Given an adjunction
there is a natural transformation (or more generally, a -morphism) , called the unit of the adjunction (in older texts, called a “front adjunction”). (A reason for the name is that is a monad, which is a kind of monoid object, and is the identity of this monoid. Since ‘identity’ in this context would suggest an identity natural transformation, we use the synonym ‘unit’.)
Similarly, there is -morphism , called the counit of the adjunction (in older texts, called a “back adjunction”). (This is the co-identity of the comonad .)
Unit and counit of an adjunction satisfy the triangle identities.
An adjunct is given by precomposition with a unit or postcomposition with a counit.
The left adjoint is fully faithful if and only if the unit is a natural isomorphism.
If the unit is a natural isomorphism, is sometimes termed lari (“left adjoint right inverse”); whilst is termed rali (“right adjoint left inverse”). Dually, if the counit is a natural isomorphism, is sometimes termed lali (“left adjoint left inverse”); whilst is termed rari (“right adjoint right inverse”). All four classes of functor are closed under composition, and contain the equivalences.
Every adjunction gives rise to a monad . The unit of this monad is the unit of the adjunction, .
unit of an adjunction
Last revised on February 26, 2021 at 11:40:22. See the history of this page for a list of all contributions to it.