Lemma 8.1.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma $ be a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, which we identify with a diagram $\varphi : \operatorname{Tw}( \Delta ^2) \rightarrow \operatorname{\mathcal{C}}$. If $\sigma $ is thin, then $\varphi $ is a colimit diagram.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Lemma 8.1.5.1. Let $Q$ denote the partially ordered set appearing in the proof of Lemma 8.1.4.4, so that we can identify $\operatorname{Tw}( \Lambda ^{2}_{1} )$ with the nerve of $Q$. Set $\varphi _0 = \varphi |_{ \operatorname{N}_{\bullet }(Q) }$. Assume that $\sigma $ is thin. We wish to show that the restriction map $\operatorname{\mathcal{C}}_{ \varphi /} \rightarrow \operatorname{\mathcal{C}}_{ \varphi _0/}$ is a trivial Kan fibration: that is, every lifting problem
8.12
\begin{equation} \begin{gathered}\label{equation:thin-in-correspondence2} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{\varphi /} \ar [d] \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}_{\varphi _0/} } \end{gathered} \end{equation}
admits a solution.
Let $K$ denote the coproduct $( \operatorname{Tw}( \Delta ^2) \star \operatorname{\partial \Delta }^ n) {\coprod }_{ ( \operatorname{N}_{\bullet }(Q) \star \operatorname{\partial \Delta }^ n) } (\operatorname{N}_{\bullet }(Q) \star \Delta ^ n)$, which we regard as a simplicial subset of $\operatorname{Tw}( \Delta ^2) \star \Delta ^ n$. Unwinding the definitions, we can identify the lifting problem (8.12) with a morphism of simplicial sets $\tau _0: K \rightarrow \operatorname{\mathcal{C}}$ satisfying $\tau _0|_{ \operatorname{Tw}( \Delta ^2) } = \varphi $. We wish to show that $\tau _0$ can be extended to a morphism $\tau : \operatorname{Tw}( \Delta ^{2} ) \star \Delta ^ n\rightarrow \operatorname{\mathcal{C}}$.
Let $\iota : \operatorname{Tw}( \Delta ^2) \star \Delta ^ n \rightarrow \operatorname{Tw}( \Delta ^{n+3} )$ be the morphism of simplicial sets given on vertices by the formula
\[ \iota (x) = \begin{cases} (i,j) & \text{ if $x = (i,j) \in \operatorname{Tw}( \Delta ^2)$} \\ (0, x+3) & \text{ if $x \in \Delta ^ n$.} \end{cases} \]
The morphism $\iota $ has a left inverse $\rho : \operatorname{Tw}( \Delta ^{n+3}) \rightarrow \operatorname{Tw}(\Delta ^2) \star \Delta ^ n$, given on vertices by the formula
\[ \rho (i,j) = \begin{cases} (i,j) \in \operatorname{Tw}(\Delta ^2) & \text{ if } j \leq 2 \\ j-3 \in \Delta ^ n & \text{ otherwise. } \end{cases} \]
We observe that $\iota $ and $\rho $ restrict to morphisms of simplicial subsets
\[ \iota _0: K \rightarrow \operatorname{Tw}( \Lambda ^{n+3}_{1} ) \quad \quad \rho _0: \operatorname{Tw}( \Lambda ^{n+3}_{1} ) \rightarrow K, \]
so that we have a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{ \iota _0 } \ar [d] & \operatorname{Tw}(\Lambda ^{n+3}_{1}) \ar [r]^-{ \rho _0} \ar [d] & K \ar [d] \\ \operatorname{Tw}( \Delta ^2) \star \Delta ^ n \ar [r]^-{ \iota } & \operatorname{Tw}(\Delta ^{n+3}) \ar [r]^-{\rho } & \operatorname{Tw}( \Delta ^2) \star \Delta ^ n} \]
where the horizontal compositions are equal to the identity.
The composition $\tau _0 \circ \rho _0$ can be identified with a morphism of simplicial sets $\psi _0: \Lambda ^{m+3}_{1} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ having the property that the composition
\[ \Delta ^2 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 < 2\} ) \xrightarrow {\psi _0} \operatorname{Cospan}(\operatorname{\mathcal{C}}) \]
coincides with $\sigma $. Since $\sigma $ is thin, we can extend $\psi _0$ to an $(n+3)$-simplex $\psi $ of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, which we identify with a map $\tau ': \operatorname{Tw}( \Delta ^{n+3} ) \rightarrow \operatorname{\mathcal{C}}$. It follows that the composition $\tau = \tau ' \circ \iota $ is a morphism $\operatorname{Tw}( \Delta ^{2} ) \star \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $\tau |_{K} = \tau _0$. $\square$