Adjoint Functors

Summary

An adjunction $F⊣G$ between functors $F:\mathcal{A}\to \mathcal{B}$ and $G:\mathcal{B}\to \mathcal{A}$ is a natural bijection $\mathcal{B}(FA,B)\cong \mathcal{A}(A,GB)$. Adjunctions arise throughout mathematics wherever a “free” construction is paired with a “forgetful” one, and they have four equivalent formulations. Left adjoints preserve colimits; right adjoints preserve limits.

Overview

Adjunctions are one of the central concepts of category theory. The slogan is: “free constructions are left adjoints to forgetful functors.” An adjunction $F⊣G$ captures the sense in which maps out of a free object $FA$ are in bijection with maps into the underlying object $GB$. This bijection must be natural in both variables — not just a random family of bijections, but one that commutes with all morphisms.

Main Content

Definition 2.1.1: Adjunction (Hom-Set Form)

Let $F:\mathcal{A}\to \mathcal{B}$ and $G:\mathcal{B}\to \mathcal{A}$ be functors. An adjunction between $F$ and $G$, written $F⊣G$, consists of a bijection

$$\overline{(-)}:\mathcal{B}(FA,B)\stackrel{\sim }{\to }\mathcal{A}(A,GB)$$

natural in $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We call $F$ the left adjoint and $G$ the right adjoint.

Naturality in $A$: for any $h:A^{\prime }\to A$, the square

$$\overline{f\circ Fh}=\bar{f}\circ h$$

Naturality in $B$: for any $k:B\to B^{\prime }$,

$$\overline{k\circ f}=Gk\circ \bar{f}$$

Notation: For $f:FA\to B$, write $\bar{f}:A\to GB$ for its transpose (or adjunct). The bijection and its inverse are mutual transposes.

Core Examples

Adjunction $F⊣G$	$F$	$G$	Bijection
Free/forgetful for groups	Free group $F(\mathbf{Set}\to \mathbf{Grp})$	Forgetful $U$	$\mathbf{Grp}(FX,G)\cong \mathbf{Set}(X,UG)$
Product/diagonal	$(-)\times B$	$(-{)}^B$ (internal hom)	$\mathcal{A}(A\times B,C)\cong \mathcal{A}(A,C^B)$
Diagonal/limit	$\Delta :\mathcal{A}\to [\mathcal{I},\mathcal{A}]$	${\lim }_{\mathcal{I}}$	Cones ↔ maps into limit
$f^{\ast }$/direct image	Inverse image $f^{-1}$	Direct image $f_{\ast }$	Preorder adjunctions in $\mathcal{O}(X)$
Abelianization/forgetful	$(-)/[-,-]:\mathbf{Grp}\to \mathbf{Ab}$	inclusion	Hom in Ab ↔ Hom in Grp
Tensor/hom for ${\mathbf{Vect}}_k$	$(-){\otimes }_{k}W$	${\mathrm{Hom}}_k(W,-)$	${\mathbf{Vect}}_k(V\otimes W,U)\cong {\mathbf{Vect}}_k(V,\mathrm{Hom}(W,U))$

Example: Adjunction for Preorders (BCT, Ch. 2.1)

In a preorder (viewed as a category), an adjunction $F⊣G$ between monotone maps $F:P\to Q$ and $G:Q\to P$ is exactly a Galois connection: $F(a)\le b$ iff $a\le G(b)$. Left adjoints in a preorder are exactly lower sets; right adjoints are upper sets. Floor and ceiling are adjoint to inclusion $\mathbb{Z}\hookrightarrow \mathbb{R}$.

Four Equivalent Definitions

Leinster presents four equivalent ways to define an adjunction $F⊣G$:

1. Hom-set bijection (Definition above): Natural isomorphism $\mathcal{B}(F-,-)\cong \mathcal{A}(-,G-)$.

2. Unit and counit (Units and Counits): Natural transformations $\eta :1_{\mathcal{A}}\Rightarrow GF$ and $\epsilon :FG\Rightarrow 1_{\mathcal{B}}$ satisfying the triangle identities.

3. Universal maps: For each $A$, a universal map ${\eta }_A:A\to GFA$ such that every $f:A\to GB$ factors uniquely through ${\eta }_A$.

4. Initial objects (Adjunctions via Initial Objects): For each $A$, ${\eta }_A:A\to GFA$ is the initial object of the comma category $(A\downarrow G)$.

All four are equivalent; the choice of definition depends on context.

Uniqueness of Adjoints

Theorem: Adjoints Are Unique Up to Isomorphism (BCT, Ch. 2.2)

If $F⊣G$ and $F⊣G^{\prime }$, then $G\cong G^{\prime }$ (natural isomorphism). Similarly, if $F⊣G$ and $F^{\prime }⊣G$, then $F\cong F^{\prime }$.

Proof sketch: The natural isomorphisms $\mathcal{A}(A,GB)\cong \mathcal{B}(FA,B)\cong \mathcal{A}(A,G^{\prime }B)$ hold for all $A$, so by the Yoneda lemma $GB\cong G^{\prime }B$ naturally.

Connections

• Units and counits (Units and Counits) give an equivalent, often more computational, formulation.
• Initial objects (Adjunctions via Initial Objects) provide the universal-property formulation.
• Adjoints preserve (co)limits (Adjoints and Limits): left adjoints preserve colimits, right adjoints preserve limits.
• Adjoint functor theorems (Adjoint Functor Theorems) give conditions under which a functor has an adjoint.
• Representability: $F⊣G$ iff for each $B$, the functor $\mathcal{B}(F-,B):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ is representable by $GB$.

See Also

• Units and Counits — Alternative formulation of adjunctions
• Adjunctions via Initial Objects — Universal maps and comma categories
• Adjoints and Limits — Preservation theorem
• Adjoint Functor Theorems — Existence theorems
• Monads - Overview — monads arise from adjunctions