Adjunctions
Table of Contents
1. Adjunctions
Let \( F \colon \mathcal C \rightarrow \mathcal D \) and \( G \colon \mathcal C \rightarrow \mathcal D \) be \( 1 \)-functors. An adjunction \( F \dashv G \) between \( F \) and \( G \) is a natural isomorphism
\begin{equation*} \mathcal D(F(-),-) \cong \mathcal C(-,G(-)) \colon \mathcal C^{op} \times \mathcal D \rightarrow \Set. \end{equation*}Let \( F \colon \mathcal C \rightarrow \mathcal D \) and \( G \colon \mathcal C \rightarrow \mathcal D \) be \( 1 \)-functors. Then an djunction \( F \dashv G \) consists of
- an isomorphism \( \Phi _{A,B} \colon \mathcal D(FA,B) \rightarrow \mathcal C(A,GB) \) for all objects \( A \in \Ob \mathcal C \) and \( B \in \Ob \mathcal D \)
such that
for all morphisms \( f \colon A \rightarrow A' \) in \( \mathcal C \) and \( g \colon B \rightarrow B' \) in \( \mathcal D \)
\begin{equation*} \xymatrix{ \mathcal D(FA',B) \ar[r] ^{\Phi _{A',B}} \ar[d] _{g \circ - Ff} & \mathcal C(A',GB) \ar[d] ^{Gg \circ - \circ f} \\ \mathcal D(FA,B') \ar[r] ^{\Phi _{A,B'}} & \mathcal C(A,GB') } \end{equation*}commutes.
1.1. Adjunctions via Unit and Counit
Let \( \Phi \colon \mathcal C \leftrightarrow \mathcal D \) be an adjunction \(F \dashv G \). The associated unit \( \eta \colon 1_{\mathcal C} \Rightarrow GF \) is defined by \( \eta_A := \Phi(1 _{FA}) \) for every \( A \in \Ob \mathcal C \). The associated counit \( \epsilon \colon FG \Rightarrow 1 _{\mathcal D}\) is defined by \( \epsilon _A := \Phi ^{-1}(1 _{GA}) \) for every \( A \in \Ob \mathcal D \).
You should think of the counit as a generalized evaluation (the is the reason why we denote the unit by “\( \epsilon \)”) and of the unit as a generalized coevaluation. As examples think of the adjunction \( - \times B \dashv \Set(B,-) \) in \( \Set \) or more generally of the \( \otimes \)-\( \Hom \) adjunction in a closed monoidal category. In fact under the identification of monoidal categories as \( 2 \)-categories with one object the definition of an adjunction in a \( 2 \)-category corresponds to an object/dual object pair with evaluation and coevaluation morphisms (these notions come from tensor categories).
Let \( F \colon \mathcal C \rightarrow \mathcal D \) and \( G \colon \mathcal D \rightarrow \mathcal C \) be \( 1 \)-functors. Then an (Hom-set) adjunction between \( F \) and \( G \) is equivalent to an adjunction in the \( 2 \)-category \( \Cat \) between \( F \) and \( G \).
In general we think that the definition via unit and counit is the more fundamental definition. It makes sense in every \( 2 \)-category and only uses elementary terms (is formulated in the first order language of category theory). But in practice the (hom-set) definition appears often. Hence it is important to know by heart how to switch between the two worlds. Therefore we spell out how to reconstruct the natural isomorphisms of an adjunction using unit and counit.
Let \( F \dashv G \) be an adjunction in \( \Cat \) where \( (\eta, \epsilon ) \) is the unit-counit pair. Then the isomorphism \( \mathcal D(FA,B) \rightarrow \mathcal C(A,GB) \) is given by
\begin{equation*} \xymatrix{FA \ar[r]^f & B} \mapsto \xymatrix{A \ar[r]^{\eta_A} & GFA \ar[r] ^{G(f)} & GB} \end{equation*}and the inverse \( \mathcal C(A,GB) \rightarrow \mathcal D(FA,B) \) is given by
\begin{equation*} \xymatrix{A \ar[r]^f & GB} \mapsto \xymatrix{FA \ar[r] ^{F(f)} & FGB \ar[r] ^{\epsilon_B} & B.} \end{equation*}In formulas we have \( \Phi _{A,B} = G(-) \circ \eta_A \) and \( \Phi _{A,B} ^{-1} = \epsilon_B \circ F(-) \).