Proposition 1.1.2.10 (0011)

Proposition 1.1.2.10. Let $S_{\bullet }$ be a simplicial set and let $\tau \in S_{n}$ be an $n$-simplex of $S_{\bullet }$ for some $n > 0$, which we will identify with a map of simplicial sets $\tau : \Delta ^{n} \rightarrow S_{\bullet }$. The following conditions are equivalent:

$(1)$: The simplex $\tau $ belongs to the image of the degeneracy operator $s^{n-1}_{i}: S_{n-1} \rightarrow S_{n}$ for some $0 \leq i < n$ (see Construction 1.1.2.1).
$(2)$: The map $\tau $ factors as a composition $\Delta ^{n} \xrightarrow {f} \Delta ^{n-1} \rightarrow S_{\bullet }$, where $f$ corresponds to a surjective map of linearly ordered sets $[n] \twoheadrightarrow [n-1]$.
$(3)$: The map $\tau $ factors as a composition $\Delta ^{n} \xrightarrow {f} \Delta ^{m} \rightarrow S_{\bullet }$, where $m < n$ and $f$ corresponds to a surjective map of linearly ordered sets $[n] \twoheadrightarrow [m]$.
$(4)$: The map $\tau $ factors as a composition $\Delta ^{n} \rightarrow \Delta ^{m} \rightarrow S_{\bullet }$, where $m < n$.
$(5)$: The map $\tau $ factors as a composition $\Delta ^{n} \xrightarrow {\tau '} \Delta ^{m} \rightarrow S_{\bullet }$, where $\tau '$ is not injective on vertices.

Proof. The implications $(1) \Leftrightarrow (2) \Rightarrow (3) \Rightarrow (4) \Rightarrow (5)$ are immediate. We will complete the proof by showing that $(5)$ implies $(1)$. Assume that $\tau $ factors as a composition $\Delta ^{n} \xrightarrow {\tau '} \Delta ^{m} \xrightarrow {\sigma '} S_{\bullet }$, where $\tau '$ is not injective on vertices. Then there exists some integer $0 \leq i < n$ satisfying $\tau '(i) = \tau '(i+1)$. It follows that $\tau '$ factors through the map $\sigma ^{i}_{n-1}: \Delta ^{n} \rightarrow \Delta ^{n-1}$ of Construction 1.1.2.1, so that $\tau $ belongs to the image of the degeneracy operator $s^{n-1}_{i}$. $\square$