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<p><strong>Basic structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/binary+linear+code">binary linear code</a></li> <li><a class="existingWikiWord" href="/nlab/show/chord+diagram">chord diagram</a></li> <li><a class="existingWikiWord" href="/nlab/show/combinatorial+design">combinatorial design</a></li> <li><a class="existingWikiWord" href="/nlab/show/graph">graph</a></li> <li><a class="existingWikiWord" href="/nlab/show/Latin+square">Latin square</a></li> <li><a class="existingWikiWord" href="/nlab/show/matroid">matroid</a></li> <li><a class="existingWikiWord" href="/nlab/show/partition">partition</a></li> <li><a class="existingWikiWord" href="/nlab/show/permutation">permutation</a></li> <li><a class="existingWikiWord" href="/nlab/show/shuffle">shuffle</a></li> <li><a class="existingWikiWord" href="/nlab/show/tree">tree</a></li> <li><a class="existingWikiWord" href="/nlab/show/Young+diagram">Young diagram</a></li> </ul> <p><strong>Generating functions</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/combinatorial+species">combinatorial species</a></li> <li><a class="existingWikiWord" href="/nlab/show/generating+function">generating function</a></li> <li><a class="existingWikiWord" href="/nlab/show/power+series">power series</a></li> </ul> <p><strong>Proof techniques</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/bijective+proof">bijective proof</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lagrange+inversion">Lagrange inversion</a></li> <li><a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+inversion">Möbius inversion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/order+polynomial">order polynomial</a></li> <li><a class="existingWikiWord" href="/nlab/show/zeta+polynomial">zeta polynomial</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/P%C3%B3lya+enumeration+theorem">Pólya enumeration theorem</a></li> </ul> <p><strong>Combinatorial identities</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/binomial+theorem">binomial theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/Catalan+number">Catalan number</a></li> <li><a class="existingWikiWord" href="/nlab/show/Chu%E2%80%93Vandermonde+identity">Chu–Vandermonde identity</a></li> </ul> <p><strong>Polytopes</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/associahedron">associahedron</a></li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/combinatorics">combinatorics</a></div></div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="content">Content</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <li><a href='#remarks'>Remarks</a></li> <ul> <li><a href='#simplicial_sets_as_spaces_built_of_simplices'>Simplicial sets as spaces built of simplices</a></li> <li><a href='#visualisation'>Visualisation</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#the_empty_simplicial_set'>The empty simplicial set</a></li> <li><a href='#simplices_yoneda_embeddings'><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>-simplices (Yoneda embeddings)</a></li> <li><a href='#the_minimal_simplicial_circle'>The minimal simplicial circle</a></li> <li><a href='#cartesian_products_of_simplices'>Cartesian products of simplices</a></li> <li><a href='#simplicial_complexes'>Simplicial complexes</a></li> <li><a href='#directed_graphs'>Directed graphs</a></li> <li><a href='#nerve_of_a_category'>Nerve of a category</a></li> <li><a href='#singular_simplicial_complex_of_a_topological_space'>Singular simplicial complex of a topological space</a></li> <li><a href='#bar_construction'>Bar construction</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#BasicProperties'>Basic properties</a></li> <li><a href='#classifying_topos'>Classifying topos</a></li> <ul> <li><a href='#simplicial_sets'>Simplicial sets</a></li> <li><a href='#cosimplicial_sets'>Cosimplicial sets</a></li> </ul> <li><a href='#as_models_in_homotopy_theory'>As models in homotopy theory</a></li> <li><a href='#relation_to_dendroidal_sets'>Relation to dendroidal sets</a></li> </ul> <li><a href='#variants'>Variants</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Simplicial sets generalize the idea of <a class="existingWikiWord" href="/nlab/show/simplicial+complexes">simplicial complexes</a>: a <em>simplicial set</em> is like a combinatorial space built up out of gluing abstract <a class="existingWikiWord" href="/nlab/show/simplex">simplices</a> to each other. Equivalently, it is an object equipped with a rule for how to consistently map the objects of the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> into it.</p> <p>More concretely, a simplicial set <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 collection 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>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_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>, so that elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math> are to be thought of as <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">simplices</a>, equipped with a rule that says:</p> <ul> <li>which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">S_{n-1}</annotation></semantics></math> are faces of which elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math>;</li> <li>which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-simplices are <a class="existingWikiWord" href="/nlab/show/thin+element">thin</a> in that they are really just <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>-simplices regarded as degenerate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-simplices.</li> </ul> <p>One of the main uses of simplicial sets is as combinatorial <em>models</em> for the (weak) <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. They can also be taken as models for <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>. This is encoded in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>. For more reasons why simplicial sets see MathOverflow <a href="http://mathoverflow.net/questions/58497/is-there-a-high-concept-explanation-for-why-simplicial-leads-to-homotopy-theor">here</a>.</p> <h2 id="Definition">Definition</h2> <p>The quick abstract definition of a simplicial set goes as follows:</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>simplicial set</strong> is a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on 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>, that is, a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Sets</mi></mrow><annotation encoding="application/x-tex">X : \Delta^{op} \to Sets</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> to the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a>; equivalently this a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>Equipped with the standard <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/presheaf">presheaves</a> as morphisms (namely <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformations</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/functors">functors</a>), simplicial sets form the category <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> (denoted both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SSet</mi></mrow><annotation encoding="application/x-tex">SSet</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>).</p> </div> <p>Explicitly this means the following.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p><strong>(simplicial set)</strong></p> <p>A <strong>simplicial set</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">X \in sSet</annotation></semantics></math> is</p> <ul> <li> <p>for each <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> a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">X_n \in Set</annotation></semantics></math> – the <strong>set 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></strong>;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/injective+map">injective map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><msub><mi>δ</mi> <mi>i</mi></msub><mo>:</mo><mspace width="mediummathspace"></mspace><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><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\delta_i :\: [n-1] \to [n]\;</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/totally+ordered+sets">totally ordered sets</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo><</mo><mn>1</mn><mo><</mo><mi>⋯</mi><mo><</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[n] \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}</annotation></semantics></math>)</p> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mi>i</mi></msub><mo>:</mo><mspace width="mediummathspace"></mspace><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;d_i :\: X_{n} \to X_{n-1}\;</annotation></semantics></math> – 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 <em><a class="existingWikiWord" href="/nlab/show/face+map">face map</a></em> on <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>-simplices (<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 \gt 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq i \leq n</annotation></semantics></math>);</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/surjective+map">surjective map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><msub><mi>σ</mi> <mi>i</mi></msub><mo>:</mo><mspace width="mediummathspace"></mspace><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><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\sigma_i :\: [n+1] \to [n]\;</annotation></semantics></math> of totally ordered sets</p> <p>a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><msub><mi>s</mi> <mi>i</mi></msub><mo>:</mo><mspace width="mediummathspace"></mspace><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;s_i :\: X_{n} \to X_{n+1}\;</annotation></semantics></math> – 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 <em><a class="existingWikiWord" href="/nlab/show/degeneracy+map">degeneracy map</a></em> on <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>-simplices (<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 \geq 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq i \leq n</annotation></semantics></math>);</p> </li> </ul> <p>such that these functions satisfy the <em><a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a></em>.</p> </div> <p id="SimplicialSetIdea"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <center> <img src="/nlab/files/SimplicialSetSchematics.jpg" width="800" /> </center> <h2 id="remarks">Remarks</h2> <h3 id="simplicial_sets_as_spaces_built_of_simplices">Simplicial sets as spaces built of simplices</h3> <ul> <li> <p>The definition is to be understood from the point of view of <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a>: a <strong>simplicial set</strong> is a space characterized by the fact that it may be <em>probed</em> by mapping standard simplices into it: the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math> assigned by a simplicial set to 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><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> is the <strong>set 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>-simplices</strong> in this space, hence the way of mapping a 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 into this space.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> a simplicial set, the <strong>face map</strong></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>i</mi></msub><mo>≔</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>δ</mi> <mi>i</mi></msup><mo stretchy="false">)</mo><mo>:</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> d_i \coloneqq S(\delta^i): S_n \rightarrow S_{n-1} </annotation></semantics></math></div> <p>is dual to the unique injection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>δ</mi> <mi>i</mi></msup><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^i : [n-1] \rightarrow [n]</annotation></semantics></math> in the category <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> whose image omits the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [n]</annotation></semantics></math>.</p> </li> <li> <p>Similarly, the <strong>degeneracy map</strong></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>i</mi></msub><mo>≔</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>σ</mi> <mi>i</mi></msup><mo stretchy="false">)</mo><mo>:</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> s_i \coloneqq S(\sigma^i) : S_n \rightarrow S_{n+1} </annotation></semantics></math></div> <p>is dual to the unique surjection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>σ</mi> <mi>i</mi></msup><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^i : [n+1] \rightarrow [n]</annotation></semantics></math> in <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [n]</annotation></semantics></math> has two elements in its preimage.</p> </li> <li> <p>The maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>δ</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\delta^i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>σ</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\sigma^i</annotation></semantics></math> satisfy certain obvious relations – the <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a> – dual to those spelled out at <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a>.</p> </li> </ul> <h3 id="visualisation">Visualisation</h3> <p>(based on <a class="existingWikiWord" href="/nlab/show/cubical+set">cubical set</a>)</p> <p>The <strong>face maps</strong> go from sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">S_{n+1}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-dimensional simplices to the corresponding set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_{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>-dimensional simplices and can be thought of as sending each simplex in the simplicial set to one of its faces, for instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math> the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S_2</annotation></semantics></math> of 2-simplices would be sent in three different ways by three different face maps to the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-simplices, for instance one of the face maps would send</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>F</mi></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>a</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>a</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ & & b \\ & \nearrow & \Downarrow^F & \searrow \\ a & & \rightarrow & & c } \right) \;\; \mapsto \;\; \left( \array{ & & b \\ & \nearrow \\ a } \right) </annotation></semantics></math></div> <p>another one would send</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>F</mi></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>a</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ & & b \\ & \nearrow & \Downarrow^F & \searrow \\ a & & \rightarrow & & c } \right) \;\; \mapsto \;\; \left( \array{ a & & \rightarrow & & c } \right) \,. </annotation></semantics></math></div> <p>On the other hand, the <strong>degeneracy maps</strong> go the other way round and send sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_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>-simplices to sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">S_{n+1}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-simplices by regarding an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex as a degenerate or “thin” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-simplex in the various different ways that this is possible. For instance, again for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math>, a degeneracy map may act by sending</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>b</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo>↗</mo> <mi>f</mi></msub></mtd> <mtd><msup><mo>⇓</mo> <mi>Id</mi></msup></mtd> <mtd><msup><mo>↘</mo> <mi>Id</mi></msup></mtd></mtr> <mtr><mtd><mi>a</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ a &\stackrel{f}{\to}& b } \right) \;\; \mapsto \;\; \left( \array{ & & b \\ & \nearrow_f & \Downarrow^{Id} & \searrow^{Id} \\ a & & \stackrel{f}\to & & b } \right) \,. </annotation></semantics></math></div> <p>Notice the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Id</mi></mrow><annotation encoding="application/x-tex">Id</annotation></semantics></math>-labels, which indicate that the edges and faces labeled by them are “<a class="existingWikiWord" href="/nlab/show/thin+element">thin</a>” in much the same way as an <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> is thin. They depend on lower dimensional features, (notice however that a simplicial set by itself is not equipped with any notion of composition of simplices, nor really, therefore, of identities. See <a class="existingWikiWord" href="/nlab/show/quasicategory">quasicategory</a> for a kind of simplicial set which does have such notions and <a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a> for more on the intuitions behind this idea of “thinness”).</p> <h2 id="Examples">Examples</h2> <h3 id="the_empty_simplicial_set">The empty simplicial set</h3> <p>The <a class="existingWikiWord" href="/nlab/show/empty+simplicial+set">empty simplicial set</a> is a simplicial set.</p> <h3 id="simplices_yoneda_embeddings"><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>-simplices (Yoneda embeddings)</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> denote the object of 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> corresponding to the totally ordered set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ 0, 1, 2,\ldots, n\}</annotation></semantics></math>. Then the represented presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(-, [n])</annotation></semantics></math>, typically written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math> is an example of a simplicial set. In particular we have <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><msub><mo stretchy="false">]</mo> <mi>m</mi></msub><mo>=</mo><msub><mi>Hom</mi> <mi>Δ</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta[n]_m=Hom_\Delta([m],[n])</annotation></semantics></math> and hence <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><msub><mo stretchy="false">]</mo> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\Delta[n]_m</annotation></semantics></math> is a finite set with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mi>n</mi></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\binom{n+m+1}{n}</annotation></semantics></math> elements.</p> <p>By the Yoneda lemma, 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>-simplices of a simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are in natural bijective correspondence to maps <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><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Delta[n] \rightarrow X</annotation></semantics></math> of simplicial sets.</p> <h3 id="the_minimal_simplicial_circle">The minimal simplicial circle</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/minimal+simplicial+circle">minimal simplicial circle</a></em>.</p> <h3 id="cartesian_products_of_simplices">Cartesian products of simplices</h3> <p>The non-degenerate <a class="existingWikiWord" href="/nlab/show/simplices">simplices</a> in the <a class="existingWikiWord" href="/nlab/show/Cartesian+product+of+simplices">Cartesian product of simplices</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \Delta[1] \times \Delta[2] </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/1-simplex">1-simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/2-simplex">2-simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[2]</annotation></semantics></math> (i.e. the canonical simplicial <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> over the <a class="existingWikiWord" href="/nlab/show/2-simplex">2-simplex</a>) is the simplicial set which looks as follows:</p> <p><img src="https://ncatlab.org/nlab/files/SimplicialCylinderOn2Simplex.jpg" width="240" /></p> <blockquote> <p>graphics grabbed from <a href="#Friedman08">Friedman 08, p. 33</a></p> </blockquote> <p>For more on this and related examples see at <em><a class="existingWikiWord" href="/nlab/show/product+of+simplices">product of simplices</a></em>.</p> <h3 id="simplicial_complexes">Simplicial complexes</h3> <p>Every <a class="existingWikiWord" href="/nlab/show/simplicial+complex">simplicial complex</a> can be viewed a simplicial set (in several different ways).</p> <p><img src="https://ncatlab.org/nlab/files/ASimplicialComplex.jpg" width="180" /></p> <blockquote> <p>graphics grabbed form <a href="https://arxiv.org/abs/1710.06129">arXiv:1710.06129</a></p> </blockquote> <p><img src="https://ncatlab.org/nlab/files/AnotherSimplicialComplex.jpg" width="400" /></p> <blockquote> <p>graphics grabbed from Maletic, 2013</p> </blockquote> <p>In particular any graph is thought of as being built of vertices and edges and so is a (1-dimensional) simplicial complex. Choosing a direction on the edges then gives a directed graph and that gives a simplicial set, as follows.</p> <h3 id="directed_graphs">Directed graphs</h3> <p>A <a class="existingWikiWord" href="/nlab/show/directed+graph">directed graph</a> (with loops and multiple edges allowed, i.e., a <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>⇉</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">E \rightrightarrows V</annotation></semantics></math> is essentially the same thing as a 1-dimensional simplicial set, by taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>0</mn></msub><mo>≔</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">S_0 \coloneqq V</annotation></semantics></math> to be the set of vertices and <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><mi>E</mi><mo>⊎</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">S_1 \coloneqq E \uplus V</annotation></semantics></math> to be the disjoint union of the set of edges with the set of vertices (the latter corresponding to the degenerate 1-simplices).</p> <h3 id="nerve_of_a_category">Nerve of a category</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a small category, 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> is a simplicial set which we denote <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>NC</mi></mrow><annotation encoding="application/x-tex">NC</annotation></semantics></math>. If we intepret the poset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> defined above as a category, we define 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>-simplices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>NC</mi></mrow><annotation encoding="application/x-tex">NC</annotation></semantics></math> to be the set of functors <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>C</mi></mrow><annotation encoding="application/x-tex">[n] \rightarrow C</annotation></semantics></math>. Equivalently, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-simplices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>NC</mi></mrow><annotation encoding="application/x-tex">NC</annotation></semantics></math> are the objects 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>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-simplices are the morphisms, and 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>-simplices are strings 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 arrows 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>. Face maps are given by composition (or omission, in the case of <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> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">d_n</annotation></semantics></math>) and degeneracy maps are given by inserting identity arrows.</p> <h3 id="singular_simplicial_complex_of_a_topological_space">Singular simplicial complex of a topological space</h3> <p>Recall from <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> or <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> the standard functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">\Delta \to Top</annotation></semantics></math> which sends <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>Δ</mi></mrow><annotation encoding="application/x-tex">[n] \in \Delta</annotation></semantics></math> to the standard topological <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><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math>. This functor induces for every <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>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the simplicial set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>X</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S X : [n] \mapsto Hom_{Top}(\Delta^n, X) </annotation></semantics></math></div> <p>called the <strong>simplicial singular complex</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. This simplicial set is always a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> and may be regarded as the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Following up on the idea of ‘’thinness’’, a singular simplex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f: \Delta^n \to X</annotation></semantics></math> may be called <strong>thin</strong> if it factors through a <a class="existingWikiWord" href="/nlab/show/retraction">retraction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><msubsup><mi>Λ</mi> <mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">r: \Delta^n \to \Lambda^{n-1}_i</annotation></semantics></math> to some horn of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math>, then the well known <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan</a> condition on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">S X</annotation></semantics></math> can be strengthened to say that every horn in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">S X</annotation></semantics></math> has a <em>thin</em> filler. This also helps to give some intuitive underpinning to the idea of <a class="existingWikiWord" href="/nlab/show/thin+element">thin element</a> in this simplicial context.</p> <h3 id="bar_construction">Bar construction</h3> <p>For the moment see <em><a class="existingWikiWord" href="/nlab/show/bar+construction">bar construction</a></em>.</p> <h2 id="properties">Properties</h2> <h3 id="BasicProperties">Basic properties</h3> <p> <div class='num_prop' id='SimplicialSetIsColimitOfItsSimplices'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of its <a class="existingWikiWord" href="/nlab/show/category+of+elements">elements</a>)</strong> <br /> Every simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">X \in sSet</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of its simplices, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>X</mi></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>el</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></munder><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>×</mo><msub><mi>X</mi> <mi>k</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} X & \;\simeq\; \underset{ \Delta[k] \in el(X) }{lim} \Delta[k] \\ & \;\simeq\; \int^{[k] \in \Delta} \Delta[k] \times X_k \end{aligned} </annotation></semantics></math></div> <p></p> </div> Here the first line shows a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> over the <a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (e.g. <a href="#GoerssJardine09">Goerss & Jardine, I, Lemma 2.1</a>) and the second line shows the corresponding <a class="existingWikiWord" href="/nlab/show/coend">coend</a>-expression (by <a href="end#CoendAsColimitOverCategoryOfElements">this Prop.</a>). <div class='proof'> <h6>Proof</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/SimplicialSets">SimplicialSets</a> is a <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> (over 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>) this is a special case of the general fact that every presheaf is a colimit of <a class="existingWikiWord" href="/nlab/show/representable+functor">representables</a> (<a href="presheaf#EveryPresheafIsColimitOfRepresentables">this Prop.</a>, the “<a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a>”).</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>In fact, the colimit in Prop. <a class="maruku-ref" href="#SimplicialSetIsColimitOfItsSimplices"></a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>, see <a href="homotopy+limit#SimplicialSetIsHomotopyColimitOverItself">there</a>.</p> </div> </p> <h3 id="classifying_topos">Classifying topos</h3> <h4 id="simplicial_sets">Simplicial sets</h4> <p>The category of simplicial sets is a <a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a>, and so in particular a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a>. In fact, it is the <a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a> of the theory of “intervals”, meaning <a class="existingWikiWord" href="/nlab/show/totally+ordered+sets">totally ordered sets</a> equipped with distinct top and bottom elements.</p> <p>Specifically, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a topos containing such an interval <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>, then we obtain a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\Delta \to E</annotation></semantics></math> sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> to the subobject</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><msub><mi>x</mi> <mn>1</mn></msub><mo>≤</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>≤</mo><mi>…</mi><mo>≤</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">}</mo><mo>↪</mo><msup><mi>I</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \{ (x_1,x_2,\dots,x_n) \;|\; x_1 \le x_2 \le \dots \le x_n \} \hookrightarrow I^n </annotation></semantics></math></div> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>/<a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>⇆</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">E \leftrightarrows Set^{\Delta^{op}}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> which classifies <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>. In particular, the generic such interval is the simplicial 1-simplex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^1</annotation></semantics></math>; see <a class="existingWikiWord" href="/nlab/show/generic+interval">generic interval</a> for more.</p> <p>The usual geometric realization into <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> cannot be obtained in this way precisely, since <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is not a topos. However, there are <a class="existingWikiWord" href="/nlab/show/Top">Top</a>-like categories which are toposes, such as <a class="existingWikiWord" href="/nlab/show/Johnstone%27s+topological+topos">Johnstone's topological topos</a>.</p> <h4 id="cosimplicial_sets">Cosimplicial sets</h4> <p>Similarly, also the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Set^{\Delta}</annotation></semantics></math> of cosimplicial sets is a classifying topos: for inhabited <a class="existingWikiWord" href="/nlab/show/linear+orders">linear orders</a>. See at <em><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></em> the section <em><a href="http://ncatlab.org/nlab/show/classifying+topos#ForLinearOrders">For (inhabited) linear orders</a></em>.</p> <h3 id="as_models_in_homotopy_theory">As models in homotopy theory</h3> <p>(…) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> (…) <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> (…) <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> (…)</p> <h3 id="relation_to_dendroidal_sets">Relation to dendroidal sets</h3> <p>For the moment see at <em><a class="existingWikiWord" href="/nlab/show/dendroidal+set">dendroidal set</a></em> the section <a href="#http://ncatlab.org/nlab/show/dendroidal+set#RelationToSimplicialSets">Relation to simplicial sets</a></p> <h2 id="variants">Variants</h2> <ul> <li>A <a class="existingWikiWord" href="/nlab/show/symmetric+set">symmetric set</a> is a simplicial set equipped with additional <em>transposition maps</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>t</mi> <mi>i</mi> <mi>n</mi></msubsup><mo>:</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">t^n_i: X_n \to X_n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i=0,\ldots,n-1</annotation></semantics></math>. These transition maps generate an <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math> and satisfy certain commutation relations with the face and degeneracy maps.</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+map">simplicial map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a></p> <ul> <li> <p><strong>simplicial set</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+simplicial+set">pointed simplicial set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+object+in+an+%28%E2%88%9E%2C1%29-category">simplicial object in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-simplicial+object">semi-simplicial object</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimplicial+set">semisimplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>, <a class="existingWikiWord" href="/nlab/show/reduced+simplicial+set">reduced simplicial set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bisimplicial+set">bisimplicial set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+set">symmetric set</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+set">cyclic set</a>, <a class="existingWikiWord" href="/nlab/show/skew-simplicial+set">skew-simplicial set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/globular+set">globular set</a>, <a class="existingWikiWord" href="/nlab/show/cubical+set">cubical set</a>, <a class="existingWikiWord" href="/nlab/show/cellular+set">cellular set</a>, <a class="existingWikiWord" href="/nlab/show/dendroidal+set">dendroidal set</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original definition:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Joseph+A.+Zilber">Joseph A. Zilber</a>, Section 8 in: <em>Semi-simplicial complexes and singular homology</em>, Annals of Mathematics 51:3 (1950), 499–513 (<a href="https://www.jstor.org/stable/1969364">jstor:1969364</a>)</li> </ul> <p>Further early discussions (aimed at <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Gabriel">Pierre Gabriel</a>, <a class="existingWikiWord" href="/nlab/show/Michel+Zisman">Michel Zisman</a>, Chapter II of <em><a class="existingWikiWord" href="/nlab/show/Calculus+of+fractions+and+homotopy+theory">Calculus of fractions and homotopy theory</a></em>, Ergebnisse der Mathematik und ihrer Grenzgebiete <strong>35</strong>, Springer (1967) [<a href="https://link.springer.com/book/10.1007/978-3-642-85844-4">doi:10.1007/978-3-642-85844-4</a>, <a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>Simplicial objects in algebraic topology</em>, University of Chicago Press 1967 (<a href="https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html">ISBN:9780226511818</a>, <a href="http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu">djvu</a>, <a class="existingWikiWord" href="/nlab/files/May_SimplicialObjectsInAlgebraicTopology.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+B.+Curtis">Edward B. Curtis</a>, <em>Simplicial homotopy theory</em>, Advances in Mathematics 6 (1971) 107–209 (<a href="https://doi.org/10.1016/0001-8708(71)90015-6">doi:10.1016/0001-8708(71)90015-6</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=279808">MR279808</a>)</p> </li> </ul> <p>Exposition and introduction:</p> <ul> <li id="Friedman08"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Friedman">Greg Friedman</a>, <em>An elementary illustrated introduction to simplicial sets</em>, Rocky Mountain J. Math. 42(2): 353-423 (2012) [<a href="http://arxiv.org/abs/0809.4221">arXiv:0809.4221</a>, <a href="https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-42/issue-2/Survey-Article-An-elementary-illustrated-introduction-to-simplicial-sets/10.1216/RMJ-2012-42-2-353.full">doi:10.1216/RMJ-2012-42-2-353</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em>A leisurely introduction to simplicial sets</em>, 2008, 14 pages (<a href="http://www.math.jhu.edu/~eriehl/ssets.pdf">pdf</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Sergeraert">Francis Sergeraert</a>, <em>Introduction to Combinatorial Homotopy Theory</em>, July 7, 2013, <a href="https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Trieste-Lecture-Notes.pdf">pdf</a>.</p> </li> <li> <p>Christian Rüschoff, <em>Simplicial Sets</em>, Lecture Notes 2017 (<a href="https://www.mathi.uni-heidelberg.de/~rueschoff/ss17sset/sset.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Rueschoff_SimplicialSets.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Further discussion in the context of <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Lamotke">Klaus Lamotke</a>, <em>Semisimpliziale algebraische Topologie</em>, Grundlehren der mathematischen Wissenschaften 147 (1968).</p> </li> <li id="GoerssJardine09"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/J.+F.+Jardine">J. F. Jardine</a>, Chapter I of: <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em>, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (<a href="https://link.springer.com/book/10.1007/978-3-0346-0189-4">doi:10.1007/978-3-0346-0189-4</a>, <a href="http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/">webpage</a>)</p> </li> <li id="Cisinski06"> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, Section 2 of: <em><a class="existingWikiWord" href="/joyalscatlab/published/Les+pr%C3%A9faisceaux+comme+type+d%27homotopie">Les préfaisceaux comme type d'homotopie</a></em>, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages (<a href="http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, <em>Notes on simplicial homotopy theory</em>, March 7, 2008, <a href="http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern47.pdf">pdf</a>.</p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/topology">topology</a></div><div class="property">category: <a class="category_link" href="/nlab/all_pages/simplicial+object">simplicial object</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on March 20, 2024 at 17:55:20. 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