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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#NerveOfACategory'>Nerve of a 1-category</a></li> <ul> <li><a href='#definition_3'>Definition</a></li> <li><a href='#examples_2'>Examples</a></li> <li><a href='#PropNerveCat'>Properties</a></li> </ul> <li><a href='#nerve_of_a_2category'>Nerve of a 2-category</a></li> <li><a href='#nerve_of_a_3category'>Nerve of a 3-category</a></li> <li><a href='#nerve_of_an_category'>Nerve of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-category</a></li> <li><a href='#nerve_of_chain_complexes'>Nerve of chain complexes</a></li> </ul> <li><a href='#remarks'>Remarks</a></li> <ul> <li><a href='#geometric_realization'>Geometric realization</a></li> <li><a href='#nerves_and_higher_categories'>Nerves and higher categories</a></li> <li><a href='#internal_nerve'>Internal nerve</a></li> <li><a href='#direct_categories_versus_finite_simplicial_sets'>Direct categories versus finite simplicial sets</a></li> </ul> <li><a href='#properties_2'>Properties</a></li> <ul> <li><a href='#PreservationOfColimits'>(Non-)Preservation of colimits</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#for_covers'>For covers</a></li> <li><a href='#for_categories'>For categories</a></li> <li><a href='#for_higher_categories'>For higher categories</a></li> </ul> </ul> </div> <p>The <em>nerve</em> is the right adjoint of a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> that exists in many situations. For the general abstract theory behind this see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>.</li> </ul> <h2 id="idea">Idea</h2> <p>As soon as any <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> comes equipped with a <a class="existingWikiWord" href="/nlab/show/simplicial+object">cosimplicial object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>C</mi></msub><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \Delta_C : \Delta \to C </annotation></semantics></math></div> <p>that we may think of as determining a <a class="existingWikiWord" href="/nlab/show/geometric+realization">realization</a> of the standard <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, we make every <a class="existingWikiWord" href="/nlab/show/object">object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/space+and+quantity">probeable</a> by <a class="existingWikiWord" href="/nlab/show/simplex">simplices</a> in that there is now a way to find the set</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>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>:</mo><mo>=</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>C</mi></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N(A)_n := Hom_C(\Delta_C[n],A) </annotation></semantics></math></div> <p>of ways to map 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> into a given object <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>These collections of sets evidently organize into a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N(A) : \Delta^{op} \to Set \,. </annotation></semantics></math></div> <p>This <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is called the <em>nerve</em> 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> (with respect to the chosen <a class="existingWikiWord" href="/nlab/show/geometric+realization">realization</a> of the standard simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>). Typically the nerve defines a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">N \colon C \to Set^{\Delta^op}</annotation></semantics></math> that has a left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>⋅</mo><mo stretchy="false">|</mo><mo lspace="verythinmathspace">:</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">|\cdot| \colon Set^{\Delta^op} \to C</annotation></semantics></math> called <a class="existingWikiWord" href="/nlab/show/realization">realization</a>.</p> <p>There are many generalizations of this procedure, some of which are described below.</p> <h2 id="definition">Definition</h2> <blockquote> <p>(notice that for the moment the following gives just one particular case of the more general notion of nerve)</p> </blockquote> <p>Let <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> be one of the categories of <a class="existingWikiWord" href="/nlab/show/geometric+shapes+for+higher+structures">geometric shapes for higher structures</a>, such as the <a class="existingWikiWord" href="/nlab/show/globe+category">globe category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/cube+category">cube category</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">\Box</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/cycle+category">cycle category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> of Connes, or certain category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> related to trees whose corresponding presheaves are <a class="existingWikiWord" href="/nlab/show/dendroidal+set">dendroidal sets</a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is any <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a> category or, more generally, 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>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> equipped with 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>i</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> i : S \to C </annotation></semantics></math></div> <p>we obtain 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>:</mo><mi>C</mi><mo>→</mo><msup><mi>V</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> N : C \to V^{S^{op}} </annotation></semantics></math></div> <p>from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/globular+sets">globular sets</a> or <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> or <a class="existingWikiWord" href="/nlab/show/cubical+sets">cubical sets</a>, respectively, (or the corresponding <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>-objects) given on an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/restricted+Yoneda+embedding">restricted Yoneda embedding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>S</mi> <mi>op</mi></msup><mover><mo>→</mo><mi>i</mi></mover><msup><mi>C</mi> <mi>op</mi></msup><mover><mo>→</mo><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mover><mi>V</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_i(c) : S^{op} \stackrel{i}\to C^{op} \stackrel{C(-,c)}{\to} V \,. </annotation></semantics></math></div> <p>This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_i(c)</annotation></semantics></math> is the <strong>nerve</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> with respect to the chosen <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">i : S \to C</annotation></semantics></math>. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><msup><mi>i</mi> <mo>*</mo></msup><mo>∘</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">N = i^* \circ Y</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Y: C \to [C^{op}, V]</annotation></semantics></math> is the curried Hom functor; if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mstyle mathvariant="sans-serif"><mi>Sets</mi></mstyle></mrow><annotation encoding="application/x-tex">V=\mathsf{Sets}</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>.</p> <p>Typically, one wants that <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> is <a class="existingWikiWord" href="/nlab/show/dense+functor">dense functor</a>, i.e. that every object <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is canonically a colimit of a diagram of objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, more precisely,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">colim</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi mathvariant="normal">pr</mi> <mi>S</mi></msub></mrow></mover><mi>S</mi><mover><mo>→</mo><mi>i</mi></mover><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C) = c, </annotation></semantics></math></div> <p>which is equivalent to the requirement that the corresponding nerve functor is <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">fully faithful</a> (in other words, if <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> is inclusion then <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 a left adequate subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in terminology of <a href="#Isbell60">Isbell 60</a>). The nerve functor may be viewed as a <span class="newWikiWord">singular functor<a href="/nlab/new/singular+functor">?</a></span> of the functor <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>.</p> <h2 id="examples">Examples</h2> <h3 id="NerveOfACategory">Nerve of a 1-category</h3> <p>For fixing notation, recall that the source and target maps of a small <a class="existingWikiWord" href="/nlab/show/category#OneCollectionOfMorphisms">category</a> form a <a class="existingWikiWord" href="/nlab/show/span">span</a> in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Span</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Span(Set)</annotation></semantics></math> where composition <a class="existingWikiWord" href="/nlab/show/span#categories_of_spans">is given by a pullback</a> (fiber product). The pairs of composable morphisms of a category are then obtained composing its source/target span with itself.</p> <div class="num_defn" id="SmallCategory"> <h6 id="definition_2">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/small+category">small category</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_\bullet</annotation></semantics></math> is</p> <ul> <li> <p>a pair of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>0</mn></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}_0 \in Set </annotation></semantics></math> (the set of <a class="existingWikiWord" href="/nlab/show/objects">objects</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}_1 \in Set</annotation></semantics></math> (the set of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>)</p> </li> <li> <p>equipped with <a class="existingWikiWord" href="/nlab/show/functions">functions</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><msub><mi>𝒞</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow></msub><msub><mi>𝒞</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mo>∘</mo></mover></mtd> <mtd><msub><mi>𝒞</mi> <mn>1</mn></msub></mtd> <mtd><mover><mover><munder><mo>→</mo><mi>s</mi></munder><mover><mo>←</mo><mi>i</mi></mover></mover><mover><mo>→</mo><mi>t</mi></mover></mover></mtd> <mtd><msub><mi>𝒞</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_1 \times_{\mathcal{C}_0} \mathcal{C}_1 &\stackrel{\circ}{\to}& \mathcal{C}_1 & \stackrel{\overset{t}{\to}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\to}}}& \mathcal{C}_0 }\,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> on the left is that over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>t</mi></mover><msub><mi>𝒞</mi> <mn>0</mn></msub><mover><mo>←</mo><mi>s</mi></mover><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_1 \stackrel{t}{\to} \mathcal{C}_0 \stackrel{s}{\leftarrow} \mathcal{C}_1</annotation></semantics></math>,</p> </li> </ul> <p>such that</p> <ul> <li> <p><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> takes values in <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a>;</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∘</mo><mi>i</mi><mo>=</mo><mi>s</mi><mo>∘</mo><mi>i</mi><mo>=</mo><msub><mi>id</mi> <mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\; </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math> defines a partial <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation which is <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unital</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(\mathcal{C}_0)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/identities">identities</a>; in particular</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s (g \circ f) = s(f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t (g \circ f) = t(g)</annotation></semantics></math>.</p> </li> </ul> </div> <h4 id="definition_3">Definition</h4> <div class="num_defn" id="NerveOfSmallCategory"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_\bullet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, def. <a class="maruku-ref" href="#SmallCategory"></a>, its <em>simplicial nerve</em> <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>𝒞</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">N(\mathcal{C}_\bullet)_\bullet</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> with</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>𝒞</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>≔</mo><msubsup><mi>𝒞</mi> <mn>1</mn> <mrow><msubsup><mo>×</mo> <mrow><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow> <mi>n</mi></msubsup></mrow></msubsup></mrow><annotation encoding="application/x-tex"> N(\mathcal{C}_\bullet)_n \coloneqq \mathcal{C}_1^{\times_{\mathcal{C}_0}^n} </annotation></semantics></math></div> <p>the set of sequences of composable morphisms of length <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>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>;</p> <p>with face maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>k</mi></msub><mo lspace="verythinmathspace">:</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> d_k \colon N(\mathcal{C}_\bullet)_{n+1} \to N(\mathcal{C}_\bullet)_{n} </annotation></semantics></math></div> <p>being</p> <ul> <li> <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo>=</mo><mi>target</mi><mo>:</mo><mi>arr</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ob</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_0= target:arr(\mathcal{C})\to ob(\mathcal{C})</annotation></semantics></math>, whilst <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">d_1</annotation></semantics></math> is similarly the domain / source function;</p> </li> <li> <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 \geq 1</annotation></semantics></math></p> <ul> <li> <p>the two outer face maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">d_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_{n+1}</annotation></semantics></math> are given by forgetting the first and the last morphism in such a sequence, respectively;</p> </li> <li> <p>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> inner face maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_{0 \lt k \lt n+1}</annotation></semantics></math> are given by composing the <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>th morphism with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k+1</annotation></semantics></math>st in the sequence.</p> </li> </ul> </li> </ul> <p>The degeneracy maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>k</mi></msub><mo lspace="verythinmathspace">:</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> s_k \colon N(\mathcal{C}_\bullet)_{n} \to N(\mathcal{C}_\bullet)_{n+1} \,. </annotation></semantics></math></div> <p>are given by inserting an <a class="existingWikiWord" href="/nlab/show/identity">identity</a> morphism on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Spelling this out in more detail: write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>n</mi></msub><mo>=</mo><mrow><mo>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>n</mi></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathcal{C}_n = \left\{ x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_2 \stackrel{f_{2,3}}{\to} \cdots \stackrel{f_{n-1,n}}{\to} x_n \right\} </annotation></semantics></math></div> <p>for the set of sequences of <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> composable morphisms. Given any element of this set and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \lt k \lt n </annotation></semantics></math>, write</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>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≔</mo><msub><mi>f</mi> <mrow><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>f</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i} </annotation></semantics></math></div> <p>for the composition of the two morphism that share the <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>th vertex.</p> <p>With this, face map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math> acts simply by “removing the index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>”:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>2</mn></msub><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>2</mn></msub><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d_0 \colon (x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto (x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mi>⋯</mi><mover><mo>→</mo><mrow></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mi>⋯</mi><mover><mo>→</mo><mrow></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d_{0\lt k \lt n} \colon ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) \mapsto ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_n \colon ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} ) \,. </annotation></semantics></math></div> <p>Similarly, writing</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>k</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>≔</mo><msub><mi>id</mi> <mrow><msub><mi>x</mi> <mi>k</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> f_{k,k} \coloneqq id_{x_k} </annotation></semantics></math></div> <p>for the identity morphism on the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math>, then the degeneracy map acts by “repeating the <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>th index”</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>k</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><mo>→</mo><msub><mi>x</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><mo>→</mo><msub><mi>x</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> s_k \colon ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \mapsto ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \,. </annotation></semantics></math></div> <p>This makes it manifest that these functions organise into a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>.</p> </div> <p>More abstractly, this construction is described as follows. Recall that</p> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo>↪</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> i \colon \Delta \hookrightarrow Cat </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> on non-empty finite <a class="existingWikiWord" href="/nlab/show/linear+orders">linear orders</a> regarded as categories, meaning that the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[n] \in Obj(\Delta)</annotation></semantics></math> may be identified with the category <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><mn>2</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[n] = \{0 \to 1 \to 2 \to \cdots \to n\}</annotation></semantics></math>. The morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> are all functors between these total linear categories.</p> </div> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/strict+category">strict category</a> its <em>nerve</em> <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(\mathcal{C})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> given by</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 lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>↪</mo><msup><mi>Cat</mi> <mi>op</mi></msup><mover><mo>→</mo><mrow><mi>Cat</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></mover><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> N(\mathcal{C}) \colon \Delta^{op} \hookrightarrow Cat^{op} \stackrel{Cat(-,\mathcal{C})}{\to} Set \,, </annotation></semantics></math></div> <p>where <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> is regarded as a <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a> with objects locally small strict categories, and morphisms being <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between these.</p> </div> <p>So the set <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><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">N(\mathcal{C})_n</annotation></semantics></math> of <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/simplices">simplices</a> of the nerve is the set of functors <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><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\{0 \to 1 \to \cdots \to n\} \to \mathcal{C}</annotation></semantics></math>. This is clearly the same as the set of sequences of composable morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> of length <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> obtained by iterated fiber product (as <a href="#NerveOfACategory">above</a> for pairs of composables):</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><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><msub><munder><mrow><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></msub><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></msub><mi>⋯</mi><msub><mo>×</mo> <mrow><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></msub><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder> <mrow><mi>n</mi><mspace width="mediummathspace"></mspace><mi>factors</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> N(\mathcal{C})_n = \underbrace{ Mor(\mathcal{C}) \times_{Obj(\mathcal{C})} Mor(\mathcal{C}) \times_{Obj(\mathcal{C})} \cdots \times_{Obj(\mathcal{C})} Mor(\mathcal{C}) }_{n \medspace factors} </annotation></semantics></math></div> <p>The collection of all functors between linear orders</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mi>⋯</mi><mo>→</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>m</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{ 0 \to 1 \to \cdots \to n \} \to \{ 0 \to 1 \to \cdots \to m \} </annotation></semantics></math></div> <p>is generated from those that map almost all generating morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>→</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \to k+1</annotation></semantics></math> to another generating morphism, except at one position, where they</p> <ul> <li> <p>map a single generating morphism to the composite of two generating morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>δ</mi> <mi>i</mi> <mi>n</mi></msubsup><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \delta^n_i : [n-1] \to [n] </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>δ</mi> <mi>i</mi> <mi>n</mi></msubsup><mo>:</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>i</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>i</mi><mo>→</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1)) </annotation></semantics></math></div></li> <li> <p>map one generating morphism to an identity morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>σ</mi> <mi>i</mi> <mi>n</mi></msubsup><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \sigma^n_i : [n+1] \to [n] </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>σ</mi> <mi>i</mi> <mi>n</mi></msubsup><mo>:</mo><mo stretchy="false">(</mo><mi>i</mi><mo>→</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↦</mo><msub><mi>Id</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> \sigma^n_i : (i \to i+1) \mapsto Id_i </annotation></semantics></math></div></li> </ul> <p>It follows that, for instance</p> <ul> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover><msub><mi>d</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow></mover><msub><mi>d</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>N</mi><mo stretchy="false">(</mo><mi>D</mi><msub><mo stretchy="false">)</mo> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{f_2}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(D)_3</annotation></semantics></math> the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub><mo>:</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mn>3</mn></msub><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d_1 := N(\mathcal{C})(\delta_1) : N(\mathcal{C})_3 \to N(\mathcal{C})_2</annotation></semantics></math> is obtained by composing the first two morphisms in this sequence: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><msub><mi>d</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow></mover><msub><mi>d</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">(d_0 \stackrel{f_2 \circ f_1}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(\mathcal{C})_2</annotation></semantics></math></p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(d_0 \stackrel{f_1}{\to} d_1) \in N(\mathcal{C})_1</annotation></semantics></math> the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>1</mn></msub><mo>:</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>σ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">s_1 := N(\mathcal{C})(\sigma_1) : N(\mathcal{C})_1 \to N(\mathcal{C})_2</annotation></semantics></math> is obtained by inserting an identity morphism: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></msub></mrow></mover><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{Id_{d_1}}{\to} d_1) \in N(\mathcal{C})_2</annotation></semantics></math>.</p> </li> </ul> <p>In this way, generally the face and degeneracy maps of the nerve of a category come from composition of morphisms and from inserting identity morphisms.</p> <p>In particular in light of their generalization to nerves of higher categories, discussed below, the cells in the 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(\mathcal{C})</annotation></semantics></math> have the following interpretation:</p> <ul> <li> <p><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><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>d</mi><mo stretchy="false">|</mo><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">N(\mathcal{C})_0 = \{d | d \in Obj(\mathcal{C})\} </annotation></semantics></math> is the collection of <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>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p><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><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>=</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>d</mi><mover><mo>→</mo><mi>f</mi></mover><mi>d</mi><mo>′</mo><mo stretchy="false">|</mo><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">N(\mathcal{C})_1 = Mor(\mathcal{C}) = \{d \stackrel{f}{\to} d' | f \in Mor(D)\}</annotation></semantics></math> is the collection of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>;</p> </li> <li> <p><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><msub><mo stretchy="false">)</mo> <mn>2</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>d</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><mo>∃</mo><mo>!</mo></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>d</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mo>|</mo></mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><msub><mrow></mrow> <mi>t</mi></msub><msub><mo>×</mo> <mi>s</mi></msub><mi>Mor</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">N(\mathcal{C})_2 = \left\{ \left. \array{ && d_1 \\ & {}^{f_1}\nearrow &\Downarrow^{\exists !}& \searrow^{f_2} \\ d_0 &&\stackrel{f_2 \circ f_1}{\to}&& d_2 } \right| (f_1, f_2) \in Mor(D) {}_t \times_s Mor(D) \right\}</annotation></semantics></math> is the collection of composable morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> as in the diagram <img src="/nlab/files/2-simplex-1-cat-nerve.svg" width="150px" /> The 2-cell itself is to be read as the <em>composition operation</em>, which is unique for an ordinary category (there is just one way to compose two morphisms);</p> </li> <li> <p><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><msub><mo stretchy="false">)</mo> <mn>3</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mrow><mtable><mtr><mtd><msub><mi>d</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>d</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">↑</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub></mrow></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>3</mn></msub><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>d</mi> <mn>3</mn></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><mover><mo>⇒</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>d</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>d</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">↑</mo></mtd> <mtd><msup><mo>↘</mo> <mrow><msub><mi>f</mi> <mn>3</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>2</mn></msub></mrow></msup></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>d</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow><mo>|</mo></mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><msub><mrow></mrow> <mi>t</mi></msub><msub><mo>×</mo> <mi>s</mi></msub><mi>Mor</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><msub><mrow></mrow> <mi>t</mi></msub><msub><mo>×</mo> <mi>s</mi></msub><mi>Mor</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">N(\mathcal{C})_3 = \left\{ \left. \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & {}^{f_2 \circ f_1}\nearrow & \downarrow^{f_3} \\ d_0 &\stackrel{f_3\circ (f_2\circ f_1)}{\to}& d_3 } \;\;\;\;\;\stackrel{\exists !}{\Rightarrow} \;\;\;\;\; \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & \searrow^{f_3\circ f_2} & \downarrow^{f_3} \\ d_0 &\stackrel{(f_3\circ f_2) \circ f_1}{\to}& d_3 } \right| (f_3,f_2, f_1) \in Mor(D) {}_t \times_s Mor(D) {}_t \times_s Mor(D) \right\} </annotation></semantics></math> is the collection of triples of composable morphisms as in the diagram <img src="/nlab/files/3-simplex-1-cat-nerve.svg" width="450px" /> to be read as the unique associators that relate one way to compose three morphisms using the above 2-cells to the other way.</p> </li> </ul> <h4 id="examples_2">Examples</h4> <div class="num_example"> <h6 id="example">Example</h6> <p><strong>(bar construction)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (for instance a <a class="existingWikiWord" href="/nlab/show/group">group</a>) with multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math>, and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} A</annotation></semantics></math> for the corresponding one-object <a class="existingWikiWord" href="/nlab/show/category">category</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">Mor(\mathbf{B} A) = A</annotation></semantics></math>. Then the nerve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mathbf{B} A)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math> is the simplicial set which is given by a <a class="existingWikiWord" href="/nlab/show/two-sided+bar+construction">two-sided bar construction</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>, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>A</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(1, A, 1)</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mi>A</mi><mo>×</mo><mi>A</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>A</mi><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> N(\mathbf{B}A) = \left( \cdots A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \stackrel{\to}{\to} {*} \right) </annotation></semantics></math></div> <p>where for example the three parallel face maps on display are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo>,</mo><mi>m</mi><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo>:</mo><mi>A</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\pi_1, m, \pi_2: A \times A \to A</annotation></semantics></math>.</p> <p>In particular, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A = G</annotation></semantics></math> is a discrete group, then the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|N(\mathbf{B} G)|</annotation></semantics></math> of the nerve of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex"> \cdots \simeq B G</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a>.</p> </div> <h4 id="PropNerveCat">Properties</h4> <p>The following lists some characteristic properties of simplicial sets that are nerves of categories.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A simplicial set is the nerve of a category precisely if it satisfies the <a class="existingWikiWord" href="/nlab/show/Segal+condition">Segal condition</a>.</p> </div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/Segal+condition">Segal condition</a></em> for more on this.</p> <p> <div class='num_prop' id='sSetIsNerveOfCategoryIffAllInnerHornsHaveUniqueFillers'> <h6>Proposition</h6> <p><br /> A <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is the nerve of a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> precisely if all <a class="existingWikiWord" href="/nlab/show/inner+horn">*inner* horns</a> have <em>unique</em> fillers.</p> </div> </p> <p>(e.g. <a class="existingWikiWord" href="/nlab/show/Kerodon">Kerodon</a>, <a href="https://kerodon.net/tag/0031">Prop. 1.2.3.1</a>; see also at <em><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></em>.)</p> <p> <div class='num_prop' id='sSetIsNerveOfGroupoidIffAllHornsHaveUniqueFillers'> <h6>Proposition</h6> <p><br /> A <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is the nerve of a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> precisely if <em>all</em> <a class="existingWikiWord" href="/nlab/show/horns">horns</a> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\gt 1</annotation></semantics></math> have <em>unique</em> fillers.</p> </div> </p> <p>(cf. e.g. <a class="existingWikiWord" href="/nlab/show/Kerodon">Kerodon</a>, <a href="https://kerodon.net/tag/0037">Prop. 1.2.4.2</a>)</p> <p>Here the point as compared to the previous statements is that in particular all the outer horns have fillers for <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 \gt 3</annotation></semantics></math>.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The 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> of a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> precisely 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 <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>.</p> </div> <p>The existence of <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> corresponds to the fact that in the <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C)</annotation></semantics></math> the “outer” <a class="existingWikiWord" href="/nlab/show/horns">horns</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><msub><mi>d</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>d</mi> <mn>1</mn></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><mi>and</mi><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><msub><mi>d</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>f</mi></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>d</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && d_0 \\ & && \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_1 } </annotation></semantics></math></div> <p>have fillers</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><msub><mi>d</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>d</mi> <mn>1</mn></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><mi>and</mi><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><msub><mi>d</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>f</mi></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>d</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && d_0 \\ & {}^{f^{-1}}\nearrow&& \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \searrow^{f^{-1}} \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 } </annotation></semantics></math></div> <p>(even unique fillers, due to the above).</p> <p>It suggests the sense that a Kan complex models an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>. The possible lack of uniqueness of fillers in general gives the ‘weakness’ needed, whilst the lack of a <a class="existingWikiWord" href="/nlab/show/coskeletal">coskeletal</a> property requirement means that the homotopy type it represents has enough generality, not being constrained to be a 1-type.</p> <p> <div class='num_prop' id='NerveOfCategoriesIsFullyFaithfulFunctor'> <h6>Proposition</h6> <p>The nerve functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo lspace="verythinmathspace">:</mo><mi>Cat</mi><mo>⟶</mo><mi>SSet</mi></mrow><annotation encoding="application/x-tex"> N \colon Cat \longrightarrow SSet </annotation></semantics></math> (on <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a>) is a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a>.</p> </div> </p> <p>(e.g <a class="existingWikiWord" href="/nlab/show/Kerodon">Kerodon</a>, <a href="https://kerodon.net/tag/002Z">Prop. 1.2.2.1</a>; <a href="#Rezk22">Rezk 2022, Prop. 4.10</a>)</p> <p>So <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small categories</a> are in <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> with morphisms of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> between their nerves.</p> <p> <div class='num_prop' id='NerveOfCategoriesPreservesFiniteProducts'> <h6>Proposition</h6> <p>The nerve functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo lspace="verythinmathspace">:</mo><mi>Cat</mi><mo>⟶</mo><mi>SSet</mi></mrow><annotation encoding="application/x-tex"> N \colon Cat \longrightarrow SSet </annotation></semantics></math> (on <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a>) <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a>, in that it sends:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a> to the terminal simplicial set,</p> </li> <li> <p>any <a class="existingWikiWord" href="/nlab/show/product+category">product category</a> to the <a class="existingWikiWord" href="/nlab/show/product+of+simplicial+sets">product of simplicial sets</a> of the nerves of the factors:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>×</mo><mi>𝒟</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>×</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N \;\colon\; \mathcal{C} \times \mathcal{D} \;\mapsto\; N(\mathcal{C}) \times N(\mathcal{D}) </annotation></semantics></math></div></li> </ol> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>By direct inspection, using that the <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in a <a class="existingWikiWord" href="/nlab/show/product+category">product category</a> are just <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> of morphisms of the two factor categories.</p> </div> </p> <p> <div class='num_prop' id='NerveOfCategoriesPreservesMappingObjects'> <h6>Proposition</h6> <p>The nerve functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo lspace="verythinmathspace">:</mo><mi>Cat</mi><mo>⟶</mo><mi>SSet</mi></mrow><annotation encoding="application/x-tex"> N \colon Cat \longrightarrow SSet </annotation></semantics></math> sends <a class="existingWikiWord" href="/nlab/show/functor+categories">functor categories</a> to the <a class="existingWikiWord" href="/nlab/show/function+complexes">function complexes</a> between the separate nerves:</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 maxsize="1.2em" minsize="1.2em">(</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>𝒳</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>𝒜</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒳</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N \big( Maps(\mathcal{X},\,\mathcal{A}) \big) \;\simeq\; Maps \big( N(\mathcal{X}) ,\, N(\mathcal{A}) \big) \,. </annotation></semantics></math></div> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> we have the following sequence of <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><msub><mo maxsize="1.8em" minsize="1.8em">)</mo> <mi>n</mi></msub></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>Cat</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒞</mi><mo>×</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>𝒟</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>×</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>×</mo><mi>N</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>n</mi></msub></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{l} \Big( N \big( Maps(\mathcal{C}, \mathcal{D}) \big) \Big)_n \\ \;\simeq\; Hom_{Cat} \big( \mathcal{C} \times [n] ,\, \mathcal{D} \big) \\ \;\simeq\; Hom_{sSet} \big( N(\mathcal{C} \times [n]) ,\, N(\mathcal{D}) \big) \\ \;\simeq\; Hom_{sSet} \big( N(\mathcal{C}) \times N([n]) ,\, N(\mathcal{D}) \big) \\ \;\simeq\; Hom_{sSet} \big( N(\mathcal{C}) \times \Delta[n] ,\, N(\mathcal{D}) \big) \\ \;\simeq\; Maps \big( N(\mathcal{C}) ,\, N(\mathcal{D}) \big)_n \end{array} </annotation></semantics></math></div> <p>Here</p> <ul> <li> <p>the first step follows as discussed at <em><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></em> (<a href="natural+transformation#InTermsOfCartMon">here</a>);</p> </li> <li> <p>the second step follow by Prop. <a class="maruku-ref" href="#NerveOfCategoriesIsFullyFaithfulFunctor"></a>;</p> </li> <li> <p>the third step follows by Prop. <a class="maruku-ref" href="#NerveOfCategoriesPreservesFiniteProducts"></a>.</p> </li> </ul> <p></p> </div> </p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is the nerve of a locally small category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> precisely if it satisfies the <a class="existingWikiWord" href="/nlab/show/Segal+conditions">Segal conditions</a>: precisely if all the <a class="existingWikiWord" href="/nlab/show/commuting+squares">commuting squares</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><msub><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>⋯</mi><mo>∘</mo><msub><mi>d</mi> <mn>0</mn></msub><mo>∘</mo><msub><mi>d</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msub><mi>S</mi> <mi>m</mi></msub></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mo></mo><mrow><mi>⋯</mi><msub><mi>d</mi> <mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub></mrow></msup></mrow></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>S</mi> <mi>n</mi></msub></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo>∘</mo><mi>⋯</mi><msub><mi>d</mi> <mn>0</mn></msub></mrow></munder></mtd> <mtd><msub><mi>S</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S_{n+m} & \overset {\cdots \circ d_0 \circ d_0} {\longrightarrow} & S_m \\ \mathllap{ ^{ \cdots d_{n+m-1}\circ d_{n+m} } } \big\downarrow && \big\downarrow \\ S_n &\underset{d_0 \circ \cdots d_0}{\longrightarrow}& S_0 } </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagrams.</p> </div> <p>Unwrapping this definition inductively 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>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+m)</annotation></semantics></math>, this says that a simplicial set is the nerve of a category if and only if all its cells in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\geq 2</annotation></semantics></math> are unique compositors, associators, pentagonators, etc of composition of 1-morphisms. No non-trivial such structure cells appear and no further higher cells appear.</p> <p>This characterization of categories in terms of nerves directly leads to the model of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> in terms of <a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces">complete Segal spaces</a> by replacing in the above discussion sets by <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (or something similar, like <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>) and pullbacks by <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullbacks</a>.</p> <p> <div class='num_prop' id='NerveOfACategoryIsTwoCoskeletal'> <h6>Proposition</h6> <p>The 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> of a category is <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletal</a>.</p> </div> (e.g. <a href="monadic+cohomology#Duskin75">Duskin 1975, §0.18(b)</a>, <a href="quasi-category#Joyal08">Joyal 2008, Cor. 1.2</a>)</p> <p>Hence in the nerve of a category, all <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>i</mi></msub><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda[n]_i \hookrightarrow \Delta[n]</annotation></semantics></math> have unique fillers for <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 \gt 3</annotation></semantics></math>, and all <a class="existingWikiWord" href="/nlab/show/boundary+of+a+simplex">boundary</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[n] \hookrightarrow \Delta[n]</annotation></semantics></math> have unique fillers for <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 \geq 3</annotation></semantics></math>.</p> <p>In summary:</p> <p> <div class='num_remark' id='CoskeletalityOfSimplicialNervesOfCategories'> <h6>Example</h6> <p><strong>(coskeletality of simplicial nerves of categories)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/simplicial+nerve">simplicial nerve</a> of a <a class="existingWikiWord" href="/nlab/show/category">category</a> (i.e. of a <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a>) is a <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletal</a> <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> (Prop. <a class="maruku-ref" href="#NerveOfACategoryIsTwoCoskeletal"></a>): The unique filler of the <a class="existingWikiWord" href="/nlab/show/boundary+of+a+simplex">boundary of an <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 \geq 3</annotation> </semantics> </math>-simplex</a> encodes the <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a>-condition on <a class="existingWikiWord" href="/nlab/show/n-tuple"><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>-tuples</a> of composable morphisms.</p> <p>Of course there is more to a category than its associativity condition, and hence the converse fails: Not every <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletal</a> <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is the <a class="existingWikiWord" href="/nlab/show/nerve+of+a+category">nerve of a category</a>. For example the <a class="existingWikiWord" href="/nlab/show/boundary+of+a+simplex">boundary of the 2-simplex</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><msup><mi>Δ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\partial \Delta^2</annotation></semantics></math>, is 2-coskeletal but not the nerve of a category, since it is missing a <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of the edges <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>, namely it is missing a filler of this <a class="existingWikiWord" href="/nlab/show/inner+horn">inner horn</a>.</p> <p>In fact, a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a <a class="existingWikiWord" href="/nlab/show/category">category</a> iff it has <em>unique</em> <a class="existingWikiWord" href="/nlab/show/inner+horn">inner <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>-horn</a>-fillers 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 \geq 2</annotation></semantics></math> (Prop. <a class="maruku-ref" href="#sSetIsNerveOfCategoryIffAllInnerHornsHaveUniqueFillers"></a>). But <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletality</a> already implies that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">k \geq 4</annotation></semantics></math>-horns have unique filles (first uniquely fill the missing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math>-face then the interior <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>)-cell. Together this implies that:</p> <p>A <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a <a class="existingWikiWord" href="/nlab/show/category">category</a> iff</p> <ol> <li> <p>it is <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletal</a>,</p> </li> <li> <p>all <em><a class="existingWikiWord" href="/nlab/show/inner+horn">inner</a></em> 2- and 3-<a class="existingWikiWord" href="/nlab/show/horns">horns</a> have unique fillers (encoding <a class="existingWikiWord" href="/nlab/show/composition">composition</a> and <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a>).</p> </li> </ol> <p>Similarly for <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> (byProp. <a class="maruku-ref" href="#sSetIsNerveOfGroupoidIffAllHornsHaveUniqueFillers"></a>):</p> <p>A <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> iff</p> <ol> <li> <p>it is <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletal</a>,</p> </li> <li> <p>all 2- and 3-<a class="existingWikiWord" href="/nlab/show/horns">horns</a> have unique fillers.</p> </li> </ol> <p>For better or worse, such a simplicial set has at times also been called a <em><a class="existingWikiWord" href="/nlab/show/1-hypergroupoid">1-hypergroupoid</a></em>, pointing to the fact that this is the first non-trivial stage in a pattern that recognizes <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>-coskeletal <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> with unique horn fillers as models for <a class="existingWikiWord" href="/nlab/show/n-groupoids"><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>-groupoids</a></p> <p>Notice that a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> which is <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletal</a> but with possibly non-unique 2-horn fillers is still a <a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a> and may still be called a <em><a class="existingWikiWord" href="/nlab/show/1-groupoid">1-groupoid</a></em> in the sense of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, but need not be the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>. It may be thought of as a <a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a> (2-<a class="existingWikiWord" href="/nlab/show/hypergroupoid">hypergroupoid</a>) which happens to be just a <a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a>.</p> </div> </p> <p><br /></p> <h3 id="nerve_of_a_2category">Nerve of a 2-category</h3> <p>For <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a> modeled as <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a> the nerve operation is called the <a class="existingWikiWord" href="/nlab/show/Duskin+nerve">Duskin nerve</a>.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>A simplicial set is the <a class="existingWikiWord" href="/nlab/show/Duskin+nerve">Duskin nerve</a> of a <a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a> precisely if it is a 2-<a class="existingWikiWord" href="/nlab/show/hypergroupoid">hypergroupoid</a>: a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> such that the horn fillers in dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\geq 3</annotation></semantics></math> are <em>unique</em>.</p> </div> <p>This is theorem 8.6 in (<a href="http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html">Duskin</a>)</p> <p>For a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-<a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a> one can apply the nerve operation for categories in stages, to obtain the <a class="existingWikiWord" href="/nlab/show/double+nerve">double nerve</a>.</p> <h3 id="nerve_of_a_3category">Nerve of a 3-category</h3> <p>One also has a nerve operation for <a class="existingWikiWord" href="/nlab/show/3-categories">3-categories</a> modeled as <a class="existingWikiWord" href="/nlab/show/tricategories">tricategories</a>: the <a class="existingWikiWord" href="/nlab/show/Street+nerve">Street nerve</a>.</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>A simplicial set is the <a class="existingWikiWord" href="/nlab/show/Street+nerve">Street nerve</a> of a <span class="newWikiWord">trigroupoid<a href="/nlab/new/trigroupoid">?</a></span> precisely if it is a 3-<a class="existingWikiWord" href="/nlab/show/hypergroupoid">hypergroupoid</a>: a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> such that the horn fillers in dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\geq 4</annotation></semantics></math> are <em>unique</em>.</p> </div> <p>This is the main result of (<a href="#Carrasco2014">Carrasco, 2014</a>).</p> <h3 id="nerve_of_an_category">Nerve of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-category</h3> <ul> <li>For <a class="existingWikiWord" href="/nlab/show/strict+omega-category">strict omega-categories</a> there is a nerve induced by the <a class="existingWikiWord" href="/nlab/show/orientals">orientals</a>; see <a class="existingWikiWord" href="/nlab/show/omega-nerve">omega-nerve</a>.</li> </ul> <h3 id="nerve_of_chain_complexes">Nerve of chain complexes</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">Ch_+</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> of abelian groups, then there is a <a class="existingWikiWord" href="/nlab/show/simplicial+object">cosimplicial chain complex</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo>:</mo><mi>Δ</mi><mo>→</mo><msub><mi>Ch</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex"> C_\bullet : \Delta \to Ch_+ </annotation></semantics></math></div> <p>given by sending the standard <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 <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> first to the free <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(\Delta[n])</annotation></semantics></math> over it and then that to the normalized <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>. This is a small version of the ordinary <a class="existingWikiWord" href="/nlab/show/homology">homology</a> <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of the standard <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>.</p> <p>The nerve induced by this cosimplicial object was first considered in</p> <ul> <li>D. Kan, <em>Functors involving c.s.s complexes</em>, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330–346 (<a href="http://www.jstor.org/stable/1993103">jstor</a>)</li> </ul> <p>The nerve/<a class="existingWikiWord" href="/nlab/show/geometric+realization">realization</a> adjunction induced from this is the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>. See there for more details.</p> <h2 id="remarks">Remarks</h2> <h3 id="geometric_realization">Geometric realization</h3> <p>Often the operation of taking the nerve of a (higher) category is followed by forming the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of the corresponding cellular set.</p> <h3 id="nerves_and_higher_categories">Nerves and higher categories</h3> <p>For many purposes it is convenient to conceive categories and especially <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-categories">∞-categories</a> entirely in terms of their nerves: those simplicial sets that arise as certain nerves are usually characterized by certain properties. So one can turn this around and <em>define</em> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a> as a simplicial set with certain properties. This is the strategy of a <a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a>. Examples for this are <a class="existingWikiWord" href="/nlab/show/complicial+set">complicial sets</a>, <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complexes</a>, <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complexes</a>,…</p> <h3 id="internal_nerve">Internal nerve</h3> <p>A variant of the nerve construction can also be applied <em>internally</em> within a category, to any internal category, see the discussion at <a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a>.</p> <h3 id="direct_categories_versus_finite_simplicial_sets">Direct categories versus finite simplicial sets</h3> <p>If a <a class="existingWikiWord" href="/nlab/show/direct+category">direct category</a> has finitely many objects then its nerve is a finite <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>. Conversely, if a finite simplicial set is the nerve of a category then the category is a direct category with finitely many objects.</p> <h2 id="properties_2">Properties</h2> <h3 id="PreservationOfColimits">(Non-)Preservation of colimits</h3> <p>While the nerve operation is a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> (<a href="nerve+and+realization#NerveAndRealizationAreAdjoint">this Prop.</a>) and <a class="existingWikiWord" href="/nlab/show/right+adjoints+preserve+limits">hence</a> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> all <a class="existingWikiWord" href="/nlab/show/limits">limits</a>, the nerve operation does not <a class="existingWikiWord" href="/nlab/show/preserved+colimit">preserve</a> all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> (Exp. <a class="maruku-ref" href="#NervesDoNotPreserveQuotientOfDeloopingByNormalSubgroup"></a>), <a class="existingWikiWord" href="/nlab/show/left+adjoints+preserve+colimits">hence</a> is not a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>.</p> <p>However, it does preserve <em>some</em> colimits (Exp. <a class="maruku-ref" href="#NervePreservesLeftQuotientsOnRightActionGroupoids"></a>); rather special ones, but of central importance in the theory of <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> constructed via <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization of simplicial topological spaces</a> (Exp. <a class="maruku-ref" href="#NerveDoesPreserveQuotientOfPairGroupoidOfGroupByGroupAction"></a>).</p> <p>(In the following Exp. <a class="maruku-ref" href="#NervesDoNotPreserveQuotientOfDeloopingByNormalSubgroup"></a> we use “card” instead of the more common notation “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert - \vert}</annotation></semantics></math>” for <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> (of <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/sets">sets</a>) in order not to clash with the notation for <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, even if the latter is not directly involved in the following examples.)</p> <p> <div class='num_remark' id='NervesDoNotPreserveQuotientOfDeloopingByNormalSubgroup'> <h6>Example</h6> <p><strong>(Nerve does not preserve quotients of delooping groupoids by normal subgroups)</strong> <br /> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi><mo>↠</mo><mi>G</mi><mo stretchy="false">/</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G \twoheadrightarrow G/N</annotation></semantics></math> be the inclusion of a <a class="existingWikiWord" href="/nlab/show/trivial+group">non-trivial</a> <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, with its <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/cardinalities">cardinalities</a> of 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>th component sets of 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>N</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Grpd</mi><mo>⟶</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> N \;\colon\; Grpd \longrightarrow sSet </annotation></semantics></math></div> <p>of their <a class="existingWikiWord" href="/nlab/show/delooping+groupoids">delooping groupoids</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Grp</mi><mo>⟶</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}(-) \;\colon\; Grp \longrightarrow Grpd </annotation></semantics></math></div> <p>satisfy, from degree <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 \geq 2</annotation></semantics></math> on, an <a class="existingWikiWord" href="/nlab/show/inequality+relation">inequality relation</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><mn>2</mn><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><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><mi>card</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>n</mi></msub><mo stretchy="false">/</mo><mi>H</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mfrac><mrow><mi>card</mi><mo stretchy="false">(</mo><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup></mrow><mrow><mi>card</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>></mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mfrac><mrow><mi>card</mi><mo stretchy="false">(</mo><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup></mrow><mrow><mi>card</mi><mo stretchy="false">(</mo><mi>H</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup></mrow></mfrac><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>card</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>n</mi></msub><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> n \geq 2 \;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\; card \Big( \big( N \mathbf{B} G \big)_n / H \Big) \;=\; \frac{ card(G)^n } { card(H) } \;\; \gt \;\; \frac{ card(G)^n } { card(H)^n } \;=\; card \Big( \big( N \mathbf{B} (G/H) \big)_n \Big) \,. </annotation></semantics></math></div> <p>But this means that it is impossible for there to be an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> (namely a degree-wise <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>) from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">N(\mathbf{B}G)/H</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">N\big(\mathbf{B}(G/H)\big)</annotation></semantics></math>, and hence that it is impossible for the nerve operation to preserve the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> which is the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/group+action">action</a>.</p> </div> </p> <p> <div class='num_remark' id='NerveDoesPreserveQuotientOfPairGroupoidOfGroupByGroupAction'> <h6>Example</h6> <p><strong>(nerve does preserve canonical quotients of chaotic groupoids of groups)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>Grp</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \,\in\, Grp(Set)</annotation></semantics></math> a (<a class="existingWikiWord" href="/nlab/show/discrete+group">discrete</a>) <a class="existingWikiWord" href="/nlab/show/group">group</a>, write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo>⇉</mo><mo>*</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \;\coloneqq\; \big( G \rightrightarrows \ast\big)</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/delooping+groupoid">delooping groupoid</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>⇉</mo><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{E}G \;\coloneqq\; \big( G \times G \rightrightarrows G \big)</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/pair+groupoid">pair groupoid</a> equipped with the usual left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/group+action">action</a> (discussed <a href="codiscrete+groupoid#UniversalGPrincipalBundle">there</a>),</p> </li> </ul> <p>so that the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <a class="existingWikiWord" href="/nlab/show/coprojection">coprojection</a> of this action is</p> <div class="maruku-equation" id="eq:QuotientCoprojectionFromPairGroupoidOfGroupToDeloopingGroupoid"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{E}G \xrightarrow{\;\;} (\mathbf{E}G)/G \;=\; \mathbf{B}G \,. </annotation></semantics></math></div> <p>Noticing that 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><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}G</annotation></semantics></math> (which is the <a class="existingWikiWord" href="/nlab/show/universal+principal+simplicial+complex">universal principal simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>W</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">N(\mathbf{E}G) \,=\, W G</annotation></semantics></math>) has component sets</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><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">{</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>g</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>G</mi> <mrow><msub><mo>×</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msup><mo maxsize="1.2em" minsize="1.2em">}</mo></mrow><annotation encoding="application/x-tex"> N(\mathbf{E}G)_n \;=\; \big\{ (g_n, g_{n-1}, \cdots, g_0) \;\in\; G^{\times_{n+1}} \big\} </annotation></semantics></math></div> <p>with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> action given degreewise by left-multiplication on <em>just the leftmost factor</em> (see also <a href="simplicial+classifying+space#SimplicialClassifyingSpaceOfAnOrdinaryGroup">this exp.</a>), we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><mi>G</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">{</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>g</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>G</mi> <mrow><msub><mo>×</mo> <mrow><mi>n</mi><mo>+</mo></mrow></msub></mrow></msup><mo maxsize="1.2em" minsize="1.2em">}</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \big( N(\mathbf{E}G)_n \big)/G \;\simeq\; \big\{ (g_{n-1}, \cdots, g_0) \;\in\; G^{\times_{n+}} \big\} \;=\; N(\mathbf{B}G)_n </annotation></semantics></math></div> <p>and hence here the nerve operation does preserve the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <a class="existingWikiWord" href="/nlab/show/coprojection">coprojection</a> <a class="maruku-eqref" href="#eq:QuotientCoprojectionFromPairGroupoidOfGroupToDeloopingGroupoid">(1)</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo>≃</mo><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> W G \;=\; N(\mathbf{E}G) \xrightarrow{\;\;} \big(N(\mathbf{E}G)\big)/G \simeq N\big( (\mathbf{E}G)/G \big) \;=\; N\big( \mathbf{B}G \big) \;=\; \overline{W} G \,. </annotation></semantics></math></div> <p>The result is the <a class="existingWikiWord" href="/nlab/show/universal+simplicial+principal+bundle">universal simplicial principal bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>Grp</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo><mover><mo>↪</mo><mrow><mi>Grp</mi><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">)</mo></mrow></mover><mi>Grp</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \,\in\, Grp(Set) \xhookrightarrow{Grp(Disc)} Grp(sSet)</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>.</p> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>The joint relevance of Exp. <a class="maruku-ref" href="#NervesDoNotPreserveQuotientOfDeloopingByNormalSubgroup"></a> and Exp. <a class="maruku-ref" href="#NerveDoesPreserveQuotientOfPairGroupoidOfGroupByGroupAction"></a> has been highlighted in <a href="equivariant+bundle#GuillouMayMerling17">Guillou, May & Merling 2017</a> (corresponding there to Exp. 2.9 and Lem. 2.10 – but Exp. 2.9 seems a little broken (?) while Lem. 2.10 does not quite get around to discussing the quotienting, for which it seems to be quoted later on).</p> </div> </p> <p>The principle behind Exp. <a class="maruku-ref" href="#NerveDoesPreserveQuotientOfPairGroupoidOfGroupByGroupAction"></a> is readily seen to be, more generally, the following:</p> <p> <div class='num_remark' id='NervePreservesLeftQuotientsOnRightActionGroupoids'> <h6>Example</h6> <p><strong>(nerve preserves left quotients of right action groupoids)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>L</mi></msub><mo>,</mo><msub><mi>G</mi> <mi>R</mi></msub><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>Grp</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_L, G_R \,\in\, Grp(Set)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mo stretchy="false">(</mo><msub><mi>G</mi> <mi>L</mi></msub><mo>×</mo><msubsup><mi>G</mi> <mi>R</mi> <mi>op</mi></msubsup><mo stretchy="false">)</mo><mi>Act</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in (G_L \times G^{op}_R) Act(Set)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/set">set</a> equipped with a <a class="existingWikiWord" href="/nlab/show/group+action">left action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">G_L</annotation></semantics></math> and a commuting right action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">G_R</annotation></semantics></math>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> of the right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">G_R</annotation></semantics></math>-action inherits the residual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">G_L</annotation></semantics></math>-action</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>×</mo><msub><mi>G</mi> <mi>R</mi></msub><munderover><mo>⇉</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></munderover><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>G</mi> <mi>L</mi></msub><mi>Act</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Grpd</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \big( X \times G_R \underoverset {(-)\cdot(-)} {pr_1} {\rightrightarrows} X \big) \;\; \in \;\; G_L Act\big( Grpd \big) </annotation></semantics></math></div> <p>and the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> by this left action is preserved by the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> operation:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>×</mo><msub><mi>G</mi> <mi>R</mi></msub><munderover><mo>⇉</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></munderover><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mo stretchy="false">/</mo><msub><mi>G</mi> <mi>L</mi></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>×</mo><msub><mi>G</mi> <mi>R</mi></msub><munderover><mo>⇉</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></munderover><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><msub><mi>G</mi> <mi>L</mi></msub><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Big( N \big( X \times G_R \underoverset {(-)\cdot(-)} {pr_1} {\rightrightarrows} X \big) \Big) /G_L \;\simeq\; N \Big( \big( X \times G_R \underoverset {(-)\cdot(-)} {pr_1} {\rightrightarrows} X \big)/ G_L \Big) \,. </annotation></semantics></math></div> <p></p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Barratt+nerve">Barratt 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/Street+nerve">Street nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-nerve">∞-nerve</a></p> </li> <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/dg-nerve">dg-nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad+with+arities">monad with arities</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="for_covers">For covers</h3> <p>The notion of the nerve of a <a class="existingWikiWord" href="/nlab/show/cover">cover</a> (in modern parlance: of its <a class="existingWikiWord" href="/nlab/show/Cech+groupoid">Cech groupoid</a>) appears in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Alexandroff">Paul Alexandroff</a>, Section 9 of: <em>Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung</em>, Mathematische Annalen 98 (1928), 617–635 (<a href="https://doi.org/10.1007/BF01451612">doi:10.1007/BF01451612</a>).</li> </ul> <h3 id="for_categories">For categories</h3> <p>The notion of the nerve of a general category already appears in</p> <ul> <li id="Grothendieck61"><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, above Proposition 4.1 of: <em>Techniques de construction et théorèmes d’existence en géométrie algébrique III : préschémas quotients</em>, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (<a href="http://www.numdam.org/item/?id=SB_1960-1961__6__99_0">numdam:SB_1960-1961__6__99_0</a>, <a href="http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf">pdf</a>)</li> </ul> <p>Another early appearance in print is:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, Section 2 of: <em>Classifying spaces and spectral sequences</em>, Publications Mathématiques de l’IHÉS, Volume 34 (1968), p. 105-112 (<a href="http://www.numdam.org/item/PMIHES_1968__34__105_0/">numdam:PMIHES_1968__34__105_0</a>)</p> <blockquote> <p>(in the context of constructing <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> for <a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>)</p> </blockquote> </li> </ul> <p>Review and exposition:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §XII.2 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kerodon">Kerodon</a>, <em>The Nerve of a Category</em> (<a href="https://kerodon.net/tag/002L">002L</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kerodon">Kerodon</a>, <em>The Nerve of a Groupoid</em> (<a href="https://kerodon.net/tag/0035">0035</a>)</p> </li> <li id="Rezk22"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, Part 1 of: <em>Introduction to quasicategories</em> (2022) [<a href="https://faculty.math.illinois.edu/~rezk/quasicats.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Rezk-IntroToQuasicategories.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, p. 117 onwards in: <em>Higher operads, higher categories</em> , London Mathematical Society Lecture Note Series, 298. Cambridge Univ. Press 2004. xiv+433 pp. ISBN: 0-521-53215-9 (<a href="https://arxiv.org/abs/math/0305049">arXiv:math.CT/0305049</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em><a href="https://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html">How I learned to love the nerve construction</a></em>, <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 Café, January 6, 2008.</p> <blockquote> <p>(an explanation of how the simplex category and the nerve construction arise canonically from the free category monad)</p> </blockquote> </li> </ul> <p>See also:</p> <ul> <li id="Isbell60"> <p><a class="existingWikiWord" href="/nlab/show/John+Isbell">John Isbell</a>, <em>Adequate subcategories</em>, Illinois J. Math. 4, 541–552 (1960) (<a href="https://www.projecteuclid.org/journals/illinois-journal-of-mathematics/volume-4/issue-4/Adequate-subcategories/10.1215/ijm/1255456274.full">doi:10.1215/ijm/1255456274</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W.+G.+Dwyer">W. G. Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/D.+M.+Kan">D. M. Kan</a>, Singular functors and realization functors, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 2, 147–153. <a href="http://www.nd.edu/~wgd/Dvi/SingularAndRealization.pdf">pdf</a></p> </li> </ul> <h3 id="for_higher_categories">For higher categories</h3> <p>For <a class="existingWikiWord" href="/nlab/show/strict+omega-categories">strict omega-categories</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>The algebra of oriented simplexes</em>, J. Pure Appl. Algebra 49 (1987) 283-335; MR89a:18019 (<a href="http://www.math.mq.edu.au/~street/aos.pdf">pdf</a>, <a href="https://doi.org/10.1016/0022-4049(87)90137-X">doi:10.1016/0022-4049(87)90137-X</a>).</li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a> and <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a>:</p> <ul> <li id="BullejosCegarra03"> <p><a class="existingWikiWord" href="/nlab/show/Manuel+Bullejos">Manuel Bullejos</a>, <a class="existingWikiWord" href="/nlab/show/Antonio+M.+Cegarra">Antonio M. Cegarra</a>, <em>On the geometry of 2-categories and their classifying spaces</em>, K-Theory <strong>29</strong> 3 (2003) 211-229 [<a href="http://dx.doi.org/10.1023/B:KTHE.0000006921.50151.00">doi:10.1023/B:KTHE.0000006921.50151.00</a>, <a href="http://www.ugr.es/\%7Ebullejos/geometryampl.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>, <a class="existingWikiWord" href="/nlab/show/Simona+Paoli">Simona Paoli</a>, <em>2-nerves for bicategories</em>, K-Theory <strong>38</strong> (2008) [<a href="https://arxiv.org/abs/math/0607271">arXiv:math/0607271</a>, <a href="http://dx.doi.org/10.1007/s10977-007-9013-2">doi:10.1007/s10977-007-9013-2</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Bressler">Paul Bressler</a>, Alexander Gorokhovsky, <a class="existingWikiWord" href="/nlab/show/Ryszard+Nest">Ryszard Nest</a>, <a class="existingWikiWord" href="/nlab/show/Boris+Tsygan">Boris Tsygan</a>, <em>Formality for algebroids I: Nerves of two-groupoids</em> [<a href="https://arxiv.org/abs/1211.6603v3">arxiv:1211.6603v3</a>]</p> </li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/3-categories">3-categories</a>:</p> <ul> <li id="Carrasco2014">Pilar Carrasco, <em>Nerves of Trigroupoids as Duskin-Glenn’s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-Hypergroupoids</em>, Applied Categorical Structures 23.5 (2015): 673-707 (<a href="https://doi.org/10.1007/s10485-014-9374-7">doi:10.1007/s10485-014-9374-7</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 5, 2024 at 08:58:10. 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