Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 5.4.8.2. Let $\operatorname{\mathcal{C}}$ be a simplicial category. Suppose that, for every pair of objects $X$ and $Y$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Let $\sigma $ be a $2$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which we identify with a (not necessarily commutative) diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z } \]

in $\operatorname{\mathcal{C}}$ together with an edge $\mu : g \circ f \rightarrow h$ in the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$. If $\mu $ is an isomorphism, then $\sigma $ is thin.

Proof. Suppose we are given integers $n \geq 3$, $0 < i < n$, and a morphism of simplicial sets $\tau _0: \Lambda ^{n}_{i} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ for which the restriction $\tau _0|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is the $2$-simplex $\sigma $. We wish to show that $\tau _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$. Let $\operatorname{Path}[n]_{\bullet }$ be the simplicial category described in Notation 2.4.3.1, and let us identify $\operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet }$ with the simplicial subcategory of $\operatorname{Path}[n]_{\bullet }$ described in Proposition 2.4.5.8. Then $\tau _0$ can be identified with a simplicial functor $F_0: \operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$, and we wish to show that $\tau _0$ can be extended to a simplicial functor $F: \operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$.

For $0 \leq j \leq n$, let $C_ j$ denote the object of $\operatorname{\mathcal{C}}$ given by $F_0(j)$. For $1 \leq j \leq n$, let $u_{j}: C_{j-1} \rightarrow C_ j$ be the morphism in $\operatorname{\mathcal{C}}$ obtained by applying $F_0$ to the unique vertex of $\operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{i} ] }( j-1, j)$, so that we have a chain of composable morphisms

\[ C_0 \xrightarrow {u_1} C_1 \xrightarrow {u_2} \cdots \xrightarrow {u_ n} C_ n \]

in the simplicial category $\operatorname{\mathcal{C}}$. Let $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ denote the simplicial cube of dimension $(n-1)$ and let ${\boldsymbol {\sqcap }}^{n-1}_{i} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1}$ denote the hollow cube of Notation 2.4.5.5, so that Remark 2.4.5.4 and Proposition 2.4.5.8 supply isomorphisms

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

Let $\lambda _0$ denote the composite map

\[ {\boldsymbol {\sqcap }}^{n-1}_{i} \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{i} ] }(0, n)_{\bullet } \xrightarrow {F_0} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_ n)_{\bullet }. \]

By virtue of Corollary 2.4.5.10, it will suffice to show that $\lambda _0$ can be extended to a morphism of simplicial sets $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet }$.

Let $I$ denote the set $\{ 1, 2, \ldots , i-1, i+1, \cdots , n-1 \} $, so that we can identify $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ with the product $\Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I}$. Under this identification, ${\boldsymbol {\sqcap }}^{n-1}_{i}$ corresponds to the pushout

\[ (\Delta ^1 \times \operatorname{\partial \raise {0.1ex}{\square }}^{I}) \coprod _{ (\{ 0\} \times \operatorname{\partial \raise {0.1ex}{\square }}^{I}) } ( \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I} ). \]

Let $v \in \operatorname{\raise {0.1ex}{\square }}^{I}$ be the initial vertex (corresponding to the empty subset of $I$), and let $e$ be the edge of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet }$ given by the composite map

\[ \Delta ^1 \times \{ v\} \hookrightarrow \Delta ^1 \times \operatorname{\partial \raise {0.1ex}{\square }}^{I} \hookrightarrow {\boldsymbol {\sqcap }}^{n-1}_{i} \xrightarrow {\lambda _0} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet }. \]

Unwinding the definitions, we see that $e$ is the image of $\mu $ under the morphism of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_{i-1}, C_{i+1})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet } \quad \quad \rho \mapsto u_ n \circ u_{n-1} \circ \cdots u_{i+2} \circ \rho \circ u_{i-1} \circ \cdots \circ u_1, \]

and is therefore an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet }$. Note that every simplex of $\operatorname{\raise {0.1ex}{\square }}^{I}$ which is not contained in the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^{I}$ has initial vertex $v$. The existence of the desired extension $\lambda $ now follows from Proposition 4.4.5.8. $\square$