Proposition 7.5.7.5 (03CS): Homotopy Invariance

Proposition 7.5.7.5 (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 $\overline{\mathscr {F}}, \overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$. 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 weak homotopy equivalence of simplicial sets. Then any two of the following conditions imply the third:

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

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 simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [r] \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \ar [d] \\ \overline{\mathscr {F}}( {\bf 1} ) \ar [r] & \overline{\mathscr {G}}( {\bf 1} ) } \]

where the upper horizontal map is a weak homotopy equivalence (Proposition 5.3.2.18). The desired result now follows from the two-out-of-three property (Remark 3.1.6.16). $\square$