nLab
unit of an adjunction

Contents

Definition

Given an adjunction

R Layer 1 XYL, 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 22-morphism) η:id XRL\eta\colon id_X \to R \circ L, called the unit of the adjunction. (This is so called because 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.)

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 (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.

Revised on December 5, 2013 05:51:54 by Urs Schreiber (89.204.138.93)