# General Topology/Simplicial complexes

There are several types of simplicial complexes.

Definition (semisimplicial set):

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

Definition (Δ-complex):

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

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