# Kerodon

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

Proposition 8.1.6.2. 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

8.10
$$\begin{gathered}\label{equation:thin-in-correspondence} \xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & & X_{2,2} \ar [dl] \\ & X_{0,1} \ar [dr] & & X_{1,2} \ar [dl] & \\ & & X_{0,2} & & } \end{gathered}$$

in the $\infty$-category $\operatorname{\mathcal{C}}$. Then $\sigma$ is thin (in the sense of Definition 2.3.2.3) if and only if the inner region is a pushout square in the $\infty$-category $\operatorname{\mathcal{C}}$.