nLab 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 (in older texts, called a “front 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 a 22-morphism ϵ:LRid Y\epsilon\colon L \circ R \to id_Y, called the counit of the adjunction (in older texts, called a “back adjunction” or “end 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 (i.e. a coreflection) if and only if the unit η:id XRL\eta : id_X \to R \circ L is a natural isomorphism (if and only if id XRLid_X \cong R \circ L by Lemma A1.1.1 of the Elephant). Dually, the right adjoint R:YXR : Y \to X is fully faithful (i.e. a reflection) if and only if the counit ϵ:LRid Y\epsilon : L \circ R \to id_Y is a natural isomorphism. (See this Prop. at adjoint functor.)

If the unit is the identity, LL is sometimes termed lari (“left adjoint right inverse”); whilst RR is termed rali (“right adjoint left inverse”). Dually, if the counit is the identity, LL is sometimes termed lali (“left adjoint left inverse”); whilst RR is termed rari (“right adjoint right inverse”). All four classes of functor are closed under composition, and contain the equivalences.

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 November 30, 2022 at 02:25:04. See the history of this page for a list of all contributions to it.