Are the degeneracy maps of a simplicial set always injective? I wonder, because I'd naively think that it wouldn't make sense to have two $n$-simplices which yield the same degenerate $n+1$-simplex. At least I haven't seen any example of a simplicial set in which this is the case.
Are degeneracy maps always injective
simplicial-stuff
Best Answer
Yes; note that, given a simplicial set $(X_n)_{n\in\mathbb{N}}$ and a degeneracy map $s_{i}:X_n\to X_{n+1}$, the face map $d_i:X_{n+1}\to X_n$ is a left inverse to $s_i$, whence $s_i$ is in particular injective. More explicitly, suppose we have $\sigma_1,\sigma_2\in X_n$ with $s_i(\sigma_1)=s_i(\sigma_2)$. By the simplicial identities, we have $d_i\circ s_i=\operatorname{id}_{X_n}$, so we obtain $\sigma_1=d_i(s_i(\sigma_1))=d_i(s_i(\sigma_2))=\sigma_2$, as desired.