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...ent [[functor]]s from either the [[category of sets]] or the category of [[simplicial set]]s to the category of [[manifold]]s. A simplicial manifold is a [[simplicial complex]] for which the [[Abstract simplicial complex|geometric realization]] is [[homeomorphic]] to a [[topological mani ...

In [[mathematics]], a '''pro-simplicial set''' is an [[inverse system]] of [[simplicial set]]s. ...icial set is called ''pro-finite'' if each term of the inverse system of [[simplicial set]]s has [[Finite group|finite]] [[homotopy groups]]. ...

{{Short description|Concept in the theory of simplicial sets.}} ...

...als]] and [[order-preserving map]]s. It is used to define [[simplicial set|simplicial]] and cosimplicial objects. ...s, which amount to inserting or deleting elements of the orderings. (See [[simplicial set]] for relations of these maps.) ...

...associated to a [[poset]] (''S'', ≤) is the [[Alexandrov topology]] (open sets are [[upper set]]s) on the poset of finite [[Total_order#Chains|chains]] of ...a set of vertices. An [[abstract simplicial complex]] Δ is a set of finite sets of vertices, known as faces <math>\sigma \subseteq V</math>, such that ...

** [[Join (mathematics)]], a least upper bound of sets orders in lattice theory ** [[Join (simplicial sets)]], an operation combining two simplicial sets ...

...r|n}}-skeleton'''}} of a [[topological space]] {{mvar|X}} presented as a [[simplicial complex]] (resp. [[CW complex]]) refers to the [[subspace (topology)|subspa == For simplicial sets == ...

...paces that are built up from [[simplex|simplices]]&mdash;that is, finite [[simplicial complex]]es. The general continuous mapping between such spaces can be repr There is a further '''simplicial approximation theorem for homotopies''', stating that a [[homotopy]] betwee ...

...h>a<c,\ b<c,\ a<d,\ b<d</math> where the [[open set]]s are downward-closed sets. ''X'' is highly [[pathological (mathematics)|pathological]] from the usual ...b''} is also the union of two [[Disjoint sets|disjoint]] contractible open sets {''a''} and {''b''}. So, like S¹, the result follows from the [[ ...

...on which to define [[homotopy group]]s. In the context of the theory of [[simplicial set]]s, the fibrant objects are known as '''Kan complexes''' after [[Daniel *P.G. Goerss and J.F. Jardine, ''Simplicial Homotopy Theory'', Progress in Math., Vol. 174, Birkhauser, Boston-Basel-Be ...

...subset]]s of {{tmath|V}} (called the '''independent sets''' or '''feasible sets''') with the following properties: Another term for an independence system is an [[abstract simplicial complex]]. ...

...he usual [[closed model category]] structure on the category of simplicial sets is known as [[Kan–Quillen model structure]], while its fibrations and fibra ...topy limits]], and his work with [[William Gerard Dwyer|William Dwyer]] on simplicial localizations of relative categories. ...

{{Short description|Any collection of sets, or subsets of a set}} ...0. Litton Educational Publishing, Inc.</ref> In some contexts, a family of sets may be allowed to contain repeated copies of any given member,<ref>{{harvnb ...

{{Short description|Simplicial set constructed from the objects and morphisms of a small category}} ...d morphisms of ''C''. The [[simplicial set|geometric realization]] of this simplicial set is a [[topological space]], called the '''classifying space of the cate ...

...implicial complex|abstract simplices]]) and maps between them. Simplicial sets are higher-dimensional generalizations of [[directed graph]]s. ...purposes of [[homotopy theory]]. Specifically, the category of simplicial sets carries a natural [[Model category|model structure]], and the corresponding ...

...example.svg|thumb|200px|Geometric realization of a 3-dimensional abstract simplicial complex]] ...angles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). ...

...cription|About the numbers of faces of different dimensions in an abstract simplicial complex}} ...em''' gives a complete characterization of the ''f''-vectors of [[abstract simplicial complex]]es. It includes as a special case the [[Erdős–Ko–Rado theorem]] an ...

...plex recording the pattern of intersections between a topological family's sets}} ...rve of an [[Good cover (algebraic topology)|open good cover]] containing 3 sets in the plane.]] ...

...r|willard}} of a [[Set (mathematics)|set]] <math>X</math> is a [[Family of sets|family]] of [[subset]]s of <math>X</math> whose union is all of <math>X</ma ...In this case <math> C </math> is said to cover <math>X</math>, or that the sets <math>U_\alpha</math> cover <math>X</math>.{{r|willard}} ...

[[File:Simplicial complex example.svg|thumb|200px|A simplicial 3-complex.]] ...an abstract simplicial complex, the former is often called a '''geometric simplicial complex'''.<ref name=":0">{{cite Matousek 2007}}, Section 4.3</ref>{{rp|pag ...