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Corollary 5.4.8.5. Let $\operatorname{\mathcal{C}}$ be a simplicial category, let $n \geq 2$ be an integer, and let $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets, which we identify with a simplicial functor $F: \operatorname{Path}[ \Lambda ^{n}_{n} ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ inducing a map of simplicial sets

\[ \lambda _0: \boldsymbol {\sqcup }^{n-1}_{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^ n_ n] }( 0, n)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }. \]

Suppose that $F$ carries the edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Lambda ^{n}_{n}$ to an isomorphism in $\operatorname{\mathcal{C}}$. Then the restriction map

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Maps $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ with $\sigma _0 = \sigma |_{ \Lambda ^ n_{n}}$} \} \ar [d]^-{\theta } \\ \{ \textnormal{Maps $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$ with $\lambda _0 = \lambda |_{ \boldsymbol {\sqcup }^{n-1}_{n-1}}$} \} } \]

is bijective.

Proof. By virtue of Corollary 2.4.6.13, we can identify $\theta $ with a pullback of the restriction map

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Maps $\sigma _1: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ with $\sigma _0 = \sigma _1|_{ \Lambda ^ n_{n}}$} \} \ar [d]^-{\theta '} \\ \{ \textnormal{Maps $\lambda _1: \operatorname{\partial \raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$ with $\lambda _0 = \lambda _1|_{ \boldsymbol {\sqcup }^{n-1}_{n-1}}$} \} .} \]

It will therefore suffice to show that $\theta '$ is bijective. Let us identify $\Delta ^{n-1}$ with a simplicial subset of $\Delta ^ n$ (via the map which is the identity on vertices), so that the boundary $\operatorname{\partial \Delta }^{n-1}$ is contained in the horn $\Lambda ^{n}_{n}$. Let $\tau _0$ denote the restriction of $\sigma _0$ to $\operatorname{\partial \Delta }^{n-1}$, let $\mu _0$ denote the $\lambda _0$ to the simplicial subset $\operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \times \{ 0\} \subseteq \boldsymbol {\sqcup }^{n-1}_{n-1}$. Note that $\mu _0$ can be written as a composition

\[ \operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \simeq \operatorname{Hom}_{ \operatorname{Path}[ \operatorname{\partial \Delta }^{n-1}]}(0,n-1)_{\bullet } \xrightarrow {\nu _0} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n-1) )_{\bullet } \xrightarrow { e \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }, \]

where $\nu _0$ is determined by $\tau _0$. Using the identifications

\[ \operatorname{\partial \Delta }^{n} \simeq \Delta ^{n-1} \coprod _{ \operatorname{\partial \Delta }^{n-1} } \Lambda ^{n}_{n} \quad \quad \operatorname{\partial \raise {0.1ex}{\square }}^{n-1} \simeq ( \operatorname{\raise {0.1ex}{\square }}^{n-2} \times \{ 0\} ) \coprod _{ ( \operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \times \{ 0\} ) } \boldsymbol {\sqcup }^{n-1}_{n-1}, \]

we can identify $\theta '$ with composition

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Maps $\tau : \Delta ^{n-1} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ with $\tau _0 = \tau |_{ \operatorname{\partial \Delta }^ n}$} \} \ar [d] \\ \{ \textnormal{Maps $\nu : \operatorname{\raise {0.1ex}{\square }}^{n-2} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n-1) )_{\bullet }$ with $\nu = \nu _0|_{ \operatorname{\partial \raise {0.1ex}{\square }}^{n-2}}$} \} \ar [d]^-{ e \circ } \\ \{ \textnormal{Maps $\mu : \operatorname{\raise {0.1ex}{\square }}^{n-2} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$ with $\mu = \mu _0|_{ \operatorname{\partial \raise {0.1ex}{\square }}^{n-2}}$} \} . } \]

Here the first map is bijective by virtue of Corollary 2.4.6.13, and the second by virtue of our assumption that $e$ is an isomorphism in the simplicial category $\operatorname{\mathcal{C}}$. $\square$