Corollary 7.5.4.11 (03BG): Homotopy Invariance

Corollary 7.5.4.11 (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a category, let $\overline{\mathscr {F}}, \overline{ \mathscr {G} }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be functors, and let $\alpha : \overline{ \mathscr {F} } \rightarrow \overline{ \mathscr {G} }$ be a natural transformation. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{C}: \overline{ \mathscr {F} }(C) \rightarrow \overline{ \mathscr {G} }(C)$ is a weak homotopy equivalence. Then any two of the following conditions imply the third:

$(1)$: The functor $\overline{ \mathscr {F} }$ is a homotopy limit diagram.
$(2)$: The functor $\overline{ \mathscr {G} }$ is a homotopy limit diagram.
$(3)$: The natural transformation $\alpha $ induces a weak homotopy equivalence $\overline{ \mathscr {F} }( {\bf 0} ) \rightarrow \overline{ \mathscr {G} }( {\bf 0} )$, where ${\bf 0}$ denotes the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$.

Proof. Using Proposition 3.1.7.1, we can choose functors $\overline{ \mathscr {F}}', \overline{ \mathscr {G}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ and a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}} \ar [r]^-{\alpha } \ar [d] & \overline{ \mathscr {G}} \ar [d] \\ \overline{ \mathscr {F} }' \ar [r]^-{\alpha '} & \overline{ \mathscr {G} }', } \]

where the vertical maps are levelwise weak homotopy equivalences. Using Proposition 7.5.4.10, we can replace $\alpha $ by the natural transformation $\alpha ': \overline{ \mathscr {F} }' \rightarrow \overline{ \mathscr {G} }'$, in which case the desired result follows from Proposition 7.5.4.4. $\square$