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href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#the_cosimplicial_category'>The cosimplicial <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='#a_categorical_description'>A categorical description</a></li> <li><a href='#a_combinatorial_description'>A combinatorial description</a></li> <li><a href='#the_homotopy_coherent_nerve'>The homotopy coherent nerve</a></li> </ul> <li><a href='#examples_and_illustrations'>Examples and illustrations</a></li> <ul> <li><a href='#IllustratOfCosimpSSetCat'>For the cosimplicial <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='#for_other_simplicial_sets'>For other simplicial sets</a></li> <li><a href='#for_the_homotopy_coherent_nerve'>For the homotopy coherent nerve</a></li> </ul> <li><a href='#Properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <ul> <li><a href='#comonadic_resolution'>Comonadic resolution</a></li> <li><a href='#WConstruction'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-Construction of topological operads</a></li> <li><a href='#relation_to_quasicategories'>Relation to quasi-categories</a></li> <li><a href='#models_for_categories'>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>-categories</a></li> <li><a href='#other_kinds_of_nerves'>Other kinds of nerves</a></li> </ul> <li><a href='#history'>History</a></li> <li><a href='#references_and_literature'>References and Literature.</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>homotopy coherent nerve</em> (also called <em>simplicial nerve</em>) of a <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> is a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> which includes information about all the higher homotopies present in the hom-spaces. It generalizes the ordinary <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of an ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <p>The homotopy coherent nerve operation</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 lspace="verythinmathspace">:</mo><mi>sSet</mi><mtext>-</mtext><mi>Cat</mi><mo>⟶</mo><mi>sSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N \colon sSet\text{-}Cat \longrightarrow sSet \,. </annotation></semantics></math></div> <p>is induced, by the general machinery of <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>, by a <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial</a> <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a>, namely a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>⟶</mo><mi>sSet</mi><mtext>-</mtext><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> \Delta \longrightarrow sSet\text{-}Cat </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> to the category of <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a> which regards each <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> as a <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> with <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> objects analogous to how the <a class="existingWikiWord" href="/nlab/show/orientals">orientals</a> regard the <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>-simplex as an <a class="existingWikiWord" href="/nlab/show/strict+omega-category">strict <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>-category</a>.</p> <h2 id="definitions">Definitions</h2> <h3 id="the_cosimplicial_category">The cosimplicial <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>We here describe the cosimplicial <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></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 lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>⟶</mo><mi>sSet</mi><mtext>-</mtext><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> S \colon \Delta \longrightarrow sSet\text{-}Cat </annotation></semantics></math></div> <p>that induces the homotopy coherent nerve.</p> <h3 id="a_categorical_description">A categorical description</h3> <p>Recall that a <a class="existingWikiWord" href="/nlab/show/graph">graph</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>, is <em><a class="existingWikiWord" href="/nlab/show/reflexive+graph">reflexive</a></em> if for each <a class="existingWikiWord" href="/nlab/show/vertex">vertex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> it is equipped with a (specified) <a class="existingWikiWord" href="/nlab/show/edge">edge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>→</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">v \to v</annotation></semantics></math>. Similarly, a <strong>reflexive <a class="existingWikiWord" href="/nlab/show/directed+graph">directed graph</a></strong> has a specified identity edge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i_X \colon X \to X</annotation></semantics></math> on each object (vertex) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/free+category">free category</a> on a reflexive directed graph has</p> <ul> <li> <p>as <a class="existingWikiWord" href="/nlab/show/objects">objects</a> the <a class="existingWikiWord" href="/nlab/show/vertices">vertices</a> of the graph,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+morphisms">identity morphisms</a> corresponding to the identity edges,</p> </li> <li> <p>non-identity morphisms consisting of sequences of non-identity edges.</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/free+category">free category</a>-construction extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ReflxDGraph</mi><mo>⟶</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> F \;\colon\; ReflxDGraph \longrightarrow Cat </annotation></semantics></math></div> <p>and as such is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Cat</mi><mo>⟶</mo><mi>ReflxDGraph</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \;\colon\; Cat \longrightarrow ReflxDGraph \,. </annotation></semantics></math></div> <p>Hence the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>F</mi><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>Cat</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">G = F U \colon Cat\to Cat</annotation></semantics></math> constitutes a <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/simplicial+resolution">simplicial resolution</a> of this comonad gives an augmented simplicial <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>⟶</mo><mi>sSet</mi><mtext>-</mtext><mi>Cat</mi></mrow><annotation encoding="application/x-tex">S \colon \Delta \longrightarrow sSet\text{-}Cat</annotation></semantics></math> with natural augmentation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mi>Id</mi></mrow><annotation encoding="application/x-tex">S\to Id</annotation></semantics></math>, and which is a <a class="existingWikiWord" href="/nlab/show/cofibrant+replacement">cofibrant replacement</a>-construction in the <a class="existingWikiWord" href="/nlab/show/Bergner+model+structure">Bergner model structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi><mo>−</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">sSet-Cat</annotation></semantics></math> (“model structure for simplicially enriched categories”).</p> <p> <div class='num_remark'> <h6>Remark</h6> <p>Some <em>words of caution</em>, as always with simplicial resolutions, there are two conventions which differ by being the opposite simplicial object of each other. In the original paper, Cordier uses a different one of these conventions from some of the other sources mentioned here. A similar problem occurs in the following combinatorial description as some sources use ‘reverse inclusion’ where others just use ‘inclusion’ for the partial order on the poset. This corresponds more or less exactly to the distinction between ‘op-lax’ and ‘lax’ functors in the theory of 2-categories.</p> <p>Because of this, it is always important to test the definition being used on a simple example, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[2]</annotation></semantics></math> will do, before commiting to the use of any specific formulae. We will see this again in another Remark later on in this entry.</p> </div> </p> <h3 id="a_combinatorial_description">A combinatorial description</h3> <p>For <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></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> the finite <a class="existingWikiWord" href="/nlab/show/ordinal+number">ordinal</a> <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><mo stretchy="false">{</mo><mn>0</mn><mo><</mo><mn>1</mn><mo><</mo><mi>⋯</mi><mo><</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[n] \coloneqq \{0 \lt 1 \lt \cdots \lt n\}</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math> be standard <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial</a> <a class="existingWikiWord" href="/nlab/show/simplex"><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>-simplex</a>, define the <a class="existingWikiWord" href="/nlab/show/sSet-enriched+category">sSet-enriched category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[n]</annotation></semantics></math> as follows:</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[n]</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1, \cdots, n\}</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">S[n]_{i,j} \in sSet</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i, j \in \{0,1,\cdots, n\}</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/nerves">nerves</a></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 stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>P</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S[n](i,j) \coloneqq N(P_{i,j}) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">P_{i,j}</annotation></semantics></math> which is equivalently</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> of <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[i,j]</annotation></semantics></math> that contain both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</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> (so in particular if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>></mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \gt j</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(i,j)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/empty+set">empty</a> and hence so is its nerve) with the partial order is given by reverse inclusion.</p> </li> <li> <p>the poset of <a class="existingWikiWord" href="/nlab/show/path+category">paths</a> in <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></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> that start at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and finish at <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> (hence is empty if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>></mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i\gt j</annotation></semantics></math>), the order relation is given by ‘subdivision’, i.e. path <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> is less than path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(i,j)</annotation></semantics></math> if <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> visits all the vertices that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> does … and perhaps some others as well.</p> <p>Of course, the way you go between the two descriptions is that a path corresponds to the set of vertices it visits and <em>vice versa</em>.</p> </li> </ol> </li> </ul> <p>Notice that the simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msub><mi>P</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(P_{i,j})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">j-i-1</annotation></semantics></math> cube in <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>:</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><msub><mi>P</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mrow><mo>×</mo><mo stretchy="false">(</mo><mi>j</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N(P_{i,j}) = (\Delta[1])^{\times (j-i-1)} \,. </annotation></semantics></math></div> <p>Under this isomorphism for instance the vertex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mrow><mo>×</mo><mo stretchy="false">(</mo><mi>j</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(0,0,1,0,1) \in (\Delta[1])^{\times (j-i-1)}</annotation></semantics></math> corresponds to the subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>i</mi><mo>+</mo><mn>5</mn><mo>,</mo><mi>j</mi><mo stretchy="false">}</mo><mo>⊂</mo><mo stretchy="false">[</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\{i,i+3,i+5,j\} \subset [i,j]</annotation></semantics></math> and to the path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>→</mo><mi>i</mi><mo>+</mo><mn>3</mn><mo>→</mo><mi>i</mi><mo>+</mo><mn>5</mn><mo>→</mo><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">i \to i+3 \to i+5 \to j=i+6</annotation></semantics></math>.</p> <p>(We will look at an example after this definition.)</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation on hom-objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∘</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>:</mo><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>×</mo><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mrow><mi>i</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \circ_{i,j,k} : S[n]_{i,j} \times S[n]_{j,k} \to S[n]_{i,k} </annotation></semantics></math></div> <p>is induced by ‘concatenation of the corresponding paths’ and thus essentially by union of the sets involved.</p> </li> </ul> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The choice to order paths by reverse inclusion agrees with constructions such as the <a class="existingWikiWord" href="/nlab/show/Duskin+nerve">Duskin nerve</a>. However, the other convention where they are ordered by inclusion also appears in the literature, such as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a> definition 1.1.5.1., and corresponds to the convention used in the original paper by Cordier (see earlier Remark). The resulting theory is more or less equivalent in as much as the results true when using one convention have analogous results when using the other.</p> </div> <h3 id="the_homotopy_coherent_nerve">The homotopy coherent nerve</h3> <p>The <strong>homotopy coherent nerve</strong> functor</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>≔</mo><msub><mi>Hom</mi> <mrow><mi>sSet</mi><mi>Cat</mi></mrow></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">[</mo><mo>•</mo><mo stretchy="false">]</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>sSet</mi><mi>Cat</mi><mo>⟶</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> N \coloneqq Hom_{sSet Cat}(S[\bullet],-) \colon sSet Cat \longrightarrow sSet </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> defined by the <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial</a> <a class="existingWikiWord" href="/nlab/show/sSet-enriched+category">sSet-enriched category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">S \colon \Delta \to sSet Cat</annotation></semantics></math> defined above.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>sSet</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">C \in sSet Cat</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/sSet-enriched+category">sSet-enriched category</a>, the homotopy coherent nerve <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> is the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> uniquely characterized as giving a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of the form</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>SSet</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mrow><mi>SSet</mi><mi>Cat</mi></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><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"> Hom_{SSet}\big(\Delta[n], N(C)\big) \;\simeq\; Hom_{SSet Cat}\big(S[n], C\big) \,. </annotation></semantics></math></div> <p>By the general logic of <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>, this functor has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></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 stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>SSet</mi><mo>→</mo><mi>SSet</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> S(-) \colon SSet \to SSet Cat </annotation></semantics></math></div> <p>the <strong>realization</strong> functor given by the <a class="existingWikiWord" href="/nlab/show/coend">coend</a> formula</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 stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><msub><mi>X</mi> <mi>n</mi></msub><mo>⋅</mo><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> S(X) \coloneqq \int^{[n] \in \Delta} X_n \cdot S[n] \,, </annotation></semantics></math></div> <p>also known as the operation of <em><a class="existingWikiWord" href="/nlab/show/rigidification+of+quasi-categories">rigidification of quasi-categories</a></em>.</p> <p>This functor does extend the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">S : \Delta \to sSet Cat</annotation></semantics></math> in that there is a canonical isomorphism</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 stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> S(\Delta[n]) \cong S[n] </annotation></semantics></math></div> <p>and hence may consistently be named <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>.</p> <h2 id="examples_and_illustrations">Examples and illustrations</h2> <h3 id="IllustratOfCosimpSSetCat">For the cosimplicial <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>We illustrate here the nature of the cosimplicial <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S : [n] \mapsto S[n]</annotation></semantics></math>, viewed from the combinatoiral viewpoint above.</p> <p>We will examine the lowest dimensional cases.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> there is nothing of note.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>}</mo></mrow><mo>=</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>=</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex"> P_{0,1} = \left\{ (0,1) \right\} = \Delta[0] = \Delta[1]^0 </annotation></semantics></math></div> <p>is the poset with a single object.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math>, there are unique paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[2]</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0]</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1]</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1]</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[2]</annotation></semantics></math>, so the corresponding homs in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[2]</annotation></semantics></math> are copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[0]</annotation></semantics></math> (or, if you prefer, of <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> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\Delta[1]^0</annotation></semantics></math>!). Things are slightly more interesting for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S[2](0,2)</annotation></semantics></math>. Looking at this from the ‘subsets’ viewpoint, as above, there clearly are two subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1,2\}</annotation></semantics></math> containing both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>, one corresponds to the direct route in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[2]</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>, the other goes via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">0\to 1\to 2</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>P</mi> <mrow><mn>0</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>}</mo></mrow><mo>=</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> P_{0,2} = \left\{ (0,1,2) \to (0,2) \right\} = \Delta[1] \,, </annotation></semantics></math></div> <p>so in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S[2](0,2)</annotation></semantics></math>, there is a 1-simplex <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> starting at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1,2\}</annotation></semantics></math> and ending at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,2\}</annotation></semantics></math>.</p> <div style="text-align: center"><svg 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<g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-4-3" x="140.721" y="79.916"></use> </g> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 5.913125 -72.210062 L 134.807656 -72.210062 " transform="matrix(1, 0, 0, -1, 5.915, 7.704)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-5-1" x="61.691" y="59.493"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-1-1" x="66.866" y="59.493"></use> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-1-5" x="70.159245" y="59.493"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-2-1" x="74.394" y="59.493"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-1-2" x="76.746" y="59.493"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-2-1" x="80.98" y="59.493"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-1-3" x="83.333" y="59.493"></use> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-1-4" x="87.567514" y="59.493"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#3WsYT-KqWhDMP6oJapyJ22DTeIs=-glyph-0-3" x="143.71" y="82.905"></use> </g> </svg></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1,2)</annotation></semantics></math> is a product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,2)</annotation></semantics></math>, this simplex can also be depicted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,2) \cdot (0,1) \Rightarrow (0,2)</annotation></semantics></math>.</p> <p>Everything else, in higher dimensions, is degenerate, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</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">S[2](0,2)\cong \Delta[1]</annotation></semantics></math>. Sometimes it is useful to think of this 1-simplex as ‘rewriting’ the direct path to that via 1, all this happening in the free category on the underlying graph of the poset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[2]</annotation></semantics></math>. (The construction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[n]</annotation></semantics></math> in general has a nice interpretation in terms of higher dimensional <a class="existingWikiWord" href="/nlab/show/rewriting">rewriting</a>. This can be given using the language of <a class="existingWikiWord" href="/nlab/show/polygraph">polygraph</a>s or <a class="existingWikiWord" href="/nlab/show/computad">computad</a>s.)</p> <p>In this example there are no significant compositions. To see examples of those, you need to look at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math>. In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[3]</annotation></semantics></math>, the simplicial hom-sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S[3](i,j)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>≠</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j) \neq (0,3)</annotation></semantics></math>, can all be analysed by the same sort of argument to the above. The new features occur in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S[3](0,3)</annotation></semantics></math>. The vertices of this simplicial set are the subsets corresponding to the direct path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">0\to 3</annotation></semantics></math> and then the three others. Rewriting the direct path can be done in two immediate ways, to go via the left or via the right route. Each of these can be ‘rewritten’ to give the longest path / largest subset. There is also, of course, an inclusion of the smallest to the largest of these, so that in total the poset here looks like:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</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>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo>=</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mrow><mo>×</mo><mn>2</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P_{0,3} = \left\{ \array{ \{0,1,2,3\}&\rightarrow & \{0,1,3\} \\ \downarrow & \searrow &\downarrow\\ \{0,2,3\}&\rightarrow &\{0,3\} } \right\} = \Delta[1]^{\times 2} \,. </annotation></semantics></math></div> <p>In addition, there will be 2-simplexes filling the two triangles, coming from the chains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo><mo>⊂</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo><mo>⊂</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1,2,3\}\subset \{0,1,3\}\subset \{0,3\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo><mo>⊂</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo><mo>⊂</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1,2,3\}\subset \{0,2,3\}\subset \{0,3\}</annotation></semantics></math> in the poset.</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></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⇑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mn>3</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⇑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⇑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mn>3</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⇑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow && \searrow^{\mathrlap{\{0,3\}}} \\ & & \Uparrow \\ 0 &&\stackrel{\{0,1,3\}}{\to}&& 3 \\ && \Uparrow \\ & \searrow && \nearrow_{\mathrlap{\{0,1,2,3\}}} } \;\;\;\;\; \;\;\;\;\; \;\;\;\;\; \,, \;\;\;\;\; \;\;\;\;\; \;\;\;\;\; \array{ & \nearrow && \searrow^{\mathrlap{\{0,3\}}} \\ & & \Uparrow \\ 0 &&\stackrel{\{0,2,3\}}{\to}&& 3 \\ && \Uparrow \\ & \searrow && \nearrow_{\mathrlap{\{0,1,2,3\}}} } \,. </annotation></semantics></math></div> <p>We thus get <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>≅</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S[3](0,3) \cong \Delta[1]^2</annotation></semantics></math>, a square.</p> <p>The composition maps</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 stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>S</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S[3](1,3)\times S[3](0,1)\to S[3](0,3)</annotation></semantics></math></div> <p>and similarly for the one with 1 replaced by 2, <em>are</em> now fairly obvious.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n = 4</annotation></semantics></math>, the corresponding diagram for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S[4](0,4)</annotation></semantics></math> gives a cube but here there is an interesting feature.</p> <p>Five of the six faces of the cube <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><msub><mi>P</mi> <mrow><mn>0</mn><mo>,</mo><mn>4</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|N(P_{0,4})|</annotation></semantics></math> correspond to the associativity of composition of triples of composable morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4]</annotation></semantics></math>. These correspond to the 5 faces of the 4-simplex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[4]</annotation></semantics></math>, as depicted for instance at <a class="existingWikiWord" href="/nlab/show/oriental">oriental</a> and at <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>.</p> <p>But the cube has one more face</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>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</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>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (0,1,2,3,4) &\to& (0,1,2,4) \\ \downarrow &\searrow& \downarrow \\ (0,2,3,4) &\to& (0,2,4) } </annotation></semantics></math></div> <p>which does not correspond to associativity: instead, this encodes the <a class="existingWikiWord" href="/nlab/show/exchange+law">exchange law</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></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>3</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>4</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>3</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>4</mn></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>3</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>4</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>4</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && 1 &&&& 3 \\ & \nearrow &\Downarrow& \searrow && \nearrow && \searrow \\ 0 &&\to&& 2 && && 4 \\ && &&&& 3 \\ & && && \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 && \to && 4 } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ && 1 &&&& 3 \\ & \nearrow && \searrow && \nearrow &\Downarrow& \searrow \\ 0 &&&& 2 && \to && 4 \\ && 1 &&&& \\ & \nearrow &\Downarrow& \searrow && && \\ 0 &&\to && 2 && \to && 4 } </annotation></semantics></math></div> <p>or, if preferred, to the fact that</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 stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo><mo>×</mo><mi>S</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S[4](2,4)\times S[4](0,2)\to S[4](0,4)</annotation></semantics></math></div> <p>is to be a simplicial map.</p> <p>A similar phenomenon occurs in higher dimensions. There are two ‘extra faces’ in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mn>5</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S[5](0,5)</annotation></semantics></math>, and so on.</p> <h3 id="for_other_simplicial_sets">For other simplicial sets</h3> <p>The description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[n]</annotation></semantics></math> works more generally for the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a <a class="existingWikiWord" href="/nlab/show/poset">poset</a>. Explicitly, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is a poset, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>NP</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(NP)</annotation></semantics></math> is isomorphic to the simplicially enriched category with the structure</p> <ul> <li>The objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>NP</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(NP)</annotation></semantics></math> are the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></li> <li>The <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>NP</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(NP)(x,y)</annotation></semantics></math> is the nerve of the poset of <a class="existingWikiWord" href="/nlab/show/chains">chains</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">S \subseteq P</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>min</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\min(S) = x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>max</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">\max(S) = y</annotation></semantics></math>, ordered by reverse inclusion</li> <li>Composition is given by taking unions</li> </ul> <p>The simplicially enriched categories constructed from spheres and <a class="existingWikiWord" href="/nlab/show/inner+horns">inner horns</a> also have simple descriptions.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>≠</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j) \neq (0,n)</annotation></semantics></math>, the hom-objects are the same as the ambient simplex: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(\Lambda^n_k)(i,j) = S(\partial \Delta^n)(i,j) = S(\Delta^n)(i,j)</annotation></semantics></math>.</p> <p>For the remaining hom-object, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≅</mo><msup><mo>□</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S(\Delta^n)(0,n) \cong \square^{n-1}</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><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-cube, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mo>∂</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∂</mo><msup><mo>□</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S(\partial \Delta^n)(0,n) = \partial \square^{n-1}</annotation></semantics></math> is the boundary, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(\Lambda^n_k)(0,n)</annotation></semantics></math> further omits the corresponding face.</p> <p>To see this, observe that the nondegenerate simplices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(\Delta^n)(0,n)</annotation></semantics></math> are either decomposable (i.e. the product of two or more simplices from other hom-sets) or are the image of a top simplex of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>m</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo>→</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(\Delta^m)(0,m) \to S(\Delta^n)(0,n)</annotation></semantics></math> for some monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>m</mi></msup><mo>→</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^m \to \Delta^n</annotation></semantics></math> preserving top and bottom elements, depending on whether the simplex contains the top vertex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,n)</annotation></semantics></math>.</p> <p>Thus, the decomposable faces of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(\Delta^n)(0,n)</annotation></semantics></math> are the ones omitting the top vertex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,n)</annotation></semantics></math>, and the remaining faces come from inner faces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^{n-1} \to \Delta^n</annotation></semantics></math>.</p> <h3 id="for_the_homotopy_coherent_nerve">For the homotopy coherent nerve</h3> <ul> <li> <p>Any <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> gives a simplicially enriched category using the embedding of <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> into <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> via the usual <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> functor. The homotopy coherent nerve of a 2-category considered in this way is, sometimes, called the <span class="newWikiWord">geometric nerve<a href="/nlab/new/geometric+nerve">?</a></span> of the 2-category. The <a class="existingWikiWord" href="/nlab/show/Duskin+nerve">Duskin nerve</a> of a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> is an extension of this construction.</p> <p>A particular case of this nerve is the nerve of a <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> <a class="existingWikiWord" href="/nlab/show/delooping">considered as</a> a 2-category.</p> </li> </ul> <h2 id="Properties">Properties</h2> <ul> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>sSet</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">C \in sSet Cat</annotation></semantics></math> is such that all <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a>s <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><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">C(x,y) \in sSet</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es, then the homotopy coherent nerve <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> is a <a class="existingWikiWord" href="/nlab/show/weak+Kan+complex">weak Kan complex</a>/<a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">f : C \to D</annotation></semantics></math> is a morphism of such Kan-complex enriched categories which is a weak equivalence (in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">model structure on sSet-categories</a>) in that</p> <ul> <li> <p>the induced functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>H</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(f) : Ho(C) \to H(D) </annotation></semantics></math></div> <p>on the ordinary underlying <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy categories</a> (obtained by taking hom-wise connected component sets) is <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a></p> </li> <li> <p>its component on each hom-object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_{x,y} : C(x,y) \to D(f(x),f(y)) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>,</p> </li> </ul> <p>then its homotopy coherent nerve</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><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N(f) : N(C) \to N(C) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">weak equivalence of</a> <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>.</p> <p>We may think of <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math> trivially as a simplicially enriched category. In the <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">model structure on sSet-categories</a> the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[n]</annotation></semantics></math> is a cofibrant replacement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math>. And Kan-complex enriched categories are fibrant. So on these the homotopy coherent nerve is given by the <a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-space</a> functor</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><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mo>•</mo><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N(C) = \mathbb{R}Hom(\Delta[\bullet], C) \,. </annotation></semantics></math></div></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/rigidification+of+quasi-categories">rigidification of quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+sSet-enriched+categories">relation between quasi-categories and sSet-enriched categories</a></p> </li> </ul> <h3 id="comonadic_resolution">Comonadic resolution</h3> <p>(To be edited)</p> <p>The use of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[A]</annotation></semantics></math>, above, extends that given at the start of this page. Here <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 related to the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> of the homotopy coherent nerve, but is defined using a <a class="existingWikiWord" href="/nlab/show/comonadic+resolution">comonadic resolution</a>. The <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> comes from the adjunction between small categories and directed graphs with distinguished ‘unit’ loops. The ‘forgetful’ part of the adjunction forgets the composition in the category, but remembers that the identity arrows are special. The left adjoint / ‘free’ part of the adjunction takes a directed graph (with distinguished ‘identity’ loops, and forms the free category on the non-identity arrows. As usual, we can form a <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> from this and hence form a functorial <a class="existingWikiWord" href="/nlab/show/simplicial+resolution">simplicial resolution</a> of any small category, <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>.</p> <p>This can also be seen to be a case of a <a class="existingWikiWord" href="/nlab/show/bar+resolution">bar resolution</a> construction, related to the <a class="existingWikiWord" href="/nlab/show/bar+construction">bar construction</a>. Here the adjoint pair also give a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> on directed graphs with distinguished ‘unit’ loops and the small category <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> is an algebra for this monad.</p> <p>Since the functors involved preserve the identities on the objects 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>, the resulting simplicial category is a simplicially enriched category, and this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[A]</annotation></semantics></math>. The <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>-dimensional arrows between objects, <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[A]</annotation></semantics></math> correspond to a path from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> 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> containing no identity arrows, together with a bracketting of the resulting string having depth <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>.</p> <h3 id="WConstruction"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-Construction of topological operads</h3> <p>By hom-wise precomposition with the singular complex functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo>:</mo><mi>Top</mi><mo>→</mo><mi>sSet</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Sing : Top \to sSet \,, </annotation></semantics></math></div> <p>which is a <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a>, the homotopy coherent nerve extends to a nerve of <a class="existingWikiWord" href="/nlab/show/Top">Top</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">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>:</mo><mi>Top</mi><mi>Cat</mi><mo>→</mo><mi>sSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N : Top Cat \to sSet \,. </annotation></semantics></math></div> <p>As such, it is a special case of the <a class="existingWikiWord" href="/nlab/show/Michael+Boardman">Boardman</a>-<a class="existingWikiWord" href="/nlab/show/Rainer+Vogt">Vogt</a> <a class="existingWikiWord" href="/nlab/show/W-construction">W-construction</a> for <a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">cofibrant replacement of</a> topological <a class="existingWikiWord" href="/nlab/show/operads">operads</a>. See also <em><a class="existingWikiWord" href="/nlab/show/dendroidal+homotopy+coherent+nerve">dendroidal homotopy coherent nerve</a></em>.</p> <p>In this construction, roughly, for <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> a <a class="existingWikiWord" href="/nlab/show/tree">tree</a> in an operad <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">O</annotation></semantics></math>, the tree is replaced with the topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">e(T) \to [0,1]</annotation></semantics></math> of maps from the set of edges of <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> to the topological unit interval.</p> <p>We may restrict this construction to the <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/category">category</a> and then trivially regarded as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>-category. Then a tree in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math> is necessarily a linear tree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo><mo>→</mo><mi>⋯</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\to \to \cdots \to</annotation></semantics></math> of some length <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 is hence mapped to the topological space of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k \to [0,1]</annotation></semantics></math>, i.e. to the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">[0,1]^k</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of the simplicial cubes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">(\Delta[1])^k</annotation></semantics></math> that we saw above.</p> <h3 id="relation_to_quasicategories">Relation to quasi-categories</h3> <p>As mentioned above, the simplicial or h.c. nerve, together with its <a class="existingWikiWord" href="/nlab/show/adjoint+functor">left adjoint</a>, serves to relate the two models of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-categories</a> given by <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> and <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched categories</a>.</p> <p>The homotopy coherent nerve extends to a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/Joyal+model+structure">Joyal model structure</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> that models <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a> and the <a class="existingWikiWord" href="/nlab/show/model+structure+on+SSet-categories">model structure on SSet-categories</a>.</p> <p>See</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation between quasi-categories and simplicial categories</a></li> </ul> <p>for details.</p> <h3 id="models_for_categories">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>-categories</h3> <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> <h3 id="other_kinds_of_nerves">Other kinds of nerves</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve">nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Duskin+nerve">Duskin nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-nerve">∞-nerve</a></p> </li> <li> <p><strong>homotopy coherent nerve</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-nerve">dg-nerve</a></p> </li> </ul> <h2 id="history">History</h2> <p>The original motivation for the introduction of the homotopy coherent nerve is that it provides a neat simplicial formulation of idea of <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+diagram">homotopy coherent diagrams</a>. Homotopy coherent algebraic structures were studied in the 1970s by <a class="existingWikiWord" href="/nlab/show/Michael+Boardman">Boardman</a> and <a class="existingWikiWord" href="/nlab/show/Rainer+Vogt">Vogt</a> in joint work, and then Vogt individually looked at homotopy coherent diagrams. The homotopy coherent nerve was first explicitly defined by <a class="existingWikiWord" href="/nlab/show/Jean-Marc+Cordier">Cordier</a> (reference below). He realised that, with a slight modification in the definition, Vogt’s definition of homotopy coherent diagram, indexed by a small category <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>, say, corresponded exactly to a simplicially enriched functor from the <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[A]</annotation></semantics></math> to the <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>. They thus also corresponded to simplicial maps from the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(Top)</annotation></semantics></math>, (although that latter object was ‘too large’ to be a simplicial ‘set’). This allowed a good definition of homotopy coherent diagrams in arbitrary simplicially enriched categories to be given.</p> <p>This definition works best when the simplicially enriched category is ‘locally Kan’, in other words it is enriched in the category of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complexes</a>. These locally Kan <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>-categories are the fibrant ones in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">model structure on sSet-categories</a>.</p> <p>Cordier and <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Porter</a> (1986) proved that 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 locally Kan simplicially enriched category then <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> is a ‘<a class="existingWikiWord" href="/nlab/show/weak+Kan+complex">weak Kan complex</a>’, in other words, a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>. Some of the main ideas behind this result can be traced to <a class="existingWikiWord" href="/nlab/show/Rainer+Vogt">Vogt</a>‘s paper of 1973.</p> <p>In more modern terminology as <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es can be considered as <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s, these locally Kan simplicially enriched categories are one particularly nice model for an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a>, and so this result is one of the earliest giving the transition from one model for <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories">(infinity,1)-categories</a> to another, the ‘weak Kan complexes’ or <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>.</p> <h2 id="references_and_literature">References and Literature.</h2> <p>The homotopy coherent nerve operation was introduced, explicitly, in</p> <ul> <li id="Cordier82"><a class="existingWikiWord" href="/nlab/show/Jean-Marc+Cordier">Jean-Marc Cordier</a>, <em>Sur la notion de diagramme homotopiquement cohérent</em>, Cahier Top. et Geom. Diff. XXIII 1 (1982) 93-112 [<a href="http://www.numdam.org/numdam-bin/feuilleter?id=CTGDC_1982__23_1">numdam:CTGDC_1982__23_1</a>]</li> </ul> <p>making a link with earlier work in</p> <ul> <li>R. D. Leitch, <em>The homotopy commutative cube</em>, J. London Math. Soc. <strong>2</strong> 9 (1974) 23-29 [<a href="https://doi.org/10.1112/jlms/s2-9.1.23">doi:10.1112/jlms/s2-9.1.23</a>]</li> </ul> <p>as well as with <a href="#Vogt73">Vogt (1973)</a> and with</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Boardman">Michael Boardman</a>, <a class="existingWikiWord" href="/nlab/show/Rainer+Vogt">Rainer Vogt</a>, <em>Homotopy invariant algebraic structures on topological spaces</em>, Lec. Notes in Math. <strong>347</strong> Springer (1973) [<a href="https://doi.org/10.1007/BFb0068547">doi:10.1007/BFb0068547</a>]</li> </ul> <p>Poof of an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> between the category obtained by inverting the ‘levelwise’ homotopy equivalence in a category of diagrams, and the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+diagrams">homotopy coherent diagrams</a>:</p> <ul> <li id="Vogt73"><a class="existingWikiWord" href="/nlab/show/Rainer+Vogt">Rainer Vogt</a>, <em>Homotopy limits and colimits</em>, Math. Z. <strong>134</strong> (1973) 11-52 [<a href="https://eudml.org/doc/171965">eudml:171965</a>, <a href="https://doi.org/10.1007/BF01219090">doi:10.1007/BF01219090</a>]</li> </ul> <p>and explicit proof that the homotopy coherent nerve of a locally Kan simplicially enriched category is a <a class="existingWikiWord" href="/nlab/show/quasicategory">quasicategory</a>, as well as the harder result on when <a class="existingWikiWord" href="/nlab/show/outer+horns">outer horns</a> in this quasicategory can be filled:</p> <ul> <li id="CordierPorter86"><a class="existingWikiWord" href="/nlab/show/Jean-Marc+Cordier">Jean-Marc Cordier</a>, <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em>Vogt’s theorem on categories of homotopy coherent diagrams</em>, Math. Proc. Cambridge Philos. Soc. <strong>100</strong> (1986) 65-90 [<a href="https://doi.org/10.1017/S0305004100065877">doi:10.1017/S0305004100065877</a>, <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/Cordier-Porter.pdf">pdf</a>]</li> </ul> <p>Two other papers are relevant to this:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marc+Cordier">Jean-Marc Cordier</a>, <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em>Maps between homotopy coherent diagrams</em>, Top. and its Appl. <strong>28</strong>, (1988), 255 – 275.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marc+Cordier">Jean-Marc Cordier</a>, <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em>Fibrant diagrams, rectifications and a construction of Loday</em>, J. Pure. Applied Alg. <strong>67</strong>, (1990), 111 – 124.</p> </li> </ul> <p>An elementary discussion of the concept of homotopy coherence forms Chapter V of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/K.+H.+Kamps">K. H. Kamps</a>, <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em>Abstract homotopy and simple homotopy theory</em>, World Scientific 1997.</li> </ul> <p>For the role played by Cordier’s simplicial nerve in the context of relating quasi-categories to simplicially enriched categories as 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>-categories see</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Theory+of+Quasi-Categories+and+its+Applications">The Theory of Quasi-Categories and its Applications</a></em>, lectures at <em><a href="https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html">Advanced Course on Simplicial Methods in Higher Categories</a></em>, CRM (2008) [<a href="http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JoyalTheoryOfQuasiCategories.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>Notes on quasi-categories</em> (2008) [<a href="http://www.math.uchicago.edu/~may/IMA/Joyal.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JoyalNotesOnQuasiCategories.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 1.1.5 in: <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em> (2009)</p> </li> </ul> <p>This emphasizes the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjunction</a> corresponding to the homotopy coherent (“simplicial”) nerve construction.</p> <p>A review of this latter aspect is also in</p> <ul> <li id="Dhand"> <p>Vivek Dhand, <em>The simplicial nerve of a simplicial category</em> (<a href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=374dc56d9a4352b8f9f71cc12dd7331e9285bb0c">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Dhand-SimplicialNerve.pdf" title="pdf">pdf</a>)</p> </li> <li> <p>Mitya Boyarchenko, <em>Notes and exercise on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</em> (<a href="http://math.uchicago.edu/~mitya/langlands/quasicategories.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Hinich">Vladimir Hinich</a>, <em>Simplicial nerve in Deformation theory</em> (<a href="http://arxiv.org/abs/0704.2503">arXiv:0704.2503</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Dendroidal sets and simplicial operads</em>, <a href="http://arxiv.org/abs/1109.1004">arxiv/1109.1004</a> (a Quillen equivalence for Segal vs. simplicial operads using coherent nerve)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em>On the structure of simplicial categories associated to quasi-categories, Math. Proc. Camb. Phil. Soc. 150 (2011), 489 - 504.</em></p> </li> </ul> <p>For more references see <a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation between quasi-categories and simplicial categories</a>.</p> <p>Two query-discussions on terminology and concrete description of the coherent/“simplicial” nerve are archived at nForum <a href="http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=754&Focus=20607#Comment_20607">here</a>. For an overview of the 2009 paper by Dugger and Spivak, see also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a href="https://golem.ph.utexas.edu/category/2010/04/understanding_the_homotopy_coh.html">Understanding the homotopy coherent nerve</a>, n-Category Café, 29 April 2010.</li> </ul> <p>Relating the <a class="existingWikiWord" href="/nlab/show/Dwyer-Kan+loop+groupoid">Dwyer-Kan loop groupoid</a>-construction to the homotopy coherent nerve-construction:</p> <ul> <li id="MinichielloRiveraZeinalian23"> <p><a class="existingWikiWord" href="/nlab/show/Emilio+Minichiello">Emilio Minichiello</a>, <a class="existingWikiWord" href="/nlab/show/Manuel+Rivera">Manuel Rivera</a>, <a class="existingWikiWord" href="/nlab/show/Mahmoud+Zeinalian">Mahmoud Zeinalian</a>, <em>Categorical models for path spaces</em>, Advances in Mathematics <strong>415</strong> (2023) 108898 [<a href="https://arxiv.org/abs/2201.03046">arXiv:2201.03046</a>, <a href="https://doi.org/10.1016/j.aim.2023.108898">doi:10.1016/j.aim.2023.108898</a>]</p> </li> <li id="Arakawa23"> <p><a class="existingWikiWord" href="/nlab/show/Kensuke+Arakawa">Kensuke Arakawa</a>, <em>Classifying Space via Homotopy Coherent Nerve</em>, Homology Homotopy Appl. 25(2), 2023, 373–381 (<a href="https://arxiv.org/abs/2208.00550">arXiv:math/2208.00550</a>,<a href="http://dx.doi.org/10.4310/HHA.2023.v25.n2.a16">doi:org/10.4310/HHA.2023.v25.n2.a16</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 29, 2024 at 10:07:15. 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