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
Screen Shot 2016-01-17 at 4.46.49 PM
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
Screen Shot 2016-01-17 at 4.47.03 PM
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
Screen Shot 2016-01-17 at 4.47.10 PM
is a push-out diagram, where f is the inclusion map and S^n=\Delta[n]/\partial\Delta[n].

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