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Theorem 5.2.5.1. Let $\pi : \operatorname{\mathcal{M}}\rightarrow \Delta ^1$ be a functor of $\infty $-categories, and suppose we are given a commutative diagram $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r] \ar [d] & \{ 1\} \times \operatorname{\mathcal{D}}\ar [d]^-{u} \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r]^-{h} & \operatorname{\mathcal{M}}} \]

in the category $(\operatorname{Set_{\Delta }})_{/\Delta ^1}$. Then $\sigma $ is a categorical pushout diagram of simplicial sets (Definition 4.5.3.1) if and only if the following conditions are satisfied:

$(1)$

The restriction $h|_{ \{ 0\} \times \operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ is a categorical equivalence of simplicial sets.

$(2)$

The morphism $u: \operatorname{\mathcal{D}}\rightarrow \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ is a categorical equivalence of simplicial sets.

$(3)$

For every vertex $C \in \operatorname{\mathcal{C}}$, the restriction $h|_{ \Delta ^1 \times \{ C\} }$ is a $\pi $-cocartesian morphism of $\operatorname{\mathcal{M}}$.

Moreover, if these conditions are satisfied, then $\pi $ is a cocartesian fibration.

Proof of Theorem 5.2.5.1 from Proposition 5.2.5.4. Let $\pi : \operatorname{\mathcal{M}}\rightarrow \Delta ^1$ be a functor of $\infty $-categories and suppose we are given a commutative diagram $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d] & \{ 1\} \times \operatorname{\mathcal{D}}\ar [d]^-{u} \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r]^-{h} & \operatorname{\mathcal{M}}} \]

in the category $(\operatorname{Set_{\Delta }})_{/\Delta ^1}$. We wish to show that $\sigma $ is a categorical pushout square if and only if conditions $(1)$ through $(4)$ of Theorem 5.2.5.1 are satisfied.

We first reduce to the case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories. Choose inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{D}}'$, where $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{D}}'$ are $\infty $-categories (Corollary 4.1.3.3). Since the fiber $\{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ is an $\infty $-category, we can extend $u$ to a functor $u': \operatorname{\mathcal{D}}' \rightarrow \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$. Similarly, the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{D}}'$ extends to a functor of $\infty $-categories $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}'$. Using Exercise 3.1.7.11, we can factor $F'$ as a composition $\operatorname{\mathcal{C}}' \xrightarrow {F''} \operatorname{\mathcal{D}}'' \xrightarrow {v} \operatorname{\mathcal{D}}'$, where $F''$ is a monomorphism and $v$ is a trivial Kan fibration. It follows from Lemma 1.4.7.5 that the inclusion map

\[ ( \Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ ( \{ 1\} \times \operatorname{\mathcal{C}}) } (\{ 1\} \times \operatorname{\mathcal{C}}' ) \hookrightarrow \Delta ^1 \times \operatorname{\mathcal{C}}' \]

is inner anodyne, so that we can extend $h$ to a functor $h': \Delta ^1 \times \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{M}}$ satisfying $h'|_{ \{ 1\} \times \operatorname{\mathcal{C}}'} = u' \circ F'$. By virtue of Proposition 4.5.3.6, $\sigma $ is a categorical pushout square if and only if the diagram $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}' \ar [r]^-{F''} \ar [d] & \{ 1\} \times \operatorname{\mathcal{D}}'' \ar [d]^-{u' \circ v} \\ \Delta ^1 \times \operatorname{\mathcal{C}}' \ar [r]^-{h'} & \operatorname{\mathcal{M}}} \]

is a categorical pushout square. We may therefore replace $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ by $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{D}}''$, and thereby reduce to the case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories and $F$ is a monomorphism.

The assumption that $F$ is a monomorphism guarantees that the natural map

\[ \iota : (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ ( \{ 1\} \times \operatorname{\mathcal{C}})} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}} \]

is also a monomorphism, and Proposition 5.2.5.4 guarantees that $\iota $ is a categorical equivalence of simplicial sets. Since $\operatorname{\mathcal{M}}$ is an $\infty $-category, Lemma 4.5.6.2 guarantees the existence of a functor $U: \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{M}}$ satisfying $U|_{ \Delta ^1 \times \operatorname{\mathcal{C}}} = h$ and $U|_{\operatorname{\mathcal{D}}} = u$. By virtue of Proposition 4.5.3.6, the diagram $\sigma $ is a categorical pushout square if and only if the functor $U$ is an equivalence of $\infty $-categories.

Note that the functor $U$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [rr]^{U} \ar [dr]^{\pi '} & & \operatorname{\mathcal{M}}\ar [dl]^{\pi } \\ & \Delta ^1, & } \]

where $\pi '$ is the cocartesian fibration of Proposition 5.2.4.15, and the functor $\pi $ is automatically an isofibration (Example 4.4.1.6). The desired result now follows by applying the criterion of Theorem 5.1.5.1 (and invoking Remark 5.1.5.8). $\square$