Adjunction

Definition

Def Adjunction An adjunction between categories and consists of functors and , with a natural transformation: with the property: For any , and , there exists a unique such that as indicated in is called the left adjoint, is called the right adjoint, and is called the unit of the adjunction. One sometimes writes for “ is left and right adjoint.” Equivalently, we can also formally define adjunction between categories and as functors and with a natural isomorphism: which specifies a family of bijections: The unit and the counit of the adjunction are then determined as

Remark

Note that the situation is a generalization of equivalence of categories, in that a pseudo-inverse is an adjoint. In that case, however, it is the relation between categories that one is interested in. Here, one is concerned with the relation between specific functors. That is to say, it is not the relation on categories “there exists an adjunction,” but rather “this functor has an adjoint” that we are concerned with.

Prop Given a function between sets, the extended image operation functor is left adjoint to the inverse image functor , where denotes the power set of set , as a poset category ordered by subset inclusion. Proof

Proposition

Prop A category has all binary products iff the diagonal functor has a right adjoint.

Proposition

Prop Adjoints are unique up to isomorphism. Specifically, given a functor and right adjoints , Proof For any , and , since and we have Thus, by Yoneda lemma, . But this isomorphism is natural in , again by adjointness.

Order Adjoints

Proposition

Prop Right adjoints preserve limits, and left adjoints preserve colimits.