Proposition 7.5.4.4 (03B9): Homotopy Invariance

Proposition 7.5.4.4 (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ be a natural transformation between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$. Assume 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 homotopy equivalence of Kan complexes. 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 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. Setting $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$ and $\mathscr {G} = \overline{\mathscr {G}}|_{\operatorname{\mathcal{C}}}$, we observe that $\alpha $ determines a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}}( {\bf 0} ) \ar [r] \ar [d] & \overline{\mathscr {G}}( {\bf 0} ) \ar [d] \\ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \ar [r] & \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {G}), } \]

where the bottom horizontal map is a homotopy equivalence (Remark 7.5.1.3). The desired result now follows from the two-out-of-three property (Remark 3.1.6.7). $\square$