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6.2.1 Adjunctions of $\infty $-Categories

We now adapt Definition 6.1.0.2 to the setting of $\infty $-categories.

Definition 6.2.1.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. We will say that a pair of natural transformations $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are compatible up to homotopy if the following conditions are satisfied:

$(Z1)$

The identity isomorphism $\operatorname{id}_{F}: F \rightarrow F$ is a composition of the natural transformations

\[ F = F \circ \operatorname{id}_{\operatorname{\mathcal{C}}} \xrightarrow { \operatorname{id}_{F} \circ \eta } F \circ G \circ F \quad \quad F \circ G \circ F \xrightarrow { \epsilon \circ \operatorname{id}_{F} } \operatorname{id}_{\operatorname{\mathcal{D}}} \circ F = F \]

in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, in the sense of Definition 1.3.4.1.

$(Z2)$

The identity isomorphism $\operatorname{id}_{G}: G \rightarrow G$ is a composition of the natural transformations

\[ G = \operatorname{id}_{\operatorname{\mathcal{D}}} \circ G \xrightarrow { \eta \circ \operatorname{id}_ G} G \circ F \circ G \quad \quad G \circ F \circ G \xrightarrow { \operatorname{id}_{G} \circ \epsilon } G \circ \operatorname{id}_{\operatorname{\mathcal{D}}} = G \]

in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$.

We say that a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is the unit of an adjunction if there exists a natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is compatible with $\eta $ up to homotopy. We say that a natural $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is the counit of an adjunction if there exists a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which is compatible with $\epsilon $ up to homotopy.

Definition 6.2.1.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. We say that $F$ is a left adjoint of $G$, or that $G$ is a right adjoint of $F$, if there exists a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which is the unit of an adjunction between $F$ and $G$. In this case, we say that $\eta $ exhibits $F$ as a left adjoint of $G$ and also that it exhibits $G$ as a right adjoint of $F$. Equivalently, $F$ is a left adjoint of $G$ if there exists a natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is the counit of an adjunction between $F$ and $G$; in this case, we say that $\epsilon $ exhibits $F$ as a left adjoint of $G$ and also that it exhibits $G$ as a right adjoint of $F$.

Notation 6.2.1.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. We say that $F$ is a left adjoint, or that $F$ admits a right adjoint, if there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which is right adjoint to $F$. We let $\operatorname{LFun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which are left adjoints.

Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between $\infty $-categories. We say that $G$ is a right adjoint, or that $G$ admits a left adjoint, if there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left adjoint to $G$. We let $\operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ spanned by those functors $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which are left adjoints.

Remark 6.2.1.4. Let $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ be the homotopy $2$-category of $\infty $-categories (see Construction 4.5.1.22). Suppose we are given functors of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, which we regard as $1$-morphisms in the $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be natural transformations and let $[\eta ]$ and $[\epsilon ]$ denote their homotopy classes, which we regard as $2$-morphisms in $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$. Then $\eta $ and $\epsilon $ are compatible up to homotopy (in the sense of Definition 6.2.1.1) if and only if the pair $( [\eta ], [\epsilon ])$ is an adjunction in the $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ (in the sense of Definition 6.1.1.1).

Remark 6.2.1.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be natural transformations. Axioms $(Z1)$ and $(Z2)$ of Definition 6.2.1.1 can be restated as follows:

$(Z1)$

There exists a $2$-simplex $\sigma $ of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ with boundary as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & F \circ G \circ F \ar [dr]^{ \epsilon \circ \operatorname{id}_ F } & \\ F \circ \operatorname{id}_{\operatorname{\mathcal{C}}} \ar [ur]^{ \operatorname{id}_ F \circ \eta } \ar [rr]^{ \operatorname{id}_ F } & & \operatorname{id}_{\operatorname{\mathcal{D}}} \circ F. } \]
$(Z2)$

There exists a $2$-simplex $\tau $ of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ with boundary as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & G \circ F \circ G \ar [dr]^{ \operatorname{id}_{G} \circ \epsilon } & \\ \operatorname{id}_{\operatorname{\mathcal{C}}} \circ G \ar [ur]^{\eta \circ \operatorname{id}_ G} \ar [rr]^{ \operatorname{id}_ G } & & G \circ \operatorname{id}_{\operatorname{\mathcal{D}}} . } \]

In this case, we will say that the $2$-simplices $\sigma $ and $\tau $ witness the axioms $(Z1)$ and $(Z2)$, respectively.

Remark 6.2.1.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. It follows from Remark 6.2.1.4 that the condition that $\eta $ is the unit of an adjunction (in the sense of Definition 6.2.1.1) depends only on the homotopy class $[\eta ]$, regarded as a morphism in the category $\mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})}$. Moreover, if $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is a counit which is compatible with $\eta $ up to homotopy, then the homotopy class $[\epsilon ]$ is uniquely determined (see Proposition 6.1.2.9). Beware that it is only the homotopy class of $\epsilon $ that is uniquely determined: if $\epsilon ': F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is homotopic to $\epsilon $, then it is also compatible with $\eta $ up to homotopy.

Remark 6.2.1.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural tranformation. Then $\eta $ is the unit of an adjunction between $F$ and $G$ if and only if the opposite natural transformation $\eta ^{\operatorname{op}}: G^{\operatorname{op}} \circ F^{\operatorname{op}} \rightarrow \operatorname{id}_{ \operatorname{\mathcal{C}}^{\operatorname{op}} }$ is the counit of an adjunction between the functors $G^{\operatorname{op}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$. Note that in this case, $\eta ^{\operatorname{op}}$ exhibits $G^{\operatorname{op}}$ as the left adjoint of $F^{\operatorname{op}}$.

Remark 6.2.1.8 (Composition of Adjoints). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $F': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories which admit right adjoints. Then the composite functor $(F' \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ also admits a right adjoint. More precisely, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $G': \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ are right adjoints of $F$ and $F'$, respectively, then the composite functor $(G \circ G'): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is right adjoint to $(F' \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ (see Corollary 6.1.5.5).

Example 6.2.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between ordinary categories, and suppose we are given natural transformations $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$. Then the pair $(\eta , \epsilon )$ is an adjunction between $F$ and $G$ (in the sense of Definition 6.1.0.2) if and only if the induced maps

\[ \operatorname{N}_{\bullet }( \eta ): \operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \rightarrow \operatorname{N}_{\bullet }(G) \circ \operatorname{N}_{\bullet }(F) \quad \quad \operatorname{N}_{\bullet }( \epsilon ): \operatorname{N}_{\bullet }(F) \circ \operatorname{N}_{\bullet }(G) \rightarrow \operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) } \]

are compatible up to homotopy, in the sense of Definition 6.2.1.1. In particular:

  • A natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is the unit of an adjunction between functors of ordinary categories $F$ and $G$ if and only if $\operatorname{N}_{\bullet }( \eta ): \operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \rightarrow \operatorname{N}_{\bullet }(G) \circ \operatorname{N}_{\bullet }(F)$ is the unit of an adjunction between functors of $\infty $-categories $\operatorname{N}_{\bullet }(F)$ and $\operatorname{N}_{\bullet }(G)$.

  • A natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ is the counit of an adjunction between functors of ordinary categories $F$ and and $G$ if and only if $\operatorname{N}_{\bullet }( \epsilon ): \operatorname{N}_{\bullet }(F) \circ \operatorname{N}_{\bullet }(G ) \rightarrow \operatorname{id}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) }$ is the unit of an adjunction between functors of $\infty $-categories $\operatorname{N}_{\bullet }(F)$ and $\operatorname{N}_{\bullet }(G)$.

  • A functor of ordinary categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint $G$ if and only if the induced functor of $\infty $-categories $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ admits a right adjoint (in which case $\operatorname{N}_{\bullet }(G)$ is a right adjoint of $\operatorname{N}_{\bullet }(F)$).

  • A functor of ordinary categories $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $F$ if and only if the induced functor of $\infty $-categories $\operatorname{N}_{\bullet }(G): \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ admits a left adjoint (in which case $\operatorname{N}_{\bullet }(F)$ is a left adjoint of $\operatorname{N}_{\bullet }(G)$).

Proposition 3.1.6.9 generalizes to the setting of $\infty $-categories:

Remark 6.2.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. The existence of natural transformations

\[ \eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F \quad \quad \epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}} \]

guarantee that $F$ and $G$ are simplicial homotopy inverses of one another, in the sense of Definition 3.1.6.1. In particular, $F$ and $G$ are homotopy equivalences of simplicial sets.

Example 6.2.1.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories, and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse of $F$. Then $G$ is also a right adjoint of $F$. More precisely, any isomorphism $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ in the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ is the unit of an adjunction between $F$ and $G$ (Proposition 6.1.4.1). Similarly, $G$ is a left adjoint of $F$.

Remark 6.2.1.12. Let $F: X \rightarrow Y$ be a morphism of Kan complexes. Then $F$ admits a right adjoint (in the sense of Notation 6.2.1.3) if and only if $F$ is a homotopy equivalence. This follows by combining Remark 6.2.1.10 with Example 6.2.1.11.

Remark 6.2.1.12 can be regarded as a special case of the following more general assertion:

Proposition 6.2.1.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let

\[ \eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F \quad \quad \epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}} \]

be natural transformations which are compatible up to homotopy. Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects $C \in \operatorname{\mathcal{C}}$ for which the unit $\eta _{C}: C \rightarrow (G \circ F)(C)$ is an isomorphism, and let $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$ be the full subcategory spanned by those objects $D \in \operatorname{\mathcal{D}}$ for which the counit $\epsilon _ D: (F \circ G)(D) \rightarrow D$ is an isomorphism. Then $F$ and $G$ restrict to functors $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}'$ and $G': \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{C}}'$ which are homotopy inverse to one another.

Proof. Let $C$ be an object of $\operatorname{\mathcal{C}}'$, so that $\eta _{C}: C \rightarrow (G \circ F)(C)$ is an isomorphism. Since $\eta $ and $\epsilon $ are compatible up to homotopy, the identity morphism $\operatorname{id}_{ F(C) }$ is a composition of $F(\eta _ C): F(C) \rightarrow (F \circ G \circ F)(C)$ with $\epsilon _{F(C)}: (F \circ G \circ F)(C) \rightarrow F(C)$ in the $\infty $-category $\operatorname{\mathcal{D}}$. It follows that $\epsilon _{F(C)}$ is an isomorphism in $\operatorname{\mathcal{D}}$ (Remark 1.3.6.3), so that $F(C)$ belongs to the full subcategory $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$. Setting $F' = F|_{\operatorname{\mathcal{C}}'}$, we obtain a functor $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}'$. A similar argument shows that we can regard $G' = G|_{\operatorname{\mathcal{D}}'}$ as a functor from $\operatorname{\mathcal{D}}'$ to $\operatorname{\mathcal{C}}'$. The unit morphism $\eta $ restricts to a natural transformation of functors $\eta ': \operatorname{id}_{\operatorname{\mathcal{C}}'} \rightarrow G' \circ F'$. By construction, $\eta '$ carries each object $C \in \operatorname{\mathcal{C}}'$ to an isomorphism, and is therefore an isomorphism in the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{C}}' )$ (Theorem 4.4.4.4). Similarly, the counit $\epsilon $ restricts to a natural isomorphism $\epsilon ': F' \circ G' \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}'}$, so that $F'$ and $G'$ are homotopy inverse to one another. $\square$

Proposition 6.2.1.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a right adjoint. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be another functor of $\infty $-categories and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. The following conditions are equivalent:

$(1)$

The natural transformation $\eta $ is the unit of an adjunction between the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.

$(2)$

The induced map $\operatorname{id}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \rightarrow \mathrm{h} \mathit{G} \circ \mathrm{h} \mathit{F}$ is the unit of an adjunction between the homotopy categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ and $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$.

Proof. The implication $(1) \Rightarrow (2)$ follows from the observation that the formation of homotopy categories defines a (strict) functor of $2$-categories

\[ \mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}} \rightarrow \mathbf{Cat} \quad \quad \operatorname{\mathcal{C}}\mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \]

and therefore carries adjunctions to adjunctions (see Exercise 6.1.1.6). We will show that $(2)$ implies $(1)$. By assumption, the functor $F$ admits a right adjoint $G': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Let $\eta ': \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow F \circ G'$ be the unit of an adjunction. Applying Corollary 6.1.3.3, we deduce that there exists a natural transformation $\gamma : G' \rightarrow G$ such that $\eta $ is a composition of the natural transformations

\[ \eta ': \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow F \circ G' \quad \quad (\operatorname{id}_ F \circ \gamma ): F \circ G' \rightarrow F \circ G \]

in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. If assumption $(2)$ is satisfied, then the image of $\gamma $ in the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{D}}}, \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is an isomorphism: that is, $\gamma $ carries each object $D \in \operatorname{\mathcal{D}}$ to an isomorphism $\gamma _{D}: G'(D) \rightarrow G(D)$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Applying Theorem 4.4.4.4, we conclude that $\gamma $ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$, so that the criterion of Corollary 6.1.3.3 guarantees that $\eta $ is also the unit of an adjunction. $\square$

Corollary 6.2.1.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ be the induced functor of homotopy categories. If $F$ admits a right adjoint $G$, then $\mathrm{h} \mathit{F}$ also admits a right adjoint, which can be identified with the functor $\mathrm{h} \mathit{G}$.

Warning 6.2.1.16. The implication $(2) \Rightarrow (1)$ of Proposition 6.2.1.14 generally fails if the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ does not have a right adjoint. For example, let $X$ be a simply connected Kan complex, let $F: \Delta ^0 \rightarrow X$ be the map corresponding to a vertex $x \in X$, and let $G: X \rightarrow \Delta ^0$ be the projection map. Since $X$ is simply connected, the functors $\mathrm{h} \mathit{F}$ and $\mathrm{h} \mathit{G}$ are equivalences of ordinary categories. In particular, the identity transformation from $\operatorname{id}_{\Delta ^0} = G \circ F$ to itself determines unit of an adjunction between $\mathrm{h} \mathit{F}$ and $\mathrm{h} \mathit{G}$. However, the functors $F$ and $G$ cannot be adjoint unless the Kan complex $X$ is contractible (see Remark 6.2.1.10)

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between $\infty $-categories and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. By virtue of Variant 6.1.2.11, the natural transformation $\eta $ exhibits $\mathrm{h} \mathit{G}$ as a right adjoint to $\mathrm{h} \mathit{F}$ if and only if, for every pair of objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, the composite map

\begin{eqnarray*} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{D}}}}( F(C), D) & = & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D ) ) \\ & \xrightarrow {G} & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (G \circ F)(C), G(D)) ) \\ & \xrightarrow { \circ [ \eta _ C ] } & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) ) \\ & = & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, G(D) )\end{eqnarray*}

is a bijection. If $\eta $ exhibits $G$ as a right adjoint to $F$, then we can say more:

Proposition 6.2.1.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between $\infty $-categories and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be the unit of an adjunction. Then, for every pair of objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (G \circ F)(C), G(D)) \xrightarrow { \circ [ \eta _ C ] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(D) ) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$; here the second map is given by the composition law of Construction 4.6.7.9.

Proof. It will suffice to show that, for every Kan complex $T$, the induced amp

\begin{eqnarray*} \pi _0( \operatorname{Fun}(T, \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) )) & = & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }(T, \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) ) \\ & \xrightarrow {\theta } & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Kan}}}(T, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) )) \\ & = & \pi _0( \operatorname{Fun}(T, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) ) ) ) \end{eqnarray*}

is bijective. Let $\underline{C} \in \operatorname{Fun}(T, \operatorname{\mathcal{C}})$ and $\underline{D} \in \operatorname{Fun}(T, \operatorname{\mathcal{D}})$ be the constant morphisms taking the values $C$ and $D$, respectively. Unwinding the definitions, we see that $\theta $ can be identified with the map

\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Fun}(T,\operatorname{\mathcal{D}})}}( F \circ \underline{C}, \underline{D} ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(T,\operatorname{\mathcal{C}})}}( \underline{C}, G \circ \underline{D} ) \]

given by the formation of right adjuncts with respect to the homotopy class $[\eta ]$ (regarded as a $2$-morphism in the category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$). The bijectivity of $\theta $ now follows from the criterion of Proposition 6.1.2.9. $\square$

Remark 6.2.1.18. We will see later that the converse of Proposition 6.2.1.17 also holds: if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors of $\infty $-categories and $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is a natural transformation which induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D))$ for every pair of objects $(C,D) \in \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$, then $\eta $ is the unit of an adjunction between $F$ and $G$ (Corollary 6.2.4.7).

Remark 6.2.1.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. It follows from Proposition 6.1.3.4 that if $F$ admits a right adjoint $G$, then $G$ is well-defined up to isomorphism as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$. We will sometimes emphasize this by referring to $G$ as the right adjoint of $F$ and denoting it by $F^{R}$. By virtue of Notation 6.1.3.8, the construction $F \mapsto F^{R}$ determines an equivalence of homotopy categories $\mathrm{h} \mathit{\operatorname{LFun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} \rightarrow \mathrm{h} \mathit{\operatorname{RFun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})}^{\operatorname{op}}$. We will see later that this construction can be upgraded to an equivalence of $\infty $-categories $\operatorname{LFun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{RFun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})^{\operatorname{op}}$ (see Proposition ).

Warning 6.2.1.20. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. The following data are essentially equivalent to one another:

  • The datum of a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which admits a right adjoint.

  • The datum of a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which admits a left adjoint.

  • The datum of a triple $(F,G,\eta )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors and $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is the unit of an adjunction between $F$ and $G$.

  • The datum of a triple $(F,G, \epsilon )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is the counit of an adjunction between $F$ and $G$.

  • The datum of a quintuple $(F, G, \eta , \epsilon , \sigma )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors, $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations which are compatible up to homotopy, and $\sigma : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is a $2$-simplex witnessing axiom $(Z1)$ of Definition 6.2.1.1 (see Remark 6.2.1.5).

  • The datum of a quintuple $(F, G, \eta , \epsilon , \tau )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors, $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations which are compatible up to homotopy, and $\tau : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ is a $2$-simplex witnessing axiom $(Z2)$ of Definition 6.2.1.1.

The following data are not equivalent to the above (or to each other):

  • The datum of a pair $(F,G)$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors which are adjoint to one another.

  • The datum of a quadruple $(F,G,\eta ,\epsilon )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors, $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations which are compatible up to homotopy,

  • The datum of a sextuple $(F, G, \eta , \epsilon , \sigma , \tau )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors, $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations, and $\sigma : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ and $\tau : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ are $2$-simplices witnessing axioms $(Z1)$ and $(Z2)$ of Definition 6.2.1.1.

To say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left adjoint to a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is somewhat imprecise: one should really specify a witness to the adjointness of $F$ and $G$, which can take the form of either a unit $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ or a counit $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$. Given both a unit $\eta $ and a counit $\epsilon $, one can further demand evidence of their compatibility, which can take the form of a $2$-simplex $\sigma : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ witnessing axiom $(Z1)$ or a $2$-simplex $\tau : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ witnessing axiom $(Z2)$. If one specifies both of the witnesses $\sigma $ and $\tau $, then one can further demand a witness to the compatibility of $\sigma $ with $\tau $; we will return to this point in ยง.