Joyal's CatLab Adjoint Functors and Monads

Contents

A first definition

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

an adjunction is a natural isomorphism

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

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

If $\mathbf{C}$ and $\mathbf{D}$ are preorders, i.e., have at most one arrow between each two objects, then the definition specialises to monotone maps $F$ and $G$, and says that $FC \leq D$ iff $C \leq GD$ for any $C \in \mathbf{C}$ and $D \in \mathbf{D}$.

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.

• For any topological space $X$, consider the orders $\mathcal{P}(X)$ of subsets of $X$ and $\mathcal{C}(X)$ of closed subsets of $X$. Topological closure defines a monotone map $F \colon \mathcal{P}(X) \to \mathcal{C}(X)$, and the inclusion $G \colon \mathcal{C}(X) \hookrightarrow \mathcal{P}(X)$ is of course monotone. $F$ is here left adjoint to $G$.

• Any category $\mathbf{C}$ of algebraic gadgets, e.g., groups, comes with a functor from sets to $\mathbf{C}$ computing the “free gadget”. For example, consider the “free group” functor $F \colon \mathbf{Set} \to \mathbf{Grp}$ sending any set $X$ to the free group $F(X)$ on $X$. There is a functor $G \colon \mathbf{Grp} \to \mathbf{Set}$ in the other direction, called “forgetful”, sending any group to its carrier. We have $F \dashv G$.

• Continuing the previous example, the full subcategory of $\mathbf{Grp}$ spanning abelian groups is [reflective] in $\mathbf{Grp}$, which means that the inclusion has a left adjoint, say $L$. This $L$ sends any group $G$ to its abelianisation.

Equivalent characterisations

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

commute.

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

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

commute.

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