# Kerodon

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

Example 8.1.4.15. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. Then $2$-simplices $\sigma$ of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) )$ can be identified with commutative diagrams

$\xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & & X_{2,2} \ar [dl] \\ & X_{0,1} \ar [dr] & & X_{1,2} \ar [dl] & \\ & & X_{0,2} & & }$

in the category $\operatorname{\mathcal{C}}$. It follows from Theorem 2.3.2.5 that the $2$-simplex $\sigma$ is thin (in the sense of Definition 2.3.2.3) if and only if the square appearing in the diagram is a pushout: that is, it induces an isomorphism $X_{0,1} \amalg _{ X_{1,1} } X_{1,2} \rightarrow X_{0,2}$ in the category $\operatorname{\mathcal{C}}$.