General Topology/Simplicial complexes

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There are several types of simplicial complexes.

Definition (semisimplicial set):

A semisimplicial set is a contravariant functor from the category of finite sets and monotone maps to .

Definition (Δ-complex):

A Δ-complex is a topological space together with a family of functions such that

  • for each , the set is the standard simplex of a certain dimension with the subspace topology induced by the topology of on it,
  • for each there exists a unique and (the interior) so that ,
  • the topology of coincides with the final topology with respect to the ,
  • and if is not the trivial simplex and is its dimension, then the map arising from mapping to any face of and then applying equals for some .