$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
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.
$\square$