Proposition 2.4.6.12 (00M7)

Proposition 2.4.6.12. Let $n$ be a positive integer and let $F: \operatorname{Path}[ \partial \Delta ^ n ]_{\bullet } \rightarrow \operatorname{Path}[ \Delta ^ n ]_{\bullet }$ be the simplicial functor induced by the boundary inclusion $\partial \Delta ^ n \hookrightarrow \Delta ^ n$. Then:

$(a)$: The functor $F$ is bijective on objects; in particular, we can identify objects of $\operatorname{Path}[ \operatorname{\partial \Delta }^ n ]_{\bullet }$ with elements of the set $[n] = \{ 0 < 1 < \cdots < n \} $.
$(b)$: For $(j,k) \neq (0,n)$, the functor $F$ induces an isomorphism of simplicial sets

\[ \operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }(j, k)_{\bullet } \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(j, k)_{\bullet }. \]
$(c)$: The functor $F$ induces a monomorphism of simplicial sets $\operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }(0,n)_{\bullet } \hookrightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(0,n)_{\bullet }$, whose image can be identified with the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[\Delta ^ n]}(0,n)_{\bullet }$ introduced in Notation 2.4.5.5.

Proof. Assertion $(a)$ is immediate from Theorem 2.4.4.10. To prove $(b)$ and $(c)$, fix an integer $m \geq 0$ and let us identify $\operatorname{Path}[ \Delta ^ n ]_{m}$ with the path category $\operatorname{Path}[G]$ of the directed graph $G$ appearing in the proof of Proposition 2.4.5.8. Using Theorem 2.4.4.10, we see that $\operatorname{Path}[ \partial \Delta ^ n ]_{m}$ can be identified with the path category of the directed subgraph $G' \subseteq G$ having the same vertices, where an edge $\overrightarrow {I} = (I_0 \supseteq \cdots \supseteq I_ m)$ of $G$ belongs to $G'$ unless $I_0 = [n]$. In particular, we see that for $(j,k) \neq (0,n)$, every edge of $G$ with source $j$ and target $k$ is contained in $G'$. It follows that the simplicial functor $F$ induces a bijection

\[ \operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }( j, k )_{m} \rightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(j, k)_{m} \]

for $(j, k) \neq (0,n)$, which proves $(b)$. Moreover, the map

\[ \operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }( 0, n )_{m} \rightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(0,n)_{m} \]

is a monomorphism, whose image consists of those chains

\[ \overrightarrow {I} = (I_0 \supseteq I_1 \supseteq \cdots \supseteq I_{m}) \]

where either $I_0 \neq [n]$ or $I_{m} \neq \{ 0, n\} $. Under the identification of $\operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(0,n)_{\bullet }$ with the cube $\operatorname{\raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{N}_{\bullet }( P( \{ 1, \ldots , n-1\} ))$ described in Remark 2.4.5.4, this is exactly the set of $m$-simplices which belong to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1}$. $\square$