View source for Simplicial complex

''simplicial scheme, abstract simplicial complex''

A set, whose elements are called vertices, in which a family of finite non-empty subsets, called simplexes or simplices, is distinguished, such that every non-empty subset of a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853601.png" /> is a simplex, called a face of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853602.png" />, and every one-element subset is a simplex.

A simplex is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853604.png" />-dimensional if it consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853605.png" /> vertices. The maximal dimension of its simplices (which may be infinite) is called the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853606.png" /> of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853607.png" />. A simplicial complex is called locally finite if each of its vertices belongs to only finitely many simplices. A simplicial complex is called ordered if its vertices admit a partial ordering that is linear on every simplex.

Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853608.png" /> be a set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853609.png" /> be a family of non-empty subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536010.png" />. A non-empty finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536011.png" /> is called a simplex if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536012.png" /> is non-empty. The resulting simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536013.png" /> is called the nerve of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536014.png" /> (cf. [[Nerve of a family of sets|Nerve of a family of sets]]).

A simplicial mapping of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536015.png" /> into a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536016.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536017.png" /> such that for every simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536019.png" />, its image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536020.png" /> is a simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536021.png" />. Simplicial complexes and their simplicial mappings form a category.

If a simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536022.png" /> is an inclusion, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536023.png" /> is called a simplicial subcomplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536024.png" />. All simplices of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536025.png" /> of dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536026.png" /> form a simplicial subcomplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536027.png" />, which is written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536028.png" /> and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536030.png" />-dimensional (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536031.png" />-) skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536033.png" />. A simplicial subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536034.png" /> of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536035.png" /> is called full if every simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536036.png" /> whose vertices all belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536037.png" /> is itself in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536038.png" />.

Every simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536039.png" /> canonically determines a [[Simplicial set|simplicial set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536040.png" />, whose simplices of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536041.png" /> are all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536042.png" />-tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536043.png" /> of vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536044.png" /> with the property that there is a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536046.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536047.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536048.png" />. The boundary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536049.png" /> and the degeneracy operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536051.png" /> are given by the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536052.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536053.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536054.png" /> denotes the omission of the symbol beneath it. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536055.png" /> is ordered one can define a simplicial subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536056.png" />, consisting of those simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536057.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536058.png" />. The (co)homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536059.png" /> are isomorphic to the (co)homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536060.png" /> and called the (co)homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536061.png" />.

To every triangulation ([[Simplicial space|simplicial space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536062.png" /> corresponds a simplicial complex, whose vertices are the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536063.png" /> and whose simplices are those non-empty finite sets of vertices which span a simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536064.png" />. For every simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536065.png" /> there is a triangulation, unique up to an isomorphism, whose simplicial complex is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536066.png" />. It is called the geometric realization (or body, or geometric simplicial complex) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536067.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536068.png" />. This yields the geometric model in the sense of Giever–Hu (see [[Simplicial set|Simplicial set]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536069.png" /> of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536070.png" />, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536071.png" /> is ordered, the geometric model in the sense of Milnor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536072.png" /> of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536073.png" />. The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536074.png" /> is a covariant functor from the category of simplicial complexes to the category of cellular spaces. A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536075.png" /> homeomorphic to the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536076.png" /> of some simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536077.png" /> is called a polyhedron (or a triangulated space, cf. [[Polyhedron, abstract|Polyhedron, abstract]]) and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536078.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536079.png" /> is the homeomorphism, is called a triangulation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536080.png" />.

The points of the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536081.png" /> can be identified with the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536082.png" /> for which the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536083.png" /> is a simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536084.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536085.png" /></td> </tr></table>

The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536086.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536087.png" />-th barycentric coordinate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536088.png" />. The formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536089.png" /></td> </tr></table>

defines a metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536090.png" />, but the corresponding metric topology is, in general, stronger than the original one. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536091.png" /> equipped with this metric topology is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536092.png" />.

A simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536093.png" /> is isomorphic to the nerve of the family of stars of vertices of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536094.png" />, that is, to the nerve of the family of open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536096.png" />.

The following statements are equivalent: 1) the simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536097.png" /> is locally finite; 2) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536098.png" /> is locally compact; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536099.png" />; 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360100.png" /> is metrizable; and 5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360101.png" /> satisfies the [[First axiom of countability|first axiom of countability]]. Moreover, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360102.png" /> is separable (compact) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360103.png" /> is at most countable (finite).

The cells of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360104.png" /> are in one-to-one correspondence with the simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360105.png" />, and the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360106.png" /> of the cell corresponding to a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360107.png" /> is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360108.png" /></td> </tr></table>

It is homeomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360109.png" />-dimensional closed ball, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360110.png" />, so that the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360111.png" /> is regular. In addition, each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360112.png" /> has a canonical linear (affine) structure, with respect to which it is isomorphic to the [[Standard simplex|standard simplex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360113.png" />. Because of this, and the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360114.png" /> for all simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360115.png" />, it turns out that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360116.png" /> can be mapped homeomorphically (can be imbedded) into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360117.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360118.png" /> may be transfinite), so that all closed cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360119.png" /> are (rectilinear) simplices. This means that the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360120.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360121.png" /> is a [[Simplicial space|simplicial space]] (a polyhedron), i.e. a union of closed simplices intersecting only on entire faces. This simplicial space is called a realization of the simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360122.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360123.png" />.

A simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360124.png" /> can only be realized in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360125.png" /> for finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360126.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360127.png" /> is locally finite, at most countable and of finite dimension. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360128.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360129.png" /> can be realized in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360130.png" />. A simplicial complex consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360131.png" /> vertices every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360132.png" />-element subset of which is a simplex cannot be realized in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360133.png" />.

From any simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360134.png" /> one can construct a new simplicial complex, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360135.png" />, whose vertices are the simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360136.png" /> and whose simplices are families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360137.png" /> of simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360138.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360139.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360140.png" /> is called the barycentric refinement (or subdivision) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360141.png" />. The cellular spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360143.png" /> are naturally homeomorphic (but not isomorphic). Under this homeomorphism, every vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360144.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360145.png" /> (that is, the zero-dimensional cell corresponding to the vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360146.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360147.png" />) is mapped onto the centre of gravity (the barycentre) of the closed simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360148.png" />.

The simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360149.png" /> is ordered in a natural way. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360150.png" /> is ordered, then the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360151.png" /> (first vertex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360152.png" />) defines a simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360153.png" /> that preserves the ordering. It is called the canonical translation. Its geometric realization (which is a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360154.png" />) is homotopic to the natural homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360155.png" />.

A simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360156.png" /> (or its geometric realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360157.png" />) is called a simplicial approximation of a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360158.png" /> if, for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360159.png" />, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360160.png" /> belongs to the minimal closed simplex containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360161.png" />, or, equivalently, if for every vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360163.png" />. The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360164.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360165.png" /> are homotopic.

The simplicial approximation theorem states that if a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360166.png" /> is finite, then for every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360167.png" /> there is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360168.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360169.png" /> there is a simplicial approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360170.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360171.png" /> (regarded as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360172.png" />).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.J. Hilton,   S. Wylie,   "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press  (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.H.C. Whitehead,   "Simplicial spaces, nuclei and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360173.png" />-groups"  ''Proc. London Math. Soc.'' , '''45'''  (1939)  pp. 243–327</TD></TR></table>



====Comments====
In the West, the concept described here is usually called an (abstract) simplicial complex; the term simplicial scheme would normally be understood to mean a simplicial object in the category of schemes (cf. [[Simplicial object in a category|Simplicial object in a category]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.R.F. Maunder,   "Algebraic topology" , v. Nostrand  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Lefshetz,   "Topology" , Chelsea, reprint  (1956)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Lamotke,   "Semisimpliziale algebraische Topologie" , Springer  (1968)</TD></TR></table>