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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="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> </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> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#InTermsOfSlices'>In terms of slice <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>-categories</a></li> <li><a href='#in_terms_of_hom_adjunction'>In terms 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>-Hom adjunction</a></li> <li><a href='#in_terms_of_products_and_equalizers'>In terms of products and equalizers</a></li> <li><a href='#TermsOfHomotopy'>In terms of homotopy limits</a></li> <li><a href='#CommutativityOfLimits'>Commutativity of limits</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#limits_of_special_shape'><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 of special shape</a></li> <ul> <li><a href='#coproduct'>Coproduct</a></li> <li><a href='#pullback__pushout'>Pullback / Pushout</a></li> <ul> <li><a href='#PushoutPasting'>Pasting law of pushouts</a></li> </ul> <li><a href='#coequalizer'>Coequalizer</a></li> <li><a href='#quotients'>Quotients</a></li> <li><a href='#Tensoring'>Tensoring and cotensoring with 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>-groupoid</a></li> <ul> <li><a href='#recap_of_the_1categorical_situation'>Recap of the 1-categorical situation</a></li> <li><a href='#definition_3'>Definition</a></li> <li><a href='#ModelsForTensoring'>Models</a></li> </ul> </ul> <li><a href='#InOvercategories'>Limits in over-<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</a></li> <li><a href='#WithValInooGrpd'>Limits and colimits with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math></a></li> <li><a href='#ColimitsInInfinityCat'>Limits and colimits with values in <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>Cat</a></li> <li><a href='#limits_in_functor_categories'>Limits in <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 categories</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <a class="existingWikiWord" href="/nlab/show/limit">limit</a> and <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> generalize from <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>. One model for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> are <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>. This entry discusses limits and colimits in quasi-categories.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and <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> two <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : K \to C</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> (a morphism of the underlying <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s) , the <strong>limit</strong> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is, if it exists, the <a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">quasi-categorical terminal object</a> in the <a class="existingWikiWord" href="/nlab/show/over+quasi-category">over quasi-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>F</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{/F}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi><mo>≔</mo><mi>TerminalObj</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>F</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C_{/F}) </annotation></semantics></math></div> <p>(well defined up to a contractible space of choices).</p> <p>A <strong>colimit</strong> in a quasi-category is accordingly a limit in the <a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite quasi-category</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Notice from the discussion at <a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a> that there are two definitions – denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♢</mo></mrow><annotation encoding="application/x-tex">\diamondsuit</annotation></semantics></math> – of join, which yield results that differ as simplicial sets, though are equivalent as quasi-categories.</p> <p>The notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>F</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{/F}</annotation></semantics></math> denotes the definition of <a class="existingWikiWord" href="/nlab/show/over+quasi-category">over quasi-category</a> induced from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math>, while the notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>F</mi></mrow></msup></mrow><annotation encoding="application/x-tex">C^{/F}</annotation></semantics></math> denotes that induced from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♢</mo></mrow><annotation encoding="application/x-tex">\diamondsuit</annotation></semantics></math>. Either can be used for the computation of limits in a quasi-category, as for quasi-categorical purposes they are weakly equivalent.</p> <p>So we also have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi><mo>≔</mo><mi>TerminalObj</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>F</mi></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C^{/F}) \,. </annotation></semantics></math></div> <p>See <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop 4.2.1.5</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="InTermsOfSlices">In terms of slice <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>-categories</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <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">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">f \colon K \to \mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> with <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>-colimiting <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>⋆</mo><msup><mi>Δ</mi> <mn>0</mn></msup><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\tilde f \colon K \star \Delta^0 \to \mathcal{C}</annotation></semantics></math>. Then the induced map of <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-categories">slice quasi-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow></msub><mo>→</mo><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{/\tilde f} \to \mathcal{C}_{f} </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/acyclic+Kan+fibration">acyclic Kan fibration</a>.</p> </div> <div class="num_prop" id="SlicingOverLimitingCone"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒦</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{K} \to \mathcal{C}</annotation></semantics></math> a diagram in 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>-category and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim} F</annotation></semantics></math> its limit, there is a natural <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>F</mi></mrow></msub><mo>≃</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{/F} \simeq \mathcal{C}_{/\underset{\leftarrow}{\lim} F} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-categories">slice (∞,1)-categories</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> (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 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>-<a class="existingWikiWord" href="/nlab/show/cones">cones</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>) and over just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim}F</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mn>0</mn></msup><mo>⋆</mo><mi>𝒦</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\tilde F \colon \Delta^0 \star \mathcal{K} \to \mathcal{C}</annotation></semantics></math> be the limiting cone. The canonical cospan 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>-functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msup><mi>Δ</mi> <mn>0</mn></msup><mo>⋆</mo><mi>𝒦</mi><mo>←</mo><mi>𝒦</mi></mrow><annotation encoding="application/x-tex"> \ast \to \Delta^0 \star \mathcal{K} \leftarrow \mathcal{K} </annotation></semantics></math></div> <p>induces a span of slice <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>-categories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi></mrow></msub><mo>←</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mover><mi>F</mi><mo stretchy="false">˜</mo></mover></mrow></msub><mo>→</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>F</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{/\underset{\leftarrow}{\lim}F} \leftarrow \mathcal{C}_{/\tilde F} \rightarrow \mathcal{C}_{/F} \,. </annotation></semantics></math></div> <p>The right functor is an equivalence by prop. <a class="maruku-ref" href="#SlicingOverLimitingCone"></a>. The left functor is induced by restriction along an op-<a class="existingWikiWord" href="/nlab/show/final+%28%E2%88%9E%2C1%29-functor">final (∞,1)-functor</a> (by the Examples discussed there) and hence is an equivalence by the discussion at <em><a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-category">slice (∞,1)-category</a></em> (<a href="#Lurie">Lurie, prop. 4.1.1.8</a>).</p> </div> <p>This appears for instance in (<a href="#Lurie">Lurie, proof of prop. 1.2.13.8</a>).</p> <h3 id="in_terms_of_hom_adjunction">In terms 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>-Hom adjunction</h3> <p>The definition of the limit in a quasi-category in terms of terminal objects in the corresponding <a class="existingWikiWord" href="/nlab/show/over+quasi-category">over quasi-category</a> is well adapted to the particular nature the incarnation 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>-categories by quasi-categories. But more intrinsically in <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, it should be true that there is an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> characterization 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>-limits : limit and colimit, should be (pointwise or global) <a class="existingWikiWord" href="/nlab/show/right+adjoint">right</a> and <a class="existingWikiWord" href="/nlab/show/left+adjoint">left</a> <a class="existingWikiWord" href="/nlab/show/adjoint+%28infinity%2C1%29-functor">adjoint (infinity,1)-functor</a> of the constant diagram <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">(\infinity,1)</annotation></semantics></math>-functor, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">const : C \to Func(K,C)</annotation></semantics></math>.</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>colim</mi><mo>⊣</mo><mi>const</mi><mo>⊣</mo><mi>lim</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mover><munder><mo>→</mo><mi>colim</mi></munder><mover><mo>←</mo><mi>const</mi></mover></mover><mover><mo>→</mo><mi>lim</mi></mover></mover><mi>Func</mi><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (colim \dashv const \dashv lim) : Func(K,C) \stackrel{\overset{lim}{\to}}{\stackrel{\overset{const}{\leftarrow}} {\underset{colim}{\to}}} Func(*,C) \simeq C \,. </annotation></semantics></math></div> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a> (<a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 5.2.2.8</a>) this requires exhibiitng a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><msub><mi>Id</mi> <mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo>→</mo><mi>const</mi><mi>colim</mi></mrow><annotation encoding="application/x-tex">\eta : Id_{Func(K,C)} \to const colim</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(Func(K,C),Func(K,C))</annotation></semantics></math> such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in Func(K,C)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Y \in C</annotation></semantics></math> the induced morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>colim</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>const</mi><mi>colim</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>η</mi><mo>,</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mover><msub><mi>Hom</mi> <mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{C}(colim(f),Y) \to Hom_{Func(K,C)}(const colim(f), const Y) \stackrel{Hom(\eta, const Y)}{\to} Hom_{Func(K,Y)}(f, const Y) </annotation></semantics></math></div> <p>is a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>.</p> <p>But first consider the following pointwise characterization.</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>. A <a class="existingWikiWord" href="/nlab/show/co-cone">co-cone</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>p</mi><mo stretchy="false">¯</mo></mover><mo>:</mo><mi>K</mi><mo>⋆</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\bar p : K \star \Delta[0] \to C</annotation></semantics></math> with cone point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/colimit">colimiting</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> (an initial object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>p</mi><mo stretchy="false">/</mo></mrow></msub></mrow><annotation encoding="application/x-tex">C_{p/}</annotation></semantics></math>) precisely if for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Y \in C</annotation></semantics></math> the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>Y</mi></msub><mo>:</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi_Y : Hom_C(X,Y) \to Hom_{Func(K,C)}(p, const Y) </annotation></semantics></math></div> <p>induced by the morpism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>→</mo><mi>const</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> p \to const X</annotation></semantics></math> that is encoded by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>p</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar p</annotation></semantics></math> is an equivalence (i.e. a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es).</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, lemma 4.2.4.3</a>.</p> <p>The key step is to realize that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{Func(K,C)}(p, const Y)</annotation></semantics></math> is given (up to equivalence) by the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><mi>p</mi><mo stretchy="false">/</mo></mrow></msup><msub><mo>×</mo> <mi>C</mi></msub><mo stretchy="false">{</mo><mi>Y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">C^{p/} \times_C \{Y\}</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>.</p> <p>Here is a detailed way to see this, using the discussion at <a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-object in a quasi-category</a>.</p> <p>We have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{Func(K,C)}(p, const Y)</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>K</mi></msup><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo stretchy="false">/</mo></mrow></msup><msub><mo>×</mo> <mrow><msup><mi>C</mi> <mi>K</mi></msup></mrow></msub><mi>const</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">(C^K)^{p/} \times_{C^K} const Y</annotation></semantics></math>. We compute</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>K</mi></msup><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo stretchy="false">/</mo></mrow></msup><msub><mo>×</mo> <mrow><msup><mi>C</mi> <mi>K</mi></msup></mrow></msub><mi>const</mi><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mrow><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><mo stretchy="false">/</mo><mi>sSet</mi></mrow></msub><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>♢</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><msup><mi>C</mi> <mi>K</mi></msup><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>K</mi></msup><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></msub><mo stretchy="false">{</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mrow><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><mo stretchy="false">/</mo><mi>sSet</mi></mrow></msub><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></munder><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><msup><mi>C</mi> <mi>K</mi></msup><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>K</mi></msup><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></msub><mo stretchy="false">{</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">{</mo><mi>p</mi><mo stretchy="false">}</mo><msub><mo>×</mo> <mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>,</mo><msup><mi>C</mi> <mi>K</mi></msup><mo stretchy="false">)</mo></mrow></msub><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>,</mo><msup><mi>C</mi> <mi>K</mi></msup><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><msup><mi>C</mi> <mi>K</mi></msup><mo stretchy="false">)</mo></mrow></msub><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><msup><mi>C</mi> <mi>K</mi></msup><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><msup><mi>C</mi> <mi>K</mi></msup><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">{</mo><mi>const</mi><mi>Y</mi><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">{</mo><mi>p</mi><mo stretchy="false">}</mo><msub><mo>×</mo> <mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mi>Hom</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>K</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi></mrow></msub><mo stretchy="false">{</mo><mi>Y</mi><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">{</mo><mi>p</mi><mo stretchy="false">}</mo><msub><mo>×</mo> <mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mi>Hom</mi><mo stretchy="false">(</mo><mi>K</mi><mo>♢</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">{</mo><mi>Y</mi><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mi>p</mi><mo stretchy="false">/</mo></mrow></msup><msub><mo>×</mo> <mi>C</mi></msub><mo stretchy="false">{</mo><mi>Y</mi><mo stretchy="false">}</mo><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} ((C^K)^{p/} \times_{C^K} const Y)_n & = Hom_{{\Delta[0]}/sSet}( \Delta[0] \diamondsuit \Delta[n] , C^K ) \times_{(C^K)_n} \{const Y\} \\ & = Hom_{{\Delta[0]}/sSet}( \Delta[0] \coprod_{\Delta[n]} \Delta[n] \times \Delta[1] , C^K ) \times_{(C^K)_n} \{const Y\} \\ & = \{p\} \times_{Hom(\Delta[0],C^K)} Hom(\Delta[0], C^K) \times_{Hom(\Delta[n], C^K)} Hom(\Delta[n] \times \Delta[1], C^K) \times_{Hom(\Delta[n], C^K)} \{const Y\} \\ & = \{p\} \times_{Hom(K,C)} Hom(K,C) \times_{Hom(\Delta[n]\times K,C)} Hom(\Delta[n]\times K \times \Delta[1], C) \times_{Hom(\Delta[n]\times K, C)} Hom(\Delta[n],C) \times_{\Delta[n],C} \{Y\} \\ &= \{p\} \times_{Hom(K,C)} Hom(K \diamondsuit \Delta[n], C) \times_{Hom(\Delta[n],C)} \{Y\} \\ &= (C^{p/}\times_C \{Y\})_n \end{aligned} </annotation></semantics></math></div> <p>Under this identification, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">\phi_Y</annotation></semantics></math> is the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msup><mi>C</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo></mrow></msup><mover><mo>→</mo><mrow><mi>ϕ</mi><mo>′</mo></mrow></mover><msup><mi>C</mi> <mrow><mover><mi>p</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">/</mo></mrow></msup><mover><mo>→</mo><mrow><mi>ϕ</mi><mo>″</mo></mrow></mover><msup><mi>C</mi> <mrow><mi>p</mi><mo stretchy="false">/</mo></mrow></msup><mo>)</mo></mrow><msub><mo>×</mo> <mi>C</mi></msub><mo stretchy="false">{</mo><mi>Y</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \left( C^{X/} \stackrel{\phi'}{\to} C^{\bar p/} \stackrel{\phi''}{\to} C^{p/} \right) \times_C \{Y\} \,, </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\phi'</annotation></semantics></math> is a section of the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><mover><mi>p</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">/</mo></mrow></msup><mo>→</mo><msup><mi>C</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">C^{\bar p/} \to C^{X/}</annotation></semantics></math>, (which one checks is an acyclic <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>) obtained by choosing composites of the co-cone components with a given morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math>.</p> <p>The morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">\phi''</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+fibration">left fibration</a> (using <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 4.2.1.6</a>)</p> <p>One finds that the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">\phi''</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+fibration">left fibration</a>.</p> <p>The strategy for the completion of the proof is: realize that the first condition of the proposition is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">\phi''</annotation></semantics></math> being an acyclic Kan fibration, and the second statement equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><msub><mo>″</mo> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">\phi''_Y</annotation></semantics></math> being an acyclic Kan fibration, then show that these two conditions in turn are equivalent.</p> </div> <h3 id="in_terms_of_products_and_equalizers">In terms of products and equalizers</h3> <p>A central theorem in ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> asserts that a <a class="existingWikiWord" href="/nlab/show/category">category</a> has <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s already if it has <a class="existingWikiWord" href="/nlab/show/product">product</a>s and <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>s. The analog statement is true here:</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/regular+cardinal">regular cardinal</a>. An <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</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> has all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-small limits precisely if it has <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>s and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-small <a class="existingWikiWord" href="/nlab/show/product">product</a>s.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 4.4.3.2</a>.</p> <h3 id="TermsOfHomotopy">In terms of homotopy limits</h3> <p>The notion of <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, which exists for <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> and in particular for <a class="existingWikiWord" href="/nlab/show/simplicial+model+categories">simplicial model categories</a> and in fact in all plain <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> – as described in more detail at <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a> – is supposed to be a model for <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>-categorical limits. In particular, under sending the Kan-complex enriched categories <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> to quasi-categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C)</annotation></semantics></math> using the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a> functor, homotopy limits should precisely corespond to quasi-categorical limits. That this is indeed the case is asserted by the following statements.</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>Let <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>J</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : J \to C</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>η</mi> <mi>i</mi></msub><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>→</mo><mi>c</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{\eta_i : F(i) \to c\}_{i \in J}</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> exhibits the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> (in the sense discussed in detail at <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a>) precisely if the induced morphism of quasi-categories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">¯</mo></mover><mo>:</mo><mi>N</mi><mo stretchy="false">(</mo><mi>J</mi><msup><mo stretchy="false">)</mo> <mo>▹</mo></msup><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \bar {N(F)} : N(J)^{\triangleright} \to N(C) </annotation></semantics></math></div> <p>is a quasi-categorical colimit <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C)</annotation></semantics></math>.</p> </div> <p>Here <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> is the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>J</mi><msup><mo stretchy="false">)</mo> <mo>▹</mo></msup></mrow><annotation encoding="application/x-tex">N(J)^{\triangleright}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a> with the point, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(F)</annotation></semantics></math> the image of the simplicial functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> under the homotopy coherent nerve and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{N(F)}</annotation></semantics></math> its extension to the join determined by the cocone maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>.</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, theorem 4.2.4.1</a></p> <p>A central ingredient in the proof is the fact, discused at <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> and at <a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a>, that <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>s do model <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s, in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/combinatorial+simplicial+model+category">combinatorial simplicial model category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau(S)</annotation></semantics></math> the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>-category under the left adjoint of the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a>, we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">equivalence of quasi-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>N</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>∘</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N(([C,A]_{proj})^\circ) \simeq Func(S, N(A^\circ)) </annotation></semantics></math></div> <p>and the same is trued for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> itself replaced by a <span class="newWikiWord">chunk<a href="/nlab/new/chunk+of+a+model+category">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">U \subset A</annotation></semantics></math>.</p> <p>With this and the discussion at <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a>, we find that the cocone components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> induce for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a \in [C,sSet]</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow></mrow></mover><mo stretchy="false">[</mo><msup><mi>J</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>j</mi><mi>F</mi><mo>,</mo><mi>const</mi><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(c,a) \stackrel{}{\to} [J^{op}, C](j F, const a) </annotation></semantics></math></div> <p>which is hence equivalently an equivalence of the corresponding <a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">quasi-categorical hom-objects</a>. The claim follows then from the above discussion of characterization of (co)limits in terms 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>-hom adjunctions.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>The quasi-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>∘</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(A^\circ)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by a <a class="existingWikiWord" href="/nlab/show/combinatorial+simplicial+model+category">combinatorial simplicial model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has all small quasi-categorical limits and colimits.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, 4.2.4.8</a>.</p> <p>It follows from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has (pretty much by definition of <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> and <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a>) all <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>s and <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>s (in fact all <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a>s) by the above proposition.</p> </div> <p>Since <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 equivalent to those of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>∘</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(A^\circ)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/combinatorial+simplicial+model+category">combinatorial simplicial model category</a> are precisely the <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-categories">locally presentable (∞,1)-categories</a>, it follows from this in particular that every locally presentable <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 has all limits and colimits.</p> <h3 id="CommutativityOfLimits">Commutativity of limits</h3> <p>The following proposition says that if for 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>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : X \times Y \to C</annotation></semantics></math> limits (colimits) over each of the two variables exist separately, then they commute.</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>Let <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s and <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> a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><msup><mi>X</mi> <mo>◃</mo></msup><mo>×</mo><msup><mi>Y</mi> <mo>◃</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">p : X^{\triangleleft} \times Y^{\triangleleft} \to C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>. If</p> <ol> <li> <p>for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>X</mi> <mo>◃</mo></msup></mrow><annotation encoding="application/x-tex">x \in X^{\triangleleft}</annotation></semantics></math> (including the cone point) the restricted diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>x</mi></msub><mo>:</mo><msup><mi>Y</mi> <mo>◃</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">p_x : Y^{\triangleleft} \to C</annotation></semantics></math> is a limit diagram;</p> </li> <li> <p>for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">y \in Y</annotation></semantics></math> (not including the cone point) the restricted diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>y</mi></msub><mo>:</mo><msup><mi>X</mi> <mo>◃</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">p_y : X^{\triangleleft} \to C</annotation></semantics></math> is a limit diagram;</p> </li> </ol> <p>then, with <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> denoting the cone point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mo>◃</mo></msup></mrow><annotation encoding="application/x-tex">Y^{\triangleleft}</annotation></semantics></math>, the restricted diagram, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>c</mi></msub><mo>:</mo><msup><mi>X</mi> <mo>◃</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">p_c : X^{\triangleleft} \to C</annotation></semantics></math> is also a limit diagram.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, lemma 5.5.2.3</a></p> </div> <p>In other words, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mi>x</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_x F(x,y)</annotation></semantics></math> exists for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mi>y</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_y F(x,y)</annotation></semantics></math> exists for all <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> and also that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mi>y</mi></msub><mo stretchy="false">(</mo><msub><mi>lim</mi> <mi>x</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_y (\lim_x F(x,y))</annotation></semantics></math> exists, then this object is also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mi>x</mi></msub><msub><mi>lim</mi> <mi>y</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_x \lim_y F(x,y)</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <h3 id="limits_of_special_shape"><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 of special shape</h3> <h4 id="coproduct">Coproduct</h4> <p>…</p> <h4 id="pullback__pushout">Pullback / Pushout</h4> <p>See also <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>.</p> <p>The non-degenerate cells of the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1] \times \Delta[1]</annotation></semantics></math> obtained as the <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> of the simplicial 1-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> with itself look like</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><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (0,0) &\to& (1,0) \\ \downarrow &\searrow& \downarrow \\ (0,1) &\to& (1,1) } </annotation></semantics></math></div> <p>A <strong>sqare</strong> in a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</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> is an image of this 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>, i.e. a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> s : \Delta[1] \times \Delta[1] \to C \,. </annotation></semantics></math></div> <p>The simplicial square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mrow><mo>×</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Delta[1]^{\times 2}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a>, as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, to the <a class="existingWikiWord" href="/nlab/show/join+of+simplicial+sets">join of simplicial sets</a> of a 2-<a class="existingWikiWord" href="/nlab/show/horn">horn</a> with the point:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo><mo>⋆</mo><mi>Λ</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mo stretchy="false">]</mo> <mn>2</mn></msub><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>v</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \Delta[1] \times \Delta[1] \simeq \{v\} \star \Lambda[2]_2 = \left( \array{ v &\to& 1 \\ \downarrow &\searrow& \downarrow \\ 0 &\to& 2 } \right) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>≃</mo><mi>Λ</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>⋆</mo><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>2</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>v</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} = \left( \array{ 0&\to& 1 \\ \downarrow &\searrow& \downarrow \\ 2 &\to& v } \right) \,. </annotation></semantics></math></div> <p>If a square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>≃</mo><mi>Λ</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>⋆</mo><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} \to C</annotation></semantics></math> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\{v\} \to C</annotation></semantics></math> as a colimit over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Λ</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : \Lambda[2]_0 \to C</annotation></semantics></math>, we say the colimit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>:</mo><mo>=</mo><munder><mi>lim</mi> <mo>→</mo></munder><mi>F</mi><mo>:</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>F</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> v := \lim_\to F := F(1) \coprod_{F(0)} F(2) </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <strong>pushout</strong> of the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>.</p> <h5 id="PushoutPasting">Pasting law of pushouts</h5> <p>We have the following <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>-categorical analog of the familiar <a href="http://ncatlab.org/nlab/show/pullback#Pasting">pasting law of pushouts</a> in ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>:</p> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>A pasting diagram of two squares is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta[2] \times \Delta[1] \to C \,. </annotation></semantics></math></div> <p>Schematically this looks like</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><mi>x</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>z</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>x</mi><mo>′</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>y</mi><mo>′</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>z</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ x &\to& y &\to& z \\ \downarrow && \downarrow && \downarrow \\ x' &\to& y' &\to& z' } \,. </annotation></semantics></math></div> <p>If the left square is a pushout diagram 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>, then the right square is precisely if the outer square is.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>A proof appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, lemma 4.4.2.1</a></p> </div> <h4 id="coequalizer">Coequalizer</h4> <p>…</p> <h4 id="quotients">Quotients</h4> <ul> <li><a class="existingWikiWord" href="/nlab/show/infinity-quotient">infinity-quotient</a></li> </ul> <h4 id="Tensoring">Tensoring and cotensoring with 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>-groupoid</h4> <h5 id="recap_of_the_1categorical_situation">Recap of the 1-categorical situation</h5> <p>An ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s is canonically <a class="existingWikiWord" href="/nlab/show/power">cotensored</a> over <a class="existingWikiWord" href="/nlab/show/Set">Set</a>:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">S, T \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>T</mi></msub><mo>:</mo><mi>S</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">const_T : S \to Set</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> parameterized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> that is constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mo>←</mo></munder><mspace width="thinmathspace"></mspace><msub><mi>const</mi> <mi>T</mi></msub><mo>≃</mo><msup><mi>T</mi> <mi>S</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lim_{\leftarrow} \, const_T \simeq T^S \,. </annotation></semantics></math></div> <p>Accordingly the cotensoring</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><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo>:</mo><msup><mi>Set</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> (-)^{(-)} : Set^{op} \times C \to C </annotation></semantics></math></div> <p>is defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>c</mi> <mi>S</mi></msup><mo>:</mo><mo>=</mo><munder><mi>lim</mi> <mo>←</mo></munder><mo stretchy="false">(</mo><mi>S</mi><mover><mo>→</mo><mrow><msub><mi>const</mi> <mi>c</mi></msub></mrow></mover><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>S</mi></munder><mi>c</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> c^S := \lim_{\leftarrow} (S \stackrel{const_c}{\to} C) = \prod_{S} c \,. </annotation></semantics></math></div> <p>And by continuity of the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> this implies the required natural isomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><msup><mi>c</mi> <mi>S</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>S</mi></msub><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>S</mi></msub><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Set</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_C(d,c^S) = Hom_C(d, {\lim_{\leftarrow}}_S c) \simeq {\lim_{\leftarrow}}_S Hom_C(d,c) \simeq Set(S,Hom_C(d,C)) \,. </annotation></semantics></math></div> <p>Correspondingly if <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> has <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s, then the <a class="existingWikiWord" href="/nlab/show/copower">tensoring</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>Set</mi><mo>×</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> (-) \otimes (-) : Set \times C \to C </annotation></semantics></math></div> <p>is given by forming <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s over constant diagrams: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mi>c</mi><mo>:</mo><mo>=</mo><msub><mrow><msub><mi>lim</mi> <mo>→</mo></msub></mrow> <mi>S</mi></msub><mi>c</mi></mrow><annotation encoding="application/x-tex">S \otimes c := {\lim_{\to}}_S c</annotation></semantics></math>, and again by continuity of the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> we have the required natural isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>⊗</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>S</mi></msub><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>S</mi></msub><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Set</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_C(S \otimes c, d) = Hom_C({\lim_{\to}}_S c,d) \simeq {\lim_{\leftarrow}}_S Hom_C(c,d) \simeq Set(S,Hom_C(c,d)) \,. </annotation></semantics></math></div> <p>Of course all the colimits appearing here are just <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s and all limits just <a class="existingWikiWord" href="/nlab/show/product">product</a>s, but for the generalization 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>-categories this is a misleading simplification, it is really the notion of limit and colimit that matters here.</p> <h5 id="definition_3">Definition</h5> <p>We expect for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">S, T \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>T</mi></msub><mo>:</mo><mi>S</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">const_T : S \to \infty Grpd</annotation></semantics></math> the constant diagram, that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mo>←</mo></munder><mspace width="thinmathspace"></mspace><msub><mi>const</mi> <mi>T</mi></msub><mo>≃</mo><msup><mi>T</mi> <mi>S</mi></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \lim_{\leftarrow} \, const_T \simeq T^S \,, </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> 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>-groupoids, which is modeled in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> by the fact that this is a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>.</p> <p>Correspondingly, 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 <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 with colimits, it is <a class="existingWikiWord" href="/nlab/show/copower">tensored</a> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> by setting</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mn>∞</mn><mi>Grpd</mi><mo>×</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> (-)\otimes (-) : \infty Grpd \times C \to C </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mi>c</mi><mo>:</mo><mo>=</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>S</mi></msub><mi>c</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> S \otimes c := {\lim_{\to}}_S c \,, </annotation></semantics></math></div> <p>where now on the right we have 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>-categorical colimit over the constant diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">const : S \to C</annotation></semantics></math> of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> 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>Then by 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>-continuity of the hom, and using the above characterization of the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> we have the required natural equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>⊗</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>S</mi></msub><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>S</mi></msub><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_C(S \otimes c, d) = Hom_C({\lim_{\to}}_S c, d) \simeq {\lim_{\leftarrow}}_S Hom_C(c,d) \simeq \infty Grpd(S,Hom_C(c,d)) \,. </annotation></semantics></math></div> <p>The following proposition should assert that this is all true</p> <div class="num_prop" id="TensoringProposition"> <h6 id="proposition_8">Proposition</h6> <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>-categorical colimit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><msub><mi>lim</mi> <mo>→</mo></msub></mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">{\lim_{\to}} c</annotation></semantics></math> over the diagram of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">S \in \infty Grpd</annotation></semantics></math> constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> is characterized by the fact that it induces natural equivalences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>S</mi></msub><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_C({\lim_{\to}}_S c, d) \simeq \infty Grpd(S, Hom_C(c,d)) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">d \in C</annotation></semantics></math>.</p> </div> <p>This is essentially <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, corollary 4.4.4.9</a>.</p> <div class="num_cor" id="EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself"> <h6 id="corollary_2">Corollary</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is 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>-colimit in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> of the constant diagram on the <a class="existingWikiWord" href="/nlab/show/point">point</a> over itself:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>S</mi></msub><msub><mi>const</mi> <mo>*</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \simeq {\lim_{\to}}_S const_* \,. </annotation></semantics></math></div></div> <p>This justifies the following definition</p> <div class="num_defn"> <h6 id="definition_4">Definition</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 <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 with colimits, the <strong>tensoring 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> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math></strong> is 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>-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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mn>∞</mn><mi>Grpd</mi><mo>×</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> (-) \otimes (-) : \infty Grpd \times C \to C </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mi>c</mi><mo>=</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><msub><mi>const</mi> <mi>c</mi></msub><mo>:</mo><mi>S</mi><mo>→</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \otimes c = \lim_{\to} (const_c : S \to C) \,. </annotation></semantics></math></div></div> <p>See <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, section 4.4.4</a>.</p> <h5 id="ModelsForTensoring">Models</h5> <p>We discuss models for <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>-(co)limits in terms of ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p> <div class='num_lemma'> <h6>Lemma</h6> <p>If <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 <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presented</a> by a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≃</mo><msup><mi>A</mi> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">C \simeq A^\circ</annotation></semantics></math>, then 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>-tensoring 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>-cotensoring 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> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is modeled by the ordinary <a class="existingWikiWord" href="/nlab/show/copower">tensoring</a> and <a class="existingWikiWord" href="/nlab/show/power">powering</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\hat c \in A</annotation></semantics></math> cofibant and representing an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">S \in sSet</annotation></semantics></math> any simplicial set, we have an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>⊗</mo><mi>S</mi><mo>≃</mo><mover><mi>C</mi><mo stretchy="false">^</mo></mover><mo>⋅</mo><mi>S</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> c \otimes S \simeq \hat C \cdot S \,. </annotation></semantics></math></div> <p></p> </div> </p> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>The powering in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> satisfies the <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sSet</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>c</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mover><mi>d</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>c</mi><mo stretchy="false">^</mo></mover><mo>⋅</mo><mi>S</mi><mo>,</mo><mover><mi>d</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> sSet(S,A(\hat c,\hat d)) \simeq A(\hat c \cdot S, \hat d) </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat c</annotation></semantics></math> a cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>d</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat d</annotation></semantics></math> a fibrant representative, we have that both sides here are <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es that are equivalent to the corresponding <a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a>s in the corresponding <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 <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>, so that this translates into an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>⋅</mo><mi>S</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_C(c \cdot S, d) \simeq \infty Grpd(S, Hom_C(c,d)) \,. </annotation></semantics></math></div> <p>The claim then follows from the above proposition.</p> </div> <h3 id="InOvercategories">Limits in over-<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</h3> <div class="num_prop"> <h6 id="proposition_9">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 <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, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \colon D \to C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C/X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-category">over-(∞,1)-category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>→</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F \colon K \to C/X</annotation></semantics></math> a diagram in the over-<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, then the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim} F</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C/X</annotation></semantics></math> coincides with 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>-limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim} F/X</annotation></semantics></math> 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>.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>Modelling <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> as a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C/X</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>X</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> C/X \;\colon\; [n] \mapsto Hom_X\big([n] \star D, C\big) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/join+of+simplicial+sets">join of simplicial sets</a>. The limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim} F</annotation></semantics></math> is the terminal object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">(C/X)/F</annotation></semantics></math>, which is the quasi-category given by the simplicial set</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>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>F</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (C/X)/F \;\colon\; [n] \mapsto Hom_{F}\big([n] \star K, C/X\big) \,. </annotation></semantics></math></div> <p>Since the join <a class="existingWikiWord" href="/nlab/show/preserved+colimit">preserves</a> <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> of simplicial sets in both arguments, we can apply the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a> to decompose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>K</mi><mo>=</mo><munder><mi>lim</mi><munder><mo>→</mo><mrow><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>K</mi></mrow></munder></munder><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n] \star K = \underset{\underset{{[r] \to [n]\star K}}{\to}}{\lim} [r]</annotation></semantics></math>, use that the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> <a class="existingWikiWord" href="/nlab/show/hom-functor+preserves+limits">sends colimits in the first argument to limits</a> and obtain</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Hom</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>Hom</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>r</mi></msub><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>r</mi></msub><mi>Hom</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>r</mi></msub><msub><mi>Hom</mi> <mi>F</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>F</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>r</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>D</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>F</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>r</mi></msub><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>⋆</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>F</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⋆</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>F</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋆</mo><mo stretchy="false">(</mo><mi>K</mi><mo>⋆</mo><mi>D</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Hom([n] \star K, C/X) &\simeq Hom\big( {\lim_{\to}}_r [r], C/X\big) \\ & \simeq {\lim_{\leftarrow}}_r Hom([r], C/X) \\ & \simeq {\lim_{\leftarrow}}_r Hom_F\big( [r] \star D, C \big) \\ & \simeq Hom_F\big({\lim_{\to}}_r ([r] \star D), C \big) \\ & \simeq Hom_F\big( ({\lim_{\to}}_r[r]) \star D, C \big) \\ & \simeq Hom_F\big(([n] \star K) \star D, C\big) \\ & \simeq Hom_F\big([n] \star (K \star D), C\big) \mathrlap{\,.} \end{aligned} </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>F</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">Hom_F\big([r]\star D,C\big)</annotation></semantics></math> is shorthand for the hom in the (ordinary) <a class="existingWikiWord" href="/nlab/show/under+category">under category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>sSet</mi> <mrow><mi>D</mi><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">sSet^{D/}</annotation></semantics></math> from the canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>→</mo><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋆</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">D \to [r] \star D</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \colon D \to C</annotation></semantics></math>. Notice that we use the 1-categorical analog of the statement that we are proving here when computing the colimit in this under-category as just the colimit in <a class="existingWikiWord" href="/nlab/show/sSet"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>sSet</mi> </mrow> <annotation encoding="application/x-tex">sSet</annotation> </semantics> </math></a>. We also use that the <a class="existingWikiWord" href="/nlab/show/join+of+simplicial+sets">join of simplicial sets</a>, being given by <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a>, is an associative tensor product.</p> <p>In conclusion we have an isomorphism of simplicial sets</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>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>F</mi><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mi>C</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (C/X)/F \,\simeq\, C/(X/F) </annotation></semantics></math></div> <p>and therefore the terminal objects of these quasi-categories coincide on both sides. This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder><mi>F</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim} F</annotation></semantics></math> is computed as an terminal object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C/(X/F)</annotation></semantics></math>.</p> </div> <h3 id="WithValInooGrpd">Limits and colimits with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math></h3> <p>Limits and colimits over a <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> with values 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%9E-Grpd">∞-Grpd</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> may be reformulated in terms of the <a class="existingWikiWord" href="/nlab/show/universal+fibration+of+%28%E2%88%9E%2C1%29-categories">universal fibration of (∞,1)-categories</a>, hence in terms of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a>.</p> <p>Let <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> be the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s. Let the <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>Z</mi><msub><mo stretchy="false">|</mo> <mi>Grpd</mi></msub><mo>→</mo><mn>∞</mn><msup><mi>Grpd</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Z|_{Grpd} \to \infty Grpd^{op}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/universal+fibration+of+%28infinity%2C1%29-categories">universal ∞-groupoid fibration</a> whose fiber over the object denoting some <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 that very <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.</p> <p>Then let <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> be any <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>X</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> F : X \to \infty Grpd </annotation></semantics></math></div> <p>an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>. Recall that the <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">coCartesian fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>F</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E_F \to X</annotation></semantics></math> classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the pullback of the <a class="existingWikiWord" href="/nlab/show/universal+fibration+of+%E2%88%9E-groupoids">universal fibration of ∞-groupoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><msub><mo stretchy="false">|</mo> <mi>Grpd</mi></msub></mrow><annotation encoding="application/x-tex">Z|_{Grpd}</annotation></semantics></math> along F:</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><msub><mi>E</mi> <mi>F</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Z</mi><msub><mo stretchy="false">|</mo> <mi>Grpd</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>F</mi></mover></mtd> <mtd><mn>∞</mn><mi>Grpd</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ E_F &\to& Z|_{Grpd} \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& \infty Grpd } </annotation></semantics></math></div> <div class="num_prop" id="InfinityGroupoidalCoLimitsViaIntegrationAndSlicing"> <h6 id="proposition_10">Proposition</h6> <p>Let the assumptions be as above. Then:</p> <ul> <li> <p>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>-colimit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">E_F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>⟶</mo></munder><mi>F</mi><mo>≃</mo><msub><mi>E</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex"> \underset{\longrightarrow}{\lim} F \simeq E_F </annotation></semantics></math></div></li> <li> <p>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>-limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid+of+sections">∞-groupoid of sections</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>F</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E_F \to 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><munder><mi>lim</mi><mo>⟵</mo></munder><mo>≃</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>F</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{\longleftarrow}{\lim} \simeq \Gamma_X(E_F) \,. </annotation></semantics></math></div></li> </ul> </div> <p>The statement for the colimit is corollary 3.3.4.6 in <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT</a>. The statement for the limit is corollary 3.3.3.4.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The form of the statement in prop. <a class="maruku-ref" href="#InfinityGroupoidalCoLimitsViaIntegrationAndSlicing"></a> is the special case of the general form of <a class="existingWikiWord" href="/nlab/show/internal+%28co-%29limits">internal (co-)limits</a>, here internal to the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Core</mi><mo stretchy="false">(</mo><msub><mo lspace="0em" rspace="thinmathspace">inftyGrpd</mo> <mi>small</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Core(\inftyGrpd_{small})</annotation></semantics></math> its small <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a>. See at <em><a href="internal+%28co-%29limit#ExamplesInfinityGroupoidal">internal (co-)limit – Groupoidal homotopy (co-)limits</a></em> for more on this.</p> </div> <h3 id="ColimitsInInfinityCat">Limits and colimits with values in <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>Cat</h3> <div class="num_prop"> <h6 id="proposition_11">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">F : D \to </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>, its <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>-colimit is given by forming the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\int F</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and then inverting the <a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a>s.</p> <p>Formally this means, with respect to the <a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a> that there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mo>→</mo></munder><mi>F</mi><mo>≃</mo><mo stretchy="false">(</mo><mo>∫</mo><mi>F</mi><msup><mo stretchy="false">)</mo> <mo>♯</mo></msup></mrow><annotation encoding="application/x-tex"> \lim_\to F \simeq (\int F)^\sharp </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of the presentation 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>-category by <a class="existingWikiWord" href="/nlab/show/marked+simplicial+sets">marked simplicial sets</a>.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, corollary 3.3.4.3</a>.</p> <p>For the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> takes values in ordinary categories see also at <a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a> the section <a href="http://ncatlab.org/nlab/show/2-limit#2ColimitsInCat">2-limits in Cat</a>.</p> <h3 id="limits_in_functor_categories">Limits in <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 categories</h3> <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 ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a> that admits small <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s and <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(D,C)</annotation></semantics></math> has all small limits and colimits, and these are computed objectwise. See <a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a>. The analogous statement is true for an <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> <div class="num_prop"> <h6 id="proposition_12">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and <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> be <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>, such that <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> has all <a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">colimits</a> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be a small quasi-category. Then</p> <ul> <li> <p>The <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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(D,C)</annotation></semantics></math> has all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-indexed colimits;</p> </li> <li> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mo>▹</mo></msup><mo>→</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K^\triangleright \to Func(D,C)</annotation></semantics></math> is a colimiting cocone precisely if for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">d \in D</annotation></semantics></math> the induced morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mo>▹</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">K^\triangleright \to C</annotation></semantics></math> is a colimiting cocone.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, corollary 5.1.2.3</a></p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>, <a class="existingWikiWord" href="/nlab/show/internal+limit">internal limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><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>-limit</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limit">finite (∞,1)-limit</a>, <a class="existingWikiWord" href="/nlab/show/relative+%28%E2%88%9E%2C1%29-limit">relative (∞,1)-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+%28%E2%88%9E%2C1%29-colimit">lax (∞,1)-colimit</a></p> </li> </ul> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The definition of limit in quasi-categories is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>Quasi-categories and Kan complexes</em> Journal of Pure and Applied Algebra 175 (2002), 207-222.</li> </ul> <p>A brief survey is on page 159 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>The theory of quasicategories and its applications</em> lectures at <a href="http://www.crm.es/HigherCategories/">Simplicial Methods in Higher Categories</a>, (<a href="http://www.crm.cat/HigherCategories/hc2.pdf">pdf</a>)</li> </ul> <p>A detailed account is in <a href="http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=48">definition 1.2.13.4, p. 48</a> in</p> <ul id="Lurie"> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial model category case?</em>, <a href="http://mathoverflow.net/questions/176111/do-homotopy-limits-compute-limits-in-the-associated-quasicategory-in-the-non-com/176142#176142">MO/176111/176142</a>.</li> </ul> <p>A discussion of <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted</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>-limits is in</p> <ul> <li>Martina Rovelli, <em>Weighted limits in an (∞,1)-category</em>, 2019, <a href="https://arxiv.org/abs/1902.00805">arxiv:1902.00805</a></li> </ul> <p>A discussion of free colimit completion constructions is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Free colimit completion in ∞-categories</em> (<a href="https://arxiv.org/abs/2210.08582">arXiv:2210.08582</a>)</li> </ul> <p>Discussion <a class="existingWikiWord" href="/nlab/show/category+internal+to+an+%28infinity%2C1%29-topos">internal to</a> any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>:</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>Limits and colimits in internal higher category theory</em> [<a href="https://arxiv.org/abs/2111.14495">arXiv:2111.14495</a>]</li> </ul> <h3 id="in_homotopy_type_theory">In homotopy type theory</h3> <p>A formalization of some aspects 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>-limits in terms of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a>-coded in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Guillaume+Brunerie">Guillaume Brunerie</a>, <em><a href="https://github.com/guillaumebrunerie/HoTT/tree/master/Coq/Limits">HoTT/Coq/Limits</a></em></li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <em>Homotopy Colimits and a Descent Theorem</em>, March 14, 2013 (<a href="http://video.ias.edu/univalent-1213-0314-EgbertRijke">video</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 26, 2024 at 06:53:26. 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