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8.2.7 Adjunctions as Profunctors

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Then, for every pair of objects $X \in \operatorname{\mathcal{C}}$ and $Y \in \operatorname{\mathcal{D}}$, Proposition 6.2.1.17 supplies a homotopy equivalence of Kan complexes

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), Y ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, G(Y) ). \]

Our goal in this section is to prove a converse of this assertion: if we can choose homotopy equivalences $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), Y ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, G(Y) )$ depending functorially on $X$ and $Y$, then the functor $G$ must be right adjoint to $F$. Using the language of profunctors, we can formulate this statement as follows:

Proposition 8.2.7.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between locally small $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor which is corepresented by $F$. Then a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is right adjoint to $F$ if and only if it represents the profunctor $\mathscr {K}$.

We will deduce Proposition 8.2.7.1 from a more precise statement (Proposition 8.2.7.7), which classifies natural transformations $\operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which are units of adjunctions between $F$ and $G$. First, we study the functoriality of the constructions introduced in ยง8.2.3. To avoid confusion, we will denote $\operatorname{Hom}$-functors on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ by $\mathscr {H}_{\operatorname{\mathcal{C}}}$ and $\mathscr {H}_{\operatorname{\mathcal{D}}}$, respectively.

Proposition 8.2.7.2 (Functoriality of $\operatorname{Hom}$-Functors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories which admit $\operatorname{Hom}$-functors $( \mathscr {H}_{\operatorname{\mathcal{C}}}, \alpha )$ and $( \mathscr {H}_{\operatorname{\mathcal{D}}}, \beta )$, respectively. Then there exists a natural transformation $\gamma : \mathscr {H}_{\operatorname{\mathcal{C}}}(-,-)\rightarrow \mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), F(-) )$ for which the diagram

8.20
\begin{equation} \begin{gathered}\label{equation:Hom-functoriality} \xymatrix@R =50pt@C=50pt{ & \underline{ \Delta ^{0} }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \ar [dl]_-{ [\alpha ] } \ar [dr]^-{ [\beta ]} \\ \mathscr {H}_{\operatorname{\mathcal{C}}}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \ar [rr]^{ [\gamma ] } & & \mathscr {H}_{\operatorname{\mathcal{D}}}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } } \end{gathered} \end{equation}

commutes (in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}) }$). Moreover, the natural transformation $\gamma $ is uniquely determined up to homotopy.

Proof. This is a special case of Proposition 7.3.6.1, since $\alpha $ exhibits $\mathscr {H}_{\operatorname{\mathcal{C}}}$ as a left Kan extension of $\underline{ \Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ along the left fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ (see Proposition 8.2.3.12). $\square$

Remark 8.2.7.3. In the situation of Proposition 8.2.7.2, suppose that we are given a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. The commutativity of (8.20) guarantees that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \ar [r]^-{F} \ar [d]^{ \sim } & \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F(X), F(Y) ) \ar [d]^{\sim } \\ \mathscr {H}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{ \gamma } & \mathscr {H}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) } \]

commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the vertical maps are the isomorphisms of Remark 8.2.3.4. We can summarize the situation more informally as follows: if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor between (locally small) $\infty $-categories, then the induced map of Kan complexes $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F(X), F(Y) )$ depends functorially on the pair $(X,Y)$ (as an object of the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$).

Proposition 8.2.7.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between locally small $\infty $-categories, and let $\gamma : \mathscr {H}_{\operatorname{\mathcal{C}}}(-,-) \rightarrow \mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), F(-) )$ be as in Proposition 8.2.7.2. Then:

  • The natural transformation $\gamma $ exhibits the functor $\mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), - ): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ as a left Kan extension of $\mathscr {H}_{\operatorname{\mathcal{C}}}$ along the functor $(\operatorname{id}\times F): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$.

  • The natural transformation $\gamma $ exhibits the functor $\mathscr {H}_{\operatorname{\mathcal{D}}}(-, F(-) ): \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as a left Kan extension of $\mathscr {H}_{\operatorname{\mathcal{C}}}$ along the functor $(F^{\operatorname{op}} \times \operatorname{id}): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.

Proof. We will prove the second assertion; the second follows by a similar argument. By virtue of the commutative diagram (8.20) and the transitivity of left Kan extensions (Proposition 7.3.7.18), it will suffice to prove the following:

$(1)$

The natural transformation $\alpha $ exhibits $\mathscr {H}_{\operatorname{\mathcal{C}}}(-,-)$ as a left Kan extension of $\underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ along the forgetful functor $\operatorname{Tw}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.

$(2)$

The natural transformation $\beta |_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ exhibits the functor $\mathscr {H}_{\operatorname{\mathcal{D}}}( -, F(-) )$ as a left Kan extension of $\underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ along the composite functor

\[ \operatorname{Tw}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\xrightarrow { F^{\operatorname{op}} \times \operatorname{id}} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. \]

Assertion $(1)$ is a special case of Proposition 8.2.3.12. Assertion $(2)$ follows from Proposition 8.2.6.13, since the natural transformation $\beta |_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ exhibits the profunctor $\mathscr {H}_{\operatorname{\mathcal{D}}}( -, F(-) )$ as represented by $F$ (see the proof of Proposition 8.2.6.15). $\square$

Corollary 8.2.7.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between locally small $\infty $-categories. Then, for any functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, precomposition with the natural transformation $\gamma $ of Proposition 8.2.7.2 induces a homotopy equivalence

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {H}_{\operatorname{\mathcal{D}}}(F(-), -), \mathscr {H}_{\operatorname{\mathcal{C}}}(-, G(-)) ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {H}_{\operatorname{\mathcal{C}}}(-,-), \mathscr {H}_{\operatorname{\mathcal{C}}}( -, (G \circ F)(-)) ). } \]

Remark 8.2.7.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be locally small $\infty $-categories, with $\operatorname{Hom}$-functors

\[ \mathscr {H}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {H}_{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}. \]

Combining Corollary 8.2.7.5 and Proposition 8.2.6.1, we obtain homotopy equivalences

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})}( \operatorname{id}_{ \operatorname{\mathcal{C}}}, G \circ F) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})}( \mathscr {H}_{\operatorname{\mathcal{C}}}(-, -), \mathscr {H}_{\operatorname{\mathcal{C}}}( -, (G \circ F)(-) ) \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})}( \mathscr {H}_{\operatorname{\mathcal{D}}}(F(-), -), \mathscr {H}_{\operatorname{\mathcal{C}}}( -, G(-) ) \ar [u] .} \]

In particular, every natural transformation of functors $\eta : \operatorname{id}_{ \operatorname{\mathcal{C}}} \rightarrow G \circ F$ determines a natural transformation of profunctors $\eta ': \mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), -) \rightarrow \mathscr {H}_{\operatorname{\mathcal{C}}}( -, G(-) )$, which is characterized up to homotopy by the requirement that the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), F(-) ) \ar [dr]^-{ [\eta '] } & \\ \mathscr {H}_{\operatorname{\mathcal{C}}}(-,-) \ar [rr]^{ [\eta ] } \ar [ur]^{ [\gamma ] } & & \mathscr {H}_{\operatorname{\mathcal{C}}}( -, (G \circ F)(-) ) } \]

commutes in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }$.

Proposition 8.2.7.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between locally small $\infty $-categories and let $\eta : \operatorname{id}_{ \operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. Then $\eta $ exhibits $G$ as a right adjoint to $F$ if and only if the morphism $\mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), -) \rightarrow \mathscr {H}_{\operatorname{\mathcal{C}}}( -, G(-) )$ of Remark 8.2.7.6 is an isomorphism of profunctors.

Proof. This is a reformulation of Corollary 6.2.4.5. $\square$

Proof of Proposition 8.2.7.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which admit $\operatorname{Hom}$-functors

\[ \mathscr {H}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {H}_{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}. \]

Suppose we are given functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. It follows from Proposition 8.2.7.7 (and Remark 8.2.7.6) that $G$ is right adjoint to $F$ if and only if the profunctors $\mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), -)$ and $\mathscr {H}_{\operatorname{\mathcal{C}}}( -, G(-) )$ are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$). In particular, if a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresented by $F$, then it is represented by $G$ if and only if $G$ is right adjoint to $F$. $\square$