nLab adjunct

Definition

A pair

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

of adjoint functors between categories $C$ and $D$, is characterized by a natural isomorphism

$$C(L X,Y) \cong D(X,R Y)$$

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

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

Properties

Proposition

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

Let $\eta_X \colon X \to R L X$ be the unit of the adjunction and $\epsilon_X \colon L R X \to X$ the counit.

Then

• the adjunct of $f \colon X \to R Y$ in $D$ is the composite

  $$\tilde f \colon L X \overset{L f}{\longrightarrow} L R Y \overset{\epsilon_Y}{\longrightarrow} Y$$

• the adjunct of $g \colon L X \to Y$ in $C$ is the composite

  $$\tilde g \colon X \stackrel{\eta_X}{\longrightarrow} R L X \overset{R g}{\longrightarrow} R Y \,.$$

For proof see this Prop. at adjoint functor.

Examples

• The process of currying is an instance of passage to adjuncts, specialized to the tensor-hom adjunction of a closed monoidal category.

Related concepts

• adjunction

• zig-zag law/triangle identity

• unit of an adjunction

• mate

• adjunct

References

• Categories Work, second edition, p. 81

• Category Theory in Context, p. 116, 124.