Proposition 6.3.4.2. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [dr]^{ F_{01} } \ar [rr]^(.55){ G } \ar [dd]^(.55){H} & & \operatorname{\mathcal{C}}_{0} \ar [dd]^(.55){H'} \ar [dr]^{ F_{0} } & \\ & \operatorname{\mathcal{D}}_{01} \ar [rr] \ar [dd] & & \operatorname{\mathcal{D}}_{0} \ar [dd] \\ \operatorname{\mathcal{C}}_{1} \ar [rr]^(.6){G'} \ar [dr]^{ F_{1} } & & \operatorname{\mathcal{C}}\ar [dr]^{ F } & \\ & \operatorname{\mathcal{D}}_{1} \ar [rr] & & \operatorname{\mathcal{D}}} \]
with the following properties:
- $(a)$
The back face
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r]^-{G} \ar [d]^{H} & \operatorname{\mathcal{C}}_0 \ar [d]^{H'} \\ \operatorname{\mathcal{C}}_{1} \ar [r]^-{G'} & \operatorname{\mathcal{C}}} \]
is a categorical pushout square of simplicial sets.
- $(b)$
The morphism of simplicial sets $F_{01}: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{D}}_{01}$ exhibits $\operatorname{\mathcal{D}}_{01}$ as a localization of $\operatorname{\mathcal{C}}_{01}$ with respect to some collection of edges $W_{01}$.
- $(c)$
The morphism of simplicial sets $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ exhibits $\operatorname{\mathcal{D}}_0$ as a localization of $\operatorname{\mathcal{C}}_0$ with respect to some collection of edges $W_0$ containing $G(W_{01})$.
- $(d)$
The morphism of simplicial sets $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{D}}_1$ exhibits $\operatorname{\mathcal{D}}_1$ as a localization of $\operatorname{\mathcal{C}}_1$ with respect to some collection of edges $W_1$ containing $H(W_{01})$.
Then the following conditions are equivalent:
- $(1)$
The front face
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{D}}_0 \ar [d] \\ \operatorname{\mathcal{D}}_1 \ar [r] & \operatorname{\mathcal{D}}} \]
is a categorical pushout square of simplicial sets.
- $(2)$
The morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to the collection of edges $W = H'( W_0 ) \cup G'( W_1)$.
Proof.
Let $\operatorname{\mathcal{E}}$ be an $\infty $-category. Assumption $(a)$ guarantees that the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}})^{\simeq } \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{C}}_1, \operatorname{\mathcal{E}})^{\simeq } \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{01}, \operatorname{\mathcal{E}})^{\simeq } } \]
is a homotopy pullback square. Applying Proposition 3.4.1.14, we deduce that the diagram of summands
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}_0[W^{-1}_0], \operatorname{\mathcal{E}})^{\simeq } \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{C}}_1[W_1^{-1}], \operatorname{\mathcal{E}})^{\simeq } \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{01}[W_{01}^{-1}], \operatorname{\mathcal{E}})^{\simeq } } \]
is also a homotopy pullback square. Invoking Corollary 3.4.1.12, we conclude that the following conditions are equivalent:
- $(1_{\operatorname{\mathcal{E}}})$
The diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}})^{\simeq } \ar [d] \\ \operatorname{Fun}(\operatorname{\mathcal{D}}_1, \operatorname{\mathcal{E}})^{\simeq } \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } } \]
is a homotopy pullback square.
- $(2_{\operatorname{\mathcal{E}}})$
Precomposition with $F$ induces a homotopy equivalence of Kan complexes
\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } \xrightarrow {\circ F} \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq }. \]
We now observe that condition $(1)$ is equivalent to the requirement that $(1_{\operatorname{\mathcal{E}}} )$ holds for every $\infty $-category $\operatorname{\mathcal{E}}$ (by definition), and condition $(2)$ is equivalent to the requirement that $(2_{\operatorname{\mathcal{E}}})$ holds for every $\infty $-category $\operatorname{\mathcal{E}}$ (Proposition 6.3.1.13).
$\square$