### Context

#### 2-Category theory

2-category theory

# Contents

## Definition

A pair

$\left(L⊣R\right):C\stackrel{\stackrel{L}{←}}{\underset{R}{\to }}D$(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

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

of hom-sets for objects $X\in D$ and $Y\in C$. Two morphisms $f:LX\to Y$ and $\stackrel{˜}{f}:X\to RY$ which correspond under this bijection are said to be adjuncts of each other. That is, $\stackrel{˜}{f}$ is the adjunct of $f$ and $f$ is the adjunct of $\stackrel{˜}{f}$.

Sometimes people call $\stackrel{˜}{f}$ the “adjoint” of $f$, and vice versa, but this is potentially confusing because it is the functors $F$ and $G$ which are adjoint. Other possible terms are conjugate and mate.

## Properties

Let ${i}_{X}:X\to RLX$ be the unit of the adjunction and ${\eta }_{X}:LRX\to X$ the counit.

Then

• the adjunct of $f:X\to RY$ in $D$ is the composite

$\stackrel{˜}{f}:LX\stackrel{Lf}{\to }LRY\stackrel{{\eta }_{Y}}{\to }Y$\tilde f : L X \stackrel{L f}{\to} L R Y \stackrel{\eta_Y}{\to} Y
• the adjunct of $g:LX\to Y$ in $C$ is the composite

$\stackrel{˜}{g}:X\stackrel{{i}_{X}}{\to }RLX\stackrel{Rg}{\to }RY$\tilde g : X \stackrel{i_X}{\to} R L X \stackrel{R g}{\to} R Y

Revised on November 17, 2010 14:58:14 by Urs Schreiber (131.211.232.104)