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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> <h4 id="topos_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#AsAGeometricEmbedding'>As a geometric embedding into a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-presheaf category</a></li> <li><a href='#GiraudAxioms'>By Giraud-Rezk-Lurie axioms</a></li> <li><a href='#morphisms'>Morphisms</a></li> </ul> <li><a href='#types_of_toposes'>Types of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</a></li> <ul> <li><a href='#topological_localizations__sheaf_toposes'>Topological localizations / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaf toposes</a></li> <li><a href='#hypercomplete_toposes'>Hypercomplete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</a></li> <li><a href='#cubical_type_theory'>Cubical type theory</a></li> </ul> <li><a href='#models'>Models</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#GlobalSectionsGeometricMorphism'>Global sections geometric morphism</a></li> <li><a href='#ClosedMonoidalStructure'>Closed monoidal structure</a></li> <li><a href='#PoweringOfInfinityToposesOverInfinityGroupoids'>Powering of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</a></li> <li><a href='#slicetoposes'>Slice-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</a></li> <li><a href='#syntax_in_univalent_homotopy_type_theory'>Syntax in univalent homotopy type theory</a></li> </ul> <li><a href='#ToposTheory'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#ReferencesGeneral'>General</a></li> <li><a href='#giraudrezklurie_axioms'>Giraud-Rezk-Lurie axioms</a></li> <li><a href='#homotopy_type_theory'>Homotopy type theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Recall the following familiar 1-categorical statement:</p> <ul> <li>Working in the 1-<a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/0-category">0-categories</a> amounts to doing <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>. The point of <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf</a> <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a> is to pass to <em>parameterized</em> <a class="existingWikiWord" href="/nlab/show/0-category">0-categories</a>, namely <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> categories. Although these <a class="existingWikiWord" href="/nlab/show/topos">topoi</a> behave much like the 1-topos <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, their objects are generalized <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> that may carry more structure. For instance, a (pre)<a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> is a <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a>.</li> </ul> <p>The idea of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes is to generalize the above situation from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math> (recall the notion of <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> and see the general discussion at <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-topos">∞-topos</a>):</p> <ul> <li>Working in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> of <a class="existingWikiWord" href="/nlab/show/infinity-groupoid">(∞,0)-categories</a> amounts to doing <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. The point of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a> is to pass to <em>parameterized</em> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-categories">(∞,0)-categories</a>, namely <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a> categories. Although these <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topoi behave much like the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>, their objects are generalized <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> with higher <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> that may carry more structure. More generally we have topoi of <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topoi of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a>. For instance, an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a> on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>.</li> </ul> <h2 id="Definition">Definition</h2> <h3 id="AsAGeometricEmbedding">As a geometric embedding into a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-presheaf category</h3> <p>Recall that <a class="existingWikiWord" href="/nlab/show/sheaf+toposes+are+equivalently+the+left+exact+reflective+subcategories+of+presheaf+toposes">sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes</a> and that the inclusion functor is necessarily an <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible functor</a>. This characterization has the following immediate generalization to a definition in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, where the only subtlety is that accessibility needs to be explicitly required:</p> <div class="num_defn" id="ToposByLocalization"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Grothendieck</a>–<a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Rezk</a>–<a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Lurie</a> <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is an <a href="reflective%20sub-%28infinity,1%29-category#AccessibleReflectiveSubcategory">accessible</a> <a href="reflective+sub-%28infinity%2C1%29-category#ExactLocalizations">left exact</a> <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a> of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mo>↪</mo><mover><mo>←</mo><mi>lex</mi></mover></mover><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,. </annotation></semantics></math></div> <p>If the above localization is a <a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is an <strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></strong>.</p> </div> <h3 id="GiraudAxioms">By Giraud-Rezk-Lurie axioms</h3> <p>Equivalently:</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p><strong>(Giraud-Rezk-Lurie axioms)</strong> <br /> An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is</p> <p>an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> that satisfies the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category-theoretic analogs of <a class="existingWikiWord" href="/nlab/show/Giraud%27s+axioms">Giraud's axioms</a>:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentable</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">(∞,1)-colimits</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/universal+colimits">are universal</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/disjoint+coproduct">disjoint</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">groupoid object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/groupoid+objects+in+an+%28%E2%88%9E%2C1%29-topos+are+effective">effective</a> (i.e. has a <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>).</p> </li> </ul> <p></p> </div> </p> <p>This is part of the statement of <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, theorem 6.1.0.6</a>.</p> <p>This is derived from the following equivalent one:</p> <div class="num_prop" id="CharacterizationByObjectClassifier"> <h6 id="proposition">Proposition</h6> <p>An <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p>that has <a class="existingWikiWord" href="/nlab/show/object+classifiers">object classifiers</a>.</p> </li> </ul> </div> <div class="num_remark" id="ReflectonOnCharacterizationByObjectClassifier"> <h6 id="remark">Remark</h6> <p>An <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a> is a (small) <em>self-reflection</em> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos inside itself (<a class="existingWikiWord" href="/nlab/show/type+of+types">type of types</a>, internal <a class="existingWikiWord" href="/nlab/show/universe">universe</a>). See also (<a href="Science+of+Logic#WesenAlsReflexionInIhmSelbst">WdL, book 2, section 1</a>).</p> </div> <p>A further equivalent one (essentially by an invocation of the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>) is:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>An <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a></p> </li> <li> <p>in which all colimits are <a class="existingWikiWord" href="/nlab/show/van+Kampen+colimits">van Kampen colimits</a>.</p> </li> </ul> </div> <h3 id="morphisms">Morphisms</h3> <p>A <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos is <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Toposes">(∞,1)Toposes</a>.</p> <h2 id="types_of_toposes">Types of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</h2> <h3 id="topological_localizations__sheaf_toposes">Topological localizations / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaf toposes</h3> <p>for the moment see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></li> </ul> <h3 id="hypercomplete_toposes">Hypercomplete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</h3> <p>for the moment see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></li> </ul> <h3 id="cubical_type_theory">Cubical type theory</h3> <p>The Cartesian cubical model of <a class="existingWikiWord" href="/nlab/show/cubical+type+theory">cubical type theory</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is <a href="https://groups.google.com/d/msg/homotopytypetheory/RQkLWZ_83kQ/s6iazlFdBgAJ">conjectured</a> to be an (∞,1)-topos not equivalent to (∞,1)-groupoids.</p> <h2 id="models">Models</h2> <p>Another main theorem about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes is that <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a> are given by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>. See there for details</p> <h2 id="properties">Properties</h2> <h3 id="GlobalSectionsGeometricMorphism">Global sections geometric morphism</h3> <p>Every <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> has a canonical <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a> to the terminal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>: the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> is the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a>s <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> is the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>⟵</mo><mi>LConst</mi></mover></munderover><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (LConst \dashv \Gamma) \;\colon\; \mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\; \bot \;\;\;} \infty Grpd \,. </annotation></semantics></math></div> <p>In fact, this is unique, up to <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence</a>: Since every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid is an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,1)</annotation> </semantics> </math>-colimit</a> (namely over itself, by <a href="infinity1-limit#EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself">this Prop.</a>) of the <a class="existingWikiWord" href="/nlab/show/point">point</a> (hence of the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>), and since the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> needs to preserve these <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-colimits (being a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>) as well as the point (being a <a class="existingWikiWord" href="/nlab/show/lex+functor">lex functor</a>).</p> <h3 id="ClosedMonoidalStructure">Closed monoidal structure</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos is a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+%28%E2%88%9E%2C1%29-category">cartesian closed (∞,1)-category</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the fact that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a> it follows that for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> X \times (-) : \mathbf{H} \to \mathbf{H} </annotation></semantics></math></div> <p>preserves all <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a>s. Since every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos is a <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a> it follows with the <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a> that there is a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right</a> <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊣</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><munder><mo>→</mo><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow></munder><mover><mo>←</mo><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X \times (-) \dashv [X,-]) : \mathbf{H} \stackrel{\overset{X \times (-)}{\leftarrow}}{\underset{[X,-]}{\to}} \mathbf{H} \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> we have that the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (<a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,-]</annotation></semantics></math> is given on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in \mathbf{H}</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>U</mi><mo>↦</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>L</mi><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [X,A] \,\colon\, U \mapsto \mathbf{H}(X \times L y(U), A) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>:</mo><mi>C</mi><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">y : C \to \mathbf{H}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+embedding">(∞,1)-Yoneda embedding</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><msub><mi>PSh</mi> <mi>C</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">L : PSh_C \to \mathbf{H}</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackification</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The argument is entirely analogous to that of the <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+sheaves">closed monoidal structure on sheaves</a>.</p> <p>We use the <a class="existingWikiWord" href="/nlab/show/full+and+faithful+%28%E2%88%9E%2C1%29-functor">full and faithful</a> <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>i</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><msub><mi>PSh</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">(L \dashv i) : \mathbf{H} \hookrightarrow PSh_C</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> to find for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">U \in C</annotation></semantics></math> the value</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [X,A](U) \simeq PSh_C(y U, [X,A]) </annotation></semantics></math></div> <p>and then the fact that <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to inclusion to get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>L</mi><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq \mathbf{H}(L y(U), [X,A]) \,. </annotation></semantics></math></div> <p>Then the defining adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊣</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X \times (-) \dashv [X,-])</annotation></semantics></math> gives</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>L</mi><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq \mathbf{H}(X \times L y(U) , A) \,. </annotation></semantics></math></div></div> <div> <h3 id="PoweringOfInfinityToposesOverInfinityGroupoids">Powering of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</h3> <p>We discuss how the <a class="existingWikiWord" href="/nlab/show/powering">powering</a> of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-toposes"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-toposes</a> over <a class="existingWikiWord" href="/nlab/show/Infinity-Grpd"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>Grpd</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">Grpd_\infty</annotation> </semantics> </math></a> is given by forming <a class="existingWikiWord" href="/nlab/show/mapping+stacks">mapping stacks</a> out of <a class="existingWikiWord" href="/nlab/show/locally+constant+infinity-stacks">locally constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</a>. All of the following formulas and their proofs hold verbatim also for <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a>, as they just use general abstract properties.</p> <p><br /></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-topos</a></p> <ul> <li> <p>with terminal <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-geometric+morphism">geometric morphism</a> denoted</p> <div class="maruku-equation" id="eq:TerminalGeometricMorphismAdjunction"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>⟵</mo><mi>LConst</mi></mover></munderover><msub><mi>Grp</mi> <mn>∞</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} Grp_\infty \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> constructs <a class="existingWikiWord" href="/nlab/show/locally+constant+infinity-stacks">locally constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</a>,</p> </li> <li> <p>and with its <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (<a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a>) <a class="existingWikiWord" href="/nlab/show/adjoint+%28infinity%2C1%29-functor">adjunction</a> denoted</p> <div class="maruku-equation" id="eq:MappingStackAdjunction"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow></mover></munderover><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathbf{H} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} { \overset{ (-) \times X }{\longleftarrow} } {\;\;\;\; \bot \;\;\;\;} \mathbf{H} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \,\in\, \mathbf{H}</annotation></semantics></math>.</p> <p>Notice that this construction is also <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-functor"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-functorial</a> in the first argument: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">Maps\big( X \xrightarrow{f} Y ,\, A \big)</annotation></semantics></math> is the morphism which under the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-Yoneda+lemma"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-Yoneda lemma</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (which is large but locally small, so that the lemma does apply) corresponds to</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>f</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Y</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \;\simeq\; \mathbf{H} \big( (-) \times X ,\, A \big) \xrightarrow{ \mathbf{H} \big( (-) \times f ,\, A \big) } \mathbf{H} \big( (-) \times Y ,\, A \big) \;\simeq\; \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \,. </annotation></semantics></math></div> <p>By definition, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><msub><mi>Grpd</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">S \in Grpd_\infty</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/powering">powering</a>] is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> over the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>K</mi></msup><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>K</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex"> X^K \,=\, {\lim_\leftarrow}_K X </annotation></semantics></math></div> <p>while the <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> over the diagram constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⋅</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>K</mi></msub><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K \cdot X \,=\, {\lim_{\to}}_K X \,. </annotation></semantics></math></div> <p> <div class="num_remark"> <h6>Remark</h6> <p>Under <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a>, the powering operations on homotopy types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> corresponds to higher order <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of suitable algebras of functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, as discussed there.</p> </div> </p> <p> <div class="num_prop"> <h6>Proposition</h6> <p>The <em>powering</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/Infinity-Grpd"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>Grpd</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">Grpd_\infty</annotation> </semantics> </math></a> is given by the <a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a> out of the <a class="existingWikiWord" href="/nlab/show/locally+constant+infinity-stack">locally constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>Grpd</mi> <mn>∞</mn> <mi>op</mi></msubsup><mo>×</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>LConst</mi> <mi>op</mi></msup><mo>×</mo><mi mathvariant="normal">id</mi></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>op</mi></msup><mo>×</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Grpd_\infty^{op} \times \mathbf{H} & \overset{ LConst^{op} \times \mathrm{id} }{\longrightarrow} & \mathbf{H}^{op} \times \mathbf{H} & \overset{Maps(-,-)}{\longrightarrow} & \mathbf{H} } </annotation></semantics></math></div> <p>in that this operation has the following properties:</p> <ol> <li> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X,\,A \,\in\, \mathbf{H}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">S \,\in\, Grpd_\infty</annotation></semantics></math> we have a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) \;\; \simeq \;\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) </annotation></semantics></math></div></li> <li> <p>In its first argument the operation</p> <ol> <li> <p>sends the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> (the <a class="existingWikiWord" href="/nlab/show/point">point</a>) to the identity:</p> <div class="maruku-equation" id="eq:MappingStackOutOfLocallyConstantPreservesLimitsInFirstArg"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>X</mi></mrow><annotation encoding="application/x-tex"> Maps \big( LConst(\ast) ,\, X \big) \;\; \simeq \;\; X </annotation></semantics></math></div></li> <li> <p>sends <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimits"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-colimits</a> to <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-limits"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-limits</a>:</p> <div class="maruku-equation" id="eq:MappingStackOutOfLConstStacksPreservesColimitInFirstArgument"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><mi>LConst</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>LConst</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Maps \Big( \underset{ \longrightarrow }{\lim} \, LConst\big(S_\bullet\big) ,\, X \Big) \;\; \simeq \;\; \underset{ \longleftarrow }{\lim} \, Maps \Big( LConst\big(S_\bullet\big) ,\, X \Big) \,, </annotation></semantics></math></div></li> </ol> <p>where all <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalences</a> shown are <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural</a>.</p> </li> </ol> <p></p> </div> </p> <p> <div class="proof"> <h6>Proof</h6> <p></p> <p>For the first statement to be proven, consider the following sequence of <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural equivalences</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left left"><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>×</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mtext><a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a></mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext><a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a></mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Γ</mi><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:TerminalGeometricMorphismAdjunction">(1)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="2.4em" minsize="2.4em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mo>*</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mo maxsize="2.4em" minsize="2.4em">)</mo></mtd> <mtd><mtext>by</mtext><mspace width="thickmathspace"></mspace><mtext><a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">this Prop.</a></mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mo>*</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>×</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext><a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a></mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Grpd</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>S</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{lll} \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) & \;\simeq\; \mathbf{H} \big( X \times LConst(S) ,\, A \big) & \text{<a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a>} \\ & \;\simeq\; \mathbf{H} \Big( LConst(S) ,\, Maps \big( X ,\, A \big) \Big) & \text{<a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \Gamma \, Maps \big( X ,\, A \big) \Big) & \text{ <a class="maruku-eqref" href="#eq:TerminalGeometricMorphismAdjunction">(1)</a> } \\ & \;\simeq\; Grpd_\infty \bigg( S ,\, \mathbf{H} \Big( \ast_{\mathbf{H}} ,\, Maps \big( X ,\, A \big) \Big) \bigg) & \text{by}\;\text{<a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">this Prop.</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( \ast_{\mathbf{H}} \times X ,\, A \big) \Big) & \text{<a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) \end{array} </annotation></semantics></math></div> <p>For the second statement, recall that <a class="existingWikiWord" href="/nlab/show/hom-functors+preserve+limits">hom-functors preserve limits</a> in that there are <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural</a> <a class="existingWikiWord" href="/nlab/show/equivalences+in+an+%28infinity%2C1%29-category">equivalences</a> of the form</p> <div class="maruku-equation" id="eq:HomFunctorPreservesLimits"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><mspace width="thinmathspace"></mspace><mo>,</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>j</mi></munder></munder><mspace width="thinmathspace"></mspace><mo>,</mo><msub><mi>A</mi> <mi>j</mi></msub><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>i</mi></munder></munder><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>j</mi></munder></munder><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mi>j</mi></msub><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \Big( \underset{\underset{i}{\longrightarrow}}{\lim} \,, X_i ,\, \underset{\underset{j}{\longleftarrow}}{\lim} \,, A_j \Big) \;\; \simeq \;\; \underset{\underset{i}{\longleftarrow}}{\lim} \, \underset{\underset{j}{\longleftarrow}}{\lim} \, \mathbf{H} \Big( X_i ,\, A_j \Big) \,, </annotation></semantics></math></div> <p>and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes have <a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a>, in particular that the <a class="existingWikiWord" href="/nlab/show/product">product</a> operation is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> and <a class="existingWikiWord" href="/nlab/show/left+adjoints+preserve+colimits">hence preserves colimits</a>:</p> <div class="maruku-equation" id="eq:ProductsPreserveColimits"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>×</mo><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>×</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-) \,\times\, \underset{{\longrightarrow}}{\lim} \, S_\bullet \;\; \simeq \;\; \underset{{\longrightarrow}}{\lim} \, \big( (-) \,\times\, S_\bullet \big) \,. </annotation></semantics></math></div> <p>With this, we get the following sequences of <a class="existingWikiWord" href="/nlab/show/natural+equivalences">natural equivalences</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left left"><mtr><mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="2.4em" minsize="2.4em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mo maxsize="2.4em" minsize="2.4em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mspace width="thinmathspace"></mspace><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><mo>⟶</mo></munder><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:ProductsPreserveColimits">(6)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:HomFunctorPreservesLimits">(5)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> </mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><mo>⟵</mo></munder><mspace width="thinmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mtext> <a class="maruku-eqref" href="#eq:HomFunctorPreservesLimits">(5)</a> </mtext><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{lll} & \mathbf{H} \bigg( (-) ,\, Maps \Big( \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) \bigg) \\ & \;\simeq\; \mathbf{H} \Big( (-) \times \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) & \text{ <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> } \\ & \;\simeq\; \mathbf{H} \Big( \underset{\longrightarrow}{\lim} \big( (-) \times LConst(S_\bullet) \big) ,\, X \Big) & \text{ <a class="maruku-eqref" href="#eq:ProductsPreserveColimits">(6)</a> } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \big( (-) \times LConst(S_\bullet) ,\, X \big) & \text{ <a class="maruku-eqref" href="#eq:HomFunctorPreservesLimits">(5)</a> } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \Big( (-) ,\, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ <a class="maruku-eqref" href="#eq:MappingStackAdjunction">(2)</a> } \\ & \;\simeq\; \mathbf{H} \Big( (-) ,\, \underset{\longleftarrow}{\lim} \, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ <a class="maruku-eqref" href="#eq:HomFunctorPreservesLimits">(5)</a> } \,. \end{array} </annotation></semantics></math></div> <p>This implies <a class="maruku-eqref" href="#eq:MappingStackOutOfLConstStacksPreservesColimitInFirstArgument">(4)</a> by the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-Yoneda+lemma"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-Yoneda lemma</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (which is large but locally small, so that the lemma does apply).</p> <p>Finally <a class="maruku-eqref" href="#eq:MappingStackOutOfLocallyConstantPreservesLimitsInFirstArg">(3)</a> is immediate from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> preserves the terminal object, by definition:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mo>*</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Maps \big( LConst(\ast) ,\, X \big) \;\simeq\; Maps \big( \ast_{\mathbf{H}} ,\, X \big) \;\simeq\; X \,. </annotation></semantics></math></div> <p></p> </div> </p> </div> <h3 id="slicetoposes">Slice-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</h3> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> an object, the <a class="existingWikiWord" href="/nlab/show/slice-%28%E2%88%9E%2C1%29-category">slice-(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> is itself an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos – an <strong><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></strong>. The projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>!</mo></msub><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\pi_! : \mathbf{H}_{/X} \to \mathbf{H}</annotation></semantics></math> part of an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mover><mover><munder><mo>→</mo><mrow><msub><mi>π</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>π</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>π</mi> <mo>!</mo></msub></mrow></mover></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi : \mathbf{H}_{/X} \stackrel{\overset{\pi_!}{\to}}{\stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}} \mathbf{H} \,. </annotation></semantics></math></div></div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 6.3.5.1</a>.</p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> could be called the <a class="existingWikiWord" href="/nlab/show/gros+topos">gros topos</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>K</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{K} \to \mathbf{H}</annotation></semantics></math> that factors as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>K</mi></mstyle><mover><mo>→</mo><mo>≃</mo></mover><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mover><mo>→</mo><mi>π</mi></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{K} \xrightarrow{\simeq} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}</annotation></semantics></math> is called an <a class="existingWikiWord" href="/nlab/show/etale+geometric+morphism">etale geometric morphism</a>.</p> <h3 id="syntax_in_univalent_homotopy_type_theory">Syntax in univalent homotopy type theory</h3> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Toposes provide <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> for <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> with a <a class="existingWikiWord" href="/nlab/show/univalence">univalent</a> Tarskian <a class="existingWikiWord" href="/nlab/show/type+of+types">type of types</a> (which inteprets as the <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a>).</p> <p>For more on this see at</p> <ul> <li> <p><em><a class="existingWikiWord" href="/homotopytypetheory/show/model+of+type+theory+in+an+%28infinity%2C1%29-topos">model of type theory in an (infinity,1)-topos</a></em></p> </li> <li> <p><em><a href="relation+between+type+theory+and+category+theory#HomotopyWithUnivalence">relation between type theory and category theory – Univalent homotopy type theory and infinity-toposes</a></em></p> </li> </ul> <h2 id="ToposTheory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos theory</h2> <p>Most of the standard constructions in <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> have or should have immediate generalizations to the context of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes, since all notions of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> exist for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>.</p> <p>For instance there are evident notions of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>s between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes, such as the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> geometric morphism to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</li> </ul> <p>Moreover, it turns out that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes come with plenty of internal structures, more than canonically present in an ordinary topos. Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos comes with its intrinsic notion of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology in an (∞,1)-topos</a></li> </ul> <p>and with an intrinsic notion of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy in an (∞,1)-topos</a>.</li> </ul> <p>In classical topos theory, cohomology and homotopy of a topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> are defined in terms of <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> with <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough point</a>s, then this classical construction is secretly really a model for the intrinsic cohomology and homotopy in the above sense of the <a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>The beginning of a list of all the structures that exist intrinsically in a big <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos is at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a>.</li> </ul> <p>But <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos theory</strong> in the style of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-analog of the <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a> is only barely beginning to be conceived.</p> <p>There are some indications as to what the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></li> </ul> <p>should be.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pretopos">(∞,1)-pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+topos">model topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a>, <a class="existingWikiWord" href="/nlab/show/coherent+%28%E2%88%9E%2C1%29-topos">coherent (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+object+in+an+%28%E2%88%9E%2C1%29-topos">category object in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-topos">tangent (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/doubly+monoidal+%28%E2%88%9E%2C1%29-topos">doubly monoidal (∞,1)-topos</a></p> </li> </ul> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher toposes</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/1-topos">1-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-topos">(1,1)-topos</a> = <a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(n,1)</annotation> </semantics> </math>-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> (<a class="existingWikiWord" href="/nlab/show/n-localic+%28infinity%2C1%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-localic</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\;\;\;</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/model+topos">model topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C2%29-topos">(2,2)-topos</a> (<a class="existingWikiWord" href="/nlab/show/n-localic+2-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-localic</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,n)</annotation> </semantics> </math>-topos</a></li> </ul> </li> </ul> </div><div> <p><strong>Locally presentable categories:</strong> <a class="existingWikiWord" href="/nlab/show/cocomplete+category">Cocomplete</a> possibly-<a class="existingWikiWord" href="/nlab/show/large+categories">large categories</a> generated under <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> by <a class="existingWikiWord" href="/nlab/show/small+object">small</a> <a class="existingWikiWord" href="/nlab/show/generators">generators</a> under <a class="existingWikiWord" href="/nlab/show/small+colimit">small</a> <a class="existingWikiWord" href="/nlab/show/relations">relations</a>. Equivalently, <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a> <a class="existingWikiWord" href="/nlab/show/reflective+localizations">reflective localizations</a> of <a class="existingWikiWord" href="/nlab/show/free+cocompletions">free cocompletions</a>. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a> localization.</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/toposes">toposes</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th>locally presentable</th><th>loc finitely pres</th><th>localization theorem</th><th><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></th><th>accessible</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locales">locales</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/suplattice">suplattice</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+lattices">algebraic lattices</a></td><td style="text-align: left;"><a href="algebraic+lattice#RelationToLocallyFinitelyPresentableCategories">Porst’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/powerset">powerset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/poset">poset</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+categories">locally finitely presentable categories</a></td><td style="text-align: left;"><a href="locally+presentable+category#AsLocalizationsOfPresheafCategories">Gabriel–Ulmer’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+toposes">model toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dugger%27s+theorem">Dugger's theorem</a></td><td style="text-align: left;">global <a class="existingWikiWord" href="/nlab/show/model+structures+on+simplicial+presheaves">model structures on simplicial presheaves</a></td><td style="text-align: left;">n/a</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-categories">locally presentable (∞,1)-categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a href="locally+presentable+infinity-category#Definition">Simpson’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf+%28%E2%88%9E%2C1%29-categories">(∞,1)-presheaf (∞,1)-categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-categories">accessible (∞,1)-categories</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="ReferencesGeneral">General</h3> <p>In retrospect, at least the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy categories</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a> have been known since</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenneth+Brown">Kenneth Brown</a>, <em><a class="existingWikiWord" href="/nlab/show/BrownAHT">Abstract homotopy theory and generalized sheaf cohomology</a></em>, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 (<a href="http://www.jstor.org/stable/1996573">jstor:1996573</a>).</li> </ul> <p>presented there via <a class="existingWikiWord" href="/nlab/show/categories+of+fibrant+objects">categories of fibrant objects</a> among <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a>. The enhancement of this to <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">of simplicial presheaves</a> originates wit:h</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, Letter to <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, 11. 4. 1984, (<a href="http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf">pdf scan</a>).</p> </li> <li id="JardineLecture"> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Jardine">John F. Jardine</a>, <em>Simplicial presheaves</em>, Journal of Pure and Applied Algebra 47 (1987), 35-87 (<a href="https://core.ac.uk/download/pdf/82485559.pdf">pdf</a>)</p> </li> </ul> <p>A more intrinsic characterization of these “<a class="existingWikiWord" href="/nlab/show/model+toposes">model toposes</a>” (<a href="#Rezk10">Rezk 2010</a>) as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes (the term seems to first appear here in <a href="#Simpson99">Simpson 1999</a>) is due to:</p> <ul> <li id="Simpson99"><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>A Giraud-type characterization of the simplicial categories associated to closed model categories as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-pretopoi</em> (<a href="http://arxiv.org/abs/math/9903167">arXiv:math/9903167</a>)</li> </ul> <p>The generalization of these <a class="existingWikiWord" href="/nlab/show/model+toposes">model toposes</a> from 1-sites to <a class="existingWikiWord" href="/nlab/show/simplicial+site">simplicial</a> <a class="existingWikiWord" href="/nlab/show/model+sites">model sites</a> is due to</p> <ul> <li id="ToenVezzosi05"><a class="existingWikiWord" href="/nlab/show/Bertrand+To%C3%ABn">Bertrand Toën</a>, <a class="existingWikiWord" href="/nlab/show/Gabriele+Vezzosi">Gabriele Vezzosi</a>, <em>Homotopical Algebraic Geometry I: Topos theory</em>, Advances in Mathematics <strong>193</strong> 2 (2005) 257-372 [<a href="http://arxiv.org/abs/math.AG/0207028">arXiv:math.AG/0207028</a>, <a href="https://doi.org/10.1016/j.aim.2004.05.004">doi:10.1016/j.aim.2004.05.004</a>]</li> </ul> <p>The term <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos</em> is due to</p> <ul> <li id="Lurie03"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Topoi</em> (<a href="https://arxiv.org/abs/math/0306109">arXiv:math/0306109</a>)</li> </ul> <p>The term <em><a class="existingWikiWord" href="/nlab/show/model+topos">model topos</a></em> was later coined in:</p> <ul> <li id="Rezk10"><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Toposes and homotopy toposes</em>, 2010 (<a href="http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Rezk_HomotopyToposes.pdf" title="pdf">pdf</a>)</li> </ul> <p>A comprehensive conceptualization and discussion of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a> is then due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 6.1 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em>, Annals of Mathematics Studies 170, Princeton University Press 2009 (<a href="https://press.princeton.edu/titles/8957.html">pup:8957</a>, <a href="https://www.math.ias.edu/~lurie/papers/HTT.pdf">pdf</a>)</li> </ul> <p>building on <a href="#Rezk10">Rezk 2010</a>. There is is also proven that the Brown-Joyal-Jardine-Toën-Vezzosi models indeed precisely model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes. Details on this relation are at <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a>.</p> <p>Overview:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Catégories supérieures et théorie des topos</em>, Séminaire Bourbaki, 21.3.2015, <a href="http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf">pdf</a>.</p> </li> <li id="Rezk19"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Lectures on Higher Topos Theory</em>, Leeds (2019) [<a href="https://rezk.web.illinois.edu/leeds-lectures-2019.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/RezkHigherToposTheory2019.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>A useful collection of facts of <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">(infinity,1)-topos theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Zhen+Lin+Low">Zhen Lin Low</a>, <em><a class="existingWikiWord" href="/nlab/show/Notes+on+homotopical+algebra">Notes on homotopical algebra</a></em></li> </ul> <p>A quick introduction to the topic is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em><a class="existingWikiWord" href="/nlab/show/A+crash+course+in+topos+theory+--+The+big+picture">A crash course in topos theory – The big picture</a></em>, lecture series at <a href="https://indico.math.cnrs.fr/event/747/">Topos à l’IHES</a>, November 2015, Paris</li> </ul> <p>On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes <a class="existingWikiWord" href="/nlab/show/category+object+in+an+%28infinity%2C1%29-category">internal to</a> other <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes;</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Louis+Martini">Louis Martini</a>, <a class="existingWikiWord" href="/nlab/show/Sebastian+Wolf">Sebastian Wolf</a>, <em>Internal higher topos theory</em> [<a href="https://arxiv.org/abs/2303.06437">arXiv:2303.06437</a>]</li> </ul> <h3 id="giraudrezklurie_axioms">Giraud-Rezk-Lurie axioms</h3> <p>A discussion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a> in terms of <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> presentations is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Fibrations and homotopy colimits of simplicial sheaves</em> (<a href="http://www.math.uiuc.edu/~rezk/rezk-sharp-maps.pdf">pdf</a>)</li> </ul> <p>More on this with an eye on <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundles">associated ∞-bundles</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Matthias+Wendt">Matthias Wendt</a>, <em>Classifying spaces and fibrations of simplicial sheaves</em> (<a href="http://arxiv.org/abs/1009.2930">arXiv</a>)</li> </ul> <h3 id="homotopy_type_theory">Homotopy type theory</h3> <p>Proof that all <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> have <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentations</a> by <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> which interpret (provide <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a>) for <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> with <a class="existingWikiWord" href="/nlab/show/univalence">univalent</a> <a class="existingWikiWord" href="/nlab/show/type+universes">type universes</a>:</p> <ul> <li id="Shulman19"><a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, <em>All <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes have strict univalent universes</em> (<a href="https://arxiv.org/abs/1904.07004">arXiv:1904.07004</a>).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 31, 2023 at 16:37:14. See the <a href="/nlab/history/%28infinity%2C1%29-topos" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/%28infinity%2C1%29-topos" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/9467/#Item_9">Discuss</a><span class="backintime"><a href="/nlab/revision/%28infinity%2C1%29-topos/87" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/%28infinity%2C1%29-topos" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/%28infinity%2C1%29-topos" accesskey="S" class="navlink" id="history" rel="nofollow">History (87 revisions)</a> <a href="/nlab/show/%28infinity%2C1%29-topos/cite" style="color: black">Cite</a> <a href="/nlab/print/%28infinity%2C1%29-topos" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/%28infinity%2C1%29-topos" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>