Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Recall that, if $F$ admits a right adjoint, then it preserves all colimit diagrams which exist in $\operatorname{\mathcal{C}}$ (Corollary 7.1.4.23). Similarly, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a functor which admits a left adjoint, then it preserves all limit diagrams which exist in $\operatorname{\mathcal{D}}$. In the setting of presentable $\infty $-categories, we have the following converse:
Theorem 9.5.6.1 (Adjoint Functor Theorem). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then:
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint if and only if it preserves small colimits.
A functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ admits a left adjoint if and only if it is accessible and preserves small limits.
Before giving the proof of Theorem 9.5.6.1, let us collect some consequences.
Corollary 9.5.6.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a presentable $\infty $-category. The following conditions are equivalent:
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a presentable fibration (Definition 9.5.3.13). That is, it is a locally cocartesian fibration having the property that, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ preserves small colimits.
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration. Moreover, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ is accessible and preserves small limits.
Proof. Using Theorem 9.5.6.1, we obtain the following reformulations of conditions $(1)$ and $(2)$:
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cocartesian fibration. Moreover, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ admits a right adjoint.
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration. Moreover, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ admits a left adjoint.
The equivalence of $(1')$ and $(2')$ is a special case of Corollary 6.2.5.5. $\square$
Recall that every presentable $\infty $-category is essentially $\operatorname{\Omega }^{+}$-small, where $\operatorname{\Omega }^{+}$ is the fixed strongly inaccessible cardinal of Remark 4.7.0.5. We have the following variant of Construction 9.5.3.9:
Construction 9.5.6.3. We define a (non-full) subcategory $\operatorname{\mathcal{QC}}^{\operatorname{RPr}} \subseteq \operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ as follows:
An object $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ if and only if $\operatorname{\mathcal{C}}$ is a presentable $\infty $-category.
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then a morphism $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ if and only if $G$ is an accessible functor which preserves small limits.
Corollary 9.5.6.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $U$ is a presentable cocartesian fibration.
The morphism $U$ is a cocartesian fibration and the covariant transport representation of $U$ takes values in the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{LPr}} \subseteq \operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ of Construction 9.5.3.9.
The morphism $U$ is a presentable cartesian fibration.
The morphism $U$ is a cartesian fibration and the contravariant transport representation of $U$ factors through the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{RPr}} \subseteq \operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ of Construction 9.5.6.3.
Proof. The equivalence $(1) \Leftrightarrow (2)$ follows immediately from the definition of presentable fibration (Example 9.5.3.15), and the equivalence $(3) \Leftrightarrow (4)$ follows from the reformulation provided by Corollary 9.5.6.2. Since every presentable fibration is both locally cartesian and locally cocartesian (Corollary 9.5.6.2), the equivalence $(1) \Leftrightarrow (3)$ is a special case of Proposition 6.2.5.7. $\square$
Corollary 9.5.6.5. There is a canonical equivalence of $\infty $-categories $\Psi : (\operatorname{\mathcal{QC}}^{\operatorname{LPr}} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{\operatorname{RPr}}$, which is characterized up to isomorphism by the following requirement:
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a presentable cocartesian fibration with covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$. Then the composition
is a contravariant transport representation of $U$ (regarded as a cartesian fibration).
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow { \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}^{\operatorname{op}} } ( \operatorname{\mathcal{QC}}^{\operatorname{LPr}} )^{\operatorname{op}} \xrightarrow {\Psi } \operatorname{\mathcal{QC}}^{\operatorname{RPr}} \]
Proof. Combine Corollary 9.5.6.4 with Proposition 8.6.8.21. $\square$
Remark 9.5.6.6. The equivalence $\Psi : (\operatorname{\mathcal{QC}}^{\operatorname{LPr}} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ of Corollary 9.5.6.5 can be described more informally as follows:
It carries each presentable $\infty $-category $\operatorname{\mathcal{C}}$ (regarded as an object of $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$) to itself (regarded as an object of $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$).
It carries each cocontinuous functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ (regarded as a morphism in $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$) to its right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ (regarded as a morphism in $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$).
More precisely, this description characterizes the equivalence of homotopy categories induced by the functor $\Psi $ (see Remark 8.6.8.19).
We now turn to the proof of Theorem 9.5.6.1. The first half is a consequence of the following more general result:
Proposition 9.5.6.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is presentable and $\operatorname{\mathcal{D}}$ is locally small. Then $F$ admits a right adjoint if and only if it is cocontinuous: that is, it preserves small colimits.
Proof. Assume that $F$ is cocontinuous; we will show that it admits a right adjoint (the reverse implication follows from Corollary 7.1.4.23). By virtue of Corollary 6.2.6.2, it will suffice to show that for every functor $h_{D}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ which is representable by an object $D \in \operatorname{\mathcal{D}}$, the composition $(h_{D} \circ F^{\operatorname{op}}): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable by an object of $\operatorname{\mathcal{C}}$. Using Theorem 9.5.5.1, we are reduced to showing that the functor $h_{D} \circ F^{\operatorname{op}}$ preserves small limits. This follows from the cocontinuity of $F$, since $h_{D}$ preserves all limits which exist in $\operatorname{\mathcal{D}}^{\operatorname{op}}$ (Corollary 7.4.1.23). $\square$
The second half of Theorem 9.5.6.1 is a consequence of the following:
Proposition 9.5.6.8. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is accessible and $\operatorname{\mathcal{D}}$ is presentable. Then $G$ admits a left adjoint if and only if it is accessible and preserves small limits.
Beware that the accessibility hypothesis in Proposition 9.5.6.8 cannot be omitted (see Warning 9.5.5.3).
Proof of Proposition 9.5.6.8. Assume that $G$ is accessible and preserves small limits; we wish to show that $G$ admits a left adjoint (the converse follows from Corollaries 7.1.4.23 and 9.4.7.15). By virtue of Corollary 6.2.6.2, it will suffice to show that for every functor $h_ C: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by an object $C \in \operatorname{\mathcal{C}}$, the composition $(h_{C} \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by an object of $\operatorname{\mathcal{D}}$. Using the criterion of Theorem 9.5.5.2, we are reduced to showing that $h_{C} \circ G$ is accessible and preserves small limits. This follows from our hypotheses on $G$, since the functor $h_{C}$ is accessible (Example 9.4.7.13) and preserves all limits which exist in $\operatorname{\mathcal{C}}$ (Corollary 7.4.1.23). $\square$
Proof of Theorem 9.5.6.1. Combine Propositions 9.5.6.7 and 9.5.6.8. $\square$
Remark 9.5.6.9. The preceding arguments show that Propositions 9.5.6.7 and 9.5.6.8 are formal consequences of Theorems 9.5.5.1 and 9.5.5.2 (respectively). The reverse is also true: assuming Propositions 9.5.6.7 and 9.5.6.8, we can immediately deduce the characterizations of representable and corepresentable functors supplied by Theorems 9.5.5.1 and 9.5.5.2. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. Then:
If $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is a continuous functor, then Proposition 9.5.6.7 guarantees that the opposite functor $\mathscr {F}^{\operatorname{op}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{\operatorname{op}}$ admits a right adjoint $T: \operatorname{\mathcal{S}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. It is then easy to see that the functor $\mathscr {F}$ is representable by the object $T( \Delta ^0 ) \in \operatorname{\mathcal{C}}$.
If $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a continuous accessible functor, then Proposition 9.5.6.8 guarantees that $\mathscr {G}$ admits a left adjoint $U: \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{C}}$. It is then easy to see that the functor $\mathscr {G}$ is corepresentable by the object $U( \Delta ^0 ) \in \operatorname{\mathcal{C}}$.
We close this section by recording more general forms of Propositions 9.5.6.7 and 9.5.6.8, which do not require any inaccessible cardinals.
Variant 9.5.6.10. Let $\kappa < \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete and $(\kappa ,\lambda )$-compactly generated, and suppose that the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ is essentially $\lambda $-small. Then a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint if and only if it satisfies the following pair of conditions:
The functor $F$ is $\lambda $-cocontinuous.
For every object $C \in \operatorname{\mathcal{C}}_{< \kappa }$ and every object $D \in \operatorname{\mathcal{D}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D )$ is essentially $\lambda $-small.
Proof. Fix an uncountable regular cardinal $\mu $ such that $\operatorname{\mathcal{D}}$ is locally $\mu $-small, so that every object $D \in \operatorname{\mathcal{D}}$ determines a representable functor $h_{D}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$. By virtue of Corollary 6.2.6.2, the functor $F$ admits a right adjoint if and only if, for every object $D \in \operatorname{\mathcal{D}}$, the composition $\operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow {F^{\operatorname{op}}} \operatorname{\mathcal{D}}^{\operatorname{op}} \xrightarrow { h_{D} } \operatorname{\mathcal{S}}_{< \mu }$ is representable by an object of $\operatorname{\mathcal{C}}$. The equivalence of this condition with $(1)$ and $(2)$ now follows from Proposition 9.5.5.4 (together with Proposition 7.4.1.22). $\square$
Variant 9.5.6.11. Let $\kappa \leq \lambda \leq \mu $ be regular cardinals, where $\lambda $ is uncountable and $\mu $ has exponential cofinality $\geq \lambda $. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are $(\kappa ,\mu )$-compactly generated and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor which is $(\kappa ,\mu )$-finitary. Assume that $\operatorname{\mathcal{D}}$ is $\lambda $-complete and that the full subcategory $\operatorname{\mathcal{D}}_{< \kappa } \subseteq \operatorname{\mathcal{D}}$ spanned by the $(\kappa ,\mu )$-compact objects is essentially $\lambda $-small. Then $G$ admits a left adjoint if and only if it satisfies the following conditions:
The functor $G$ is $\lambda $-continuous: that is, it preserves $\lambda $-small limits.
For every pair of objects $C \in \operatorname{\mathcal{C}}_{< \kappa }$ and $D \in \operatorname{\mathcal{D}}_{< \kappa }$, the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, G(D) )$ is essentially $\lambda $-small.
Proof. Assume first that $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. In this case, condition $(1)$ follows from Corollary 7.1.4.23. To prove $(2)$, note that our assumption that $G$ is $(\kappa ,\mu )$-finitary guarantees that $F$ is $(\kappa ,\mu )$-compact (Example 9.4.2.17). In particular, if $C \in \operatorname{\mathcal{C}}_{< \kappa }$, then $F(C) \in \operatorname{\mathcal{D}}_{< \kappa }$, so that the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, G(D) ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D )$ is essentially $\lambda $-small for any $D \in \operatorname{\mathcal{D}}_{< \kappa }$ by virtue of our assumption that the $\infty $-category $\operatorname{\mathcal{D}}_{< \kappa }$ is essentially $\lambda $-small.
We now prove the converse. Assume that conditions $(1)$ and $(2)$ are satisfied; we wish to show that $G$ admits a left adjoint. Fix a regular cardinal $\nu $ of exponential cofinality $\geq \mu $ such that $\operatorname{\mathcal{C}}$ is locally $\nu $-small. For each object $C \in \operatorname{\mathcal{C}}$, let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \nu }$ be the functor corepresented by $C$. By virtue of Corollary 6.2.6.2, it will suffice to show that each of the functors $(h^{C} \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \nu }$ is corepresentable by an object of $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ is $(\kappa ,\mu )$-cocomplete, the collection of objects $C \in \operatorname{\mathcal{C}}$ which satisfy this condition is closed under the formation of $\mu $-small $\kappa $-filtered colimits. We may therefore assume without loss of generality that the object $C \in \operatorname{\mathcal{C}}_{< \kappa }$ is $(\kappa ,\mu )$-compact. In this case, we will show that the functor $\mathscr {F} = h^{C} \circ G$ is corepresentable by a $(\kappa ,\mu )$-compact object of $\operatorname{\mathcal{D}}$. Since $C$ is $(\kappa ,\mu )$-compact and the functor $G$ is $(\kappa ,\mu )$-finitary, the functor $\mathscr {F}$ is also $(\kappa ,\mu )$-finitary. It follows from assumption $(2)$ that the space $\mathscr {F}(D) = \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, G(D) )$ is essentially $\lambda $-small for $D \in \operatorname{\mathcal{D}}_{< \kappa }$, and is therefore essentially $\mu $-small for any object $D \in \operatorname{\mathcal{D}}$. Invoking the criterion of Proposition 9.5.5.5, we are reduced to showing that the functor $\mathscr {F}$ is $\lambda $-continuous. This follows from assumption $(1)$, since the corepresentable functor $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \nu }$ preserves all limits which exist in $\operatorname{\mathcal{C}}$ (Corollary 7.4.1.23). $\square$