2016 – Mathtuition88

A simplicial map $f:X\to Y$ is a family of functions $f:X_n\to Y_n$ that commutes with $d_i$ and $s_i$ . If each $X_n$ is a subset of $Y_n$ such that the inclusions $X_n\hookrightarrow Y_n$ is a simplicial map, then $X$ is said to be a simplicial subset of $Y$ .

The $n$ -simplex $\Delta[n]$ is defined as follows:
$\Delta[n]_k:=\{(i_0,i_1,\dots,i_k)\mid 0\leq i_0\leq i_1\leq\dots\leq i_k\leq n\}$
where $k\leq n$ .
The face $d_j:\Delta[n]_k\to\Delta[n]_{k-1}$ is defined by $d_j(i_0,i_1,\dots,i_k)=(i_0,i_1,\dots\i_{j-1},i_{j+1},\dots,i_k)$ , i.e. deleting $i_j$ . The degeneracy $s_j:\Delta[n]_k\to\Delta[n]_{k+1}$ is given by $s_j(i_0,i_1,\dots,i_k)=(i_0,i_1,\dots,i_j,i_j,\dots,i_k)$ , i.e. repeating $i_j$ . Let $\sigma_n=(0,1,\dots,n)\in\Delta[n]_n$ . Any element in $\Delta[n]$ can be written as iterated compositions of faces and degeneracies of $\sigma_n$ .