Simplex category

Remarks

Every morphism of $\Delta$ is a composite of two kinds of basic maps. The face map $\delta_i \colon [n-1] \to [n]$, for $0 \le i \le n$, is the injective order-preserving map whose image misses $i$. The degeneracy map $\sigma_i \colon [n+1] \to [n]$, for $0 \le i \le n$, is the surjective order-preserving map that takes the value $i$ twice. For example, the map $[2] \to [1]$ with $0 \mapsto 0$, $1 \mapsto 0$, $2 \mapsto 1$ is $\sigma_0$. One pictures $[n]$ as the standard $n$-simplex with vertices $0, \dots, n$: face maps include a face, and degeneracy maps collapse an edge. Functors from the opposite category $\Delta^{\mathrm{op}}$ are the simplicial objects, and functors $\Delta^{\mathrm{op}} \to \mathbf{Set}$ are the simplicial sets.