Definition 7.6.2.12 (03FT)

Definition 7.6.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will say that $\operatorname{\mathcal{C}}$ admits pullbacks if, for every pair of morphisms $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ having the same target, there exists a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]

We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves pullbacks if, for every pullback square $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the composition $(F \circ \sigma ): \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{D}}$ is a pullback square in the $\infty $-category $\operatorname{\mathcal{D}}$.

We say that $\operatorname{\mathcal{C}}$ admits pushouts if, for every pair of morphisms $g_0: Y \rightarrow Y_0$ and $g_1: Y \rightarrow Y_1$ having the same source, there exists a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ Y \ar [r]^-{g_0} \ar [d]^-{g_1} & Y_0 \ar [d] \\ Y_1 \ar [r] & Y_{01}. } \]

We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves pushouts if, for every pushout square $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the composition $(F \circ \sigma ): \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{D}}$ is a pushout square in the $\infty $-category $\operatorname{\mathcal{D}}$.