unit of an adjunction




Given an adjunction

(LR):XRLY (L \dashv R) \colon\; X \underoverset{R}{L}{\rightleftarrows} Y

there is a natural transformation (or more generally, a 22-morphism) η:id XRL\eta\colon id_X \to R \circ L, called the unit of the adjunction. (A reason for the name is that RLR \circ L is a monad, which is a kind of monoid object, and η\eta 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 22-morphism ϵ:LRid Y\epsilon\colon L \circ R \to id_Y, called the counit of the adjunction. (This is the co-identity of the comonad LRL \circ R.)



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 L:XYL : X \to Y is fully faithful if and only if the unit η:id XRL\eta : id_X \to R \circ L is a natural isomorphism.

Relation to monads

Every adjunction (LR)(L \dashv R) gives rise to a monad TRLT \coloneqq R \circ L. The unit of this monad idTid \to T is the unit of the adjunction, idRLid \to R \circ L.

Last revised on March 30, 2020 at 09:28:00. See the history of this page for a list of all contributions to it.