# 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 .