Proposition 4.5.2.19 (Homotopy Invariance). Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [dr]^{ F_{01} } \ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}_{0} \ar [dd] \ar [dr]^{F_0} & \\ & \operatorname{\mathcal{D}}_{01} \ar [rr] \ar [dd] & & \operatorname{\mathcal{D}}_{0} \ar [dd] \\ \operatorname{\mathcal{C}}_{1} \ar [rr] \ar [dr]^{ F_1 } & & \operatorname{\mathcal{C}}\ar [dr]^{F} & \\ & \operatorname{\mathcal{D}}_{1} \ar [rr] & & \operatorname{\mathcal{D}}, } \]
where $F_0$, $F_1$, and $F$ are equivalences of $\infty $-categories. Then any two of the following conditions imply the third:
- $(1)$
The back face
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}} \]is a categorical pullback square.
- $(2)$
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 pullback square.
- $(3)$
The functor $F_{01}$ is an equivalence of $\infty $-categories.