View source for Simplicial set

''(formerly called semi-simplicial complex, full semi-simplicial complex)''

A simplicial object in the category of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s0854001.png" /> (cf. [[Simplicial object in a category|Simplicial object in a category]]), that is, a system of sets (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s0854003.png" />-fibres) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s0854004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s0854005.png" />, connected by mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s0854006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s0854007.png" /> (boundary operators), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s0854008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s0854009.png" /> (degeneracy operators), satisfying the conditions

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The elements of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540011.png" /> are called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540013.png" />-dimensional simplices of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540014.png" />. If only the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540015.png" /> are given, satisfying the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540017.png" />, then the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540018.png" /> is called a semi-simplicial set.

A simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540019.png" /> between two simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540021.png" /> is a morphism of functors, i.e. a sequence of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540023.png" />, satisfying the relations

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Simplicial sets and their simplicial mappings form a category, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540025.png" />. If all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540026.png" /> are imbeddings, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540027.png" /> is called a simplicial subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540028.png" />. In this case, the boundary and degeneracy operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540029.png" /> are the restrictions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540030.png" /> of the corresponding operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540031.png" />.

Given any topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540032.png" />, one can define a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540033.png" />, called the singular simplicial set of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540034.png" />. Its simplices are the singular simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540035.png" /> (see [[Singular homology|Singular homology]]), i.e. continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540037.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540038.png" />-dimensional geometric [[Standard simplex|standard simplex]]:

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The boundary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540040.png" /> and degeneracy operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540041.png" /> of this simplicial set are defined by the formulas

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The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540045.png" /> is a functor (called the singular functor) from the category of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540046.png" /> into the category of simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540047.png" />.

An arbitrary [[Simplicial complex|simplicial complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540048.png" /> determines a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540049.png" />. Its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540050.png" />-dimensional simplices are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540051.png" />-tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540052.png" /> of vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540053.png" /> with the property that there is a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540056.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540057.png" />. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540059.png" /> for this simplicial set are given by

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540062.png" /> means that the symbol below it is omitted. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540063.png" /> is ordered, then the simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540064.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540065.png" /> form a simplicial subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540067.png" />. The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540068.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540069.png" />) is a functor from the category of simplicial complexes (ordered simplicial complexes) into the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540070.png" />.

For an arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540071.png" /> one can define a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540072.png" />. Its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540073.png" />-simplices are equivalence classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540074.png" />-tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540076.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540077.png" /> if there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540078.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540079.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540080.png" />). The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540082.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540083.png" /> are given by

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The simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540086.png" /> is actually a simplicial group.

Given an arbitrary Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540087.png" /> and any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540088.png" />, one can define a simplicial set (in fact, a simplicial Abelian group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540089.png" />. Its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540090.png" />-dimensional simplices are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540091.png" />-dimensional cochains of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540092.png" />-dimensional geometric standard simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540093.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540094.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540095.png" />). Denoting the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540096.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540098.png" />, one defines the simplicial mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s08540099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400100.png" /> by the formulas

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The induced homomorphisms of cochain groups

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are, by definition, the boundary and degeneracy operators of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400105.png" />. The simplices that are cocycles form a simplicial subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400106.png" />, called the Eilenberg–MacLane simplicial set and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400107.png" />. The coboundary operator on the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400108.png" /> defines a canonical simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400109.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400110.png" />. Since the concept of a one-dimensional cocycle also makes sense when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400111.png" /> is non-Abelian (see [[Non-Abelian cohomology|Non-Abelian cohomology]]), the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400112.png" /> can be defined without the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400113.png" /> is Abelian. This simplicial set is isomorphic to the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400114.png" /> (by assigning to every simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400115.png" /> the values at the vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400116.png" /> of a zero-dimensional cochain whose coboundary is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400117.png" />).

By assigning to every fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400118.png" /> of a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400119.png" /> the free Abelian group generated by it, one obtains a simplicial Abelian group and thus a chain complex. This complex is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400120.png" /> and is called the chain complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400121.png" />. The (co)homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400122.png" /> (with coefficients in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400123.png" />) are called the (co) homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400125.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400126.png" />. The (co)homology groups of a singular simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400127.png" /> are the (co)homology groups of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400128.png" />. The (co)homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400130.png" /> are isomorphic and are called the (co) homology groups of the simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400131.png" />. The (co)homology groups of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400132.png" /> are the (co) homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400133.png" />.

A simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400134.png" /> of a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400135.png" /> is called degenerate if there is a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400136.png" /> and a degeneracy operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400137.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400138.png" />. The Eilenberg–Zil'ber lemma states that any simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400139.png" /> can be uniquely written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400140.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400141.png" /> is a certain epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400142.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400143.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400144.png" /> is a non-degenerate simplex. The smallest simplicial subset of a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400145.png" /> containing all its non-degenerate simplices of dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400146.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400147.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400148.png" />, and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400150.png" />-dimensional skeleton or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400152.png" />-skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400153.png" />.

The standard geometric simplices (cf. [[Standard simplex|Standard simplex]])

<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/s085400/s085400154.png" /></td> </tr></table>

form a co-simplicial topological space with respect to the co-boundary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400155.png" /> and co-degeneracy operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400156.png" />, defined 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/s085400/s085400157.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/s085400/s085400158.png" /></td> </tr></table>

In the disjoint union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400159.png" />, where all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400160.png" /> are regarded as discrete sets, 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/s085400/s085400161.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/s085400/s085400162.png" /></td> </tr></table>

generate an equivalence relation, the quotient space by which is a complex (a cellular space) whose cells are in one-to-one correspondence with the non-degenerate simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400163.png" />. This complex is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400164.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400165.png" /> and is called the geometric realization in the sense of Milnor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400166.png" />. Any simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400167.png" /> induces a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400168.png" />, 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/s085400/s085400169.png" /></td> </tr></table>

and the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400170.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400171.png" /> defines a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400172.png" />. This functor is left adjoint to the singular functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400173.png" />. The corresponding natural isomorphisms

<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/s085400/s085400174.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/s085400/s085400175.png" /></td> </tr></table>

are defined 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/s085400/s085400176.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/s085400/s085400177.png" /></td> </tr></table>

where

<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/s085400/s085400178.png" /></td> </tr></table>

For any topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400179.png" /> the adjunction morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400180.png" /> is a weak homotopy equivalence (which proves that any topological space is weakly homotopy equivalent to a complex).

The construction of the geometric realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400181.png" /> extends to the case of a simplicial topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400182.png" />. One can also define the geometric realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400183.png" /> in the sense of Giever–Hu by taking only the boundary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400184.png" /> into account (in this model there are cells for all the simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400185.png" />, not just for the non-degenerate ones). If every degeneracy operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400186.png" /> is a closed cofibration (a condition which holds automatically in the case of a simplicial set), then the natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400187.png" /> is a homotopy equivalence.

The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400188.png" /> admits products: given simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400189.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400190.png" />, their product is the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400191.png" /> for which

<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/s085400/s085400192.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/s085400/s085400193.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/s085400/s085400194.png" /></td> </tr></table>

In particular, given any simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400195.png" />, one can define its product with the simplicial segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400196.png" />. The projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400197.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400198.png" /> define a bijective mapping

<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/s085400/s085400199.png" /></td> </tr></table>

which is a homeomorphism if the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400200.png" /> is a complex (for example, if both simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400202.png" /> are countable or if one of the complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400204.png" /> is locally finite). In particular, it follows that the geometric realization of any countable simplicial monoid (group, Abelian group) is a topological monoid (group, Abelian group).

Two simplicial mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400205.png" /> are called homotopic if there is a simplicial mapping (a homotopy) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400206.png" /> such that

<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/s085400/s085400207.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/s085400/s085400208.png" /></td> </tr></table>

for any simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400209.png" /> and for any composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400210.png" /> (of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400211.png" />) of degeneracy operators. This definition (modelled on the usual definition of homotopy of continuous mappings) is equivalent to the interpretation in simplicial sets of the general definition of homotopy of simplicial mappings between arbitrary simplicial objects (see [[Simplicial object in a category|Simplicial object in a category]]).

Given the notion of homotopy, it is possible to develop a homotopy theory for simplicial sets similar to that for polyhedra. It turns out that these two theories are completely parallel; this finds expression in the fact that the corresponding homotopy categories are equivalent (the equivalence being induced by the geometric realization functor). In particular, geometric realizations of homotopic simplicial mappings are homotopic and, for example, the geometric realization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400212.png" /> is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400213.png" />. However, the actual construction of the homotopy theory for simplicial sets differs slightly in its details from the construction of the homotopy theory for topological spaces. The main difference is that the relation of homotopy for simplicial mappings is not, in general, an equivalence relation. This difficulty is overcome in the following way.

A simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400214.png" /> of the standard horn (see [[Standard simplex|Standard simplex]]) into a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400215.png" /> is called a horn in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400216.png" />. Every horn is uniquely defined by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400217.png" />-tuple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400218.png" />-simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400219.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400220.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400221.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400222.png" />. One says that a horn fills out if one can find an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400223.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400224.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400225.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400226.png" />. The simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400227.png" /> is said to be full (or to satisfy the Kan condition) if all its horns fill out.

The singular simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400228.png" /> of an arbitrary topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400229.png" /> is always full, and so is every simplicial group; in particular, the Eilenberg–MacLane simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400230.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400231.png" /> are full. The importance of full simplicial sets lies in the fact that the relation of homotopy between simplicial mappings from an arbitrary simplicial set to a full simplicial set is an equivalence relation. Therefore, in the subcategory of full simplicial sets, the construction of a homotopy theory involves no major difficulties. Moreover, there is a functor (see [[#References|[4]]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400232.png" /> assigning to every simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400233.png" /> a full simplicial set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400234.png" />, whose geometric realization is homotopy equivalent to the geometric realization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400235.png" /> and which can therefore be used in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400236.png" /> in all questions of homotopy.

Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400237.png" />-simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400238.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400239.png" /> of a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400240.png" /> are called comparable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400241.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400242.png" />. Two such simplices are said to be homotopic if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400243.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400244.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400245.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400246.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400247.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400248.png" />. For full simplicial sets this is an equivalence relation, and two simplices are homotopic if and only if their characteristic simplicial mappings are homotopic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400249.png" />.

A simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400250.png" /> is said to be pointed if it contains a distinguished zero-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400251.png" /> (where the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400252.png" /> is also used to denote all degenerations of this simplex as well as the simplicial set generated by it, which is usually referred to as the distinguished point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400253.png" />). For a full pointed simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400254.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400255.png" /> of homotopy classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400256.png" />-dimensional simplices comparable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400257.png" /> is a group when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400258.png" />. This group is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400260.png" />-dimensional homotopy group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400261.png" />; this terminology is justified by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400262.png" /> and, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400263.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400264.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400265.png" />. A simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400266.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400267.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400268.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400270.png" />-connected set; in particular, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400271.png" />-connected simplicial set is called connected, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400272.png" />-connected simplicial set simply connected. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400273.png" />, the addition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400274.png" /> is induced by the operation which assigns to two simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400275.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400276.png" /> (comparable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400277.png" />) the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400278.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400279.png" /> is a simplex of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400280.png" />, filling the horn <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400281.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400282.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400283.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400284.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400285.png" /> is a simplicial monoid with unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400286.png" />, then the addition is also induced by the multiplication in this monoid (the product of two simplices comparable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400287.png" /> is comparable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400288.png" />).

Since any simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400289.png" /> comparable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400290.png" /> is a cycle (of the chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400291.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400292.png" />), there is a natural Hurewicz homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400293.png" />, which induces an isomorphism

<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/s085400/s085400294.png" /></td> </tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400295.png" /> (Poincaré's theorem), and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400296.png" /> it is an isomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400297.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400298.png" />-connected (Hurewicz' theorem). For full simplicial sets both variants of Whitehead's theorem hold, that is, a simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400299.png" /> of full simplicial sets is a homotopy equivalence if and only if it induces an isomorphism of homotopy groups; in the simply-connected case this condition is equivalent to the induced homomorphisms of the homology groups being isomorphisms.

In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400300.png" /> is a simplicial group, the homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400301.png" /> is isomorphic to the homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400302.png" /> of the (not necessarily Abelian) chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400303.png" /> for which

<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/s085400/s085400304.png" /></td> </tr></table>

and the boundary operator is the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400305.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400306.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400307.png" /> is Abelian, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400308.png" /> is a subcomplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400309.png" />, regarded as a chain complex, and also a chain deformation retract of it, and hence a direct summand of it. It turns out that the subcomplex generated by the degenerate simplices can be taken as the other direct summand. Therefore, the corresponding quotient complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400310.png" /> is chainwise equivalent to it. For example, it follows that the cohomology groups of an arbitrary simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400311.png" /> are isomorphic to the normalized cohomology groups (the normalization theorem), that is, the groups obtained from the cochains that vanish on all degenerate simplices. Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400312.png" />.

The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400313.png" /> induces an equivalence between the homotopy theory of simplicial Abelian groups and the homology theory of chain complexes. In particular, it follows that any connected simplicial Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400314.png" /> is homotopy equivalent to a product of Eilenberg–MacLane simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400315.png" />.

A full simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400316.png" /> is called minimal when comparable simplices are homotopic if and only if they coincide. The simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400317.png" /> is minimal. Every homotopy equivalence of minimal simplicial sets is an isomorphism. Every full simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400318.png" /> has a minimal subset. It is a deformation retract, and is thus uniquely defined up to isomorphism.

A simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400319.png" /> is called a Kan fibration if any horn <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400320.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400321.png" /> can be filled whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400322.png" /> can be, and for any filling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400323.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400324.png" /> there is a filling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400325.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400326.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400327.png" />. Kan fibrations are the simplicial analogue of Serre fibrations (cf. [[Serre fibration|Serre fibration]]), and they satisfy the following homotopy lifting theorem: If the simplicial mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400328.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400329.png" /> satisfy the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400330.png" />, then there is a simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400331.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400332.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400333.png" />. If the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400334.png" /> is surjective, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400335.png" /> is full if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400336.png" /> is full. The fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400337.png" /> is the (automatically full) simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400338.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400339.png" /> is the distinguished point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400340.png" />. For any Serre fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400341.png" /> the simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400342.png" /> is a Kan fibration, and for any Kan fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400343.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400344.png" /> is a Serre fibration (see [[#References|[5]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400345.png" /> be a full pointed simplicial set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400346.png" />. Write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400347.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400348.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400349.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400350.png" />, that is, when

<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/s085400/s085400351.png" /></td> </tr></table>

(see [[Standard simplex|Standard simplex]]). This is an equivalence relation, and the quotient sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400352.png" /> form a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400353.png" /> (with respect to the induced boundary and degeneracy operators), called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400355.png" />-co-skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400356.png" />. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400357.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400358.png" />, the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400359.png" /> is full and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400360.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400361.png" />. Moreover, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400362.png" /> the natural surjective simplicial mapping

<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/s085400/s085400363.png" /></td> </tr></table>

is a fibration inducing an isomorphism of homotopy groups in dimensions less than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400364.png" />. In particular, the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400365.png" /> is homotopy equivalent to the Eilenberg–MacLane simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400366.png" />. The sequence of fibrations

<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/s085400/s085400367.png" /></td> </tr></table>

is called the [[Postnikov system|Postnikov system]] of a full simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400368.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400369.png" /> is minimal, then this sequence is a resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400370.png" /> (see [[Homotopy type|Homotopy type]]).

The construction of the Postnikov system admits a direct generalization to an arbitrary fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400371.png" /> of a full simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400372.png" /> over a full simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400373.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400374.png" /> be the simplicial set whose fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400375.png" /> are the quotient sets of the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400376.png" /> by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400377.png" />, which holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400378.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400379.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400380.png" />. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400381.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400382.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400383.png" /> the natural simplicial mapping

<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/s085400/s085400384.png" /></td> </tr></table>

is a fibration inducing an isomorphism of homotopy groups in dimensions less than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400385.png" /> or greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400386.png" />. In particular, the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400387.png" /> is homotopy equivalent to the Eilenberg–MacLane simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400388.png" />. The fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400389.png" /> is the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400390.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400391.png" /> is the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400392.png" />. The sequence of fibrations

<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/s085400/s085400393.png" /></td> </tr></table>

is called the Moore–Postnikov system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400394.png" />.

It is convenient to define spectra in the language of simplicial sets. A simplicial spectrum is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400395.png" /> of pointed sets (whose elements are called simplices, and the distinguished simplex is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400396.png" />) defined for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400397.png" />, and equipped with mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400398.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400399.png" /> (boundary operators), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400400.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400401.png" /> (degeneracy operators), which satisfy the relations (*) together with the following condition: For every simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400402.png" /> there is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400403.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400404.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400405.png" />. To any spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400406.png" /> and integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400407.png" /> one can assign the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400408.png" /> defined 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/s085400/s085400409.png" /></td> </tr></table>

These simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400410.png" /> are equipped with imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400411.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400412.png" /> is the [[Suspension|suspension]] functor. From the sequence of simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400413.png" /> and imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400414.png" />, the simplicial spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400415.png" /> can in turn be uniquely recovered. If every member of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400416.png" /> is full, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400417.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400418.png" /> is the [[Loop|loop]] functor. The geometric realization functor gives an equivalence of the category of simplicial spectra and the category of topological spectra. Simplicial spectra can be defined for an arbitrary category. The category of Abelian group spectra is isomorphic to the category of (Abelian) chain complexes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Gabriel,   M. Zisman,   "Calculus of fractions and homotopy theory" , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.P. May,   "Simplicial objects in algebraic topology" , v. Nostrand  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Lamotke,   "Semisimpliziale algebraische Topologie" , Springer  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D.M. Kan,   "On c.s.s. complexes"  ''Amer. J. Math.'' , '''79'''  (1957)  pp. 449–476</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D.G. Quillen,   "The geometric realization of a Kan fibration is a Serre fibration"  ''Proc. Amer. Math. Soc.'' , '''19'''  (1968)  pp. 1499–1500</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.H. Brown,   "Finite computability of Postnikov complexes"  ''Ann. of Math. (2)'' , '''65'''  (1957)  pp. 1–20</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D.M. Kan,   "A combinatorial definition of homotopy groups"  ''Ann. of Math. (2)'' , '''67'''  (1958)  pp. 282–312</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  D.M. Kan,   "On homotopy theory and c.s.s. groups"  ''Ann. of Math. (2)'' , '''68'''  (1958)  pp. 38–53</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  D.M. Kan,   "An axiomatization of the homotopy groups"  ''Illinois J. Math.'' , '''2'''  (1958)  pp. 548–566</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D.M. Kan,   "A relation between CW-complexes and free c.s.s. groups"  ''Amer. J. Math.'' , '''81'''  (1959)  pp. 512–528</TD></TR></table>



====Comments====
The  "Kan condition"  that every horn fills out is also called the extension condition.

A simplicial set or simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400419.png" /> is called a Kan complex if it satisfies the Kan condition, [[#References|[2]]], p. 2.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400420.png" /> be the set of all monomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400421.png" /> of horns.

A class of monomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400422.png" /> in a category is called saturated if it satisfies the following conditions:

i) all isomorphisms belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400423.png" />;

ii) let

<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/s085400/s085400424.png" /></td> </tr></table>

be a co-Cartesian square. Then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400425.png" />, also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400426.png" /> (stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400427.png" /> under pushouts; a co-Cartesian square is a [[Cartesian square|Cartesian square]] in the dual category);

iii) given a commutative diagram

<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/s085400/s085400428.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400429.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400430.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400431.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400432.png" /> (stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400433.png" /> under retractions);

iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400434.png" /> is stable under countable compositions and arbitrary direct sums.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400435.png" /> be the saturated closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400436.png" />, i.e. the intersection of all saturated classes containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400437.png" />. These are called the anodyne extensions in [[#References|[1]]].

A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400438.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400439.png" /> is called a Kan fibration if for each anodyne extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400440.png" /> and commutative square

<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/s085400/s085400441.png" /></td> </tr></table>

there exists a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400442.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400443.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400444.png" />. A simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400445.png" /> is a Kan complex if and only if the unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400446.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085400/s085400447.png" /> is the standard zero simplex, is a Kan fibration.