nLab left adjoint

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

• left adjoints preserve colimits

• left adjoints preserve epimorphisms.

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

• The left adjoint of the nerve functor $N: \mathsf{Grpd} \to \mathsf{Set}^{\Delta^{op}}$ from the category of groupoids to the category of simplicial sets is the fundamental groupoid functor.

Related entries

• Galois connection for more on left adjoints of monotone functions.
• adjoint functor for more on left adjoints of functors.
• adjunction for more on left adjoints in $2$-categories.
• examples of adjoint functors for examples.
• pro-left adjoint
• 2-adjunction for a categorified notion of adjunction.