nLab unit of an adjunction

Definition

Given an adjunction

$$(L \dashv R) \colon\; X \underoverset{R}{L}{\rightleftarrows} Y$$

there is a natural transformation (or more generally, a $2$-morphism) $\eta\colon id_X \to R \circ L$, called the unit of the adjunction (in older texts, called a “front adjunction”). (A reason for the name is that $R \circ L$ is a monad, which is a kind of monoid object, and $\eta$ 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 $2$-morphism $\epsilon\colon L \circ R \to id_Y$, called the counit of the adjunction (in older texts, called a “back adjunction” or “end adjunction”). (This is the co-identity of the comonad $L \circ R$.)

Properties

General

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 $L : X \to Y$ is fully faithful (i.e. a coreflection) if and only if the unit $\eta : id_X \to R \circ L$ is a natural isomorphism (if and only if $id_X \cong R \circ L$ by Lemma A1.1.1 of the Elephant). Dually, the right adjoint $R : Y \to X$ is fully faithful (i.e. a reflection) if and only if the counit $\epsilon : L \circ R \to id_Y$ is a natural isomorphism. (See this Prop. at adjoint functor.)

If the unit is the identity, $L$ is sometimes termed lari (“left adjoint right inverse”); whilst $R$ is termed rali (“right adjoint left inverse”). Dually, if the counit is the identity, $L$ is sometimes termed lali (“left adjoint left inverse”); whilst $R$ 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 $(L \dashv R)$ gives rise to a monad $T \coloneqq R \circ L$. The unit of this monad $id \to T$ is the unit of the adjunction, $id \to R \circ L$.

Related concepts

• adjunction

• zig-zag law/triangle identity

• unit

  • unit object

  • unit of an adjunction

    derived adjunction unit

  • unit of a monad

• adjunct