# Kerodon

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Corollary 7.5.8.6 (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 categorical equivalence of simplicial sets. Then any two of the following conditions imply the third:

$(1)$

The functor $\overline{\mathscr {F}}$ is a categorical colimit diagram.

$(2)$

The functor $\overline{\mathscr {G}}$ is a categorical colimit diagram.

$(3)$

The natural transformation $\alpha$ induces a categorical 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. By virtue of Proposition 7.5.8.4 (and Proposition 4.5.3.8), it will suffice to show that for every $\infty$-category $\operatorname{\mathcal{D}}$, any two of the following conditions imply the third:

$(1_{\operatorname{\mathcal{D}}} )$

The functor

$(\operatorname{\mathcal{C}}^{\triangleright })^{\operatorname{op}} \rightarrow \operatorname{QCat}\quad \quad C \mapsto \operatorname{Fun}( \overline{\mathscr {F}}(C), \operatorname{\mathcal{D}})$

is a categorical limit diagram.

$(2_{\operatorname{\mathcal{D}}})$

The functor

$(\operatorname{\mathcal{C}}^{\triangleright })^{\operatorname{op}} \rightarrow \operatorname{QCat}\quad \quad C \mapsto \operatorname{Fun}( \overline{\mathscr {G}}(C), \operatorname{\mathcal{D}})$

is a categorical limit diagram.

$(3_{\operatorname{\mathcal{D}}})$

The natural transformation $\alpha$ induces an equivalence of $\infty$-categories $\operatorname{Fun}( \overline{\mathscr {G}}( {\bf 1} ), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \overline{\mathscr {F}}( {\bf 1} ), \operatorname{\mathcal{D}})$.

This follows from Remark 7.5.5.6. $\square$