# Kerodon

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Remark 2.4.4.17. Let $S_{\bullet }$ be a simplicial set. For each $m \geq 0$, Theorem 2.4.4.10 guarantees that $\operatorname{Path}[S]_ m$ can be realized as the path category of a directed graph $G_ m$ (Construction 1.2.6.1), which can be described explicitly as follows:

• The vertices of $G_{m}$ are the vertices of the simplicial set $S_{\bullet }$.

• The edges of $G_{m}$ are the elements of the set $E(S,m)$ of Notation 2.4.4.9.

It follows that we can regard the construction $[m] \mapsto \operatorname{Path}[ G_ m ]$ as a simplicial object of $\operatorname{Cat}$. The face and degeneracy operators on this simplicial object can be described as follows:

• For $0 \leq i \leq m$, the degeneracy operator $s_{i}: \operatorname{Path}[ G_ m ] \rightarrow \operatorname{Path}[ G_{m+1} ]$ is induced by a map of directed graphs from $G_ m$ to $G_{m+1}$, which is the identity on vertices and given on edges by the construction

$(\sigma , I_0 \supseteq \cdots \supseteq I_{m} ) \mapsto (\sigma , I_{0} \supseteq \cdots \supseteq I_{i-1} \supseteq I_{i} \supseteq I_{i} \supseteq I_{i+1} \supseteq \cdots \supseteq I_ m ).$
• For $0 < i < m$, the face map $d_{i}: \operatorname{Path}[G_ m] \rightarrow \operatorname{Path}[G_{m-1} ]$ is induced by a map of directed graphs from $G_{m}$ to $G_{m-1}$, which is the identity on vertices and given on edges by the construction

$(\sigma , I_0 \supseteq \cdots \supseteq I_{m} ) \mapsto (\sigma , I_{0} \supseteq \cdots \supseteq I_{i-1} \supseteq I_{i+1} \supseteq \cdots \supseteq I_ m ).$
• Each of the face maps $d_{0}: \operatorname{Path}[G_ m] \rightarrow \operatorname{Path}[G_{m-1}]$ is induced by a morphism directed graphs $f: G_ m \rightarrow G_{m-1}$ which is the identity on vertices. Let $(\sigma , \overrightarrow {I})$ be an edge of $G_{m}$, given by a nondegenerate simplex $\sigma : \Delta ^ n \rightarrow S_{\bullet }$ and a chain of subsets $\overrightarrow {I} = (I_0 \supseteq \cdots \supseteq I_{m})$ of $[n]$. Then the subset $I_{1} \subseteq I_{0} = [n]$ is the image of a unique monotone injection $\alpha : [n'] \hookrightarrow [n]$, and the composite map $\Delta ^{n'} \xrightarrow {\alpha } \Delta ^{n} \xrightarrow {\sigma } S_{\bullet }$ factors uniquely as a composition $\Delta ^{n'} \twoheadrightarrow \Delta ^{n''} \xrightarrow {\tau } S_{\bullet }$, where the first map is surjective on vertices and $\tau$ is a nondegenerate $n''$-simplex of $S_{\bullet }$. For $0 \leq i < m$, let $J_{i} \subseteq [n'']$ denote the image of the composite map $I_{i+1} \hookrightarrow I_{1} \simeq [n'] \twoheadrightarrow [n'']$, and set $\overrightarrow {J} = (J_{0} \supseteq J_1 \supseteq \cdots \supseteq J_{m-1} )$. In the case $n'' = 0$, the morphism $f$ carries $(\sigma , \overrightarrow {I} )$ to the vertex $\tau \in \operatorname{Vert}(G_{m-1} )$. In the case $n'' > 0$ the morphism $f$ carries $(\sigma , \overrightarrow {I} )$ to the edge $( \tau , \overrightarrow {J}) \in \operatorname{Edge}( G_{m-1})$.

• The face maps $d_ m: \operatorname{Path}[G_ m] \rightarrow \operatorname{Path}[G_{m-1} ]$ are generally not induced by maps of directed graphs $G_{m} \rightarrow G_{m-1}$: that is, they do not carry indecomposable morphisms of $\operatorname{Path}[G_ m]$ to indecomposable morphisms of $\operatorname{Path}[G_{m-1}]$. More precisely, if $(\sigma , \overrightarrow {I} )$ is an edge of $G_{n}$ with $I_{m-1} = \{ 0 = i_0 < i_1 < \cdots < i_ k = m \}$, then $d_0$ carries $(\sigma , \overrightarrow {I} )$ to a path of length $k$ in the category $\operatorname{Path}[ G_{m-1} ]$.