Category theory MOC

Adjoint functor

An adjunction of functors is an adjunction in Category of small categories. cat Let , be categories. A pair of functors form an adjunction, written

or compactly , iff there is a natural isomorphism in [[Functor category|]] 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 .

Proof of uniqueness

By duality, it suffices to prove right adjoints are unique up to natural isomorphism. Suppose . Then by adjunction

hence for any object we have

naturally and thus

naturally (see Yoneda embedding).

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

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Footnotes

1. 2010. Category theory, §9 ↩

2. 2020. From categories to homotopy theory, p. 40 ↩