General Topology/Simplicial complexes

There are several types of simplicial complexes.

Definition (semisimplicial set):

A semisimplicial set is a contravariant functor from the category ${\mathcal {N}}$ of finite sets and monotone maps to ${\textbf {Set}}$.

Definition (Δ-complex):

A Δ-complex is a topological space $X$ together with a family of functions $\sigma _{\alpha }:\Delta _{\alpha }\to X$ such that

• for each $\alpha$, the set $\Delta _{\alpha }$ is the standard simplex of a certain dimension $n$ with the subspace topology induced by the topology of $\mathbb {R} ^{n+1}$ on it,
• for each $x\in X$ there exists a unique $\alpha$ and $z\in {\overset {\circ }{\Delta }}_{\alpha }$ (the interior) so that $x=\sigma _{\alpha }(z)$,
• the topology of $X$ coincides with the final topology with respect to the $\sigma _{\alpha }$,
• and if $\Delta _{\alpha }$ is not the trivial simplex and $n\geq 1$ is its dimension, then the map arising from mapping $\Delta _{n-1}$ to any face of $\Delta _{\alpha }$ and then applying $\sigma _{\alpha }$ equals $\sigma _{\beta }$ for some $\beta$.