Definitions
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 $2$-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 $L \circ R$.)
Unit and counit of an adjunction satisfy the triangle identities.
An adjunct is given by precomposition with a unit or postcomposition with a counit.
By this Prop. at adjoint functor:
The left adjoint $L \colon X \to Y$ 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 $R \colon Y \to X$ is fully faithful (i.e. a reflection) if and only if the counit $\epsilon \colon L \circ R \to id_Y$ is a natural isomorphism.
If the unit is the identity morphism then sometimes
$L$ is termed lari (“left adjoint right inverse”);
$R$ is termed rali (“right adjoint left inverse”).
Dually, if the counit is the identity morphism then sometimes
$L$ is termed lali (“left adjoint left inverse”);
$R$ is termed rari (“right adjoint right inverse”).
All four classes of functor are closed under composition, and contain the equivalences.
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, $id \to R \circ L$.
unit of an adjunction
In the following, a rali with invertible counit is called a surjective equivalence:
Last revised on June 19, 2024 at 19:04:51. See the history of this page for a list of all contributions to it.