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 .
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 .