Kerodon

Example 7.7.5.3. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces. Then $\operatorname{\mathcal{S}}$ satisfies both the first and second Mather cube theorem. To prove this, we can assume without loss of generality that (7.88) arises from a commutative diagram in the ordinary category of Kan complexes (Corollary 5.6.5.18). In this case, a face of the diagram (7.88) is a pullback square in $\operatorname{\mathcal{S}}$ if and only if it is a homotopy pullback square (Example 7.6.3.2), and a pushout square in $\operatorname{\mathcal{S}}$ if and only if it is a homotopy pushout square (Example 7.6.3.3). The first Mather cube theorem is now a restatement of Theorem 3.4.4.4, while the second cube theorem is a restatement of Theorem 3.4.3.3.