nLab
adjunct

Contents

Definition

A pair

(LR):CRLD (L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D

of adjoint functors between categories CC and DD, is characterized by a natural isomorphism

C(LX,Y)D(X,RY) C(L X,Y) \cong D(X,R Y)

of hom-sets for objects XDX\in D and YCY\in C. Two morphisms f:LXYf:L X \to Y and g:XRYg : X \to R Y which correspond under this bijection are said to be adjuncts of each other. That is, gg is the (right-)adjunct of ff, and ff is the (left-)adjunct of gg. Sometimes one writes g=f g = f^\sharp and f=g f = g^\flat, as in musical notation.

Sometimes people call f˜\tilde f the “adjoint” of ff, and vice versa, but this is potentially confusing because it is the functors FF and GG which are adjoint. Other possible terms are conjugate, transpose, and mate.

Properties

Proposition

(adjuncts in terms of adjunction (co-)unit)

Let i X:XRLXi_X : X \to R L X be the unit of the adjunction and η X:LRXX\eta_X : L R X \to X the counit.

Then

  • the adjunct of f:XRYf : X \to R Y in DD is the composite

    f˜:LXLfLRYη YY \tilde f : L X \stackrel{L f}{\to} L R Y \stackrel{\eta_Y}{\to} Y
  • the adjunct of g:LXYg : L X \to Y in CC is the composite

    g˜:Xi XRLXRgRY \tilde g : X \stackrel{i_X}{\to} R L X \stackrel{R g}{\to} R Y

Examples

References

Revised on July 21, 2017 10:13:57 by Todd Trimble (24.146.226.222)