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6.1 Adjunctions in $2$-Categories

We begin by reviewing the theory of adjoint functors in the setting of classical category theory, originally introduced in [MR131451].

Definition 6.1.0.1 (Kan). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors. A $\operatorname{Hom}$-adjunction between $F$ and $G$ is a collection of bijections

\[ \rho _{C,D}: \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(C), D) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) ) \]

which depend functorially on $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$ (that is, the construction $(C,D) \mapsto \rho _{C,D}$ is an isomorphism in the functor category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{Set})$). In this case, we say that the construction $(C,D) \mapsto \rho _{C,D}$ exhibits $F$ as a left adjoint to $G$ and $G$ as a right adjoint to $F$.

In the situation of Definition 6.1.0.1, functoriality imposes strong constraints on the construction $(C,D) \mapsto \rho _{C,D}$. For each object $C \in \operatorname{\mathcal{C}}$, let $\eta _ C: C \rightarrow (G \circ F)(C)$ be the morphism of $\operatorname{\mathcal{C}}$ given by the image of the identity morphism $\operatorname{id}_{F(C)}$ under the bijection

\[ \rho _{C,F(C)}: \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), F(C) ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{D}}}( C, (G \circ F)(C) ). \]

For every morphism $f: F(C) \rightarrow D$ in $\operatorname{\mathcal{D}}$, the commutativity of the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(C), F(C) ) \ar [r]^-{ \rho _{C,F(C)}}_{\sim } \ar [d]^{f \circ } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( C, (G \circ F)(C) ) \ar [d]^{ G(f) \circ } \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \ar [r]^-{ \rho _{C,D} }_{\sim } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(D) ) } \]

supplies an equality

\[ \rho _{C,D}(f) = \rho _{C,D}( f \circ \operatorname{id}_{F(C)} ) = G(f) \circ \rho _{C,F(C)}( \operatorname{id}_{F(C)} ) = G(f) \circ \eta _ C. \]

In particular, the bijection $\rho _{C,D}$ is completely determined by the morphism $\eta _ C$. Moreover, the functoriality of $\rho _{\bullet , \bullet }$ in the first variable guarantees that the construction $C \mapsto \eta _ C$ is a natural transformation from the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ to the composition $G \circ F$. Similarly, the inverse bijections $\rho _{C,D}^{-1}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(C), D)$ can be recovered from the collection of morphisms $\{ \epsilon _ D = \rho _{ G(D), D}^{-1}( \operatorname{id}_{G(D)} ) \} _{D \in \operatorname{\mathcal{D}}}$, which comprise a natural transformation of functors $\epsilon : (F \circ G) \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$. This leads to a reformulation of Definition 6.1.0.1:

Definition 6.1.0.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories. An adjunction between $F$ and $G$ is a pair $(\eta , \epsilon )$, where $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations satisfying the following compatibility conditions:

$(Z1)$

For each object $C \in \operatorname{\mathcal{C}}$, the composite morphism

\[ F(C) \xrightarrow { F( \eta _ C) } (F \circ G \circ F)(C) \xrightarrow { \epsilon _{F(C)} } F(C) \]

is equal to the identity $\operatorname{id}_{ F(C)}$.

$(Z2)$

For each object $D \in \operatorname{\mathcal{D}}$, the composite morphism

\[ G(D) \xrightarrow { \eta _{ G(D)} } (G \circ F \circ G)(D) \xrightarrow { G(\epsilon _ D)} G(D) \]

is equal to the identity $\operatorname{id}_{ G(D) }$.

If these conditions are satisfied, then we will refer to $\eta $ as the unit of the adjunction $(\eta , \epsilon )$ and to $\epsilon $ as the counit of the adjunction $(\eta , \epsilon )$. In this case, we will say that $(\eta , \epsilon )$ exhibits $F$ as a left adjoint to $G$ and also that it exhibits $G$ as a right adjoint to $F$.

Example 6.1.0.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories, and let $\{ \rho _{C,D} \} _{C \in \operatorname{\mathcal{C}}, D \in \operatorname{\mathcal{D}}}$ be a $\operatorname{Hom}$-adjunction between $F$ and $G$ (in the sense of Definition 6.1.0.1). Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be the natural transformations given by the formulae

\[ \eta _ C = \rho _{C,F(C)}( \operatorname{id}_{F(C)} ) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, (G \circ F)(C) ) \]
\[ \epsilon _ D = \rho _{G(D),D}^{-1}( \operatorname{id}_{G(D)} ) \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (F \circ G)(D), D). \]

Then the pair $(\eta , \epsilon )$ is an adjunction between $F$ and $G$ (in the sense of Definition 6.1.0.2). Condition $(Z1)$ follows from the observation that for each object $C \in \operatorname{\mathcal{C}}$, we have

\begin{eqnarray*} \operatorname{id}_{F(C)} & = & \rho _{C, F(C)}^{-1}(\rho _{C,F(C)}( \operatorname{id}_{F(C)} )) \\ & = & \rho _{C, F(C)}^{-1}( \eta _ C ) \\ & = & \rho _{C, F(C)}^{-1}( \operatorname{id}_{(G \circ F)(C)} \circ \eta _ C) \\ & = & \rho _{(G \circ F)(C), F(C)}^{-1}( \operatorname{id}_{(G \circ F)(C)}) \circ F(\eta _ C) \\ & = & \epsilon _{ F(C)} \circ F(\eta _ C). \end{eqnarray*}

The verification of $(Z2)$ is similar.

Exercise 6.1.0.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories. Show that every adjunction $(\eta , \epsilon )$ between $F$ and $G$ can be obtained by applying the construction of Example 6.1.0.3 to a unique $\operatorname{Hom}$-adjunction $\{ \rho _{C,D} \} _{C \in \operatorname{\mathcal{C}}, D \in \operatorname{\mathcal{D}}}$ between $F$ and $G$ (see Example 6.1.2.7).

It follows from Exercise 6.1.0.4 that Definitions 6.1.0.1 and 6.1.0.2 are essentially equivalent to one another. However, an advantage of Definition 6.1.0.2 is that it can be formulated entirely in the language of functors and natural transformations: that is, it uses only the structure of the $2$-category $\mathbf{Cat}$ of Example 2.2.0.4. In §6.1.1, we exploit this observation to generalize the notion of adjunction to an arbitrary $2$-category. Given a $2$-category $\operatorname{\mathcal{C}}$ containing $1$-morphisms $f: C \rightarrow D$ and $g: D \rightarrow C$, we define an adjunction between $f$ and $g$ to be a pair of $2$-morphisms

\[ \eta : \operatorname{id}_{C} \Rightarrow g \circ f \quad \quad \epsilon : f \circ g \Rightarrow \operatorname{id}_ D \]

satisfying analogues of the compatibility conditions $(Z1)$ and $(Z2)$ above (Definition 6.1.1.1).

Our first goal is to adapt Exercise 6.1.0.4 to the setting of a general $2$-category $\operatorname{\mathcal{C}}$. Suppose we are given $1$-morphisms $f: C \rightarrow D$, $g: D \rightarrow C$, $c: T \rightarrow C$, and $d: T \rightarrow D$ in $\operatorname{\mathcal{C}}$. In §6.1.2, we show that every adjunction $(\eta , \epsilon )$ between $f$ and $g$ determines a bijection

\[ \operatorname{Hom}_{\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,D)}( f \circ c, d ) \simeq \operatorname{Hom}_{\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,C)}(c, g \circ d ), \]

depending functorially on $c$ and $d$ (see Corollary 6.1.2.6 and Remark 6.1.2.4). Here the map from right to left is constructed using the unit map $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$, and from left to right using the counit $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$. As an application, we show that an adjunction $(\eta , \epsilon )$ is completely determined by the unit $\eta $ (or the counit $\epsilon $), and give a criterion which can be used to test if an arbitrary $2$-morphism $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is the unit of an adjunction (see Proposition 6.1.2.9, Variant 6.1.2.12, and Proposition 6.1.2.13)

Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: C \rightarrow D$ be a $1$-morphism in $\operatorname{\mathcal{C}}$. In §6.1.3, we show that if $f$ admits a right adjoint $g$, then $g$ is uniquely determined up to (canonical) isomorphism (Corollary 6.1.3.3). Moreover, the formation of right adjoints can be regarded as a (partially defined) functor from $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)^{\operatorname{op}}$ to $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)$ (Notation 6.1.3.5), with a (partially defined) inverse given by the formation of left adjoints (Notation 6.1.3.8). In §6.1.4, we consider the special case where $f: C \xrightarrow {\sim } D$ is an isomorphism in $\operatorname{\mathcal{C}}$: in this case, $f$ automatically admits a right adjoint (and a left adjoint), which can be identified with a homotopy inverse isomorphism $D \xrightarrow {\sim } C$ (Proposition 6.1.4.1).

In §6.1.5, we show that the formation of adjoints is compatible with composition. More precisely, if $f: C \rightarrow D$ and $f': D \rightarrow E$ are $1$-morphisms in a $2$-category $\operatorname{\mathcal{C}}$ which admit right adjoints $g: D \rightarrow C$ and $g': E \rightarrow D$, respectively, then the composition $(f' \circ f): C \rightarrow E$ also admits a right adjoint, which is canonically isomorphic to the composition $(g \circ g'): E \rightarrow C$ (Corollary 6.1.5.5).

The theory of adjunctions can be usefully applied to many $2$-categories $\operatorname{\mathcal{C}}$ other than $\mathbf{Cat}$ (for example, we will use it in § to generalize the theory of adjoint functors to the setting of $\infty $-categories). In §6.1.6, we consider the case $\operatorname{\mathcal{C}}$ has a single object $X$, and can therefore be identified with the monoidal category $\operatorname{\mathcal{E}}= \underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ (see Example 2.2.2.4). Specializing the theory of adjunctions to this situation, we recover the classical notion of a duality datum in $\operatorname{\mathcal{E}}$ (Definition 6.1.6.1).

Structure

  • Subsection 6.1.1: Adjunctions
  • Subsection 6.1.2: Adjuncts
  • Subsection 6.1.3: Uniqueness of Adjoints
  • Subsection 6.1.4: Adjoints of Isomorphisms
  • Subsection 6.1.5: Composition of Adjunctions
  • Subsection 6.1.6: Duality in Monoidal Categories