Adjoint Functors

A homomorphism out of a free group is just a choice of images for the generators — one bijection of hom-sets, natural in both variables, and the same pattern everywhere.

The idea

An adjunction relates two functors running in opposite directions, and it records a pattern that recurs across algebra. The pattern is visible in a familiar instance: to define a homomorphism out of a free group, it is enough to say where the generators go. Pick one element of the target for each letter, and every word is forced, because a homomorphism sends a product to the product of the images. Writing $F(S)$ for the free group on a set $S$ and $U(G)$ for the underlying set of a group $G$, that convenience is a bijection $\mathbf{Grp}(F(S), G) \cong \mathbf{Set}(S, U(G)),$ one for each $S$ and each $G$, where $\mathbf{Grp}(X, Y)$ denotes the homomorphisms from $X$ to $Y$ and $\mathbf{Set}(X, Y)$ the functions.

The definition records the pattern.

Definition (Adjunction).

Let $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D} \to \mathcal{C}$ be functors. Then $F$ is left adjoint to $G$, and $G$ is right adjoint to $F$, written $F \dashv G$, when there is a bijection $\mathcal{D}(F(A), B) \cong \mathcal{C}(A, G(B))$ for every object $A$ of $\mathcal{C}$ and every object $B$ of $\mathcal{D}$, compatible with composing arrows on either side.

Which functor is applied to which argument carries the whole content: the left adjoint acts on the source of the arrow, the right adjoint on the target. Free construction on the left, forgetful functor on the right, is the case to keep in mind; it covers most of the examples in algebra.

Ways to work on it

• Walkthrough. The definition of an adjunction as a natural bijection of hom-sets, with free and forgetful functors as the running example.
• Proof. Prove that the free group functor is left adjoint to the forgetful functor: restricting a homomorphism to the generators is a natural bijection.
• Practice. Count the arrows out of a free object, say which functor of a pair is the left adjoint, and decide whether a claimed adjunction is genuine.
• Hardest. Pick the genuine adjunction out of four, compose two adjunctions to identify a group, and meet floor and ceiling as adjoints.

Not sure where to start? Take the ten-question placement test.