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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="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id="modelcategorical">Model-categorical</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-limits+-+contents">Edit this sidebar</a> </p> </div></div></div> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#computation'>Computation</a></li> <ul> <li><a href='#reduction_to_semisimplicial_objects'>Reduction to semisimplicial objects</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#simplicial_sets'>Simplicial sets</a></li> <li><a href='#chain_complexes_of_abelian_groups'>Chain complexes of abelian groups</a></li> <li><a href='#topological_spaces'>Topological spaces</a></li> </ul> <li><a href='#related_notions'>Related notions</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><strong>Homotopy realizations</strong> are a special case of <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a>, when the indexing diagram is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^{op}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the <a class="existingWikiWord" href="/nlab/show/category+of+simplices">category of simplices</a>.</p> <p>Homotopy realizations can be defined in any <a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, just like <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a>, but practical computations are typically carried out in presence of additional structures such as <a class="existingWikiWord" href="/nlab/show/model+structures">model structures</a>, in fact, <a class="existingWikiWord" href="/nlab/show/enriched+model+categories">enriched model categories</a> are the most common setup.</p> <h2 id="computation">Computation</h2> <p>In any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a>, the homotopy realization of a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</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 lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X\colon \Delta^{op}\to C</annotation></semantics></math></div> <p>can be computed in three different ways, all of which use the notion of <a class="existingWikiWord" href="/nlab/show/tensor+product+of+functors">tensor product of functors</a> (i.e., <a class="existingWikiWord" href="/nlab/show/weighted+colimits">weighted colimits</a>).</p> <p>Specifically, consider the <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊗</mo><mo lspace="verythinmathspace">:</mo><msup><mi>V</mi> <mi>Δ</mi></msup><mo>×</mo><msup><mi>C</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\otimes\colon V^\Delta \times C^{\Delta^{op}} \to C</annotation></semantics></math></div> <p>that takes the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+functors">tensor product of functors</a>, i.e., the <a class="existingWikiWord" href="/nlab/show/weighted+colimit">weighted colimit</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> over which <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> is enriched.</p> <p>If we set the first argument to the <a class="existingWikiWord" href="/nlab/show/constant+functor">constant functor</a> with value <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> (the <a class="existingWikiWord" href="/nlab/show/monoidal+unit">monoidal unit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>), then the resulting functor is the <a class="existingWikiWord" href="/nlab/show/colimit+functor">colimit functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">C^{\Delta^{op}}\to C</annotation></semantics></math>.</p> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/left+Quillen+bifunctor">left Quillen bifunctor</a> if we equip <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">V^\Delta</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">C^{\Delta^{op}}</annotation></semantics></math> with one of the three following pairs of model structures:</p> <ul> <li>injective and projective;</li> <li>projective and injective;</li> <li>Reedy and Reedy.</li> </ul> <p>Accordingly, the homotopy realization of a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X\colon \Delta^{op}\to C</annotation></semantics></math> can be computed as follows.</p> <ul> <li>Cofibrantly resolve the constant weight <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> in one of the three model structures listed above.</li> <li>Cofibrantly resolve <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> in the other model structure in the same pair.</li> <li>Compute <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mn>1</mn><mo>⊗</mo><mi>Q</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Q 1 \otimes Q X</annotation></semantics></math>, which is the homotopy realization 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>.</li> </ul> <p>Cofibrant resolutions in the <a class="existingWikiWord" href="/nlab/show/injective+model+structure">injective model structure</a> can be computed by applying some <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution+functor">cofibrant resolution functor</a> of <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> objectwise.</p> <p>Cofibrant resolutions in the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> can be computed inductively, by repeatedly factoring the <a class="existingWikiWord" href="/nlab/show/latching+map">latching map</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> as a <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a> followed by a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> and adjusting <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> accordingly.</p> <p>Cofibrant resolutions in the <a class="existingWikiWord" href="/nlab/show/projective+model+structure">projective model structure</a> can be computed explicitly in some practical examples.</p> <h3 id="reduction_to_semisimplicial_objects">Reduction to semisimplicial objects</h3> <p>The inclusion of the <a class="existingWikiWord" href="/nlab/show/category+of+semisimplices">category of semisimplices</a> (i.e., fininite inhabited totally ordered sets and injective order-preserving maps) into the <a class="existingWikiWord" href="/nlab/show/category+of+simplices">category of simplices</a> (with the injectivity condition dropped) is a <a class="existingWikiWord" href="/nlab/show/homotopy+initial+functor">homotopy initial functor</a>, i.e., restricting along the inclusion of opposite categories preserves <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a>.</p> <p>Thus, homotopy realizations can be computed as <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a> over the opposite <a class="existingWikiWord" href="/nlab/show/category+of+semisimplices">category of semisimplices</a>. The latter category is a <a class="existingWikiWord" href="/nlab/show/direct+category">direct category</a>, which makes cofibrancy conditions particularly easy.</p> <h2 id="examples">Examples</h2> <h3 id="simplicial_sets">Simplicial sets</h3> <p>A Reedy <a class="existingWikiWord" href="/nlab/show/cofibrant+replacement">cofibrant replacement</a> of the constant weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">1\colon \Delta\to sSet</annotation></semantics></math> can be computed as the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mi>Δ</mi></msub><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">Y_\Delta\colon\Delta\to sSet</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+objects">simplicial objects</a> in <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> with <a class="existingWikiWord" href="/nlab/show/simplicial+weak+equivalences">simplicial weak equivalences</a> coincides with the <a class="existingWikiWord" href="/nlab/show/injective+model+structure">injective model structure</a>, as explained in the article <a class="existingWikiWord" href="/nlab/show/elegant+Reedy+category">elegant Reedy category</a>. In particular, all objects are <a class="existingWikiWord" href="/nlab/show/cofibrant">cofibrant</a>.</p> <p>Thus, the homotopy realization of</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 lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">X\colon \Delta^{op} \to sSet</annotation></semantics></math></div> <p>can be computed as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mi>Δ</mi></msub><mo>⊗</mo><mi>X</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">Y_\Delta\otimes X,</annotation></semantics></math></div> <p>which is isomorphic to the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</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>.</p> <h3 id="chain_complexes_of_abelian_groups">Chain complexes of abelian groups</h3> <p>For <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> with <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> (which we equip with the <a class="existingWikiWord" href="/nlab/show/injective+model+structure+on+chain+complexes">injective model structure on chain complexes</a>), a computation analogous to the one for simplicial sets above Reedy cofibrantly resolves the constant weight as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">N</mi><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>→</mo><mi>Ch</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">\mathrm{N} \mathbf{Z} [-]\colon \Delta \to Ch,</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">N</mi></mrow><annotation encoding="application/x-tex">\mathrm{N}</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/normalized+chains+functor">normalized chains functor</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{Z}[-]</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/free+simplicial+abelian+group+functor">free simplicial abelian group functor</a>.</p> <p>Once again, all <a class="existingWikiWord" href="/nlab/show/simplicial+objects">simplicial objects</a> in <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> are Reedy cofibrant.</p> <p>Thus, the homotopy realization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Ch</mi></mrow><annotation encoding="application/x-tex">X\colon \Delta^{op} \to Ch</annotation></semantics></math> can be computed as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">N</mi><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>⊗</mo><mi>X</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">\mathrm{N} \mathbf{Z} [-] \otimes X,</annotation></semantics></math></div> <p>which is isomorphic to the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> of the <a class="existingWikiWord" href="/nlab/show/double+chain+complex">double chain complex</a> obtained by applying the <a class="existingWikiWord" href="/nlab/show/Dold%E2%80%93Kan+correspondence">Dold–Kan correspondence</a> to the <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Ch</mi></mrow><annotation encoding="application/x-tex">X\colon \Delta^{op}\to Ch</annotation></semantics></math>.</p> <h3 id="topological_spaces">Topological spaces</h3> <p>Consider <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> with <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>. Below, we use the <a class="existingWikiWord" href="/nlab/show/Serre+model+structure">Serre model structure</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/topological+simplex">topological simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">\mathbf{\Delta}\colon \Delta\to Top</annotation></semantics></math> is Reedy cofibrant as a <a class="existingWikiWord" href="/nlab/show/cosimplicial+topological+space">cosimplicial topological space</a>.</p> <p>Not all <a class="existingWikiWord" href="/nlab/show/simplicial+objects">simplicial objects</a> in <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> are Reedy <a class="existingWikiWord" href="/nlab/show/cofibrant">cofibrant</a>, since the <a class="existingWikiWord" href="/nlab/show/latching+map">latching map</a> need not be a <a class="existingWikiWord" href="/nlab/show/cofibration+of+topological+spaces">cofibration of topological spaces</a>, i.e., a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a relative cellular map.</p> <p>However, we can pass to the semisimplicial setting, as explained above. In this, case Reedy cofibrancy boils down to the objectwise cofibrancy.</p> <p>Thus, the homotopy realization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X\colon \Delta^{op} \to Top</annotation></semantics></math> can be computed as</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>Δ</mi></mstyle><mo>⊗</mo><mi>Q</mi><mi>X</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Delta}\otimes Q X,</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> denotes objectwise <a class="existingWikiWord" href="/nlab/show/cofibrant+replacement">cofibrant replacement</a>. This is precisely the classical <a class="existingWikiWord" href="/nlab/show/fat+geometric+realization">fat geometric realization</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+topological+spaces">simplicial topological spaces</a>.</p> <h2 id="related_notions">Related notions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/realization">realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> </ul> <h2 id="references">References</h2> <p>For the case of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kensuke+Arakawa">Kensuke Arakawa</a>, <em>Homotopy Limits and Homotopy Colimits of Chain Complexes</em>, <a href="https://arxiv.org/abs/2310.00201">arXiv:2310.00201</a>.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 5, 2023 at 18:01:24. 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