Proposition 9.4.5.10 (05MJ)

Proposition 9.4.5.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$: The morphism $F$ is a Morita equivalence.
$(2)$: The morphism $F^{\operatorname{op}}$ is a Morita equivalence.
$(3)$: For every uncountable cardinal $\kappa $, precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
$(4)$: There exists an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\kappa $-small and precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.

Proof. The implications $(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (4)$ are immediate. We complete the proof by showing that $(4)$ implies $(1)$. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories. Fix a regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\kappa $-small and assume that precomposition with $F$ induces an equivalence of $\infty $-categories

\[ G: \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ). \]

Let $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be the full subcategory spanned by the atomic object (see Proposition 8.5.5.5) and define $\widehat{\operatorname{\mathcal{D}}} \subseteq \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}^{< \kappa } )$ similarly. Then $G$ induces an equivalence of $\infty $-categories $\widehat{\operatorname{\mathcal{D}}} \rightarrow \widehat{\operatorname{\mathcal{C}}}$, which admits a homotopy inverse $\widehat{F}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{D}}}$. It follows from Example 8.4.4.5 that the diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{h^{\bullet }_{\operatorname{\mathcal{C}}}} & \operatorname{\mathcal{D}}\ar [d]^{h^{\bullet }_{\operatorname{\mathcal{D}}} } \\ \widehat{\operatorname{\mathcal{C}}} \ar [r]^-{ \widehat{F} }_{\sim } & \widehat{\operatorname{\mathcal{D}}} } \]

commutes up to isomorphism. The vertical maps exhibit $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{D}}}$ as idempotent completions of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively (Proposition 8.5.5.5), and are therefore Morita equivalences (Example 9.4.5.3). Combining this observation with Remarks 9.4.5.6 and 9.4.5.5, we conclude that $F$ is a Morita equivalence. $\square$