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In mathematics, specifically category theory, adjunction is a possible relationship between two functors.

Adjunction is ubiquitous in mathematics, as it specifies intuitive notions of optimization and efficiency.

In the most concise symmetric definition, an adjunction between categories C and D is a pair of functors,

${\displaystyle F:{\mathcal {D}}\rightarrow {\mathcal {C}}}$   and   ${\displaystyle G:{\mathcal {C}}\rightarrow {\mathcal {D}}}$

and a family of bijections

${\displaystyle \mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)}$

which is natural in the variables X and Y. The functor F is called a left adjoint functor, while G is called a right adjoint functor. The relationship “F is left adjoint to G” (or equivalently, “G is right adjoint to F”) is sometimes written

${\displaystyle F\dashv G.}$

This definition and others are made precise below.

Introduction

“The slogan is ‘Adjoint functors arise everywhere’.” (Saunders Mac Lane, Categories for the working mathematician)

The long list of examples in this article is only a partial indication of how often an interesting mathematical construction is an adjoint functor. As a result, general theorems about left/right adjoint functors, such as the equivalence of their various definitions or the fact that they respectively preserve colimits/limits (which are also found in every area of mathematics), can encode the details of many useful and otherwise non-trivial results.

Spelling (or morphology)

In Mac Lane, Categories for the working mathematician, chap. 4, "Adjoints", one can verify the following usage.

The hom-set bijection ${\displaystyle \varphi }$ is an "adjunction".

The functor ${\displaystyle F}$ is left "adjoint" for ${\displaystyle G}$.

Motivation

Solutions to optimization problems

It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. For example, an elementary problem in ring theory is how to turn a rng (which is like a ring that might not have a multiplicative identity) into a ring. The most efficient way is to adjoin an element '1' to the rng, adjoin all (and only) the elements which are necessary for satisfying the ring axioms (e.g. r+1 for each r in the ring), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaic in the sense that it works in essentially the same way for any rng.

This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them.

The idea of using an initial property is to set up the problem in terms of some auxiliary category E, and then identify that what we want is to find an initial object of E. This has an advantage that the optimization — the sense that we are finding the most efficient solution — means something rigorous and is recognisable, rather like the attainment of a supremum. Picking the right category E is something of a knack: for example, take the given rng R, and make a category E whose objects are rng homomorphisms RS, with S a ring having a multiplicative identity. The morphisms in E between RS1 and RS2 are commutative triangles of the form (RS1,RS2, S1S2) where S1 → S2 is a ring map (which preserves the identity). The existence of a morphism between RS1 and RS2 implies that S1 is at least as efficient a solution as S2 to our problem: S2 can have more adjoined elements and/or more relations not imposed by axioms than S1. Therefore, the assertion that an object RR* is initial in E, that is, that there is a morphism from it to any other element of E, means that the ring R* is a most efficient solution to our problem.

The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor.

Symmetry of optimization problems

Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution?

Problems formulations

Mathematicians do not generally need the full adjoint functor concept. Concepts can be judged according to their use in solving problems, as well as for their use in building theories. The tension between these two motivations was especially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, who used category theory to take compass bearings in other work — in functional analysis, homological algebra and finally algebraic geometry.

It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role of adjunction was inherent in Grothendieck's approach. For example, one of his major achievements was the formulation of Serre duality in relative form — loosely, in a continuous family of algebraic varieties. The entire proof turned on the existence of a right adjoint to a certain functor. This is something undeniably abstract, and non-constructive, but also powerful in its own way.

Posets

Every partially ordered set can be viewed as a category (with a single morphism between x and y if and only if xy). A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection). See that article for a number of examples: the case of Galois theory of course is a leading one. Any Galois connection gives rise to closure operators and to inverse order-preserving bijections between the corresponding closed elements.

As is the case for Galois groups, the real interest lies often in refining a correspondence to a duality (i.e. antitone order isomorphism). A treatment of Galois theory along these lines by Kaplansky was influential in the recognition of the general structure here.

The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes:

• adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status
• closure operators may indicate the presence of adjunctions, as corresponding monads (cf. the Kuratowski closure axioms)
• a very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G.
• division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint.

Together these observations provide explanatory value all over mathematics.

Examples

Free groups

The construction of free groups is a common and illuminating example.

Suppose that F : GrpSet is the functor assigning to each set Y the free group generated by the elements of Y, and that G : GrpSet is the forgetful functor, which assigns to each group X its underlying set. Then F is left adjoint to G:

Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Let  ${\displaystyle \varepsilon _{X}:FGX\to X}$  be the group homomorphism which sends the generators of FGX to the elements of X they correspond to, which exists by the universal property of free groups. Then each  ${\displaystyle (GX,\varepsilon _{X})}$  is a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through  ${\displaystyle \varepsilon _{X}:FGX\to X}$  via a unique set map from Z to GX. This means that (F,G) is an adjoint pair.

Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Let  ${\displaystyle \eta _{Y}:Y\to GFY}$  be the set map given by "inclusion of generators". Then each  ${\displaystyle (FY,\eta _{Y})}$  is an initial morphism from Y to G, because any set map from Y to the underlying set GW of a group will factor through  ${\displaystyle \eta _{Y}:Y\to GFY}$  via a unique group homomorphism from FY to W. This also means that (F,G) is an adjoint pair.

Hom-set adjunction. Maps from the free group FY to a group X correspond precisely to maps from the set Y to the set GX: each homomorphism from FY to X is fully determined by its action on generators. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G).

Counit-unit adjunction. One can also verify directly that ε and η are natural. Then, a direct verification that they form a counit-unit adjunction  ${\displaystyle (\varepsilon ,\eta ):F\dashv G}$  is as follows:

The first counit-unit equation  ${\displaystyle 1_{F}=\varepsilon F\circ F\eta }$  says that for each set Y the composition

${\displaystyle FY{\xrightarrow {\;F(\eta _{Y})\;}}FGFY{\xrightarrow {\;\varepsilon _{FY}\,}}FY}$

should be the identity. The intermediate group FGFY is the free group generated freely by the words of the free group FY. (Think of these words as placed in parentheses to indicate that they are independent generators.) The arrow  ${\displaystyle F(\eta _{Y})}$  is the group homomorphism from FY into FGFY sending each generator y of FY to the corresponding word of length one (y) as a generator of FGFY. The arrow  ${\displaystyle \varepsilon _{FY}}$  is the group homomorphism from FGFY to FY sending each generator to the word of FY it corresponds to (so this map is "dropping parentheses"). The composition of these maps is indeed the identity on FY.

The second counit-unit equation  ${\displaystyle 1_{G}=G\varepsilon \circ \eta G}$  says that for each group X the composition

${\displaystyle GX{\xrightarrow {\;\eta _{GX}\;}}GFGX{\xrightarrow {\;G(\varepsilon _{X})\,}}GX}$

should be the identity. The intermediate set GFGX is just the underlying set of FGX. The arrow  ${\displaystyle \eta _{GX}}$  is the "inclusion of generators" set map from the set GX to the set GFGX. The arrow  ${\displaystyle G(\varepsilon _{X})}$  is the set map from GFGX to GX which underlies the group homomorphism sending each generator of FGX to the element of X it corresponds to ("dropping parentheses"). The composition of these maps is indeed the identity on GX.

Free constructions and forgetful functors

Free objects are all examples of a left adjoint to a forgetful functor which assigns to an algebraic object its underlying set. These algebraic free functors have generally the same description as in the detailed description of the free group situation above.

Diagonal functors and limits

Products, fibred products, equalizers, and kernels are all examples of the categorical notion of a limit. Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and the counit of the adjunction provides the defining maps from the limit object (i.e. from the diagonal functor on the limit, in the functor category). Below are some specific examples.

• Products Let Π : Grp2Grp the functor which assigns to each pair (X1, X2) the product group X1×X2, and let Δ : Grp2Grp be the diagonal functor which assigns to every group X the pair (X, X) in the product category Grp2. The universal property of the product group shows that Π is right-adjoint to Δ. The counit of this adjunction is the defining pair of projection maps from X1×X2 to X1 and X2 which define the limit, and the unit is the diagonal inclusion of a group X into X1×X2 (mapping x to (x,x)).
The cartesian product of sets, the product of rings, the product of topological spaces etc. follow the same pattern; it can also be extended in a straightforward manner to more than just two factors. More generally, any type of limit is right adjoint to a diagonal functor.
• Kernels. Consider the category D of homomorphisms of abelian groups. If f1 : A1B1 and f2 : A2B2 are two objects of D, then a morphism from f1 to f2 is a pair (gA, gB) of morphisms such that gBf1 = f2gA. Let G : DAb be the functor which assigns to each homomorphism its kernel and let F : DAb be the functor which maps the group A to the homomorphism A → 0. Then G is right adjoint to F, which expresses the universal property of kernels. The counit of this adjunction is the defining embedding of a homomorphism's kernel into the homomorphism's domain, and the unit is the morphism identifying a group A with the kernel of the homomorphism A → 0.
A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left adjoints.

Colimits and diagonal functors

Coproducts, fibred coproducts, coequalizers, and cokernels are all examples of the categorical notion of a colimit. Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object. Below are some specific examples.

• Coproducts. If F : AbAb2 assigns to every pair (X1, X2) of abelian groups their direct sum, and if G : AbAb2 is the functor which assigns to every abelian group Y the pair (Y, Y), then F is left adjoint to G, again a consequence of the universal property of direct sums. The unit of this adjoint pair is the defining pair of inclusion maps from X1 and X2 into the direct sum, and the counit is the additive map from the direct sum of (X,X) to back to X (sending an element (a,b) of the direct sum to the element a+b of X).
Analogous examples are given by the direct sum of vector spaces and modules, by the free product of groups and by the disjoint union of sets.

Further examples

Algebra

• Adjoining an identity to a rng. This example was discussed in the motivation section above. Given a rng R, a multiplicative identity element can be added by taking RxZ and defining a Z-bilinear product with (r,0)(0,1) = (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking a ring to the underlying rng.
• Ring extensions. Suppose R and S are rings, and ρ : RS is a ring homomorphism. Then S can be seen as a (left) R-module, and the tensor product with S yields a functor F : R-ModS-Mod. Then F is left adjoint to the forgetful functor G : S-ModR-Mod.
• Tensor products. If R is a ring and M is a right R module, then the tensor product with M yields a functor F : R-ModAb. The functor G : AbR-Mod, defined by G(A) = homZ(M,A) for every abelian group A, is a right adjoint to F.
• From monoids and groups to rings The integral monoid ring construction gives a functor from monoids to rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid. Similarly, the integral group ring construction yields a functor from groups to rings, left adjoint to the functor that assigns to a given ring its group of units. One can also start with a field K and consider the category of K-algebras instead of the category of rings, to get the monoid and group rings over K.
• Field of fractions. Consider the category Domm of integral domains with injective morphisms. The forgetful functor FieldDomm from fields has a left adjoint - it assigns to every integral domain its field of fractions.
• Polynomial rings. Let Ring* be the category of pointed commutative rings with unity (pairs (A,a) where A is a ring, ${\displaystyle a\in A}$ and morphisms preserve the distinguished elements). The forgetful functor G:Ring*Ring has a left adjoint - it assigns to every ring R the pair (R[x],x) where R[x] is the polynomial ring with coefficients from R.
• The Grothendieck group. In K-theory, the point of departure is to observe that the category of vector bundles on a topological space has a commutative monoid structure under direct sum. One may make an abelian group out of this monoid, the Grothendieck group, by formally adding an additive inverse for each bundle (or equivalence class). Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third section discussion above. That is, one can imitate the construction of negative numbers; but there is the other option of an existence theorem. For the case of finitary algebraic structures, the existence by itself can be referred to universal algebra, or model theory; naturally there is also a proof adapted to category theory, too.

Topology

• A functor with a left and a right adjoint. Let G be the functor from topological spaces to sets that associates to every topological space its underlying set (forgetting the topology, that is). G has a left adjoint F, creating the discrete space on a set Y, and a right adjoint H creating the trivial topology on Y.
• Suspensions and loop spaces Given topological spaces X and Y, the space [SX, Y] of homotopy classes of maps from the suspension SX of X to Y is naturally isomorphic to the space [X, ΩY] of homotopy classes of maps from X to the loop space ΩY of Y. This is an important fact in homotopy theory.
• Stone-Čech compactification. Let KHaus be the category of compact Hausdorff spaces and G : KHausTop be the inclusion functor to the category of topological spaces. Then G has a left adjoint F : TopKHaus, the Stone–Čech compactification. The unit of this adjoint pair yields a continuous map from every topological space X into its Stone-Čech compactification. This map is an embedding (i.e. injective, continuous and open) if and only if X is a Tychonoff space.
• Direct and inverse images of sheaves Every continuous map f : XY between topological spaces induces a functor f from the category of sheaves (of sets, or abelian groups, or rings...) on X to the corresponding category of sheaves on Y, the direct image functor. It also induces a functor f −1 from the category of sheaves of abelian groups on Y to the category of sheaves of abelian groups on X, the inverse image functor. f −1 is left adjoint to f. Here a more subtle point is that the left adjoint for coherent sheaves will differ from that for sheaves (of sets).
• Soberification. The article on Stone duality describes an adjunction between the category of topological spaces and the category of sober spaces that is known as soberification. Notably, the article also contains a detailed description of another adjunction that prepares the way for the famous duality of sober spaces and spatial locales, exploited in pointless topology.

Category theory

• A series of adjunctions. The functor π0 which assigns to a category its set of connected components is left-adjoint to the functor D which assigns to a set the discrete category on that set. Moreover, D is left-adjoint to the object functor U which assigns to each category its set of objects, and finally U is left-adjoint to A which assigns to each set the indiscrete category on that set.
• Exponential object. In a cartesian closed category the endofunctor CC given by –×A has a right adjoint –A.

Categorical logic

In the category of sets, if we choose subsets as the canonical subobjects, then these functions are given by:
${\displaystyle (T\subseteq Y)\;\mapsto \;f^{*}(T)=f^{-1}\lbrack T\rbrack }$
${\displaystyle (S\subseteq X)\;\mapsto \;\exists _{f}S=\{\;y\in Y\;\mid \;\exists x\in f^{-1}\lbrack \{y\}\rbrack ,x\in S\;\}=f\lbrack S\rbrack }$
${\displaystyle (S\subseteq X)\;\mapsto \;\forall _{f}S=\{\;y\in Y\;\mid \;\forall x\in f^{-1}\lbrack \{y\}\rbrack ,x\in S\;\}}$

Properties

Existence

Not every functor G : CD admits a left adjoint. If C is a complete category, then the functors with left adjoints can be characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object Y of D there exists a family of morphisms

fi : YG(Xi)

where the indices i come from a set I, not a proper class, such that every morphism

h : YG(X)

can be written as

h = G(t) o fi

for some i in I and some morphism

t : XiX in C.

An analogous statement characterizes those functors with a right adjoint.

Uniqueness

If the functor F : CD has two right adjoints G and G′, then G and G′ are naturally isomorphic. The same is true for left adjoints.

Conversely, if F is left adjoint to G, and G is naturally isomorphic to G′ then F is also left adjoint to G′. More generally, if 〈F, G, ε, η〉 is an adjunction (with counit-unit (ε,η)) and

σ : FF
τ : GG

are natural isomorphisms then 〈F′, G′, ε′, η′〉 is an adjunction where

{\displaystyle {\begin{aligned}\eta '&=(\tau \ast \sigma )\circ \eta \\\varepsilon '&=\varepsilon \circ (\sigma ^{-1}\ast \tau ^{-1}).\end{aligned}}}

Here ${\displaystyle \circ }$ denotes vertical composition of natural transformations, and ${\displaystyle \ast }$ denotes horizontal composition.

Composition

Adjunctions can be composed in a natural fashion. Specifically, if 〈F, G, ε, η〉 is an adjunction between C and D and 〈F′, G′, ε′, η′〉 is an adjunction between D and E then the functor

${\displaystyle F'\circ F:{\mathcal {C}}\leftarrow {\mathcal {E}}}$

${\displaystyle G\circ G':{\mathcal {C}}\to {\mathcal {E}}.}$

More precisely, there is an adjunction between FF and G G′ with unit and counit given by the compositions:

{\displaystyle {\begin{aligned}&1_{\mathcal {E}}{\xrightarrow {\eta }}GF{\xrightarrow {G\eta 'F}}GG'F'F\\&F'FGG'{\xrightarrow {F'\varepsilon G'}}F'G'{\xrightarrow {\varepsilon '}}1_{\mathcal {C}}.\end{aligned}}}

This new adjunction is called the composition of the two given adjunctions.

One can then form a category whose objects are all small categories and whose morphisms are adjunctions.

Limit preservation

The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is a right adjoint) is continuous (i.e. commutes with limits in the category theoretical sense); every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (i.e. commutes with colimits).

Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example:

• applying a right adjoint functor to a product of objects yields the product of the images;
• applying a left adjoint functor to a coproduct of objects yields the coproduct of the images;
• every right adjoint functor is left exact;
• every left adjoint functor is right exact.

If C and D are preadditive categories and F : CD is an additive functor with a right adjoint G : CD, then G is also an additive functor and the hom-set bijections

${\displaystyle \Phi _{Y,X}:\mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)}$

are, in fact, isomorphisms of abelian groups. Dually, if G is additive with a left adjoint F, then F is also additive.

Moreover, if both C and D are additive categories (i.e. preadditive categories with all finite biproducts), then any pair of adjoint functors between them are automatically additive.

Relationships

Universal constructions

As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G : CD from every object of D, then G has a left adjoint.

However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D (equivalently, every object of C).

Equivalences of categories

If a functor F: CD is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

Every adjunction 〈F, G, ε, η〉 extends an equivalence of certain subcategories. Define C1 as the full subcategory of C consisting of those objects X of C for which εX is an isomorphism, and define D1 as the full subcategory of D consisting of those objects Y of D for which ηY is an isomorphism. Then F and G can be restricted to D1 and C1 and yield inverse equivalences of these subcategories.

In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of F (i.e. a functor G such that FG is naturally isomorphic to 1D) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses.

Every adjunction 〈F, G, ε, η〉 gives rise to an associated monadT, η, μ〉 in the category D. The functor

${\displaystyle T:{\mathcal {D}}\to {\mathcal {D}}}$

is given by T = GF. The unit of the monad

${\displaystyle \eta :1_{\mathcal {D}}\to T}$

is just the unit η of the adjunction and the multiplication transformation

${\displaystyle \mu :T^{2}\to T\,}$

is given by μ = GεF. Dually, the triple 〈FG, ε, FηG〉 defines a comonad in C.

Every monad arises from some adjunction—in fact, typically from many adjunctions—in the above fashion. Two constructions, called the category of Eilenberg–Moore algebras and the Kleisli category are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad.

References

1. William Lawvere, Adjointness in foundations, Dialectica, 1969, available here. The notation is different nowadays; an easier introduction by Peter Smith in these lecture notes, which also attribute the concept to the article cited.
2. Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 See page 58
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