id="S6.Ex85.m1.5.5.1.1.1.1.1.4.2.2" stretchy="false" xref="S6.Ex85.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex85.m1.6.6.2.2.2.2.4" xref="S6.Ex85.m1.6.6.2.2.2.3.cmml">,</mo><mrow id="S6.Ex85.m1.6.6.2.2.2.2.2" xref="S6.Ex85.m1.6.6.2.2.2.2.2.cmml"><msubsup id="S6.Ex85.m1.6.6.2.2.2.2.2.2" xref="S6.Ex85.m1.6.6.2.2.2.2.2.2.cmml"><mi id="S6.Ex85.m1.6.6.2.2.2.2.2.2.2.2" xref="S6.Ex85.m1.6.6.2.2.2.2.2.2.2.2.cmml">i</mi><mi id="S6.Ex85.m1.6.6.2.2.2.2.2.2.2.3" xref="S6.Ex85.m1.6.6.2.2.2.2.2.2.2.3.cmml">d</mi><mo id="S6.Ex85.m1.6.6.2.2.2.2.2.2.3" xref="S6.Ex85.m1.6.6.2.2.2.2.2.2.3.cmml">∗</mo></msubsup><mo id="S6.Ex85.m1.6.6.2.2.2.2.2.1" xref="S6.Ex85.m1.6.6.2.2.2.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex85.m1.6.6.2.2.2.2.2.3" xref="S6.Ex85.m1.6.6.2.2.2.2.2.3.cmml">ℳ</mi></mrow><mo id="S6.Ex85.m1.6.6.2.2.2.2.5" stretchy="false" xref="S6.Ex85.m1.6.6.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex85.m1.8.8.4.5" stretchy="false" xref="S6.Ex85.m1.8.8.4.5.cmml">→</mo><mrow id="S6.Ex85.m1.8.8.4.4" xref="S6.Ex85.m1.8.8.4.4.cmml"><msup id="S6.Ex85.m1.8.8.4.4.4" xref="S6.Ex85.m1.8.8.4.4.4.cmml"><mi id="S6.Ex85.m1.8.8.4.4.4.2" xref="S6.Ex85.m1.8.8.4.4.4.2.cmml">H</mi><mo id="S6.Ex85.m1.8.8.4.4.4.3" xref="S6.Ex85.m1.8.8.4.4.4.3.cmml">∗</mo></msup><mo id="S6.Ex85.m1.8.8.4.4.3" xref="S6.Ex85.m1.8.8.4.4.3.cmml">⁢</mo><mrow id="S6.Ex85.m1.8.8.4.4.2.2" xref="S6.Ex85.m1.8.8.4.4.2.3.cmml"><mo id="S6.Ex85.m1.8.8.4.4.2.2.3" stretchy="false" xref="S6.Ex85.m1.8.8.4.4.2.3.cmml">(</mo><mrow id="S6.Ex85.m1.7.7.3.3.1.1.1" xref="S6.Ex85.m1.7.7.3.3.1.1.1.cmml"><mi id="S6.Ex85.m1.7.7.3.3.1.1.1.2" xref="S6.Ex85.m1.7.7.3.3.1.1.1.2.cmml">N</mi><mo id="S6.Ex85.m1.7.7.3.3.1.1.1.1" xref="S6.Ex85.m1.7.7.3.3.1.1.1.1.cmml">⁢</mo><msub id="S6.Ex85.m1.7.7.3.3.1.1.1.3" xref="S6.Ex85.m1.7.7.3.3.1.1.1.3.cmml"><mi id="S6.Ex85.m1.7.7.3.3.1.1.1.3.2" xref="S6.Ex85.m1.7.7.3.3.1.1.1.3.2.cmml">F</mi><mn id="S6.Ex85.m1.7.7.3.3.1.1.1.3.3" xref="S6.Ex85.m1.7.7.3.3.1.1.1.3.3.cmml">1</mn></msub><mo id="S6.Ex85.m1.7.7.3.3.1.1.1.1a" xref="S6.Ex85.m1.7.7.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.Ex85.m1.7.7.3.3.1.1.1.4.2" xref="S6.Ex85.m1.7.7.3.3.1.1.1.cmml"><mo id="S6.Ex85.m1.7.7.3.3.1.1.1.4.2.1" stretchy="false" xref="S6.Ex85.m1.7.7.3.3.1.1.1.cmml">(</mo><mi id="S6.Ex85.m1.3.3" xref="S6.Ex85.m1.3.3.cmml">d</mi><mo id="S6.Ex85.m1.7.7.3.3.1.1.1.4.2.2" stretchy="false" xref="S6.Ex85.m1.7.7.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex85.m1.8.8.4.4.2.2.4" xref="S6.Ex85.m1.8.8.4.4.2.3.cmml">;</mo><mrow id="S6.Ex85.m1.8.8.4.4.2.2.2" xref="S6.Ex85.m1.8.8.4.4.2.2.2.cmml"><mi id="S6.Ex85.m1.8.8.4.4.2.2.2.2" xref="S6.Ex85.m1.8.8.4.4.2.2.2.2.cmml">N</mi><mo id="S6.Ex85.m1.8.8.4.4.2.2.2.1" xref="S6.Ex85.m1.8.8.4.4.2.2.2.1.cmml">⁢</mo><mi id="S6.Ex85.m1.8.8.4.4.2.2.2.3" xref="S6.Ex85.m1.8.8.4.4.2.2.2.3.cmml">U</mi><mo id="S6.Ex85.m1.8.8.4.4.2.2.2.1a" xref="S6.Ex85.m1.8.8.4.4.2.2.2.1.cmml">⁢</mo><msup id="S6.Ex85.m1.8.8.4.4.2.2.2.4" xref="S6.Ex85.m1.8.8.4.4.2.2.2.4.cmml"><mrow id="S6.Ex85.m1.8.8.4.4.2.2.2.4.2.2" xref="S6.Ex85.m1.8.8.4.4.2.2.2.4.cmml"><mo id="S6.Ex85.m1.8.8.4.4.2.2.2.4.2.2.1" stretchy="false" xref="S6.Ex85.m1.8.8.4.4.2.2.2.4.cmml">(</mo><mi id="S6.Ex85.m1.4.4" xref="S6.Ex85.m1.4.4.cmml">d</mi><mo id="S6.Ex85.m1.8.8.4.4.2.2.2.4.2.2.2" stretchy="false" xref="S6.Ex85.m1.8.8.4.4.2.2.2.4.cmml">)</mo></mrow><mo id="S6.Ex85.m1.8.8.4.4.2.2.2.4.3" xref="S6.Ex85.m1.8.8.4.4.2.2.2.4.3.cmml">∗</mo></msup><mo id="S6.Ex85.m1.8.8.4.4.2.2.2.1b" xref="S6.Ex85.m1.8.8.4.4.2.2.2.1.cmml">⁢</mo><msubsup id="S6.Ex85.m1.8.8.4.4.2.2.2.5" xref="S6.Ex85.m1.8.8.4.4.2.2.2.5.cmml"><mi id="S6.Ex85.m1.8.8.4.4.2.2.2.5.2.2" xref="S6.Ex85.m1.8.8.4.4.2.2.2.5.2.2.cmml">i</mi><mi id="S6.Ex85.m1.8.8.4.4.2.2.2.5.2.3" xref="S6.Ex85.m1.8.8.4.4.2.2.2.5.2.3.cmml">d</mi><mo id="S6.Ex85.m1.8.8.4.4.2.2.2.5.3" 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id="S6.Ex85.m1.1.1.cmml" xref="S6.Ex85.m1.1.1">𝑑</ci><times id="S6.Ex85.m1.8.8.6.4.3.cmml" xref="S6.Ex85.m1.8.8.6.4.3"></times></apply></apply><apply id="S6.Ex85.m1.8.8.4.cmml" xref="S6.Ex85.m1.8.8.4"><ci id="S6.Ex85.m1.8.8.4.5.cmml" xref="S6.Ex85.m1.8.8.4.5">→</ci><apply id="S6.Ex85.m1.6.6.2.2.cmml" xref="S6.Ex85.m1.6.6.2.2"><times id="S6.Ex85.m1.6.6.2.2.3.cmml" xref="S6.Ex85.m1.6.6.2.2.3"></times><apply id="S6.Ex85.m1.6.6.2.2.4.cmml" xref="S6.Ex85.m1.6.6.2.2.4"><csymbol cd="ambiguous" id="S6.Ex85.m1.6.6.2.2.4.1.cmml" xref="S6.Ex85.m1.6.6.2.2.4">superscript</csymbol><ci id="S6.Ex85.m1.6.6.2.2.4.2.cmml" xref="S6.Ex85.m1.6.6.2.2.4.2">𝐻</ci><times id="S6.Ex85.m1.6.6.2.2.4.3.cmml" xref="S6.Ex85.m1.6.6.2.2.4.3"></times></apply><interval closure="open" id="S6.Ex85.m1.6.6.2.2.2.3.cmml" xref="S6.Ex85.m1.6.6.2.2.2.2"><apply id="S6.Ex85.m1.5.5.1.1.1.1.1.cmml" xref="S6.Ex85.m1.5.5.1.1.1.1.1"><times id="S6.Ex85.m1.5.5.1.1.1.1.1.1.cmml" xref="S6.Ex85.m1.5.5.1.1.1.1.1.1"></times><ci 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xref="S6.Ex85.m1.8.8.4.4.2.2.2.5.2.3">𝑑</ci></apply><times id="S6.Ex85.m1.8.8.4.4.2.2.2.5.3.cmml" xref="S6.Ex85.m1.8.8.4.4.2.2.2.5.3"></times></apply><ci id="S6.Ex85.m1.8.8.4.4.2.2.2.6.cmml" xref="S6.Ex85.m1.8.8.4.4.2.2.2.6">ℳ</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex85.m1.8c">NU(d)^{*}:H^{*}(NF_{2}(d),i_{d}^{*}\mathcal{M})\to H^{*}(NF_{1}(d);NU(d)^{*}i_% {d}^{*}\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S6.Ex85.m1.8d">italic_N italic_U ( italic_d ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_N italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_d ) , italic_i start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M ) → italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_N italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_d ) ; italic_N italic_U ( italic_d ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmtheorem6.p1.6"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem6.p1.6.1">is an isomorphism, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex86"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathfrak{u}^{*}:H^{*}(\operatorname*{hocolim}_{\mathcal{D}}F_{2};\mathcal{M})% \to H^{*}(\operatorname*{hocolim}_{\mathcal{D}}F_{1};\mathfrak{u}^{*}\mathcal{% M})" class="ltx_Math" display="block" id="S6.Ex86.m1.4"><semantics id="S6.Ex86.m1.4a"><mrow id="S6.Ex86.m1.4.4" xref="S6.Ex86.m1.4.4.cmml"><msup id="S6.Ex86.m1.4.4.5" xref="S6.Ex86.m1.4.4.5.cmml"><mi id="S6.Ex86.m1.4.4.5.2" xref="S6.Ex86.m1.4.4.5.2.cmml">𝔲</mi><mo id="S6.Ex86.m1.4.4.5.3" xref="S6.Ex86.m1.4.4.5.3.cmml">∗</mo></msup><mo id="S6.Ex86.m1.4.4.4" lspace="0.278em" rspace="0.278em" xref="S6.Ex86.m1.4.4.4.cmml">:</mo><mrow id="S6.Ex86.m1.4.4.3" xref="S6.Ex86.m1.4.4.3.cmml"><mrow id="S6.Ex86.m1.2.2.1.1" xref="S6.Ex86.m1.2.2.1.1.cmml"><msup id="S6.Ex86.m1.2.2.1.1.3" xref="S6.Ex86.m1.2.2.1.1.3.cmml"><mi id="S6.Ex86.m1.2.2.1.1.3.2" xref="S6.Ex86.m1.2.2.1.1.3.2.cmml">H</mi><mo id="S6.Ex86.m1.2.2.1.1.3.3" xref="S6.Ex86.m1.2.2.1.1.3.3.cmml">∗</mo></msup><mo id="S6.Ex86.m1.2.2.1.1.2" xref="S6.Ex86.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S6.Ex86.m1.2.2.1.1.1.1" xref="S6.Ex86.m1.2.2.1.1.1.2.cmml"><mo id="S6.Ex86.m1.2.2.1.1.1.1.2" stretchy="false" xref="S6.Ex86.m1.2.2.1.1.1.2.cmml">(</mo><mrow id="S6.Ex86.m1.2.2.1.1.1.1.1" xref="S6.Ex86.m1.2.2.1.1.1.1.1.cmml"><munder id="S6.Ex86.m1.2.2.1.1.1.1.1.1" xref="S6.Ex86.m1.2.2.1.1.1.1.1.1.cmml"><mo 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xref="S6.Ex86.m1.3.3.2.2.1.1.1.2.3.cmml">1</mn></msub></mrow><mo id="S6.Ex86.m1.4.4.3.3.2.2.4" xref="S6.Ex86.m1.4.4.3.3.2.3.cmml">;</mo><mrow id="S6.Ex86.m1.4.4.3.3.2.2.2" xref="S6.Ex86.m1.4.4.3.3.2.2.2.cmml"><msup id="S6.Ex86.m1.4.4.3.3.2.2.2.2" xref="S6.Ex86.m1.4.4.3.3.2.2.2.2.cmml"><mi id="S6.Ex86.m1.4.4.3.3.2.2.2.2.2" xref="S6.Ex86.m1.4.4.3.3.2.2.2.2.2.cmml">𝔲</mi><mo id="S6.Ex86.m1.4.4.3.3.2.2.2.2.3" xref="S6.Ex86.m1.4.4.3.3.2.2.2.2.3.cmml">∗</mo></msup><mo id="S6.Ex86.m1.4.4.3.3.2.2.2.1" xref="S6.Ex86.m1.4.4.3.3.2.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex86.m1.4.4.3.3.2.2.2.3" xref="S6.Ex86.m1.4.4.3.3.2.2.2.3.cmml">ℳ</mi></mrow><mo id="S6.Ex86.m1.4.4.3.3.2.2.5" stretchy="false" xref="S6.Ex86.m1.4.4.3.3.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex86.m1.4b"><apply id="S6.Ex86.m1.4.4.cmml" xref="S6.Ex86.m1.4.4"><ci id="S6.Ex86.m1.4.4.4.cmml" xref="S6.Ex86.m1.4.4.4">:</ci><apply id="S6.Ex86.m1.4.4.5.cmml" 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encoding="application/x-tex" id="S6.Ex86.m1.4c">\mathfrak{u}^{*}:H^{*}(\operatorname*{hocolim}_{\mathcal{D}}F_{2};\mathcal{M})% \to H^{*}(\operatorname*{hocolim}_{\mathcal{D}}F_{1};\mathfrak{u}^{*}\mathcal{% M})</annotation><annotation encoding="application/x-llamapun" id="S6.Ex86.m1.4d">fraktur_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; caligraphic_M ) → italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; fraktur_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmtheorem6.p1.7"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem6.p1.7.1">is an isomorphism.</span></p> </div> </div> <div class="ltx_proof" id="S6.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.5.p1"> <p class="ltx_p" id="S6.5.p1.5">Let <math alttext="\mathcal{M}^{\prime}" class="ltx_Math" display="inline" id="S6.5.p1.1.m1.1"><semantics id="S6.5.p1.1.m1.1a"><msup id="S6.5.p1.1.m1.1.1" xref="S6.5.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.5.p1.1.m1.1.1.2" xref="S6.5.p1.1.m1.1.1.2.cmml">ℳ</mi><mo id="S6.5.p1.1.m1.1.1.3" xref="S6.5.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.5.p1.1.m1.1b"><apply id="S6.5.p1.1.m1.1.1.cmml" xref="S6.5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.5.p1.1.m1.1.1.1.cmml" xref="S6.5.p1.1.m1.1.1">superscript</csymbol><ci id="S6.5.p1.1.m1.1.1.2.cmml" xref="S6.5.p1.1.m1.1.1.2">ℳ</ci><ci id="S6.5.p1.1.m1.1.1.3.cmml" xref="S6.5.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p1.1.m1.1c">\mathcal{M}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p1.1.m1.1d">caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be a coefficient system for the bisimplicial set <math alttext="N(\mathcal{D};F_{2})" class="ltx_Math" display="inline" id="S6.5.p1.2.m2.2"><semantics id="S6.5.p1.2.m2.2a"><mrow id="S6.5.p1.2.m2.2.2" xref="S6.5.p1.2.m2.2.2.cmml"><mi id="S6.5.p1.2.m2.2.2.3" xref="S6.5.p1.2.m2.2.2.3.cmml">N</mi><mo id="S6.5.p1.2.m2.2.2.2" xref="S6.5.p1.2.m2.2.2.2.cmml">⁢</mo><mrow id="S6.5.p1.2.m2.2.2.1.1" xref="S6.5.p1.2.m2.2.2.1.2.cmml"><mo id="S6.5.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S6.5.p1.2.m2.2.2.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.5.p1.2.m2.1.1" xref="S6.5.p1.2.m2.1.1.cmml">𝒟</mi><mo id="S6.5.p1.2.m2.2.2.1.1.3" xref="S6.5.p1.2.m2.2.2.1.2.cmml">;</mo><msub id="S6.5.p1.2.m2.2.2.1.1.1" xref="S6.5.p1.2.m2.2.2.1.1.1.cmml"><mi id="S6.5.p1.2.m2.2.2.1.1.1.2" xref="S6.5.p1.2.m2.2.2.1.1.1.2.cmml">F</mi><mn id="S6.5.p1.2.m2.2.2.1.1.1.3" xref="S6.5.p1.2.m2.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S6.5.p1.2.m2.2.2.1.1.4" stretchy="false" xref="S6.5.p1.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.5.p1.2.m2.2b"><apply id="S6.5.p1.2.m2.2.2.cmml" xref="S6.5.p1.2.m2.2.2"><times id="S6.5.p1.2.m2.2.2.2.cmml" xref="S6.5.p1.2.m2.2.2.2"></times><ci id="S6.5.p1.2.m2.2.2.3.cmml" xref="S6.5.p1.2.m2.2.2.3">𝑁</ci><list id="S6.5.p1.2.m2.2.2.1.2.cmml" xref="S6.5.p1.2.m2.2.2.1.1"><ci id="S6.5.p1.2.m2.1.1.cmml" xref="S6.5.p1.2.m2.1.1">𝒟</ci><apply id="S6.5.p1.2.m2.2.2.1.1.1.cmml" xref="S6.5.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.5.p1.2.m2.2.2.1.1.1.1.cmml" xref="S6.5.p1.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S6.5.p1.2.m2.2.2.1.1.1.2.cmml" xref="S6.5.p1.2.m2.2.2.1.1.1.2">𝐹</ci><cn id="S6.5.p1.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S6.5.p1.2.m2.2.2.1.1.1.3">2</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p1.2.m2.2c">N(\mathcal{D};F_{2})</annotation><annotation encoding="application/x-llamapun" id="S6.5.p1.2.m2.2d">italic_N ( caligraphic_D ; italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> such that <math alttext="J^{*}(\mathcal{M}^{\prime})=\mathcal{M}" class="ltx_Math" display="inline" id="S6.5.p1.3.m3.1"><semantics id="S6.5.p1.3.m3.1a"><mrow id="S6.5.p1.3.m3.1.1" xref="S6.5.p1.3.m3.1.1.cmml"><mrow id="S6.5.p1.3.m3.1.1.1" xref="S6.5.p1.3.m3.1.1.1.cmml"><msup id="S6.5.p1.3.m3.1.1.1.3" xref="S6.5.p1.3.m3.1.1.1.3.cmml"><mi id="S6.5.p1.3.m3.1.1.1.3.2" xref="S6.5.p1.3.m3.1.1.1.3.2.cmml">J</mi><mo id="S6.5.p1.3.m3.1.1.1.3.3" xref="S6.5.p1.3.m3.1.1.1.3.3.cmml">∗</mo></msup><mo id="S6.5.p1.3.m3.1.1.1.2" xref="S6.5.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S6.5.p1.3.m3.1.1.1.1.1" xref="S6.5.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S6.5.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S6.5.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><msup id="S6.5.p1.3.m3.1.1.1.1.1.1" xref="S6.5.p1.3.m3.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.5.p1.3.m3.1.1.1.1.1.1.2" xref="S6.5.p1.3.m3.1.1.1.1.1.1.2.cmml">ℳ</mi><mo id="S6.5.p1.3.m3.1.1.1.1.1.1.3" xref="S6.5.p1.3.m3.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.5.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S6.5.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.5.p1.3.m3.1.1.2" xref="S6.5.p1.3.m3.1.1.2.cmml">=</mo><mi class="ltx_font_mathcaligraphic" id="S6.5.p1.3.m3.1.1.3" xref="S6.5.p1.3.m3.1.1.3.cmml">ℳ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.5.p1.3.m3.1b"><apply id="S6.5.p1.3.m3.1.1.cmml" xref="S6.5.p1.3.m3.1.1"><eq id="S6.5.p1.3.m3.1.1.2.cmml" xref="S6.5.p1.3.m3.1.1.2"></eq><apply id="S6.5.p1.3.m3.1.1.1.cmml" xref="S6.5.p1.3.m3.1.1.1"><times id="S6.5.p1.3.m3.1.1.1.2.cmml" xref="S6.5.p1.3.m3.1.1.1.2"></times><apply id="S6.5.p1.3.m3.1.1.1.3.cmml" xref="S6.5.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S6.5.p1.3.m3.1.1.1.3.1.cmml" xref="S6.5.p1.3.m3.1.1.1.3">superscript</csymbol><ci id="S6.5.p1.3.m3.1.1.1.3.2.cmml" xref="S6.5.p1.3.m3.1.1.1.3.2">𝐽</ci><times id="S6.5.p1.3.m3.1.1.1.3.3.cmml" xref="S6.5.p1.3.m3.1.1.1.3.3"></times></apply><apply id="S6.5.p1.3.m3.1.1.1.1.1.1.cmml" xref="S6.5.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.5.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S6.5.p1.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S6.5.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S6.5.p1.3.m3.1.1.1.1.1.1.2">ℳ</ci><ci id="S6.5.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S6.5.p1.3.m3.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S6.5.p1.3.m3.1.1.3.cmml" xref="S6.5.p1.3.m3.1.1.3">ℳ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p1.3.m3.1c">J^{*}(\mathcal{M}^{\prime})=\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p1.3.m3.1d">italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = caligraphic_M</annotation></semantics></math>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S6.Thmtheorem5" title="Lemma 6.5. ‣ 6. Cohomology of bisimplicial sets with nontrivial coefficients ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">6.5</span></a>, the natural transformation <math alttext="U:F_{1}\to F_{2}" class="ltx_Math" display="inline" id="S6.5.p1.4.m4.1"><semantics id="S6.5.p1.4.m4.1a"><mrow id="S6.5.p1.4.m4.1.1" xref="S6.5.p1.4.m4.1.1.cmml"><mi id="S6.5.p1.4.m4.1.1.2" xref="S6.5.p1.4.m4.1.1.2.cmml">U</mi><mo id="S6.5.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.5.p1.4.m4.1.1.1.cmml">:</mo><mrow id="S6.5.p1.4.m4.1.1.3" xref="S6.5.p1.4.m4.1.1.3.cmml"><msub id="S6.5.p1.4.m4.1.1.3.2" xref="S6.5.p1.4.m4.1.1.3.2.cmml"><mi id="S6.5.p1.4.m4.1.1.3.2.2" xref="S6.5.p1.4.m4.1.1.3.2.2.cmml">F</mi><mn id="S6.5.p1.4.m4.1.1.3.2.3" xref="S6.5.p1.4.m4.1.1.3.2.3.cmml">1</mn></msub><mo id="S6.5.p1.4.m4.1.1.3.1" stretchy="false" xref="S6.5.p1.4.m4.1.1.3.1.cmml">→</mo><msub id="S6.5.p1.4.m4.1.1.3.3" xref="S6.5.p1.4.m4.1.1.3.3.cmml"><mi id="S6.5.p1.4.m4.1.1.3.3.2" xref="S6.5.p1.4.m4.1.1.3.3.2.cmml">F</mi><mn id="S6.5.p1.4.m4.1.1.3.3.3" xref="S6.5.p1.4.m4.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.5.p1.4.m4.1b"><apply id="S6.5.p1.4.m4.1.1.cmml" xref="S6.5.p1.4.m4.1.1"><ci id="S6.5.p1.4.m4.1.1.1.cmml" xref="S6.5.p1.4.m4.1.1.1">:</ci><ci id="S6.5.p1.4.m4.1.1.2.cmml" xref="S6.5.p1.4.m4.1.1.2">𝑈</ci><apply id="S6.5.p1.4.m4.1.1.3.cmml" xref="S6.5.p1.4.m4.1.1.3"><ci id="S6.5.p1.4.m4.1.1.3.1.cmml" xref="S6.5.p1.4.m4.1.1.3.1">→</ci><apply id="S6.5.p1.4.m4.1.1.3.2.cmml" xref="S6.5.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S6.5.p1.4.m4.1.1.3.2.1.cmml" xref="S6.5.p1.4.m4.1.1.3.2">subscript</csymbol><ci id="S6.5.p1.4.m4.1.1.3.2.2.cmml" xref="S6.5.p1.4.m4.1.1.3.2.2">𝐹</ci><cn id="S6.5.p1.4.m4.1.1.3.2.3.cmml" type="integer" xref="S6.5.p1.4.m4.1.1.3.2.3">1</cn></apply><apply id="S6.5.p1.4.m4.1.1.3.3.cmml" xref="S6.5.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S6.5.p1.4.m4.1.1.3.3.1.cmml" xref="S6.5.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S6.5.p1.4.m4.1.1.3.3.2.cmml" xref="S6.5.p1.4.m4.1.1.3.3.2">𝐹</ci><cn id="S6.5.p1.4.m4.1.1.3.3.3.cmml" type="integer" xref="S6.5.p1.4.m4.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p1.4.m4.1c">U:F_{1}\to F_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p1.4.m4.1d">italic_U : italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> induces a bisimplicial map <math alttext="\mathfrak{U}:N(\mathcal{D};F_{1})\to N(\mathcal{D};F_{2})" class="ltx_Math" display="inline" id="S6.5.p1.5.m5.4"><semantics id="S6.5.p1.5.m5.4a"><mrow id="S6.5.p1.5.m5.4.4" xref="S6.5.p1.5.m5.4.4.cmml"><mi id="S6.5.p1.5.m5.4.4.4" xref="S6.5.p1.5.m5.4.4.4.cmml">𝔘</mi><mo id="S6.5.p1.5.m5.4.4.3" lspace="0.278em" rspace="0.278em" xref="S6.5.p1.5.m5.4.4.3.cmml">:</mo><mrow id="S6.5.p1.5.m5.4.4.2" xref="S6.5.p1.5.m5.4.4.2.cmml"><mrow id="S6.5.p1.5.m5.3.3.1.1" xref="S6.5.p1.5.m5.3.3.1.1.cmml"><mi id="S6.5.p1.5.m5.3.3.1.1.3" xref="S6.5.p1.5.m5.3.3.1.1.3.cmml">N</mi><mo id="S6.5.p1.5.m5.3.3.1.1.2" xref="S6.5.p1.5.m5.3.3.1.1.2.cmml">⁢</mo><mrow id="S6.5.p1.5.m5.3.3.1.1.1.1" xref="S6.5.p1.5.m5.3.3.1.1.1.2.cmml"><mo id="S6.5.p1.5.m5.3.3.1.1.1.1.2" stretchy="false" xref="S6.5.p1.5.m5.3.3.1.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.5.p1.5.m5.1.1" xref="S6.5.p1.5.m5.1.1.cmml">𝒟</mi><mo id="S6.5.p1.5.m5.3.3.1.1.1.1.3" xref="S6.5.p1.5.m5.3.3.1.1.1.2.cmml">;</mo><msub id="S6.5.p1.5.m5.3.3.1.1.1.1.1" xref="S6.5.p1.5.m5.3.3.1.1.1.1.1.cmml"><mi id="S6.5.p1.5.m5.3.3.1.1.1.1.1.2" xref="S6.5.p1.5.m5.3.3.1.1.1.1.1.2.cmml">F</mi><mn id="S6.5.p1.5.m5.3.3.1.1.1.1.1.3" xref="S6.5.p1.5.m5.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.5.p1.5.m5.3.3.1.1.1.1.4" stretchy="false" xref="S6.5.p1.5.m5.3.3.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.5.p1.5.m5.4.4.2.3" stretchy="false" xref="S6.5.p1.5.m5.4.4.2.3.cmml">→</mo><mrow id="S6.5.p1.5.m5.4.4.2.2" xref="S6.5.p1.5.m5.4.4.2.2.cmml"><mi id="S6.5.p1.5.m5.4.4.2.2.3" xref="S6.5.p1.5.m5.4.4.2.2.3.cmml">N</mi><mo id="S6.5.p1.5.m5.4.4.2.2.2" xref="S6.5.p1.5.m5.4.4.2.2.2.cmml">⁢</mo><mrow id="S6.5.p1.5.m5.4.4.2.2.1.1" xref="S6.5.p1.5.m5.4.4.2.2.1.2.cmml"><mo id="S6.5.p1.5.m5.4.4.2.2.1.1.2" stretchy="false" xref="S6.5.p1.5.m5.4.4.2.2.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.5.p1.5.m5.2.2" xref="S6.5.p1.5.m5.2.2.cmml">𝒟</mi><mo id="S6.5.p1.5.m5.4.4.2.2.1.1.3" xref="S6.5.p1.5.m5.4.4.2.2.1.2.cmml">;</mo><msub id="S6.5.p1.5.m5.4.4.2.2.1.1.1" xref="S6.5.p1.5.m5.4.4.2.2.1.1.1.cmml"><mi id="S6.5.p1.5.m5.4.4.2.2.1.1.1.2" xref="S6.5.p1.5.m5.4.4.2.2.1.1.1.2.cmml">F</mi><mn id="S6.5.p1.5.m5.4.4.2.2.1.1.1.3" xref="S6.5.p1.5.m5.4.4.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S6.5.p1.5.m5.4.4.2.2.1.1.4" stretchy="false" xref="S6.5.p1.5.m5.4.4.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.5.p1.5.m5.4b"><apply id="S6.5.p1.5.m5.4.4.cmml" xref="S6.5.p1.5.m5.4.4"><ci id="S6.5.p1.5.m5.4.4.3.cmml" xref="S6.5.p1.5.m5.4.4.3">:</ci><ci id="S6.5.p1.5.m5.4.4.4.cmml" xref="S6.5.p1.5.m5.4.4.4">𝔘</ci><apply id="S6.5.p1.5.m5.4.4.2.cmml" xref="S6.5.p1.5.m5.4.4.2"><ci id="S6.5.p1.5.m5.4.4.2.3.cmml" xref="S6.5.p1.5.m5.4.4.2.3">→</ci><apply id="S6.5.p1.5.m5.3.3.1.1.cmml" xref="S6.5.p1.5.m5.3.3.1.1"><times id="S6.5.p1.5.m5.3.3.1.1.2.cmml" xref="S6.5.p1.5.m5.3.3.1.1.2"></times><ci id="S6.5.p1.5.m5.3.3.1.1.3.cmml" xref="S6.5.p1.5.m5.3.3.1.1.3">𝑁</ci><list id="S6.5.p1.5.m5.3.3.1.1.1.2.cmml" xref="S6.5.p1.5.m5.3.3.1.1.1.1"><ci id="S6.5.p1.5.m5.1.1.cmml" xref="S6.5.p1.5.m5.1.1">𝒟</ci><apply id="S6.5.p1.5.m5.3.3.1.1.1.1.1.cmml" xref="S6.5.p1.5.m5.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.5.p1.5.m5.3.3.1.1.1.1.1.1.cmml" xref="S6.5.p1.5.m5.3.3.1.1.1.1.1">subscript</csymbol><ci id="S6.5.p1.5.m5.3.3.1.1.1.1.1.2.cmml" xref="S6.5.p1.5.m5.3.3.1.1.1.1.1.2">𝐹</ci><cn id="S6.5.p1.5.m5.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S6.5.p1.5.m5.3.3.1.1.1.1.1.3">1</cn></apply></list></apply><apply id="S6.5.p1.5.m5.4.4.2.2.cmml" xref="S6.5.p1.5.m5.4.4.2.2"><times id="S6.5.p1.5.m5.4.4.2.2.2.cmml" xref="S6.5.p1.5.m5.4.4.2.2.2"></times><ci id="S6.5.p1.5.m5.4.4.2.2.3.cmml" xref="S6.5.p1.5.m5.4.4.2.2.3">𝑁</ci><list id="S6.5.p1.5.m5.4.4.2.2.1.2.cmml" xref="S6.5.p1.5.m5.4.4.2.2.1.1"><ci id="S6.5.p1.5.m5.2.2.cmml" xref="S6.5.p1.5.m5.2.2">𝒟</ci><apply id="S6.5.p1.5.m5.4.4.2.2.1.1.1.cmml" xref="S6.5.p1.5.m5.4.4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.5.p1.5.m5.4.4.2.2.1.1.1.1.cmml" xref="S6.5.p1.5.m5.4.4.2.2.1.1.1">subscript</csymbol><ci id="S6.5.p1.5.m5.4.4.2.2.1.1.1.2.cmml" xref="S6.5.p1.5.m5.4.4.2.2.1.1.1.2">𝐹</ci><cn id="S6.5.p1.5.m5.4.4.2.2.1.1.1.3.cmml" type="integer" xref="S6.5.p1.5.m5.4.4.2.2.1.1.1.3">2</cn></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p1.5.m5.4c">\mathfrak{U}:N(\mathcal{D};F_{1})\to N(\mathcal{D};F_{2})</annotation><annotation encoding="application/x-llamapun" id="S6.5.p1.5.m5.4d">fraktur_U : italic_N ( caligraphic_D ; italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) → italic_N ( caligraphic_D ; italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. Consider the following commuting diagram:</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex87"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><svg class="ltx_picture ltx_markedasmath" height="80.22" id="S6.Ex87.m1.1.1.pic1" overflow="visible" version="1.1" width="298.57"><g transform="matrix(1 0 0 -1 63.45 19.67) translate(63.45,0)"><g transform="translate(-54.51,0) translate(4.15,0)"><foreignobject height="15.07" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="100.72"><math alttext="\textstyle{H^{*}(N(\mathcal{D};F_{2});\mathcal{M}^{\prime})\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}" class="ltx_Math" display="inline" id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3"><semantics id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3a"><mrow id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3" xref="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.cmml"><msup id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.4" xref="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.4.cmml"><mi id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.4.2" xref="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.4.2.cmml">H</mi><mo id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.4.3" xref="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.4.3.cmml">∗</mo></msup><mo id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.3" xref="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.3.cmml">⁢</mo><mrow id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.2.2" xref="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.2.3.cmml"><mo id="S6.Ex87.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.2.2.3" stretchy="false" 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encoding="application/x-tex" id="S6.Ex87.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.2c">\textstyle{H^{*}(\operatorname*{hocolim}_{\mathcal{D}}F_{1};\mathfrak{u}^{*}% \mathcal{M})}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex87.m1.1.1.pic1.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.m1.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_hocolim start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; fraktur_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M )</annotation></semantics></math></foreignobject></g></g></svg></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.5.p1.8">where horizontal maps are defined as in Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3.3. Induced Maps on Cohomology ‣ 3. Cohomology of simplicial sets and Thomason Cohomology ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">3.4</span></a> and the vertical maps are the maps defined in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S6.Thmtheorem3" title="Proposition 6.3. ‣ 6. Cohomology of bisimplicial sets with nontrivial coefficients ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">6.3</span></a>. By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14659v1#S6.Thmtheorem3" title="Proposition 6.3. ‣ 6. Cohomology of bisimplicial sets with nontrivial coefficients ‣ Thomason cohomology and Quillen’s Theorem A"><span class="ltx_text ltx_ref_tag">6.3</span></a>, vertical maps <math alttext="J^{*}" class="ltx_Math" display="inline" id="S6.5.p1.6.m1.1"><semantics id="S6.5.p1.6.m1.1a"><msup id="S6.5.p1.6.m1.1.1" xref="S6.5.p1.6.m1.1.1.cmml"><mi id="S6.5.p1.6.m1.1.1.2" xref="S6.5.p1.6.m1.1.1.2.cmml">J</mi><mo id="S6.5.p1.6.m1.1.1.3" xref="S6.5.p1.6.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S6.5.p1.6.m1.1b"><apply id="S6.5.p1.6.m1.1.1.cmml" xref="S6.5.p1.6.m1.1.1"><csymbol cd="ambiguous" id="S6.5.p1.6.m1.1.1.1.cmml" xref="S6.5.p1.6.m1.1.1">superscript</csymbol><ci id="S6.5.p1.6.m1.1.1.2.cmml" xref="S6.5.p1.6.m1.1.1.2">𝐽</ci><times id="S6.5.p1.6.m1.1.1.3.cmml" xref="S6.5.p1.6.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p1.6.m1.1c">J^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p1.6.m1.1d">italic_J start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> are isomorphisms. Hence <math alttext="\mathfrak{u}^{*}" class="ltx_Math" display="inline" id="S6.5.p1.7.m2.1"><semantics id="S6.5.p1.7.m2.1a"><msup id="S6.5.p1.7.m2.1.1" xref="S6.5.p1.7.m2.1.1.cmml"><mi id="S6.5.p1.7.m2.1.1.2" xref="S6.5.p1.7.m2.1.1.2.cmml">𝔲</mi><mo id="S6.5.p1.7.m2.1.1.3" xref="S6.5.p1.7.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S6.5.p1.7.m2.1b"><apply id="S6.5.p1.7.m2.1.1.cmml" xref="S6.5.p1.7.m2.1.1"><csymbol cd="ambiguous" id="S6.5.p1.7.m2.1.1.1.cmml" xref="S6.5.p1.7.m2.1.1">superscript</csymbol><ci id="S6.5.p1.7.m2.1.1.2.cmml" xref="S6.5.p1.7.m2.1.1.2">𝔲</ci><times id="S6.5.p1.7.m2.1.1.3.cmml" xref="S6.5.p1.7.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p1.7.m2.1c">\mathfrak{u}^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p1.7.m2.1d">fraktur_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is an isomorphism if <math alttext="\mathfrak{U}^{*}" class="ltx_Math" display="inline" id="S6.5.p1.8.m3.1"><semantics id="S6.5.p1.8.m3.1a"><msup id="S6.5.p1.8.m3.1.1" xref="S6.5.p1.8.m3.1.1.cmml"><mi id="S6.5.p1.8.m3.1.1.2" xref="S6.5.p1.8.m3.1.1.2.cmml">𝔘</mi><mo id="S6.5.p1.8.m3.1.1.3" xref="S6.5.p1.8.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S6.5.p1.8.m3.1b"><apply id="S6.5.p1.8.m3.1.1.cmml" xref="S6.5.p1.8.m3.1.1"><csymbol cd="ambiguous" id="S6.5.p1.8.m3.1.1.1.cmml" xref="S6.5.p1.8.m3.1.1">superscript</csymbol><ci id="S6.5.p1.8.m3.1.1.2.cmml" xref="S6.5.p1.8.m3.1.1.2">𝔘</ci><times id="S6.5.p1.8.m3.1.1.3.cmml" xref="S6.5.p1.8.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p1.8.m3.1c">\mathfrak{U}^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p1.8.m3.1d">fraktur_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is an isomorphism.</p> </div> <div class="ltx_para" id="S6.6.p2"> <p class="ltx_p" id="S6.6.p2.6">Let <math alttext="X_{i}=N(\mathcal{D};F_{i})" class="ltx_Math" display="inline" id="S6.6.p2.1.m1.2"><semantics id="S6.6.p2.1.m1.2a"><mrow id="S6.6.p2.1.m1.2.2" xref="S6.6.p2.1.m1.2.2.cmml"><msub id="S6.6.p2.1.m1.2.2.3" xref="S6.6.p2.1.m1.2.2.3.cmml"><mi id="S6.6.p2.1.m1.2.2.3.2" xref="S6.6.p2.1.m1.2.2.3.2.cmml">X</mi><mi id="S6.6.p2.1.m1.2.2.3.3" xref="S6.6.p2.1.m1.2.2.3.3.cmml">i</mi></msub><mo id="S6.6.p2.1.m1.2.2.2" xref="S6.6.p2.1.m1.2.2.2.cmml">=</mo><mrow id="S6.6.p2.1.m1.2.2.1" xref="S6.6.p2.1.m1.2.2.1.cmml"><mi id="S6.6.p2.1.m1.2.2.1.3" xref="S6.6.p2.1.m1.2.2.1.3.cmml">N</mi><mo id="S6.6.p2.1.m1.2.2.1.2" xref="S6.6.p2.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S6.6.p2.1.m1.2.2.1.1.1" xref="S6.6.p2.1.m1.2.2.1.1.2.cmml"><mo id="S6.6.p2.1.m1.2.2.1.1.1.2" stretchy="false" xref="S6.6.p2.1.m1.2.2.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.6.p2.1.m1.1.1" xref="S6.6.p2.1.m1.1.1.cmml">𝒟</mi><mo id="S6.6.p2.1.m1.2.2.1.1.1.3" xref="S6.6.p2.1.m1.2.2.1.1.2.cmml">;</mo><msub id="S6.6.p2.1.m1.2.2.1.1.1.1" xref="S6.6.p2.1.m1.2.2.1.1.1.1.cmml"><mi id="S6.6.p2.1.m1.2.2.1.1.1.1.2" xref="S6.6.p2.1.m1.2.2.1.1.1.1.2.cmml">F</mi><mi id="S6.6.p2.1.m1.2.2.1.1.1.1.3" xref="S6.6.p2.1.m1.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.6.p2.1.m1.2.2.1.1.1.4" stretchy="false" xref="S6.6.p2.1.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p2.1.m1.2b"><apply id="S6.6.p2.1.m1.2.2.cmml" xref="S6.6.p2.1.m1.2.2"><eq id="S6.6.p2.1.m1.2.2.2.cmml" xref="S6.6.p2.1.m1.2.2.2"></eq><apply id="S6.6.p2.1.m1.2.2.3.cmml" xref="S6.6.p2.1.m1.2.2.3"><csymbol cd="ambiguous" id="S6.6.p2.1.m1.2.2.3.1.cmml" xref="S6.6.p2.1.m1.2.2.3">subscript</csymbol><ci id="S6.6.p2.1.m1.2.2.3.2.cmml" xref="S6.6.p2.1.m1.2.2.3.2">𝑋</ci><ci id="S6.6.p2.1.m1.2.2.3.3.cmml" xref="S6.6.p2.1.m1.2.2.3.3">𝑖</ci></apply><apply id="S6.6.p2.1.m1.2.2.1.cmml" xref="S6.6.p2.1.m1.2.2.1"><times id="S6.6.p2.1.m1.2.2.1.2.cmml" xref="S6.6.p2.1.m1.2.2.1.2"></times><ci id="S6.6.p2.1.m1.2.2.1.3.cmml" xref="S6.6.p2.1.m1.2.2.1.3">𝑁</ci><list id="S6.6.p2.1.m1.2.2.1.1.2.cmml" xref="S6.6.p2.1.m1.2.2.1.1.1"><ci id="S6.6.p2.1.m1.1.1.cmml" xref="S6.6.p2.1.m1.1.1">𝒟</ci><apply id="S6.6.p2.1.m1.2.2.1.1.1.1.cmml" xref="S6.6.p2.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.6.p2.1.m1.2.2.1.1.1.1.1.cmml" xref="S6.6.p2.1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S6.6.p2.1.m1.2.2.1.1.1.1.2.cmml" xref="S6.6.p2.1.m1.2.2.1.1.1.1.2">𝐹</ci><ci id="S6.6.p2.1.m1.2.2.1.1.1.1.3.cmml" xref="S6.6.p2.1.m1.2.2.1.1.1.1.3">𝑖</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p2.1.m1.2c">X_{i}=N(\mathcal{D};F_{i})</annotation><annotation encoding="application/x-llamapun" id="S6.6.p2.1.m1.2d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_N ( caligraphic_D ; italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="i=1,2" class="ltx_Math" display="inline" id="S6.6.p2.2.m2.2"><semantics id="S6.6.p2.2.m2.2a"><mrow id="S6.6.p2.2.m2.2.3" xref="S6.6.p2.2.m2.2.3.cmml"><mi id="S6.6.p2.2.m2.2.3.2" xref="S6.6.p2.2.m2.2.3.2.cmml">i</mi><mo id="S6.6.p2.2.m2.2.3.1" xref="S6.6.p2.2.m2.2.3.1.cmml">=</mo><mrow id="S6.6.p2.2.m2.2.3.3.2" xref="S6.6.p2.2.m2.2.3.3.1.cmml"><mn id="S6.6.p2.2.m2.1.1" xref="S6.6.p2.2.m2.1.1.cmml">1</mn><mo id="S6.6.p2.2.m2.2.3.3.2.1" xref="S6.6.p2.2.m2.2.3.3.1.cmml">,</mo><mn id="S6.6.p2.2.m2.2.2" xref="S6.6.p2.2.m2.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p2.2.m2.2b"><apply id="S6.6.p2.2.m2.2.3.cmml" xref="S6.6.p2.2.m2.2.3"><eq id="S6.6.p2.2.m2.2.3.1.cmml" xref="S6.6.p2.2.m2.2.3.1"></eq><ci id="S6.6.p2.2.m2.2.3.2.cmml" xref="S6.6.p2.2.m2.2.3.2">𝑖</ci><list id="S6.6.p2.2.m2.2.3.3.1.cmml" xref="S6.6.p2.2.m2.2.3.3.2"><cn id="S6.6.p2.2.m2.1.1.cmml" type="integer" xref="S6.6.p2.2.m2.1.1">1</cn><cn id="S6.6.p2.2.m2.2.2.cmml" type="integer" xref="S6.6.p2.2.m2.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p2.2.m2.2c">i=1,2</annotation><annotation encoding="application/x-llamapun" id="S6.6.p2.2.m2.2d">italic_i = 1 , 2</annotation></semantics></math>. The homomorphism <math alttext="\mathfrak{U}^{*}" class="ltx_Math" display="inline" id="S6.6.p2.3.m3.1"><semantics id="S6.6.p2.3.m3.1a"><msup id="S6.6.p2.3.m3.1.1" xref="S6.6.p2.3.m3.1.1.cmml"><mi id="S6.6.p2.3.m3.1.1.2" xref="S6.6.p2.3.m3.1.1.2.cmml">𝔘</mi><mo id="S6.6.p2.3.m3.1.1.3" xref="S6.6.p2.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S6.6.p2.3.m3.1b"><apply id="S6.6.p2.3.m3.1.1.cmml" xref="S6.6.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S6.6.p2.3.m3.1.1.1.cmml" xref="S6.6.p2.3.m3.1.1">superscript</csymbol><ci id="S6.6.p2.3.m3.1.1.2.cmml" xref="S6.6.p2.3.m3.1.1.2">𝔘</ci><times id="S6.6.p2.3.m3.1.1.3.cmml" xref="S6.6.p2.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p2.3.m3.1c">\mathfrak{U}^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.6.p2.3.m3.1d">fraktur_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is defined on the cochain level as the chain map <math alttext="\mathfrak{U}^{*}:C^{*}(X_{2};\mathcal{M}^{\prime})\to C^{*}(X_{1};\mathfrak{U}% ^{*}\mathcal{M}^{\prime})" class="ltx_Math" display="inline" id="S6.6.p2.4.m4.4"><semantics id="S6.6.p2.4.m4.4a"><mrow id="S6.6.p2.4.m4.4.4" xref="S6.6.p2.4.m4.4.4.cmml"><msup id="S6.6.p2.4.m4.4.4.6" xref="S6.6.p2.4.m4.4.4.6.cmml"><mi id="S6.6.p2.4.m4.4.4.6.2" xref="S6.6.p2.4.m4.4.4.6.2.cmml">𝔘</mi><mo id="S6.6.p2.4.m4.4.4.6.3" xref="S6.6.p2.4.m4.4.4.6.3.cmml">∗</mo></msup><mo id="S6.6.p2.4.m4.4.4.5" lspace="0.278em" rspace="0.278em" xref="S6.6.p2.4.m4.4.4.5.cmml">:</mo><mrow id="S6.6.p2.4.m4.4.4.4" xref="S6.6.p2.4.m4.4.4.4.cmml"><mrow id="S6.6.p2.4.m4.2.2.2.2" xref="S6.6.p2.4.m4.2.2.2.2.cmml"><msup id="S6.6.p2.4.m4.2.2.2.2.4" xref="S6.6.p2.4.m4.2.2.2.2.4.cmml"><mi id="S6.6.p2.4.m4.2.2.2.2.4.2" xref="S6.6.p2.4.m4.2.2.2.2.4.2.cmml">C</mi><mo id="S6.6.p2.4.m4.2.2.2.2.4.3" xref="S6.6.p2.4.m4.2.2.2.2.4.3.cmml">∗</mo></msup><mo id="S6.6.p2.4.m4.2.2.2.2.3" xref="S6.6.p2.4.m4.2.2.2.2.3.cmml">⁢</mo><mrow id="S6.6.p2.4.m4.2.2.2.2.2.2" xref="S6.6.p2.4.m4.2.2.2.2.2.3.cmml"><mo id="S6.6.p2.4.m4.2.2.2.2.2.2.3" stretchy="false" xref="S6.6.p2.4.m4.2.2.2.2.2.3.cmml">(</mo><msub id="S6.6.p2.4.m4.1.1.1.1.1.1.1" xref="S6.6.p2.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S6.6.p2.4.m4.1.1.1.1.1.1.1.2" xref="S6.6.p2.4.m4.1.1.1.1.1.1.1.2.cmml">X</mi><mn id="S6.6.p2.4.m4.1.1.1.1.1.1.1.3" xref="S6.6.p2.4.m4.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.6.p2.4.m4.2.2.2.2.2.2.4" xref="S6.6.p2.4.m4.2.2.2.2.2.3.cmml">;</mo><msup id="S6.6.p2.4.m4.2.2.2.2.2.2.2" xref="S6.6.p2.4.m4.2.2.2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.6.p2.4.m4.2.2.2.2.2.2.2.2" xref="S6.6.p2.4.m4.2.2.2.2.2.2.2.2.cmml">ℳ</mi><mo id="S6.6.p2.4.m4.2.2.2.2.2.2.2.3" xref="S6.6.p2.4.m4.2.2.2.2.2.2.2.3.cmml">′</mo></msup><mo id="S6.6.p2.4.m4.2.2.2.2.2.2.5" stretchy="false" xref="S6.6.p2.4.m4.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.6.p2.4.m4.4.4.4.5" stretchy="false" xref="S6.6.p2.4.m4.4.4.4.5.cmml">→</mo><mrow id="S6.6.p2.4.m4.4.4.4.4" xref="S6.6.p2.4.m4.4.4.4.4.cmml"><msup id="S6.6.p2.4.m4.4.4.4.4.4" xref="S6.6.p2.4.m4.4.4.4.4.4.cmml"><mi id="S6.6.p2.4.m4.4.4.4.4.4.2" xref="S6.6.p2.4.m4.4.4.4.4.4.2.cmml">C</mi><mo id="S6.6.p2.4.m4.4.4.4.4.4.3" xref="S6.6.p2.4.m4.4.4.4.4.4.3.cmml">∗</mo></msup><mo id="S6.6.p2.4.m4.4.4.4.4.3" xref="S6.6.p2.4.m4.4.4.4.4.3.cmml">⁢</mo><mrow id="S6.6.p2.4.m4.4.4.4.4.2.2" xref="S6.6.p2.4.m4.4.4.4.4.2.3.cmml"><mo id="S6.6.p2.4.m4.4.4.4.4.2.2.3" stretchy="false" xref="S6.6.p2.4.m4.4.4.4.4.2.3.cmml">(</mo><msub id="S6.6.p2.4.m4.3.3.3.3.1.1.1" xref="S6.6.p2.4.m4.3.3.3.3.1.1.1.cmml"><mi id="S6.6.p2.4.m4.3.3.3.3.1.1.1.2" xref="S6.6.p2.4.m4.3.3.3.3.1.1.1.2.cmml">X</mi><mn id="S6.6.p2.4.m4.3.3.3.3.1.1.1.3" xref="S6.6.p2.4.m4.3.3.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S6.6.p2.4.m4.4.4.4.4.2.2.4" xref="S6.6.p2.4.m4.4.4.4.4.2.3.cmml">;</mo><mrow id="S6.6.p2.4.m4.4.4.4.4.2.2.2" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.cmml"><msup 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id="S6.6.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.6.p2.4.m4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.6.p2.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S6.6.p2.4.m4.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.6.p2.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S6.6.p2.4.m4.1.1.1.1.1.1.1.2">𝑋</ci><cn id="S6.6.p2.4.m4.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.6.p2.4.m4.1.1.1.1.1.1.1.3">2</cn></apply><apply id="S6.6.p2.4.m4.2.2.2.2.2.2.2.cmml" xref="S6.6.p2.4.m4.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.6.p2.4.m4.2.2.2.2.2.2.2.1.cmml" xref="S6.6.p2.4.m4.2.2.2.2.2.2.2">superscript</csymbol><ci id="S6.6.p2.4.m4.2.2.2.2.2.2.2.2.cmml" xref="S6.6.p2.4.m4.2.2.2.2.2.2.2.2">ℳ</ci><ci id="S6.6.p2.4.m4.2.2.2.2.2.2.2.3.cmml" xref="S6.6.p2.4.m4.2.2.2.2.2.2.2.3">′</ci></apply></list></apply><apply id="S6.6.p2.4.m4.4.4.4.4.cmml" xref="S6.6.p2.4.m4.4.4.4.4"><times id="S6.6.p2.4.m4.4.4.4.4.3.cmml" xref="S6.6.p2.4.m4.4.4.4.4.3"></times><apply id="S6.6.p2.4.m4.4.4.4.4.4.cmml" xref="S6.6.p2.4.m4.4.4.4.4.4"><csymbol cd="ambiguous" id="S6.6.p2.4.m4.4.4.4.4.4.1.cmml" xref="S6.6.p2.4.m4.4.4.4.4.4">superscript</csymbol><ci id="S6.6.p2.4.m4.4.4.4.4.4.2.cmml" xref="S6.6.p2.4.m4.4.4.4.4.4.2">𝐶</ci><times id="S6.6.p2.4.m4.4.4.4.4.4.3.cmml" xref="S6.6.p2.4.m4.4.4.4.4.4.3"></times></apply><list id="S6.6.p2.4.m4.4.4.4.4.2.3.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2"><apply id="S6.6.p2.4.m4.3.3.3.3.1.1.1.cmml" xref="S6.6.p2.4.m4.3.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.6.p2.4.m4.3.3.3.3.1.1.1.1.cmml" xref="S6.6.p2.4.m4.3.3.3.3.1.1.1">subscript</csymbol><ci id="S6.6.p2.4.m4.3.3.3.3.1.1.1.2.cmml" xref="S6.6.p2.4.m4.3.3.3.3.1.1.1.2">𝑋</ci><cn id="S6.6.p2.4.m4.3.3.3.3.1.1.1.3.cmml" type="integer" xref="S6.6.p2.4.m4.3.3.3.3.1.1.1.3">1</cn></apply><apply id="S6.6.p2.4.m4.4.4.4.4.2.2.2.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2"><times id="S6.6.p2.4.m4.4.4.4.4.2.2.2.1.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.1"></times><apply id="S6.6.p2.4.m4.4.4.4.4.2.2.2.2.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.6.p2.4.m4.4.4.4.4.2.2.2.2.1.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.2">superscript</csymbol><ci id="S6.6.p2.4.m4.4.4.4.4.2.2.2.2.2.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.2.2">𝔘</ci><times id="S6.6.p2.4.m4.4.4.4.4.2.2.2.2.3.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.2.3"></times></apply><apply id="S6.6.p2.4.m4.4.4.4.4.2.2.2.3.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S6.6.p2.4.m4.4.4.4.4.2.2.2.3.1.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.3">superscript</csymbol><ci id="S6.6.p2.4.m4.4.4.4.4.2.2.2.3.2.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.3.2">ℳ</ci><ci id="S6.6.p2.4.m4.4.4.4.4.2.2.2.3.3.cmml" xref="S6.6.p2.4.m4.4.4.4.4.2.2.2.3.3">′</ci></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p2.4.m4.4c">\mathfrak{U}^{*}:C^{*}(X_{2};\mathcal{M}^{\prime})\to C^{*}(X_{1};\mathfrak{U}% ^{*}\mathcal{M}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.6.p2.4.m4.4d">fraktur_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) → italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; fraktur_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> which sends <math alttext="f\in C^{n}(X_{2};\mathcal{M}^{\prime})" class="ltx_Math" display="inline" id="S6.6.p2.5.m5.2"><semantics id="S6.6.p2.5.m5.2a"><mrow id="S6.6.p2.5.m5.2.2" xref="S6.6.p2.5.m5.2.2.cmml"><mi id="S6.6.p2.5.m5.2.2.4" xref="S6.6.p2.5.m5.2.2.4.cmml">f</mi><mo id="S6.6.p2.5.m5.2.2.3" xref="S6.6.p2.5.m5.2.2.3.cmml">∈</mo><mrow id="S6.6.p2.5.m5.2.2.2" xref="S6.6.p2.5.m5.2.2.2.cmml"><msup id="S6.6.p2.5.m5.2.2.2.4" xref="S6.6.p2.5.m5.2.2.2.4.cmml"><mi id="S6.6.p2.5.m5.2.2.2.4.2" xref="S6.6.p2.5.m5.2.2.2.4.2.cmml">C</mi><mi id="S6.6.p2.5.m5.2.2.2.4.3" xref="S6.6.p2.5.m5.2.2.2.4.3.cmml">n</mi></msup><mo id="S6.6.p2.5.m5.2.2.2.3" xref="S6.6.p2.5.m5.2.2.2.3.cmml">⁢</mo><mrow id="S6.6.p2.5.m5.2.2.2.2.2" xref="S6.6.p2.5.m5.2.2.2.2.3.cmml"><mo id="S6.6.p2.5.m5.2.2.2.2.2.3" stretchy="false" xref="S6.6.p2.5.m5.2.2.2.2.3.cmml">(</mo><msub id="S6.6.p2.5.m5.1.1.1.1.1.1" xref="S6.6.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S6.6.p2.5.m5.1.1.1.1.1.1.2" xref="S6.6.p2.5.m5.1.1.1.1.1.1.2.cmml">X</mi><mn id="S6.6.p2.5.m5.1.1.1.1.1.1.3" xref="S6.6.p2.5.m5.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.6.p2.5.m5.2.2.2.2.2.4" xref="S6.6.p2.5.m5.2.2.2.2.3.cmml">;</mo><msup id="S6.6.p2.5.m5.2.2.2.2.2.2" xref="S6.6.p2.5.m5.2.2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.6.p2.5.m5.2.2.2.2.2.2.2" xref="S6.6.p2.5.m5.2.2.2.2.2.2.2.cmml">ℳ</mi><mo id="S6.6.p2.5.m5.2.2.2.2.2.2.3" xref="S6.6.p2.5.m5.2.2.2.2.2.2.3.cmml">′</mo></msup><mo id="S6.6.p2.5.m5.2.2.2.2.2.5" stretchy="false" xref="S6.6.p2.5.m5.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p2.5.m5.2b"><apply id="S6.6.p2.5.m5.2.2.cmml" xref="S6.6.p2.5.m5.2.2"><in id="S6.6.p2.5.m5.2.2.3.cmml" xref="S6.6.p2.5.m5.2.2.3"></in><ci id="S6.6.p2.5.m5.2.2.4.cmml" xref="S6.6.p2.5.m5.2.2.4">𝑓</ci><apply id="S6.6.p2.5.m5.2.2.2.cmml" xref="S6.6.p2.5.m5.2.2.2"><times id="S6.6.p2.5.m5.2.2.2.3.cmml" xref="S6.6.p2.5.m5.2.2.2.3"></times><apply id="S6.6.p2.5.m5.2.2.2.4.cmml" xref="S6.6.p2.5.m5.2.2.2.4"><csymbol cd="ambiguous" id="S6.6.p2.5.m5.2.2.2.4.1.cmml" xref="S6.6.p2.5.m5.2.2.2.4">superscript</csymbol><ci id="S6.6.p2.5.m5.2.2.2.4.2.cmml" xref="S6.6.p2.5.m5.2.2.2.4.2">𝐶</ci><ci id="S6.6.p2.5.m5.2.2.2.4.3.cmml" xref="S6.6.p2.5.m5.2.2.2.4.3">𝑛</ci></apply><list id="S6.6.p2.5.m5.2.2.2.2.3.cmml" xref="S6.6.p2.5.m5.2.2.2.2.2"><apply id="S6.6.p2.5.m5.1.1.1.1.1.1.cmml" xref="S6.6.p2.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.6.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S6.6.p2.5.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S6.6.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S6.6.p2.5.m5.1.1.1.1.1.1.2">𝑋</ci><cn id="S6.6.p2.5.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.6.p2.5.m5.1.1.1.1.1.1.3">2</cn></apply><apply id="S6.6.p2.5.m5.2.2.2.2.2.2.cmml" xref="S6.6.p2.5.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.6.p2.5.m5.2.2.2.2.2.2.1.cmml" xref="S6.6.p2.5.m5.2.2.2.2.2.2">superscript</csymbol><ci id="S6.6.p2.5.m5.2.2.2.2.2.2.2.cmml" xref="S6.6.p2.5.m5.2.2.2.2.2.2.2">ℳ</ci><ci id="S6.6.p2.5.m5.2.2.2.2.2.2.3.cmml" xref="S6.6.p2.5.m5.2.2.2.2.2.2.3">′</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p2.5.m5.2c">f\in C^{n}(X_{2};\mathcal{M}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.6.p2.5.m5.2d">italic_f ∈ italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> to <math alttext="\mathfrak{U}^{*}(f)\in C^{n}(X_{1};\mathfrak{U}^{*}\mathcal{M}^{\prime})" class="ltx_Math" display="inline" id="S6.6.p2.6.m6.3"><semantics id="S6.6.p2.6.m6.3a"><mrow id="S6.6.p2.6.m6.3.3" xref="S6.6.p2.6.m6.3.3.cmml"><mrow id="S6.6.p2.6.m6.3.3.4" xref="S6.6.p2.6.m6.3.3.4.cmml"><msup id="S6.6.p2.6.m6.3.3.4.2" xref="S6.6.p2.6.m6.3.3.4.2.cmml"><mi id="S6.6.p2.6.m6.3.3.4.2.2" xref="S6.6.p2.6.m6.3.3.4.2.2.cmml">𝔘</mi><mo id="S6.6.p2.6.m6.3.3.4.2.3" xref="S6.6.p2.6.m6.3.3.4.2.3.cmml">∗</mo></msup><mo id="S6.6.p2.6.m6.3.3.4.1" xref="S6.6.p2.6.m6.3.3.4.1.cmml">⁢</mo><mrow id="S6.6.p2.6.m6.3.3.4.3.2" xref="S6.6.p2.6.m6.3.3.4.cmml"><mo id="S6.6.p2.6.m6.3.3.4.3.2.1" stretchy="false" xref="S6.6.p2.6.m6.3.3.4.cmml">(</mo><mi id="S6.6.p2.6.m6.1.1" xref="S6.6.p2.6.m6.1.1.cmml">f</mi><mo id="S6.6.p2.6.m6.3.3.4.3.2.2" stretchy="false" xref="S6.6.p2.6.m6.3.3.4.cmml">)</mo></mrow></mrow><mo id="S6.6.p2.6.m6.3.3.3" xref="S6.6.p2.6.m6.3.3.3.cmml">∈</mo><mrow id="S6.6.p2.6.m6.3.3.2" xref="S6.6.p2.6.m6.3.3.2.cmml"><msup id="S6.6.p2.6.m6.3.3.2.4" xref="S6.6.p2.6.m6.3.3.2.4.cmml"><mi id="S6.6.p2.6.m6.3.3.2.4.2" xref="S6.6.p2.6.m6.3.3.2.4.2.cmml">C</mi><mi id="S6.6.p2.6.m6.3.3.2.4.3" xref="S6.6.p2.6.m6.3.3.2.4.3.cmml">n</mi></msup><mo id="S6.6.p2.6.m6.3.3.2.3" xref="S6.6.p2.6.m6.3.3.2.3.cmml">⁢</mo><mrow id="S6.6.p2.6.m6.3.3.2.2.2" xref="S6.6.p2.6.m6.3.3.2.2.3.cmml"><mo id="S6.6.p2.6.m6.3.3.2.2.2.3" stretchy="false" xref="S6.6.p2.6.m6.3.3.2.2.3.cmml">(</mo><msub id="S6.6.p2.6.m6.2.2.1.1.1.1" xref="S6.6.p2.6.m6.2.2.1.1.1.1.cmml"><mi id="S6.6.p2.6.m6.2.2.1.1.1.1.2" xref="S6.6.p2.6.m6.2.2.1.1.1.1.2.cmml">X</mi><mn id="S6.6.p2.6.m6.2.2.1.1.1.1.3" xref="S6.6.p2.6.m6.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.6.p2.6.m6.3.3.2.2.2.4" xref="S6.6.p2.6.m6.3.3.2.2.3.cmml">;</mo><mrow id="S6.6.p2.6.m6.3.3.2.2.2.2" xref="S6.6.p2.6.m6.3.3.2.2.2.2.cmml"><msup id="S6.6.p2.6.m6.3.3.2.2.2.2.2" xref="S6.6.p2.6.m6.3.3.2.2.2.2.2.cmml"><mi id="S6.6.p2.6.m6.3.3.2.2.2.2.2.2" xref="S6.6.p2.6.m6.3.3.2.2.2.2.2.2.cmml">𝔘</mi><mo id="S6.6.p2.6.m6.3.3.2.2.2.2.2.3" xref="S6.6.p2.6.m6.3.3.2.2.2.2.2.3.cmml">∗</mo></msup><mo id="S6.6.p2.6.m6.3.3.2.2.2.2.1" xref="S6.6.p2.6.m6.3.3.2.2.2.2.1.cmml">⁢</mo><msup id="S6.6.p2.6.m6.3.3.2.2.2.2.3" xref="S6.6.p2.6.m6.3.3.2.2.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.6.p2.6.m6.3.3.2.2.2.2.3.2" xref="S6.6.p2.6.m6.3.3.2.2.2.2.3.2.cmml">ℳ</mi><mo id="S6.6.p2.6.m6.3.3.2.2.2.2.3.3" xref="S6.6.p2.6.m6.3.3.2.2.2.2.3.3.cmml">′</mo></msup></mrow><mo id="S6.6.p2.6.m6.3.3.2.2.2.5" stretchy="false" xref="S6.6.p2.6.m6.3.3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p2.6.m6.3b"><apply id="S6.6.p2.6.m6.3.3.cmml" xref="S6.6.p2.6.m6.3.3"><in id="S6.6.p2.6.m6.3.3.3.cmml" xref="S6.6.p2.6.m6.3.3.3"></in><apply id="S6.6.p2.6.m6.3.3.4.cmml" xref="S6.6.p2.6.m6.3.3.4"><times id="S6.6.p2.6.m6.3.3.4.1.cmml" xref="S6.6.p2.6.m6.3.3.4.1"></times><apply id="S6.6.p2.6.m6.3.3.4.2.cmml" xref="S6.6.p2.6.m6.3.3.4.2"><csymbol cd="ambiguous" id="S6.6.p2.6.m6.3.3.4.2.1.cmml" xref="S6.6.p2.6.m6.3.3.4.2">superscript</csymbol><ci id="S6.6.p2.6.m6.3.3.4.2.2.cmml" xref="S6.6.p2.6.m6.3.3.4.2.2">𝔘</ci><times id="S6.6.p2.6.m6.3.3.4.2.3.cmml" xref="S6.6.p2.6.m6.3.3.4.2.3"></times></apply><ci id="S6.6.p2.6.m6.1.1.cmml" xref="S6.6.p2.6.m6.1.1">𝑓</ci></apply><apply id="S6.6.p2.6.m6.3.3.2.cmml" xref="S6.6.p2.6.m6.3.3.2"><times id="S6.6.p2.6.m6.3.3.2.3.cmml" xref="S6.6.p2.6.m6.3.3.2.3"></times><apply id="S6.6.p2.6.m6.3.3.2.4.cmml" xref="S6.6.p2.6.m6.3.3.2.4"><csymbol cd="ambiguous" id="S6.6.p2.6.m6.3.3.2.4.1.cmml" xref="S6.6.p2.6.m6.3.3.2.4">superscript</csymbol><ci id="S6.6.p2.6.m6.3.3.2.4.2.cmml" xref="S6.6.p2.6.m6.3.3.2.4.2">𝐶</ci><ci id="S6.6.p2.6.m6.3.3.2.4.3.cmml" xref="S6.6.p2.6.m6.3.3.2.4.3">𝑛</ci></apply><list id="S6.6.p2.6.m6.3.3.2.2.3.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2"><apply id="S6.6.p2.6.m6.2.2.1.1.1.1.cmml" xref="S6.6.p2.6.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.6.p2.6.m6.2.2.1.1.1.1.1.cmml" xref="S6.6.p2.6.m6.2.2.1.1.1.1">subscript</csymbol><ci id="S6.6.p2.6.m6.2.2.1.1.1.1.2.cmml" xref="S6.6.p2.6.m6.2.2.1.1.1.1.2">𝑋</ci><cn id="S6.6.p2.6.m6.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.6.p2.6.m6.2.2.1.1.1.1.3">1</cn></apply><apply id="S6.6.p2.6.m6.3.3.2.2.2.2.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2"><times id="S6.6.p2.6.m6.3.3.2.2.2.2.1.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.1"></times><apply id="S6.6.p2.6.m6.3.3.2.2.2.2.2.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.6.p2.6.m6.3.3.2.2.2.2.2.1.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.2">superscript</csymbol><ci id="S6.6.p2.6.m6.3.3.2.2.2.2.2.2.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.2.2">𝔘</ci><times id="S6.6.p2.6.m6.3.3.2.2.2.2.2.3.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.2.3"></times></apply><apply id="S6.6.p2.6.m6.3.3.2.2.2.2.3.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.3"><csymbol cd="ambiguous" id="S6.6.p2.6.m6.3.3.2.2.2.2.3.1.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.3">superscript</csymbol><ci id="S6.6.p2.6.m6.3.3.2.2.2.2.3.2.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.3.2">ℳ</ci><ci id="S6.6.p2.6.m6.3.3.2.2.2.2.3.3.cmml" xref="S6.6.p2.6.m6.3.3.2.2.2.2.3.3">′</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p2.6.m6.3c">\mathfrak{U}^{*}(f)\in C^{n}(X_{1};\mathfrak{U}^{*}\mathcal{M}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.6.p2.6.m6.3d">fraktur_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_f ) ∈ italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; fraktur_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex88"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(\mathfrak{U}^{*}f)(\sigma,\tau)=(\sigma,NU(d_{0})(\tau))" class="ltx_Math" display="block" id="S6.Ex88.m1.6"><semantics id="S6.Ex88.m1.6a"><mrow id="S6.Ex88.m1.6.6" xref="S6.Ex88.m1.6.6.cmml"><mrow id="S6.Ex88.m1.5.5.1" xref="S6.Ex88.m1.5.5.1.cmml"><mrow id="S6.Ex88.m1.5.5.1.1.1" xref="S6.Ex88.m1.5.5.1.1.1.1.cmml"><mo id="S6.Ex88.m1.5.5.1.1.1.2" stretchy="false" xref="S6.Ex88.m1.5.5.1.1.1.1.cmml">(</mo><mrow id="S6.Ex88.m1.5.5.1.1.1.1" xref="S6.Ex88.m1.5.5.1.1.1.1.cmml"><msup id="S6.Ex88.m1.5.5.1.1.1.1.2" xref="S6.Ex88.m1.5.5.1.1.1.1.2.cmml"><mi id="S6.Ex88.m1.5.5.1.1.1.1.2.2" xref="S6.Ex88.m1.5.5.1.1.1.1.2.2.cmml">𝔘</mi><mo id="S6.Ex88.m1.5.5.1.1.1.1.2.3" xref="S6.Ex88.m1.5.5.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S6.Ex88.m1.5.5.1.1.1.1.1" xref="S6.Ex88.m1.5.5.1.1.1.1.1.cmml">⁢</mo><mi id="S6.Ex88.m1.5.5.1.1.1.1.3" xref="S6.Ex88.m1.5.5.1.1.1.1.3.cmml">f</mi></mrow><mo id="S6.Ex88.m1.5.5.1.1.1.3" stretchy="false" xref="S6.Ex88.m1.5.5.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex88.m1.5.5.1.2" xref="S6.Ex88.m1.5.5.1.2.cmml">⁢</mo><mrow id="S6.Ex88.m1.5.5.1.3.2" xref="S6.Ex88.m1.5.5.1.3.1.cmml"><mo id="S6.Ex88.m1.5.5.1.3.2.1" stretchy="false" xref="S6.Ex88.m1.5.5.1.3.1.cmml">(</mo><mi id="S6.Ex88.m1.1.1" xref="S6.Ex88.m1.1.1.cmml">σ</mi><mo id="S6.Ex88.m1.5.5.1.3.2.2" xref="S6.Ex88.m1.5.5.1.3.1.cmml">,</mo><mi id="S6.Ex88.m1.2.2" xref="S6.Ex88.m1.2.2.cmml">τ</mi><mo id="S6.Ex88.m1.5.5.1.3.2.3" stretchy="false" xref="S6.Ex88.m1.5.5.1.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex88.m1.6.6.3" xref="S6.Ex88.m1.6.6.3.cmml">=</mo><mrow id="S6.Ex88.m1.6.6.2.1" xref="S6.Ex88.m1.6.6.2.2.cmml"><mo id="S6.Ex88.m1.6.6.2.1.2" stretchy="false" xref="S6.Ex88.m1.6.6.2.2.cmml">(</mo><mi id="S6.Ex88.m1.4.4" xref="S6.Ex88.m1.4.4.cmml">σ</mi><mo id="S6.Ex88.m1.6.6.2.1.3" xref="S6.Ex88.m1.6.6.2.2.cmml">,</mo><mrow 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xref="S6.Ex88.m1.6.6.2.1.1.cmml"><mo id="S6.Ex88.m1.6.6.2.1.1.5.2.1" stretchy="false" xref="S6.Ex88.m1.6.6.2.1.1.cmml">(</mo><mi id="S6.Ex88.m1.3.3" xref="S6.Ex88.m1.3.3.cmml">τ</mi><mo id="S6.Ex88.m1.6.6.2.1.1.5.2.2" stretchy="false" xref="S6.Ex88.m1.6.6.2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex88.m1.6.6.2.1.4" stretchy="false" xref="S6.Ex88.m1.6.6.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex88.m1.6b"><apply id="S6.Ex88.m1.6.6.cmml" xref="S6.Ex88.m1.6.6"><eq id="S6.Ex88.m1.6.6.3.cmml" xref="S6.Ex88.m1.6.6.3"></eq><apply id="S6.Ex88.m1.5.5.1.cmml" xref="S6.Ex88.m1.5.5.1"><times id="S6.Ex88.m1.5.5.1.2.cmml" xref="S6.Ex88.m1.5.5.1.2"></times><apply id="S6.Ex88.m1.5.5.1.1.1.1.cmml" xref="S6.Ex88.m1.5.5.1.1.1"><times id="S6.Ex88.m1.5.5.1.1.1.1.1.cmml" xref="S6.Ex88.m1.5.5.1.1.1.1.1"></times><apply id="S6.Ex88.m1.5.5.1.1.1.1.2.cmml" xref="S6.Ex88.m1.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex88.m1.5.5.1.1.1.1.2.1.cmml" 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xref="S6.Ex88.m1.6.6.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex88.m1.6.6.2.1.1.1.1.1.1.cmml" xref="S6.Ex88.m1.6.6.2.1.1.1.1">subscript</csymbol><ci id="S6.Ex88.m1.6.6.2.1.1.1.1.1.2.cmml" xref="S6.Ex88.m1.6.6.2.1.1.1.1.1.2">𝑑</ci><cn id="S6.Ex88.m1.6.6.2.1.1.1.1.1.3.cmml" type="integer" xref="S6.Ex88.m1.6.6.2.1.1.1.1.1.3">0</cn></apply><ci id="S6.Ex88.m1.3.3.cmml" xref="S6.Ex88.m1.3.3">𝜏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex88.m1.6c">(\mathfrak{U}^{*}f)(\sigma,\tau)=(\sigma,NU(d_{0})(\tau))</annotation><annotation encoding="application/x-llamapun" id="S6.Ex88.m1.6d">( fraktur_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_f ) ( italic_σ , italic_τ ) = ( italic_σ , italic_N italic_U ( italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_τ ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.6.p2.8">for every simplex <math alttext="(\sigma,\tau)" class="ltx_Math" display="inline" id="S6.6.p2.7.m1.2"><semantics id="S6.6.p2.7.m1.2a"><mrow id="S6.6.p2.7.m1.2.3.2" xref="S6.6.p2.7.m1.2.3.1.cmml"><mo id="S6.6.p2.7.m1.2.3.2.1" stretchy="false" xref="S6.6.p2.7.m1.2.3.1.cmml">(</mo><mi id="S6.6.p2.7.m1.1.1" xref="S6.6.p2.7.m1.1.1.cmml">σ</mi><mo id="S6.6.p2.7.m1.2.3.2.2" xref="S6.6.p2.7.m1.2.3.1.cmml">,</mo><mi id="S6.6.p2.7.m1.2.2" xref="S6.6.p2.7.m1.2.2.cmml">τ</mi><mo id="S6.6.p2.7.m1.2.3.2.3" stretchy="false" xref="S6.6.p2.7.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p2.7.m1.2b"><interval closure="open" id="S6.6.p2.7.m1.2.3.1.cmml" xref="S6.6.p2.7.m1.2.3.2"><ci id="S6.6.p2.7.m1.1.1.cmml" xref="S6.6.p2.7.m1.1.1">𝜎</ci><ci id="S6.6.p2.7.m1.2.2.cmml" xref="S6.6.p2.7.m1.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p2.7.m1.2c">(\sigma,\tau)</annotation><annotation encoding="application/x-llamapun" id="S6.6.p2.7.m1.2d">( italic_σ , italic_τ )</annotation></semantics></math> in <math alttext="X_{1}=N(\mathcal{D};F_{1})" class="ltx_Math" display="inline" id="S6.6.p2.8.m2.2"><semantics id="S6.6.p2.8.m2.2a"><mrow id="S6.6.p2.8.m2.2.2" xref="S6.6.p2.8.m2.2.2.cmml"><msub id="S6.6.p2.8.m2.2.2.3" xref="S6.6.p2.8.m2.2.2.3.cmml"><mi id="S6.6.p2.8.m2.2.2.3.2" xref="S6.6.p2.8.m2.2.2.3.2.cmml">X</mi><mn id="S6.6.p2.8.m2.2.2.3.3" xref="S6.6.p2.8.m2.2.2.3.3.cmml">1</mn></msub><mo id="S6.6.p2.8.m2.2.2.2" xref="S6.6.p2.8.m2.2.2.2.cmml">=</mo><mrow id="S6.6.p2.8.m2.2.2.1" xref="S6.6.p2.8.m2.2.2.1.cmml"><mi id="S6.6.p2.8.m2.2.2.1.3" xref="S6.6.p2.8.m2.2.2.1.3.cmml">N</mi><mo id="S6.6.p2.8.m2.2.2.1.2" xref="S6.6.p2.8.m2.2.2.1.2.cmml">⁢</mo><mrow id="S6.6.p2.8.m2.2.2.1.1.1" xref="S6.6.p2.8.m2.2.2.1.1.2.cmml"><mo id="S6.6.p2.8.m2.2.2.1.1.1.2" stretchy="false" xref="S6.6.p2.8.m2.2.2.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.6.p2.8.m2.1.1" xref="S6.6.p2.8.m2.1.1.cmml">𝒟</mi><mo id="S6.6.p2.8.m2.2.2.1.1.1.3" xref="S6.6.p2.8.m2.2.2.1.1.2.cmml">;</mo><msub id="S6.6.p2.8.m2.2.2.1.1.1.1" xref="S6.6.p2.8.m2.2.2.1.1.1.1.cmml"><mi id="S6.6.p2.8.m2.2.2.1.1.1.1.2" xref="S6.6.p2.8.m2.2.2.1.1.1.1.2.cmml">F</mi><mn id="S6.6.p2.8.m2.2.2.1.1.1.1.3" xref="S6.6.p2.8.m2.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.6.p2.8.m2.2.2.1.1.1.4" stretchy="false" xref="S6.6.p2.8.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p2.8.m2.2b"><apply id="S6.6.p2.8.m2.2.2.cmml" xref="S6.6.p2.8.m2.2.2"><eq id="S6.6.p2.8.m2.2.2.2.cmml" xref="S6.6.p2.8.m2.2.2.2"></eq><apply id="S6.6.p2.8.m2.2.2.3.cmml" xref="S6.6.p2.8.m2.2.2.3"><csymbol cd="ambiguous" id="S6.6.p2.8.m2.2.2.3.1.cmml" xref="S6.6.p2.8.m2.2.2.3">subscript</csymbol><ci id="S6.6.p2.8.m2.2.2.3.2.cmml" xref="S6.6.p2.8.m2.2.2.3.2">𝑋</ci><cn id="S6.6.p2.8.m2.2.2.3.3.cmml" type="integer" xref="S6.6.p2.8.m2.2.2.3.3">1</cn></apply><apply id="S6.6.p2.8.m2.2.2.1.cmml" xref="S6.6.p2.8.m2.2.2.1"><times id="S6.6.p2.8.m2.2.2.1.2.cmml" xref="S6.6.p2.8.m2.2.2.1.2"></times><ci id="S6.6.p2.8.m2.2.2.1.3.cmml" xref="S6.6.p2.8.m2.2.2.1.3">𝑁</ci><list id="S6.6.p2.8.m2.2.2.1.1.2.cmml" xref="S6.6.p2.8.m2.2.2.1.1.1"><ci id="S6.6.p2.8.m2.1.1.cmml" xref="S6.6.p2.8.m2.1.1">𝒟</ci><apply id="S6.6.p2.8.m2.2.2.1.1.1.1.cmml" xref="S6.6.p2.8.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.6.p2.8.m2.2.2.1.1.1.1.1.cmml" xref="S6.6.p2.8.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S6.6.p2.8.m2.2.2.1.1.1.1.2.cmml" xref="S6.6.p2.8.m2.2.2.1.1.1.1.2">𝐹</ci><cn id="S6.6.p2.8.m2.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.6.p2.8.m2.2.2.1.1.1.1.3">1</cn></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p2.8.m2.2c">X_{1}=N(\mathcal{D};F_{1})</annotation><annotation encoding="application/x-llamapun" id="S6.6.p2.8.m2.2d">italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_N ( caligraphic_D ; italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.7.p3"> <p class="ltx_p" id="S6.7.p3.4">Recall that for a bisimplicial set <math alttext="X" class="ltx_Math" display="inline" id="S6.7.p3.1.m1.1"><semantics id="S6.7.p3.1.m1.1a"><mi id="S6.7.p3.1.m1.1.1" xref="S6.7.p3.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.7.p3.1.m1.1b"><ci id="S6.7.p3.1.m1.1.1.cmml" xref="S6.7.p3.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p3.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.7.p3.1.m1.1d">italic_X</annotation></semantics></math> and a coefficient system <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S6.7.p3.2.m2.1"><semantics id="S6.7.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S6.7.p3.2.m2.1.1" xref="S6.7.p3.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S6.7.p3.2.m2.1b"><ci id="S6.7.p3.2.m2.1.1.cmml" xref="S6.7.p3.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p3.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S6.7.p3.2.m2.1d">caligraphic_M</annotation></semantics></math>, the cochain complex <math alttext="C^{*}(X;\mathcal{M})" class="ltx_Math" display="inline" id="S6.7.p3.3.m3.2"><semantics id="S6.7.p3.3.m3.2a"><mrow id="S6.7.p3.3.m3.2.3" xref="S6.7.p3.3.m3.2.3.cmml"><msup id="S6.7.p3.3.m3.2.3.2" xref="S6.7.p3.3.m3.2.3.2.cmml"><mi id="S6.7.p3.3.m3.2.3.2.2" xref="S6.7.p3.3.m3.2.3.2.2.cmml">C</mi><mo id="S6.7.p3.3.m3.2.3.2.3" xref="S6.7.p3.3.m3.2.3.2.3.cmml">∗</mo></msup><mo id="S6.7.p3.3.m3.2.3.1" xref="S6.7.p3.3.m3.2.3.1.cmml">⁢</mo><mrow id="S6.7.p3.3.m3.2.3.3.2" xref="S6.7.p3.3.m3.2.3.3.1.cmml"><mo id="S6.7.p3.3.m3.2.3.3.2.1" stretchy="false" xref="S6.7.p3.3.m3.2.3.3.1.cmml">(</mo><mi id="S6.7.p3.3.m3.1.1" xref="S6.7.p3.3.m3.1.1.cmml">X</mi><mo id="S6.7.p3.3.m3.2.3.3.2.2" xref="S6.7.p3.3.m3.2.3.3.1.cmml">;</mo><mi class="ltx_font_mathcaligraphic" id="S6.7.p3.3.m3.2.2" xref="S6.7.p3.3.m3.2.2.cmml">ℳ</mi><mo id="S6.7.p3.3.m3.2.3.3.2.3" stretchy="false" xref="S6.7.p3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p3.3.m3.2b"><apply id="S6.7.p3.3.m3.2.3.cmml" xref="S6.7.p3.3.m3.2.3"><times id="S6.7.p3.3.m3.2.3.1.cmml" xref="S6.7.p3.3.m3.2.3.1"></times><apply id="S6.7.p3.3.m3.2.3.2.cmml" xref="S6.7.p3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S6.7.p3.3.m3.2.3.2.1.cmml" xref="S6.7.p3.3.m3.2.3.2">superscript</csymbol><ci id="S6.7.p3.3.m3.2.3.2.2.cmml" xref="S6.7.p3.3.m3.2.3.2.2">𝐶</ci><times id="S6.7.p3.3.m3.2.3.2.3.cmml" xref="S6.7.p3.3.m3.2.3.2.3"></times></apply><list id="S6.7.p3.3.m3.2.3.3.1.cmml" xref="S6.7.p3.3.m3.2.3.3.2"><ci id="S6.7.p3.3.m3.1.1.cmml" xref="S6.7.p3.3.m3.1.1">𝑋</ci><ci id="S6.7.p3.3.m3.2.2.cmml" xref="S6.7.p3.3.m3.2.2">ℳ</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p3.3.m3.2c">C^{*}(X;\mathcal{M})</annotation><annotation encoding="application/x-llamapun" id="S6.7.p3.3.m3.2d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ; caligraphic_M )</annotation></semantics></math> has <math alttext="n" class="ltx_Math" display="inline" id="S6.7.p3.4.m4.1"><semantics id="S6.7.p3.4.m4.1a"><mi id="S6.7.p3.4.m4.1.1" xref="S6.7.p3.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.7.p3.4.m4.1b"><ci id="S6.7.p3.4.m4.1.1.cmml" xref="S6.7.p3.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p3.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.7.p3.4.m4.1d">italic_n</annotation></semantics></math>-cochains</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex89"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C^{n}(X;\mathcal{M})=\mathrm{Hom}_{R\Delta(X)}(\mathrm{Tot}(P_{*,*})_{n};%