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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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=""><a class="existingWikiWord" href="/nlab/show/categories+of+categories+-+contents">categories of categories</a></h4> <div class="hide"><div> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1,r+1)</annotation></semantics></math>-categories of <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pos">Pos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Set">Set</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rel">Rel</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ho%28Cat%29">Ho(Cat)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AccCat">AccCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PrCat">PrCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/LexCat">LexCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MonCat">MonCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/VCat">VCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CatAdj">CatAdj</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Prof">Prof</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Operad">Operad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ModCat">ModCat</a>, <a class="existingWikiWord" href="/nlab/show/CombModCat">CombModCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pr%28%E2%88%9E%2C1%29Cat">Pr(∞,1)Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Operad">(∞,1)Operad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29Cat">(∞,n)Cat</a></p> </li> </ul> </div></div> </div> </div> <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><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)Cat</annotation></semantics></math></strong> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a> of all small <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>.</p> <p>Its full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> on <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s is <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#the_category'>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>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-category</a></li> <ul> <li><a href='#as_an_category'>As an <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</a></li> <li><a href='#as_an_enriched_model_category'>As an enriched model category</a></li> </ul> <li><a href='#the_category_2'>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</a></li> <ul> <li><a href='#as_an_category_2'>As an <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</a></li> <li><a href='#as_an_enriched_model_category_2'>As an enriched model category</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#LimitsAndColimits'>Limits and colimits 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='#Automorphisms'>Automorphisms</a></li> </ul> </ul> <li><a href='#presentations'>Presentations</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="the_category">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>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-category</h2> <h3 id="as_an_category">As an <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</h3> <p>One incarnation of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-categories">(∞,2)-categories</a> is given by <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>-enriched categories (see there for details). As such <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><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)Cat</annotation></semantics></math> is the full <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a> on those <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s that are <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>. By the fact described 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> this is indeed a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>-enriched category.</p> <h3 id="as_an_enriched_model_category">As an enriched model category</h3> <p>The <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> presenting this <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a> is the Joyal <a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Joyal</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Joyal}</annotation></semantics></math>. Its full <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> is the <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> enriched category of quasi-categories from above.</p> <h2 id="the_category_2">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</h2> <p>Sometimes it is useful to consider inside the full <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>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-catgeory 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 just the maximal <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 discarding all non-invertible <a class="existingWikiWord" href="/nlab/show/k-morphism">2-morphisms</a>. This is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-categories">(∞,1)-category of (∞,1)-categories</a>.</p> <h3 id="as_an_category_2">As an <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</h3> <p>As an <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-categories">(∞,1)-category of (∞,1)-categories</a> is obtained from the quasi-category-enriched version by picking in each <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a> simplicial set 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><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)Cat</annotation></semantics></math> the maximal <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>.</p> <h3 id="as_an_enriched_model_category_2">As an enriched model category</h3> <p>One <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure presenting this is the <a class="existingWikiWord" href="/nlab/show/model+structure+on+marked+simplicial+over-sets">model structure on marked simplicial sets</a>. As a plain <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> this is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Joyal</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Joyal}</annotation></semantics></math>, but as an <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> it is <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> enriched, so that its full <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>-subcategory on fibrant-cofibrant objects presents 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><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> <h3 id="properties">Properties</h3> <h4 id="LimitsAndColimits">Limits and colimits 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</h4> <p><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">Limits and colimits</a> over a <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> 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><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)Cat</annotation></semantics></math> may be reformulation in terms of the <a class="existingWikiWord" href="/nlab/show/universal+fibration+of+%28infinity%2C1%29-categories">universal fibration of (infinity,1)-categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>Cat</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Z \to (\infty,1)Cat^{op}</annotation></semantics></math></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/%28%E2%88%9E%2C1%29-category">(∞,1)-category</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><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> F : X \to (\infty,1)Cat </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+%28%E2%88%9E%2C1%29-categories">universal fibration of (∞,1)-categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</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></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><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ E_F &\to& Z \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& (\infty,1)Cat } </annotation></semantics></math></div> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>Let the assumptions be as above. Then:</p> <ul> <li> <p>The 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 <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><msub><mi>E</mi> <mi>F</mi></msub><mo>≃</mo><mi>colim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex"> E_F \simeq colim F </annotation></semantics></math></div></li> <li> <p>The 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/%28infinity%2C1%29-category+of+cartesian+section">(infinity,1)-category of cartesian section</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><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>F</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><mi>F</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(E_F) \simeq lim F \,. </annotation></semantics></math></div></li> </ul> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, section 3.3</a>.</p> </div> <h4 id="Automorphisms">Automorphisms</h4> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>The full subcategory of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-categories">(∞,1)-category of (∞,1)-categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo>,</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func((\infty,1)Cat, (\infty,1)Cat)</annotation></semantics></math> on those <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s that are equivalences is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>Id</mi><mo>,</mo><mi>op</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Id, op\}</annotation></semantics></math>: it contains only the identity functor and the one that sends 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 to its <a class="existingWikiWord" href="/nlab/show/opposite+%28infinity%2C1%29-category">opposite (infinity,1)-category</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+Toen">Bertrand Toen</a>, <em>Vers une axiomatisation de la théorie des catégories supérieures</em> , K-theory 34 (2005), no. 3, <p>233-263.</p> </li> </ul> <p>It appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, theorem 5.2.9.1</a> (<a href="http://arxiv.org/abs/math.CT/0608040">arxiv v4+</a> only)</p> <p>First of all the statement is true for the ordinary category of <a class="existingWikiWord" href="/nlab/show/poset">poset</a>s. This is <a href="http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=311">prop. 5.2.9.14</a>.</p> <p>From this the statement is deduced 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> -categories by observing that posets are characterized by the fact that two parallel functors into them that are objectwise equivalent are already equivalent, <a href="http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=310">prop. 5.2.9.11</a>, which means that posets <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> are characterized by the fact that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_0 (\infty,1)Cat(D,C) \to Hom_{Set}( \pi_0 (\infty,1)Cat(*,D) , \pi_0 (\infty,1)Cat(*,C) ) </annotation></semantics></math></div> <p>is an injection for all <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><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">D \in (\infty,1)Cat</annotation></semantics></math>.</p> <p>This is preserved under automorphisms 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><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)Cat</annotation></semantics></math>, hence any such automorphism preserves posets, hence restricts to an automorphism of the category of posets, hence must be either the identity or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">(-)^{op}</annotation></semantics></math> there, by the above statement for posets.</p> <p>Now finally the main point of the proof is to see that the linear posets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>⊂</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\Delta \subset (\infty,1)Cat</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/dense+functor">dense</a> 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><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)Cat</annotation></semantics></math>, i.e. that the identity transformation of the inclusion functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>↪</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\Delta \hookrightarrow (\infty,1)Cat</annotation></semantics></math> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Id_{(\infty,1)Cat}</annotation></semantics></math> as the left <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Δ</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>Lan</mi><mo>=</mo><mpadded width="0"><mi>Id</mi></mpadded></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Delta &\hookrightarrow& (\infty,1)Cat \\ \downarrow & \nearrow_{Lan = \mathrlap{Id}} \\ (\infty,1)Cat } \,. </annotation></semantics></math></div></div> <h2 id="presentations">Presentations</h2> <div> <p>The entries of the following table display <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> and <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> between these that <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">present</a> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> (second table), of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operads">(∞,1)-operads</a> (third table) and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-categories">monoidal (∞,1)-categories</a> (fourth table).</p> <table><thead><tr><th></th><th>general pattern</th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;">strict <a class="existingWikiWord" href="/nlab/show/enriched+%28%E2%88%9E%2C1%29-category">enrichment</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/enriched+%28%E2%88%9E%2C1%29-category">enriched (∞,1)-category</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+category+in+an+%28%E2%88%9E%2C1%29-category">internal (∞,1)-category</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">SimplicialCategories</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math><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><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">SimplicialSets</a>/<a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">RelativeSimplicialSets</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/nerve">simplicial nerve</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+categories">SegalCategories</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">CompleteSegalSpaces</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Operad">(∞,1)Operad</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+operads">SimplicialOperads</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+dendroidal+nerve">homotopy coherent dendroidal nerve</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">DendroidalSets</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">RelativeDendroidalSets</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dendroidal+set">dendroidal nerve</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">SegalOperads</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">DendroidalCompleteSegalSpaces</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mon%28%E2%88%9E%2C1%29Cat">Mon(∞,1)Cat</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">DendroidalCartesianFibrations</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+%28infinity%2C1%29-category+theory">formal <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</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pos">Pos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Set">Set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, <a class="existingWikiWord" href="/nlab/show/Operad">Operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</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>Cat</strong>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Operad">(∞,1)Operad</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+2-category+of+%28%E2%88%9E%2C1%29-categories">homotopy 2-category of (∞,1)-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29Cat">(∞,n)Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28infinity%2C+1%29Prof">(infinity, 1)Prof</a></p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/category">category</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on October 13, 2021 at 15:32:54. See the <a href="/nlab/history/%28infinity%2C1%29Cat" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/%28infinity%2C1%29Cat" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/%28infinity%2C1%29Cat/18" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/%28infinity%2C1%29Cat" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/%28infinity%2C1%29Cat" accesskey="S" class="navlink" id="history" rel="nofollow">History (18 revisions)</a> <a href="/nlab/show/%28infinity%2C1%29Cat/cite" style="color: black">Cite</a> <a href="/nlab/print/%28infinity%2C1%29Cat" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/%28infinity%2C1%29Cat" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>