is a distinguished simplex.

We will not put any further restrictions on the representation of simplicial sets in a computer – the above algorithms will be sufficient. On the other hand, it is important that such representations exist. We will describe one possibility for finite simplicial sets and complexes.

Let $X$ be a finite simplicial set with a free action of $G$. Let us choose arbitrarily one simplex from each orbit of the non-degenerate simplices; these simplices together with all of their degeneracies are the distinguished ones. Then every simplex $x\in X$ can be represented uniquely as $x=as_Iy$, where $a\in G$, $s_I$ is an iterated degeneracy operator (i.e. a composition $s_{{{i}}_{{m}}}\cdots s_{{{i}}_1}$ with ${{i}}_1<\cdots<{{i}}_{{m}}$), and $y$ is a non-degenerate distinguished simplex. With this representation, it is possible to compute the action of $G$ and the degeneracy operators easily, while face operators are computed using the relations among the face and degeneracy operators and a table of faces of non-degenerate distinguished simplices. This table is finite and it can be provided on the input.

A special case is that of a finite simplicial complex. Here, one can prescribe a simplex (degenerate or not) uniquely by a finite sequence of its vertices.

