Adjoint functors

Relationship between two functors abstracting many common constructions

In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

The terms adjoint and adjunct are both used, and are cognates: one is taken directly from Latin, the other from Latin via French. In the classic text Categories for the working mathematician, Mac Lane makes a distinction between the two. Given a family

The long list of examples in this article indicates that common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve colimits/limits (which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

In a sense, 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 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, so that the problem at hand corresponds to finding an initial object of E. This has an advantage that the optimization—the sense that the process finds the most efficient solution—means something rigorous and is recognisable, rather like the attainment of a supremum. The category E is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.

Back to our 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). (Note that this is precisely the definition of the comma category of R over the inclusion of unitary rings into rng.) 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. More explicitly: Let F denote the above process of adjoining an identity to a rng, so F(R)=R*. Let G denote the process of “forgetting″ whether a ring S has an identity and considering it simply as a rng, so essentially G(S)=S. Then F is the left adjoint functor of G.

Note however that we haven't actually constructed R* yet; it is an important and not altogether trivial algebraic fact that such a left adjoint functor RR* actually exists.

It is also possible to start with the functor F, and pose the following (vague) question: is there a problem to which F is the most efficient solution?

The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves.

This gives the intuition behind the fact that adjoint functors occur in pairs: if F is left adjoint to G, then G is right adjoint to F.

The equivalency of these definitions is quite useful. Adjoint functors arise everywhere, in all areas of mathematics. Since the structure in any of these definitions gives rise to the structures in the others, switching between them makes implicit use of a great deal of tedious details that would otherwise have to be repeated separately in every subject area.

The theory of adjoints has the terms left and right at its foundation, and there are many components which live in one of two categories C and D which are under consideration. Therefore it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category C or the "righthand" category D, and also to write them down in this order whenever possible.

In this article for example, the letters X, F, f, ε will consistently denote things which live in the category C, the letters Y, G, g, η will consistently denote things which live in the category D, and whenever possible such things will be referred to in order from left to right (a functor F : DC can be thought of as "living" where its outputs are, in C).

The latter equation is expressed by the following commutative diagram:

These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem.

A hom-set adjunction between two categories C and D consists of two functors F : DC and G : CD and a natural isomorphism

In this situation, F is left adjoint to G and G is right adjoint to F .

This definition is a logical compromise in that it is somewhat more difficult to satisfy than the universal morphism definitions, and has fewer immediate implications than the counit–unit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions.

In order to interpret Φ as a natural isomorphism, one must recognize homC(F–, –) and homD(–, G–) as functors. In fact, they are both bifunctors from Dop × C to Set (the category of sets). For details, see the article on hom functors. Explicitly, the naturality of Φ means that for all morphisms f : XX′ in C and all morphisms g : Y Y in D the following diagram commutes:

The vertical arrows in this diagram are those induced by composition. Formally, Hom(Fg, f) : HomC(FY, X) → HomC(FY′, X′) is given by hf o h o Fg for each h in HomC(FY, X). Hom(g, Gf) is similar.

A counit–unit adjunction between two categories C and D consists of two functors F : DC and G : CD and two natural transformations

respectively called the counit and the unit of the adjunction (terminology from universal algebra), such that the compositions

In equation form, the above conditions on (ε,η) are the counit–unit equations

Note: The use of the prefix "co" in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies an initial property whereas the counit morphisms will satisfy terminal properties, and dually. The term unit here is borrowed from the theory of monads where it looks like the insertion of the identity 1 into a monoid.

The idea of adjoint functors was introduced by Daniel Kan in 1958.[2] Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as

If one starts looking for these adjoint pairs of functors, they turn out to be very common in abstract algebra, and elsewhere as well. The example section below provides evidence of this; furthermore, universal constructions, which may be more familiar to some, give rise to numerous adjoint pairs of functors.

In accordance with the thinking of Saunders Mac Lane, any idea, such as adjoint functors, that occurs widely enough in mathematics should be studied for its own sake.[citation needed]

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[discuss], but also powerful in its own way.

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

Hom-set adjunction. Group homomorphisms 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, another restatement of the universal property of free groups. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G).

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.

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.

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.

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.

Every partially ordered set can be viewed as a category (where the elements of the poset become the category's objects and we have a single morphism from x to 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:

There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.

An equivalent formulation, where X denotes any object of C and Y denotes any object of D, is as follows:

In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functors F and G alone are in general not sufficient to determine the adjunction. The equivalence of these situations is demonstrated below.

Given a right adjoint functor G : CD; in the sense of initial morphisms, one may construct the induced hom-set adjunction by doing the following steps.

A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor. (The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairs is a trivially defined inclusion or forgetful functor.)

hence ΦΨ is the identity transformation. Thus Φ is a natural isomorphism with inverse Φ−1 = Ψ.

Given functors F : DC, G : CD, and a hom-set adjunction Φ : homC(F-,-) → homD(-,G-), one can construct a counit–unit adjunction

which defines families of initial and terminal morphisms, in the following steps:

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

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

An analogous statement characterizes those functors with a right adjoint.

If the functor F : DC 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

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

More precisely, there is an adjunction between F F' and G' G with unit and counit given respectively by the compositions:

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

Since there is also a natural way to define an identity adjunction between a category C and itself, one can then form a category whose objects are all small categories and whose morphisms are adjunctions.

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:

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

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.

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).

If a functor F : DC 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

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

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.