Joyal's CatLab
Adjoint Functors and Monads

A first definition

An adjunction is the category-theoretical analogue of a Galois connection in order theory. Given categories C\mathbf{C} and D\mathbf{D}, and functors

an adjunction is a natural isomorphism

D(FC,D)C(C,GD)\mathbf{D}(FC,D) \cong \mathbf{C}(C,GD)

of functors C op×DSet\mathbf{C}^{\mathrm{op}} \times \mathbf{D} \to \mathrm{Set}.

In this situation, we say that FF is left adjoint to GG and GG is right adjoint to FF, referring to their positions in D(F,)\mathbf{D}(F-,-) and C(,G)\mathbf{C}(-,G-). This is abbreviated as FGF \dashv G.

If C\mathbf{C} and D\mathbf{D} are preorders, i.e., have at most one arrow between each two objects, then the definition specialises to monotone maps FF and GG, and says that FCDFC \leq D iff CGDC \leq GD for any CCC \in \mathbf{C} and DDD \in \mathbf{D}.

Examples

Here are a few examples of adjunctions. They are meant to give a flavour of what adjunctions can be, and the reader should probably not yet try to prove that they are. Indeed, the various characterisations of adjunctions which we’ll describe below will allow possibly more intuitive proofs.

It would be nice to have here an example of a coreflection, but the few intuitive examples of coreflections I know are too long to be exposed here. Anyone has an idea?

Equivalent characterisations

Proposition. Functors FF and GG as above form an adjunction iff there is a natural transformation η:id CGF\eta \colon \mathrm{id}_{\mathbf{C}} \to GF, such that for any objects CCC \in \mathbf{C} and DDD \in \mathbf{D}, for any arrow f:CGDf \colon C \to G D, there is a unique arrow f:FCDf' \colon F C \to D making the diagram

commute.

This characterisation probably allows easy proofs that the examples above form adjunctions. There is a dual one:

Proposition. Functors FF and GG as above form an adjunction iff there is a natural transformation ϵ:FGid D\epsilon \colon FG \to \mathrm{id}_{\mathbf{D}}, such that for any objects CCC \in \mathbf{C} and DDD \in \mathbf{D}, for any arrow g:FCDg \colon F C \to D, there is a unique arrow f:CGDf' \colon C \to G D making the diagram

commute.

Proposition. Functors FF and GG as above form an adjunction iff there are natural transformations η:id CGF\eta \colon \mathrm{id}_{\mathbf{C}} \to GF and ϵ:FGid D\epsilon \colon FG \to \mathrm{id}_{\mathbf{D}} satisfying the so-called “zig-zag” identities, namely ϵ FF(η)=id C\epsilon_F \circ F(\eta) = \mathrm{id}_{\mathbf{C}} and G(ϵ)η G=id DG(\epsilon) \circ \eta_G = \mathrm{id}_{\mathbf{D}}.

I tried to make this into a pretty 2-categorical diagram, but couldn’t in 30 minutes, then gave up.

Revised on December 22, 2012 at 03:11:50 by Jacques Carette