Adjoint functors


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, 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.
By definition, an adjunction between categories C and D is a pair of functors
and, for all objects X in C and Y in D a bijection between the respective morphism sets
such that this family of bijections is natural in X and Y. Naturality here means that there are natural isomorphisms between the pair of functors and for a fixed X in C, and also the pair of functors and for a fixed Y in D.
The functor F is called a left adjoint functor or left adjoint to G , while G is called a right adjoint functor or right adjoint to F .
An adjunction between categories C and D is somewhat akin to a "weak form" of an equivalence between C and D, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.

Terminology and notations

Two different roots are being used: "adjunct" and "adjoint". From Oxford shorter English dictionary, "adjunct" is from Latin, "adjoint" is from French.
In Mac Lane, Categories for the working mathematician, chap. 4, "Adjoints", one can verify the following usage. Given a family
of hom-set bijections, we call an "adjunction" or an "adjunction between and ". If is an arrow in, is the right "adjunct" of . The functor is left "adjoint" to, and is right adjoint to.
In general, the phrases " is a left adjoint" and " has a right adjoint" are equivalent.
If F is left adjoint to G, we also write
The terminology comes from the Hilbert space idea of adjoint operators T, U with, which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.

Introduction and Motivation

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, and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

Solutions to optimization problems

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 into a ring. The most efficient way is to adjoin an element '1' to the rng, adjoin all the elements which are necessary for satisfying the ring axioms, 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 R → S, with S a ring having a multiplicative identity. The morphisms in E between R → S1 and R → S2 are commutative triangles of the form where S1 → S2 is a ring map. The existence of a morphism between R → S1 and R → S2 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 R → R* 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*. 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. 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 R → R* actually exists.

Symmetry of optimization problems

It is also possible to start with the functor F, and pose the following 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.

Formal definitions

There are various equivalent definitions for adjoint functors:
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.

Conventions

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.

Definition via universal morphisms

By definition, a functor
is a left adjoint functor if for each object in there exists a universal morphism
from to. Spelled out, this means that for each object in there exists an object
in and a morphism such that for every object
in and every morphism there exists a unique morphism
with.
The latter equation is expressed by the following commutative diagram:
In this situation, one can show that can be turned into a functor in a unique way such that
for all morphisms in ; is then called a left adjoint to.
Similarly, we may define right-adjoint functors. A functor is a right adjoint functor if for each object in,
there exists a universal morphism from to. Spelled out, this means that for each object in,
there exists an object in and a morphism such that for every object in
and every morphism there exists a unique morphism with.
Again, this can be uniquely turned into a functor such that for a morphism in ; is then called a right adjoint to.
It is true, as the terminology implies, that is left adjoint to if and only if is right adjoint to.
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.

Definition via Hom-set adjunction

A hom-set adjunction between two categories C and D consists of two functors F : D → C and and a natural isomorphism
This specifies a family of bijections
for all objects X in C and Y in D.
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 and as functors. In fact, they are both bifunctors from to Set. For details, see the article on hom functors. Explicitly, the naturality of Φ means that for all morphisms in C and all morphisms in D the following diagram commutes:
The vertical arrows in this diagram are those induced by composition. Formally, Hom : HomC → HomC is given by h → f o h o Fg for each h in HomC. Hom is similar.

Definition via counit–unit adjunction

A counit–unit adjunction between two categories C and D consists of two functors F : D → C and G : C → D and two natural transformations
respectively called the counit and the unit of the adjunction, such that the compositions
are the identity transformations 1F and 1G on F and G respectively.
In this situation we say that F is left adjoint to G and G is right adjoint to F , and may indicate this relationship by writing , or simply .
In equation form, the above conditions on are the counit–unit equations
which mean that for each X in C and each Y in D,
Note that denotes the identify functor on the category, denotes the identity natural transformation from the functor F to itself, and denotes the identity morphism of the object FY.
These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimes called the triangle identities, or sometimes the zig-zag equations because of the appearance of the corresponding string diagrams. A way to remember them is to first write down the nonsensical equation and then fill in either F or G in one of the two simple ways which make the compositions defined.
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.

History

The idea of adjoint functors was introduced by Daniel Kan in 1958. 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
in the category of abelian groups, where F was the functor , and G was the functor hom.
The use of the equals sign is an abuse of notation; those two groups are not really identical but there is a way of identifying them that is natural. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the bilinear mappings from X × A to Y. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a natural isomorphism.

Ubiquity

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.
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.

Examples

Free groups

The construction of free groups is a common and illuminating example.
Let F : Set → Grp be the functor assigning to each set Y the free group generated by the elements of Y, and let G : Grp → Set be the forgetful functor, which assigns to each group X its underlying set. Then F is left adjoint to G:
Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Let be the set map given by "inclusion of generators". This is an initial morphism from Y to G, because any set map from Y to the underlying set GW of some group W will factor through via a unique group homomorphism from FY to W. This is precisely the universal property of the free group on Y.
Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Let 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 is a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through via a unique set map from Z to GX. This means that is an adjoint pair.
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.
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 is as follows:
The first counit–unit equation says that for each set Y the composition
should be the identity. The intermediate group FGFY is the free group generated freely by the words of the free group FY. The arrow is the group homomorphism from FY into FGFY sending each generator y of FY to the corresponding word of length one as a generator of FGFY. The arrow is the group homomorphism from FGFY to FY sending each generator to the word of FY it corresponds to. The composition of these maps is indeed the identity on FY.
The second counit–unit equation says that for each group X the composition
should be the identity. The intermediate set GFGX is just the underlying set of FGX. The arrow is the "inclusion of generators" set map from the set GX to the set GFGX. The arrow 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. The composition of these maps is indeed the identity on GX.

Free constructions and forgetful functors

s 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

, 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, and the counit of the adjunction provides the defining maps from the limit object. Below are some specific examples.
s, 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, and the unit of the adjunction provides the defining maps into the colimit object. Below are some specific examples.

Algebra

Every partially ordered set can be viewed as a category. A pair of adjoint functors between two partially ordered sets is called a 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. 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 adjunction between categories C and D consists of
An equivalent formulation, where X denotes any object of C and Y denotes any object of D, is as follows:
From this assertion, one can recover that:
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.

Universal morphisms induce hom-set adjunction

Given a right adjoint functor G : C → D; 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.

counit–unit adjunction induces hom-set adjunction

Given functors F : D → C, G : C → D, and a counit–unit adjunction : F G, we can construct a hom-set adjunction by finding the natural transformation Φ : homC → homD in the following steps:
Given functors F : D → C, G : C → D, and a hom-set adjunction Φ : homC → homD, one can construct a counit–unit adjunction
which defines families of initial and terminal morphisms, in the following steps:

Existence

Not every functor G : C → D 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
can be written as
for some i in I and some morphism
An analogous statement characterizes those functors with a right adjoint.
An important special case is that of locally presentable categories. If is a functor between locally presentable categories, then
If the functor F : D → C 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 and
are natural isomorphisms then 〈F′, G′, ε′, η′〉 is an adjunction where
Here denotes vertical composition of natural transformations, and 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
is left adjoint to
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.

Limit preservation

The most important property of adjoints is their continuity: every functor that has a left adjoint is continuous ; every functor that has a right adjoint is cocontinuous.
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 : D → C is an additive functor with a right adjoint G : C → D, 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, 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 : C → D 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.

Equivalences of categories

If a functor F : D → C 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 need not be a right adjoint of F. Adjoints generalize two-sided inverses.

Monads

Every adjunction 〈F, G, ε, η〉 gives rise to an associated monad 〈T, η, μ〉 in the category D. The functor
is given by T = GF. The unit of the monad
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.