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Proposition 2.4.5.8. Let $0 < i < n$ be positive integers and let $F: \operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet } \rightarrow \operatorname{Path}[ \Delta ^ n ]_{\bullet }$ be the simplicial functor induced by the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$. Then:

$(a)$

The functor $F$ is bijective on objects; in particular, we can identify the objects of $\operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet }$ 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

\[ \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{i} ] }(j, k)_{\bullet } \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(j, k)_{\bullet }. \]
$(c)$

The functor $F$ induces a monomorphism of simplicial sets

\[ \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{i} ] }(0,n)_{\bullet } \hookrightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(0,n)_{\bullet }, \]

whose image can be identified with the hollow cube

\[ {\boldsymbol {\sqcap }}^{n-1}_{i} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[\Delta ^ n]}(0,n)_{\bullet } \]

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 $m \geq 0$. 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 $0 \leq j < k \leq n$, an edge of $G$ with source $j$ and target $k$ is a chain of subsets

    \[ \{ j, j+1, \ldots , k-1, k\} = I_0 \supseteq \cdots \supseteq I_ m = \{ j, k \} \]

Using Theorem 2.4.4.10, we see that $\operatorname{Path}[ \Lambda ^ n_ i ]_{m}$ 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 $\Lambda ^{n}_{i}$: that is, if and only if $[n] \setminus \{ i\} \nsubseteq I_{0}$. 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 $\operatorname{Hom}_{ \operatorname{Path}[\Lambda ^{n}_{i} ] }( j, k )_{m} \rightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(j, k)_{m}$ for $(j, k) \neq (0,n)$, which proves $(b)$. Moreover, the map $\operatorname{Hom}_{ \operatorname{Path}[\Lambda ^{n}_{i} ] }( 0, n )_{m} \rightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(0,n)_{m}$ is a monomorphism, whose image consists of those chains

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

where either $I_ m \neq \{ 0,n\} $ or $([n] \setminus \{ i\} ) \nsubseteq I_0$. 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 corresponds to collection of $m$-simplices of $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ given by chains of subsets

\[ J_0 \subseteq J_1 \subseteq \cdots \subseteq J_{m} \subseteq \{ 1, \ldots , n-1 \} \]

where either $J_0 \nsubseteq \{ i\} $ or $J_ m \subsetneq \{ 1, \ldots , n-1 \} $, which is exactly the set of $m$-simplices which belong to the hollow cube $\boldsymbol {\sqcap }^{n-1}_{i}$. $\square$