Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 1.1.3.1. 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 map $s_{i}: S_{n-1} \rightarrow S_{n}$ for some $0 \leq i \leq n-1$ (see Notation 1.1.1.9).

$(2)$

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 $[n] \rightarrow [n-1]$.

$(3)$

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 $[n] \rightarrow [m]$.

$(4)$

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 $\sigma ^{i}: \Delta ^{n} \rightarrow \Delta ^{n-1}$ of Notation 1.1.1.9, so that $\tau$ belongs to the image of the degeneracy map $s_{i}$. $\square$