Category theory MOC

Adjoint functor

An adjunction of functors is an adjunction in . #m/def/cat Let , be categories. A pair of functors form an adjunction, written

or compactly , iff there is a natural isomorphism in of hom-sets¹

When adjoints exist they are unique up to natural isomorphism, hence we call the left adjoint of , and the right adjoint of .

The name comes from an analogy to the Adjoint operator. In the archetypal examples, we think of as forgetful and as free — See Free-forgetful adjunction.

Unit and coünit

We can equivalently rephrase the condition for an adjunction in terms of a unit or coünit, so named since they form the corresponding data for a monad or comonad induced by the adjunction respectively.

• There exists a natural transformation called the unit of adjunction such that for any objects , , and morphism , there exists a unique adjunct such that .

  
  

• There exists a natural transformation called the coünit of adjunction such that for any objects , , and morphism , there exists a unique adjunct such that .

  
  

To see that either of these are necessary and sufficient, note²

This gives us another perspective on adjunctions: They are a weakening of Equivalence of categories.

Properties

• Right adjoint functors are continuous
• Left adjoint functors are cocontinuous

#state/tidy | #lang/en | #SemBr