Definition 9.4.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. We say that $F$ is a Morita equivalence if, for every idempotent complete $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
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9.4.5 Digression: Morita Equivalence
Recall that a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a categorical equivalence if, for every $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ (Definition 4.5.3.1). We now consider a slightly weaker version of this condition.
Example 9.4.5.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. If $F$ is a categorical equivalence, then it is a Morita equivalence. Beware that the converse is false in general.
Example 9.4.5.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$. Then $F$ is a Morita equivalence (Proposition 8.5.5.2).
Remark 9.4.5.4. Let $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ be a monomorphism of simplicial sets and let $\operatorname{\mathcal{E}}$ be an idempotent complete $\infty $-category. If $\iota $ is a Morita equivalence, then the functor $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ \iota } \operatorname{Fun}(\operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}})$ is both an isofibration (Corollary 4.4.5.3) and an equivalence of $\infty $-categories, and therefore a trivial Kan fibration (Proposition 4.5.5.20). In particular, every diagram $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{E}}$ can be extended to $\operatorname{\mathcal{C}}$.
Remark 9.4.5.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. If two of the morphisms $F$, $G$, and $G \circ F$ are Morita equivalences, then so is the third. In particular, the collection of Morita equivalences is closed under composition.
Remark 9.4.5.6 (Isomorphism Invariance). Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\operatorname{\mathcal{D}}$ be an $\infty $-category, and suppose we are given a pair of diagrams $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then $F$ is a Morita equivalence if and only if $F'$ is a Morita equivalence.
Remark 9.4.5.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ is a Morita equivalence if and only if it satisfies the following condition:
For every idempotent complete $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection of sets
\[ \pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ). \]
The necessity of condition $(\ast )$ is immediate. Conversely, suppose that condition $(\ast )$ is satisfied, and let $\operatorname{\mathcal{E}}$ be an idempotent complete $\infty $-category; we wish to show that the functor $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is an equivalence of $\infty $-categories. By virtue of Proposition 4.5.1.22, it will suffice to show that for every simplicial set $K$, the induced map
\[ \pi _0( \operatorname{Fun}(K, \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( K, \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ) \]
is a bijection. This follows by applying condition $(\ast )$ to the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{E}})$ (which is idempotent complete by virtue of Corollary 8.5.4.10).
Proposition 9.4.5.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is a Morita equivalence if and only if it satisfies the following pair of conditions:
The functor $F$ is fully faithful.
For every object $Y \in \operatorname{\mathcal{D}}$, there exists an object $X \in \operatorname{\mathcal{C}}$ such that $Y$ is a retract of $F(X)$.
Proof. Using Corollary 8.5.5.3, we can choose a functor $H: \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$ which exhibits $\widehat{\operatorname{\mathcal{D}}}$ as an idempotent completion of $\operatorname{\mathcal{D}}$. Then $H$ is a Morita equivalence (Example 9.4.5.3). By virtue of Remark 9.4.5.5, we can replace $F$ by the composite functor $H \circ F$ and thereby reduce to proving Proposition 9.4.5.8 in the special case where $\operatorname{\mathcal{D}}$ is idempotent complete. In this case, the desired result is a reformulation of Proposition 8.5.5.2. $\square$
Corollary 9.4.5.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ is a categorical equivalence if and only if it is a Morita equivalence and the induced map of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is essentially surjective.
Proof. Using Remark 9.4.5.5 (and Proposition 4.1.3.2), we can reduce to the case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, in which case the desired result follows from the criterion of Proposition 9.4.5.8. $\square$
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:
The morphism $F$ is a Morita equivalence.
The morphism $F^{\operatorname{op}}$ is a Morita equivalence.
For every uncountable cardinal $\kappa $, precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
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. $\square$
Proposition 9.4.5.11. Suppose we are given a commutative diagram of simplicial sets with the following properties:
The morphisms $U$ and $U'$ are cartesian fibrations.
The morphism $F$ carries $U$-cartesian edges of $\operatorname{\mathcal{E}}$ to $U'$-cartesian edges of $\operatorname{\mathcal{E}}'$.
The morphism $\overline{F}$ is a Morita equivalence of simplicial sets
For each vertex $C \in \operatorname{\mathcal{C}}$ having image $C' = \overline{F}(C)$, the functor
is a Morita equivalence of $\infty $-categories.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{F} \ar [d]^{U} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}' } \]
\[ F_ C: \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \{ C' \} \times _{ \operatorname{\mathcal{C}}' } \operatorname{\mathcal{E}}' = \operatorname{\mathcal{E}}'_{ C' } \]
Then $F$ is a Morita equivalence of simplicial sets.
Proof. Using Corollary 5.6.7.3, we can choose a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r]^-{G} \ar [d]^{U'} & \operatorname{\mathcal{E}}'' \ar [d]^{ U'' } \\ \operatorname{\mathcal{C}}' \ar [r]^-{\overline{G}} & \operatorname{\mathcal{C}}'', } \]
where $U''$ is a cartesian fibration, $\overline{G}$ is inner anodyne, and $\operatorname{\mathcal{C}}''$ is an $\infty $-category. It follows from Corollary 5.6.7.6 that $G$ is a categorical equivalence of simplicial sets; in particular, it is a Morita equivalence (Example 9.4.5.2). Consequently, to show that $F$ is a Morita equivalence, it will suffice to show that the composite map $G \circ F$ is a Morita equivalence (Remark 9.4.5.5). We may therefore replace $U'$ by $U''$, and thereby reduce to proving Proposition 9.4.5.11 in the special case where $\operatorname{\mathcal{C}}'$ is an $\infty $-category. Similarly, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty $-category. To complete the proof, it will suffice to show that the functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ satisfies the conditions of Proposition 9.4.5.8:
- $(a)$
Since $\overline{F}$ is a Morita equivalence of $\infty $-categories, it is fully faithful. Similarly, for each object $C \in \operatorname{\mathcal{C}}$ having image $C' = \overline{F}(C)$, condition $(4)$ guarantees that the functor $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C'}$ is fully faithful. Using condition $(2)$ and Proposition 5.1.6.7, we conclude that $F$ is fully faithful.
- $(b)$
Let $Y$ be an object of $\operatorname{\mathcal{E}}'$; we wish to show that $Y$ is a retract of $F(X)$, for some object $X \in \operatorname{\mathcal{E}}$. Set $\overline{Y} = U'(Y)$. Since $\overline{F}$ is a Morita equivalence, $\overline{Y}$ is a retract of $C' = \overline{F}( C )$, for some object $C \in \operatorname{\mathcal{C}}$. Choose a retraction diagram $\overline{\sigma }:$
\[ \xymatrix@R =50pt@C=50pt{ & C' \ar [dr]^{ \overline{r} } & \\ \overline{Y} \ar [ur]^{ \overline{i} } \ar [rr]^-{ \operatorname{id}} & & \overline{Y} } \]in the $\infty $-category $\operatorname{\mathcal{C}}$. Our assumption that $U'$ is a cartesian fibration guarantees that we can lift $\overline{\sigma }$ to a retraction diagram
\[ \xymatrix@R =50pt@C=50pt{ & X' \ar [dr]^{r} & \\ Y \ar [rr]^-{\operatorname{id}} \ar [ur]^{i} & & Y } \]in the $\infty $-category $\operatorname{\mathcal{E}}'$. Since the functor $F_{ C}$ is a Morita equivalence, there exists an object $X \in \operatorname{\mathcal{E}}_{C}$ such that $X'$ is a retract of $F(X)$ in the $\infty $-category $\operatorname{\mathcal{E}}'_{C'}$, and therefore also in the $\infty $-category $\operatorname{\mathcal{E}}'$. Applying Remark 8.5.1.6, we conclude that $Y$ is a retract of $F(X)$.