Simplicial set

Remarks

Concretely, a simplicial set is a family of sets $X_n$ with face maps $d_i \colon X_n \to X_{n-1}$ and degeneracy maps $s_i \colon X_n \to X_{n+1}$, $0 \le i \le n$, that satisfy the simplicial identities. An $n$-simplex is degenerate when it lies in the image of some $s_i$. Example: the standard $n$-simplex $\Delta^n = \Hom_{\Delta}(-, [n])$ is a representable functor, so it is a simplicial set; its nondegenerate $m$-simplices are the injective order-preserving maps $[m] \to [n]$. Non-example: a family of sets $X_n$ with only face maps and no degeneracy maps is not a simplicial set; such data form a semi-simplicial set. The category of simplicial sets is often written $\mathbf{sSet}$.