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.

## Properties

### In homotopy type theory

Note: the HoTT book calls a internal category in HoTT a “precategory” and a univalent category a “category”, but here we shall refer to the standard terminology of “category” and “univalent category” respectively.

###### Lemma

(Lemma 9.3.2 in the HoTT book)
If $A$ is an univalent category and $B$ is a category then the type “$F$ is a left adjoint” is a mere proposition.

###### Proof

Suppose we are given $(G, \eta, \epsilon)$ with the triangle identities and also $(G', \eta', \epsilon')$. Define $\gamma: G \to G'$ to be $(G' \epsilon )(\eta G')$. Then

\begin{aligned} \delta \gamma &= (G \epsilon')(\eta G')(G' \epsilon) (\eta' G)\\ &= (G \epsilon')(G F G' \epsilon_))\eta G' F G)(\eta' G)\\ &= (G \epsilon ')(G \epsilon' F G)(G F \eta' G)(\eta G)\\ &= (G \epsilon)(\eta G)\\ &= 1_G \end{aligned}

using Lemma 9.2.8 (see natural transformation) and the triangle identities. Similarly, we show $\gamma \delta=1_{G'}$, so $\gamma$ is a natural isomorphism $G \cong G'$. By Theorem 9.2.5 (see functor category), we have an identity $G=G'$.

Now we need to know that when $\eta$ and $\epsilon$ are transported? along this identity, they become equal to $\eta'$ and $\epsilon '$. By Lemma 9.1.9,

Lemma 9.1.9 needs to be included. For now as transports are not yet written up I didn’t bother including a reference to the page category. -Ali

this transport is given by composing with $\gamma$ or $\delta$ as appropriate. For $\eta$, this yields

$(G' \epsilon F)(\eta' G F)\eta = (G' \epsilon F)(G' F \eta)\eta'=\eta'$

using Lemma 9.2.8 (see natural transformation) and the traingle identity. The case of $\epsilon$ is similar. FInally, the triangle identities transport correctly automatically, since hom-sets are sets.

## Examples

Last revised on June 9, 2022 at 08:21:49. See the history of this page for a list of all contributions to it.