Kerodon

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Corollary 8.1.4.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ admit pushouts. The following conditions are equivalent:

$(1)$

The functor $F$ carries pushout squares in $\operatorname{\mathcal{C}}$ to pushout squares in $\operatorname{\mathcal{D}}$.

$(2)$

The induced map $\operatorname{Cospan}(F): \operatorname{Cospan}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$ is a functor of $(\infty ,2)$-categories, in the sense of Definition 5.4.7.1.

Proof. The implication $(1) \Rightarrow (2)$ follows immediately from the criterion of Proposition 8.1.4.2. For the converse implication, suppose that $\operatorname{Cospan}(F): \operatorname{Cospan}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$ is a functor of $2$-categories, and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a pushout square, which we display as a diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X_{01}. }$

Let $\rho : \operatorname{Tw}( \Delta ^2 ) \rightarrow \Delta ^1 \times \Delta ^1$ denote the morphism of simplicial sets given on vertices by the formula $\rho (i,j) = (\max (0, 1-i) , \max (0,j-1) )$. Then $\sigma \circ \rho$ can be identified with a $2$-simplex $\tau$ of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, corresponding to a diagram

$\xymatrix@R =50pt@C=50pt{ X_0 \ar [dr]_{\operatorname{id}} & & X \ar [dl] \ar [dr] & & X_1 \ar [dl]^{\operatorname{id}} \\ & X_0 \ar [dr] & & X_1 \ar [dl] \\ & & X_{01} & & }$

in the $\infty$-category $\operatorname{\mathcal{C}}$. It follows from the criterion of Proposition 8.1.4.2 that $\tau$ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\operatorname{Cospan}(F)$ is a functor of $(\infty ,2)$-categories, then it carries $\tau$ to a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{D}})$. Applying the criterion of Proposition 8.1.4.2 again, we conclude that $F(\sigma )$ is a pushout square in $\operatorname{\mathcal{D}}$. $\square$