## Push-out

Let $f:A\to B$ and $g:A\to C$ be simplicial maps. We define $X_n$ to be the push-out in the diagram

i.e. $X_n=B_n\coprod C_n/\sim$, where $\sim$ is the equivalence relation such that $f(a)\sim g(a)$ for $a\in A_n$. Then $X=\{X_n\}_{n\geq 0}$ with faces and degeneracies induced from that in $B$ and $C$ forms a simplicial set with a push-out diagram of simplicial sets

For example, let $\partial\Delta[n]=\langle d_i(0,1,\dots,n)\mid 0\leq i\leq n\rangle\subseteq \Delta[n]$ be the simplicial subset of $\Delta[n]$ generated by all of the faces of the $n$-simplex $\sigma_n=(0,1,\dots,n)$. Then

is a push-out diagram, where $f$ is the inclusion map and $S^n=\Delta[n]/\partial\Delta[n]$.