Pushouts in animae are homotopy pushouts

Remark 1.7.12. For \(C = \An \), these notions recover the classical homotopy pushouts and homotopy pullbacks of algebraic topology. We will show in Section 2.3 that the functor \(\Pi _{\infty }\colon \Top \to \An \) sends homotopy pullbacks of spaces to pullbacks of animae, and homotopy pushouts along relative cell complexes to pushouts of animae, see Proposition 2.3.10, Proposition 2.3.22. This is one of the main ways in which the abstract theory of this chapter connects to classical homotopy theory.