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6.1.1 Adjunctions

Our goal in this section is to generalize the notion of an adjunction to an arbitrary $2$-category $\operatorname{\mathcal{C}}$. Here Definition 6.1.0.2 adapts without essential change; the only additional complications are the fact that the associativity and unit constraints of $\operatorname{\mathcal{C}}$ need not be strict.

Definition 6.1.1.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $C$ and $D$ be objects of $\operatorname{\mathcal{C}}$, and let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms in $\operatorname{\mathcal{C}}$. An adjunction between $f$ and $g$ is a pair of $2$-morphisms $( \eta , \epsilon )$, where $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is a morphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ is a morphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,D)$, which satisfy the following compatibility conditions:

$(Z1)$

The composition

\[ f \xRightarrow [\sim ]{\rho _{f}^{-1}} f \circ \operatorname{id}_{C} \xRightarrow {\operatorname{id}_ f \circ \eta } f \circ (g \circ f) \xRightarrow [\sim ]{\alpha _{f,g,f}} (f \circ g) \circ f \xRightarrow {\epsilon \circ \operatorname{id}_ f} \operatorname{id}_{D} \circ f \xRightarrow [\sim ]{\lambda _{f}} f \]

is the identity $2$-morphism from $f$ to itself. Here $\lambda _{f}$ and $\rho _{f}$ are the left and right unit constraints of the $2$-category $\operatorname{\mathcal{C}}$ (Construction 2.2.1.11) and $\alpha _{f,g,f}$ is the associativity constraint for the $\operatorname{\mathcal{C}}$.

$(Z2)$

The composition

\[ g \xRightarrow [\sim ]{\lambda _{g}^{-1}} \operatorname{id}_{C} \circ g \xRightarrow {\eta \circ \operatorname{id}_ g} (g \circ f) \circ g \xRightarrow [\sim ]{\alpha _{g,f,g}^{-1}} g \circ (f \circ g) \xRightarrow {\operatorname{id}_{g} \circ \epsilon } g \circ \operatorname{id}_ D \xRightarrow [\sim ]{\rho _{g}} g \]

is the identity $2$-morphism from $g$ to itself.

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 say that $(\eta , \epsilon )$ exhibits $f$ as a left adjoint of $g$, and also that it exhibits $g$ as a right adjoint of $f$.

Example 6.1.1.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories, which we regard as $1$-morphisms in the strict $2$-category $\mathbf{Cat}$ of Example 2.2.0.4. An adjunction between $F$ and $G$ in the $2$-category $\mathbf{Cat}$ is an adjunction between $F$ and $G$ in the usual category-theoretic sense: that is, a pair of natural transformations $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which satisfy the requirements of Definition 6.1.0.2.

Remark 6.1.1.3. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ be $2$-morphisms of $\operatorname{\mathcal{C}}$. Then the pair $(\eta , \epsilon )$ is an adjunction between $f$ and $g$ in the $2$-category $\operatorname{\mathcal{C}}$ if and only if the pair $(\eta ^{\operatorname{op}}, \epsilon ^{\operatorname{op}})$ it is an adjunction between $g^{\operatorname{op}}$ and $f^{\operatorname{op}}$ in the opposite $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (Construction 2.2.3.1). Note that in this case, $g^{\operatorname{op}}$ is the left adjoint, while $f^{\operatorname{op}}$ is the right adjoint.

Remark 6.1.1.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ be $2$-morphisms of $\operatorname{\mathcal{C}}$. Then the pair $(\eta , \epsilon )$ is an adjunction between $f$ and $g$ in the $2$-category $\operatorname{\mathcal{C}}$ if and only if the pair $(\epsilon ^{\operatorname{c}}, \eta ^{\operatorname{c}})$ is an adjunction between $g^{\operatorname{c}}$ and $f^{\operatorname{c}}$ in the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ (Construction 2.2.3.4). Note that in this case, $\epsilon ^{\operatorname{c}}$ is the unit of the adjunction and $\eta ^{\operatorname{c}}$ is the counit. Similarly, $g^{\operatorname{c}}$ is the left adjoint and $f^{\operatorname{c}}$ is the right adjoint.

Remark 6.1.1.5 (Isomorphism Invariance). Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f,f': C \rightarrow D$ and $g,g': D \rightarrow C$ be $1$-morphisms in $\operatorname{\mathcal{C}}$, and let $(\eta , \epsilon )$ be an adjunction between $f$ and $g$. Suppose we are given invertible $2$-morphisms $\beta : g \xRightarrow {\sim } g'$ and $\gamma : f \xRightarrow {\sim } f'$. Let $\eta '$ denote the composition $\operatorname{id}_{C} \xRightarrow { \eta } g \circ f \xRightarrow [\sim ]{ \beta \circ \gamma } g' \circ f'$ and let $\epsilon '$ denote the composition $f' \circ g' \xRightarrow [\sim ]{ \gamma ^{-1} \circ \beta ^{-1} } f \circ g \xRightarrow {} \operatorname{id}_{D}$. Then the pair $(\eta ', \epsilon ')$ is an adjunction between $f'$ and $g'$.

Exercise 6.1.1.6 (Functoriality). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $2$-categories. Suppose we are given $1$-morphisms $f: C \rightarrow D$ and $g: D \rightarrow C$ in $\operatorname{\mathcal{C}}$. Let $(\eta , \epsilon )$ be an adjunction between $f$ and $g$ in the $2$-category $\operatorname{\mathcal{C}}$, let $\eta '$ denote the composition

\[ \operatorname{id}_{ F(C) } \xRightarrow [\sim ]{} F( \operatorname{id}_{C} ) \xRightarrow { F(\eta ) } F( g \circ f ) \xRightarrow [\sim ]{\mu ^{-1}_{g,f} } F(g) \circ F(f), \]

and let $\epsilon '$ denote the composition

\[ F(f) \circ F(g) \xRightarrow [\sim ]{ \mu _{f,g} } F(f \circ g) \xRightarrow {F(\epsilon ) } F( \operatorname{id}_{D} ) \xRightarrow [\sim ]{} \operatorname{id}_{F(D)}, \]

where $\mu _{f,g}$ and $\mu _{g,f}$ are the composition constraints of the functor $F$ and the unlabeled isomorphisms are the identity constraints of $F$. Show that the pair $(\eta ', \epsilon ')$ is an adjunction between $F(f)$ and $F(g)$ in the $2$-category $\operatorname{\mathcal{D}}$.

Example 6.1.1.7. Let $\operatorname{\mathcal{C}}$ be an ordinary category which admits fiber products, and let $\operatorname{Corr}(\operatorname{\mathcal{C}})$ denote the $2$-category of correspondences in $\operatorname{\mathcal{C}}$ (Example 2.2.2.1). Every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ determines diagrams

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dl]_{\operatorname{id}_ X} \ar [dr]^{f} & & & X \ar [dl]_{f} \ar [dr]^{\operatorname{id}_{X}} & \\ X & & Y & Y & & X } \]

which we can regard as $1$-morphisms $f_{!}: X \rightarrow Y$ and $f^{!}: Y \rightarrow X$ in the $2$-category $\operatorname{Corr}(\operatorname{\mathcal{C}})$. Unwinding the definitions, we see that the compositions $f^{!} \circ f_{!}$ and $f_{!} \circ f^{!}$ are given (up to isomorphism) by the diagrams

\[ \xymatrix@R =50pt@C=50pt{ & X \times _{Y} X \ar [dl]_{\pi _0} \ar [dr]^{\pi _1} & & & X \ar [dl]_{f} \ar [dr]^{f} & \\ X & & X & Y & & Y, } \]

where $\pi _0, \pi _1: X \times _{Y} X \rightarrow X$ are the projection maps. We can therefore regard the diagonal map $\delta : X \rightarrow X \times _{Y} X$ as a $2$-morphism from $\operatorname{id}_ X$ to $f^{!} \circ f_{!}$ in $\operatorname{Corr}(\operatorname{\mathcal{C}})$, and the morphism $f: X \rightarrow Y$ as a $2$-morphism from $f_{!} \circ f^{!}$ to $\operatorname{id}_{Y}$ in $\operatorname{Corr}(\operatorname{\mathcal{C}})$. The pair $( \delta , f )$ is an adjunction between $f_{!}$ and $f^{!}$.