unit of an adjunction
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Given an adjunction
there is a natural transformation (or more generally, a -morphism) , called the unit of the adjunction. (This is so called because 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. (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.
Relation to monads
Every adjunction gives rise to a monad . The unit of this monad is the unit of the adjunction, .
Revised on December 5, 2013 05:51:54
by Urs Schreiber