Contents

category theory

# Contents

## Idea

The left part of a pair of adjoint functors is one of two best approximations to a weak inverse of the other functor of the pair. (The other best approximation is the functor’s right adjoint, if it exists.) Note that a weak inverse itself, if it exists, must be a left adjoint, forming an adjoint equivalence.

A left adjoint to a forgetful functor is called a free functor. Many left adjoints can be constructed as quotients of free functors.

The concept generalises immediately to enriched categories and in 2-categories.

## Definitions

### For categories

###### Definition

Given categories $\mathcal{C}$ and $\mathcal{D}$ and a functor $R: \mathcal{D} \to \mathcal{C}$, a left adjoint of $R$ is a functor $L: \mathcal{C} \to \mathcal{D}$ together with natural transformations $\iota: id_\mathcal{C} \to R \circ L$ and $\epsilon: L \circ R \to id_\mathcal{D}$ such that the following diagrams (known as the triangle identities) commute, where $\cdot$ denotes whiskering of a functor with a natural transformation.

###### Remark

Definition is equivalent to requiring that there is a natural isomorphism between the Hom functors

$Hom_\mathcal{C}\left(L(-),-\right), Hom_\mathcal{D}\left(-,R(-)\right): D^{op} \times C \to \mathsf{Set}.$

Depending upon one’s interpretation of $\mathsf{Set}$, the category of sets, one may strictly speaking need to restrict to locally small categories for this equivalence to parse.

### For enriched categories

The equivalent formulation of Definition given in Remark generalises immediately to the setting of enriched categories.

###### Definition

Given $\mathbb{V}$-enriched categories $\mathcal{C}$ and $\mathcal{D}$ and a $\mathbb{V}$-enriched functor $R: \mathcal{D} \to \mathcal{C}$, a left adjoint of $R$ is a $\mathbb{V}$-enriched functor $L: \mathcal{C} \to \mathcal{D}$ together with a $\mathbb{V}$-enriched natural isomorphism between the Hom functors

$Hom_\mathcal{C}\left((L(-),-\right), Hom_\mathcal{D}\left(-,R(-)\right): D^{op} \times C \to \mathbb{V}.$

### In a 2-category

Definition generalises immediately from CAT, the 2-category of (large) categories, to any 2-category.

###### Definition

Let $\mathcal{A}$ be a 2-category. Given objects $\mathcal{C}$ and $\mathcal{D}$ and a 1-arrow $R: \mathcal{D} \to \mathcal{C}$ of $\mathcal{A}$, a left adjoint of $R$ is a 1-arrow $L: \mathcal{C} \to \mathcal{D}$ together with 2-arrows $\iota: id_\mathcal{C} \to R \circ L$ and $\epsilon: L \circ R \to id_\mathcal{D}$ such that the following diagrams commute, where $\cdot$ denotes whiskering in $\mathcal{A}$.

###### Remark

If one assumes that one’s ambient 2-category has more structure, bringing it closer to being a 2-topos, for example a Yoneda structure, one should be able to give an equivalent formulation of Definition akin to that of Remark .

### For preorders and posets

Restricted to preorders or posets, Definition in its equivalent formulation of Remark can be expressed in the following terminology.

###### Definition

Given posets or preorders $\mathcal{C}$ and $\mathcal{D}$ and a monotone function $R: \mathcal{D} \to \mathcal{C}$, a left adjoint of $R$ is a monotone function $L: \mathcal{C} \to \mathcal{D}$ such that, for all $x$ in $\mathcal{D}$ and $y$ in $\mathcal{C}$, we have that $L(x) \leq y$ holds if and only if $x \leq R(y)$ holds.

## Examples

Last revised on February 27, 2021 at 03:50:04. See the history of this page for a list of all contributions to it.