The standard topological simplex and its affine face maps

For each integer $n\ge 0$, the standard topological $n$-simplex is ${\mathrm{\Delta }}^n:=\{(t_0,\dots ,t_n)\in {\mathbb{R}}^{n+1}:t_i\ge 0\text{ for all }i,{\sum }_{i=0}^n{t}_i=1\},$ with the subspace topology from ${\mathbb{R}}^{n+1}$. Its vertices are the standard basis vectors $v_0,\dots ,v_n$.

For $n\ge 1$ and $0\le i\le n$, the $i$th affine face map ${\delta }_i:{\mathrm{\Delta }}^{n-1}\to {\mathrm{\Delta }}^n$ is the continuous affine map that inserts a zero in the $i$th coordinate: ${\delta }_i(t_0,\dots ,t_{n-1})=(t_0,\dots ,t_{i-1},0,t_i,\dots ,t_{n-1})\mathrm{.}$ Equivalently, ${\delta }_i$ carries the vertex $v_k$ of ${\mathrm{\Delta }}^{n-1}$ to $v_k$ for $k<i$ and to $v_{k+1}$ for $k\ge i$, so its image is the face opposite the vertex $v_i$.

For $n=0$ there are no face maps because ${\mathrm{\Delta }}^{-1}$ is not part of the chain-level indexing used on this page.