$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 5.4.8.4. Let $n \geq 2$ be an integer and let $F: \operatorname{Path}[ \Lambda ^{n}_{n} ]_{\bullet } \rightarrow \operatorname{Path}[ \Delta ^ n ]_{\bullet }$ be the simplicial functor induced by the horn inclusion $\Lambda ^{n}_{n} \hookrightarrow \Delta ^ n$. Then:
- $(a)$
The functor $F$ is bijective on objects; in particular, we can identify the objects of $\operatorname{Path}[ \Lambda ^{n}_{n} ]_{\bullet }$ with elements of the set $[n] = \{ 0 < 1 < \cdots < n \} $.
- $(b)$
For $(0,n-1) \neq (i,j) \neq (0,n)$, the functor $F$ induces an isomorphism of simplicial sets
\[ \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{n} ] }(i, j)_{\bullet } \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(i, j)_{\bullet }. \]
- $(c)$
The functor $F$ induces a monomorphism of simplicial sets
\[ \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{n} ] }(0,n-1)_{\bullet } \hookrightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(0,n-1)_{\bullet }, \]
whose image can be identified with the boundary
\[ \operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-2} \simeq \operatorname{Hom}_{ \operatorname{Path}[\Delta ^ n]}(0,n-1)_{\bullet } \]
introduced in Notation 2.4.5.5.
- $(d)$
The functor $F$ induces a monomorphism of simplicial sets
\[ \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{n} ] }(0,n)_{\bullet } \hookrightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(0,n)_{\bullet }, \]
whose image can be identified with the hollow cube
\[ {\boldsymbol {\sqcup }}^{n-1}_{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[\Delta ^ n]}(0,n)_{\bullet } \]
introduced in Notation 2.4.5.5.
Proof.
Assertion $(a)$ is immediate from Theorem 2.4.4.10. To prove the remaining assertions, fix an integer $m \geq 0$. Using Lemma 2.4.4.16, we see that $\operatorname{Path}[ \Delta ^ n ]_{m}$ can be identified with the path category $\operatorname{Path}[G]$ of a directed graph $G$ which can be described concretely as follows:
The vertices of $G$ are the elements of the set $[n] = \{ 0 < 1 < \cdots < n \} $.
For $0 \leq i < j \leq n$, an edge of $G$ with source $j$ and target $k$ is a chain of subsets
\[ \{ i, i+1, \ldots , j-1, j\} \supseteq I_0 \supseteq \cdots \supseteq I_ m = \{ i, j \} \]
Using Theorem 2.4.4.10, we see that $\operatorname{Path}[ \Lambda ^ n_ n ]_{m}$ 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 $G'$ if and only if the subset $I_0 \subseteq [n]$ corresponds to a simplex of $\Delta ^ n$ which belongs to the horn $\Lambda ^{n}_{n}$: that is, if and only if $[n-1] \nsubseteq I_{0}$. We now argue as follows:
For $(0, n-1) \neq (i,j) \neq (0,n)$, every path from $i$ to $j$ in the graph $G$ is also a path in the graph $G'$. This proves $(b)$.
Let $\tau $ be a morphism from $0$ to $n-1$ in the category $\operatorname{Path}[ n]_{m}$, which we identify with a chain of subsets
\[ [n-1] \supseteq I_0 \supseteq I_1 \supseteq \cdots \supseteq I_{m} \supseteq \{ 0, n-1 \} . \]
Then $\tau $ belongs to $\operatorname{Path}[ \Lambda ^{n}_{n} ]_{m}$ if and only if $I_0 \neq [n-1]$ or $I_ m \neq \{ 0, n-1 \} $: that is, if and only if $\tau $ corresponds to an $m$-simplex of the cube $\operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-2}$. This proves $(c)$.
Let $\tau $ be a morphism from $0$ to $n$ in the category $\operatorname{Path}[n]_ m$, which we identify with a chain of subsets
\[ [n] \supseteq I_0 \supseteq I_1 \supseteq \cdots \supseteq I_{m} \supseteq \{ 0, n \} . \]
Then $\tau $ belongs to $\operatorname{Path}[ \Lambda ^{n}_{n} ]_{m}$ if and only if $I_0 \neq [n]$ or $\{ 0, n \} \neq I_ m \neq \{ 0, n-1, n\} $: that is, if and only if $\tau $ corresponds to an $m$-simplex of the hollow cube ${\boldsymbol {\sqcup }}^{n-1}_{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1}$. This proves $(d)$.
$\square$