Kerodon

Remark 7.5.1.3 (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$. Then $\alpha $ induces a morphism of weighted nerves $T: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})$, and therefore a morphism of Kan complexes $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha ): \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {G} )$. If $\alpha $ is a levelwise homotopy equivalence, then $T$ is an equivalence of left fibrations over $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Corollary 5.3.3.20), so $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha )$ is a homotopy equivalence.