nLab
unit of an adjunction

Definition

Given an adjunction

$$R\begin{array}{c}\underset{X}{\overset{Y}{ Layer 1 }}\end{array}⊢L,$$R \mathrlap{\begin{matrix}\underoverset{X}{Y}{\begin{svg} <svg width="19" height="94" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="73447"> <g> <title>Layer 1</title> <path fill="none" stroke="#000000" d="m5.738871,0c-7.27908,34.015419 -6.27507,69.082848 -0.3765,90.999992" id="svg_73447_1" marker-end="url(#se_marker_end_svg_73447_1)"/> <path fill="none" stroke="#000000" d="m11.606452,93c9.406131,-33.735249 8.109031,-68.513321 0.48652,-90.25001" marker-end="url(#se_marker_end_svg_73447_2)" id="svg_73447_2"/> </g> <defs> <marker id="se_marker_end_svg_73447_1" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50"> <path id="svg_73447_3" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#000000" stroke="#000000" stroke-width="10"/> </marker> <marker id="se_marker_end_svg_73447_2" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50"> <path id="svg_73447_4" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#000000" stroke="#000000" stroke-width="10"/> </marker> </defs> </svg> \end{svg}\includegraphics[width=14]{updownarrows}}\end{matrix}}\vdash L\, ,

there is a natural transformation (or more generally, a $2$-morphism) $\eta :{\mathrm{id}}_X\to R\circ L$, called the unit of the adjunction. (This is so called because $R\circ L$ 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 $2$-morphism $ϵ:L\circ R\to {\mathrm{id}}_Y$, called the counit of the 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.

Relation to monads

Every adjunction $(L⊣R)$ gives rise to a monad $T≔R\circ L$. The unit of this monad $\mathrm{id}\to T$ is the unit of the adjunction, $\mathrm{id}\to R\circ L$.

Related concepts

• adjunction

• zig-zag law/triangle identity

• unit

  • unit of an adjunction

  • unit of a monad

• adjunct