Adjoint functors

The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. (Category theory discusses the structure concept in mathematics as a whole; see also algebraic structure, structure (category theory).) Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians.

Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some of these.

Motivation

Ubiquity of adjoint functors

The idea of an adjoint functor was formulated 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

Hom (FB, C) = Hom (B, GC)

in the category of abelian groups, where F was the functor 'take the tensor product with A', and G was the functor Hom(A, .). Here

Hom (X,Y)

means 'all homomorphisms of abelian groups'. The use of the equals sign is an abuse of notation; those two groups aren't 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 BxA to C. That's something particular to the case of tensor product, though. What category theory teaches is that 'natural' is a well-defined term of art in mathematics: natural equivalence.

The terminology comes from the Hilbert space idea of adjoint operators T, U with <Tx,y> = <x,Uy>, which is formally similar to the above Hom relation. We say that F is left adjoint to G, and G is right adjoint to F. Since G may have itself a right adjoint, quite different from F (see below for an example), the analogy breaks down at that point.

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 MacLane, any idea such as adjoint functors that occurs widely enough in mathematics should be studied for its own sake.

Deep problems formulated with adjoint functors

By itself, the generality of the adjoint functor concept isn't a recommendation to most mathematicians. Concepts are judged according to their use in solving problems, at least as much as for their use in building theories. The tension between these two potential motivations for developing a mathematical concept was especially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, who used category theory to take compass bearings in foundational, axiomatic 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 - one could say 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.

Adjoint functors as solving optimization problems

One good way to motivate adjoint functors is to explain what problem they solve, and how they solve it.

That can only be done, in some sense, by what mathematicians call 'hand-waving'. It can be said, however, that adjoint functors pin down the concept of the best structure of a type one is interested in constructing. For example, an elementary question in ring theory is how to add a multiplicative identity to a ring that doesn't have one (the Wikipedia definition actually assumes one: see ring (mathematics) and glossary of ring theory). The best way is to add an element '1' to the ring, add nothing extra you don't need (you will need to have r+1 for r in the ring, clearly), and add no relations in the new ring that aren't forced by axioms. This is rather vague, though suggestive.

There are several ways to make precise this concept of best structure. Adjoint functors are one method; the notion of universal properties provide another, essentially equivalent but arguably more concrete approach.

Universal properties are also based on category theory. The idea is to set up the problem in terms of some auxiliary category C; and then identify what we want to do as showing that C has an initial object. This has an advantage that the optimisation — the sense that we are finding the best solution - is singled out and recognisable rather like the attainment of a supremum. To do it is something of a knack: for example, take the given ring R, and make a category C whose objects are ring homomorphisms R → S, with S a ring having a multiplicative identity. The morphisms in C must fill in triangles that are commutative diagrams, and preserve multiplicative identity. The assertion is that C has an initial object R → R*, and R* is then the sought-after ring.

The adjoint functor method for defining an multiplicative identity for rings is to look at two categories, C₀ and C₁, of rings, respectively without and with assumption of multiplicative identity. There is a functor from C₁ to C₀ that forgets about the 1. We are seeking a left adjoint to it. This is a clear, if dry, formulation.

One way to see what is achieved by using either formulation is to try a direct method. (This is favoured, for example, by John Conway.) One simply adds to R a new element 1, and calculates on the basis that any equation resulting is valid if and only if it holds for all rings that we can create from R and 1. This is the impredicative method: meaning that the ring we are trying to construct is one of the rings quantified over in 'all rings'. This overt use of impredicativity is honest, in a way that category theory has no intention of being.

The answer regarding the way to get a (unital) ring from one that is not unital is simple enough (see examples below); this section has been discussion how to formulate the question.

The major argument in favour of adjoint functors is probably this: if one goes through the universal property or impredicative reasoning often enough, it seems like repeating the same kind of steps.

The case of partial orders

Every partially ordered set can be viewed as a category (with a single morphism between x and y if and only if x ≤ y). A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it 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 Martin Hyland 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 division by ring ideals, can be recognised as the attempt to provide an adjoint.

Together these observations provide explanatory value all over mathematics.

Formal definitions

A pair of adjoint functors between two categories C and D consists of two functors F : C → D and G : D → C and a natural isomorphism consisting of bijective functions

φ_X,Y: Mor_D(F(X), Y) → Mor_C(X, G(Y))

for all objects X in C and Y in D. We then say that F is left-adjoint to G and G is right-adjoint to F.

Every adjoint pair of functors defines a unit η, a natural transformation from the functor Id_C to GF consisting of morphisms

η_X : X -> GF(X)

for every X in C. η_X is defined as φ_X,F(X) (id_F(X)). Analogously, one may define a co-unit ε, a natural transformation consisting of morphisms

ε_Y : FG(Y) → Y.

for every Y in D.

Examples

Free objects. If F : Set → Group is the functor assigning to each set X the free group over X, and if G : Group → Set is the forgetful functor assigning to each group its underlying set, then the universal property of the free group shows that F is left adjoint to G. The unit of this adjoint pair is the embedding of a set X into the free group over X.

Free rings, free abelian groups, and free modules follow the same pattern.

Products. Let F : Group → Group² be the functor which assigns to every group X the pair (X, X) in the product category Group², and G : Group² → Group the functor which assigs to each pair (Y₁, Y₂) the product group Y₁×Y₂. The universal property of the product group shows that G is right-adjoint to F. The co-unit gives the natural projections from the product to the factors.

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.

Coproducts. If F : Ab² → Ab assigns to every pair (X₁, X₂) of abelian groups their direct sum and if G : Ab → Ab² 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 the adjoint pair provides the natural embeddings from the factors into the direct sum.

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.

Kernels. Consider the category D of homomorphisms of abelian groups. If f₁ : A₁ → B₁ and f₂ : A₂ → B₂ are two objects of D, then a morphism from f₁ to f₂ is a pair (g_A, g_B) of morphisms such that g_Bf₁ = f₂g_A. Let G : D → Ab be the functor which assigns to each homomorphism its kernel and let F : Ab → D be the morphism which maps the group A to the homomorphism A → 0. Then G is right adjoint to F, which expresses the universal property of kernels, and the co-unit of this adjunction yields the natural embedding of a homomorphism's kernel into the homomorphism's domain.

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.

Making a ring unital This example was discussed in section 1.3 above. Given a non-unitary ring 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 non-unital ring.

Ring extensions. Suppose R and S are rings, and ρ : R → S 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-Mod → S-Mod. Then F is left adjoint to the forgetful functor G : S-Mod → R-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-Mod → Ab. The functor G : Ab → R-Mod, defined by G(A) = Hom_Z(A, M) for every abelian group A, is a right adjoint to F.

From monoids and groups to rings The 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 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.

Direct and inverse images of sheaves Every continuous map f : X → Y 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^* from the category of sheaves on Y to the category of sheaves on X, the inverse image functor. f^* is left adjoint to f_*.

The Grothendieck construction. 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. To make an abelian group out of this monoid, one can follow the method of making a presentation of a group, adding formally 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.

Frobenius reciprocity in the representation theory of groups: see induced representation. This example foreshadowed the general theory by about half a century.

Stone-Čech compactification. Let D be the category of compact Hausdorff spaces and G : D → Top be the forgetful functor which treats every compact Hausdorff space as a topological space. Then G has a left adjoint F : Top → D, 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.

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 spacial locales, exploited in pointless 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.

Properties

Relation to universal constructions

All pairs of adjoint functors arise from universal constructions. The example constructions above can all be spelled out with a univeral property, and in fact some of the relevant articles do so.

Universal constructions are more general than adjoint functor pairs: as mentioned earlier, 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.

Uniqueness of adjoints

If the functor F : C → D had two right-adjoints G₁ and G₂, then G₁ and G₂ are naturally isomorphic. The same is true for left-adjoints.

Adjoints preserve certain limits

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.

Additivity

If the functor F : C → D is left adjoint to G : D → C and both C and D are additive categories, then both F and G are additive functors.

Composition

If the functor F₁ : C → D has G₁ : D → C as right adjoint and the functor F₂ : D → E has G₂ : E → D as right adjoint, then the composition F₂oF₁ : C → E has G₁oG₂ : E → C as right adjoint.

Characterization via unit and co-unit

The unit η : 1_C → GF and the co-unit ε : FG → 1_D have the following properties: the composition (εF)o(Fη), a natural transformation F→FGF→F, is equal to 1_F, and the composition (Gε)o(ηG) : G→GFG→G is equal to 1_G.

Conversely, given two natural transformations η : 1_C → GF and ε : FG → 1_D with these properties, then the functors F and G form an adjoint pair.

Adjoint pairs extend equivalences

Every adjoint pair extends an equivalence of certain subcategories. Specifically, if F : C → D is left adjoint to G : D → C with unit η and co-unit ε, define C₁ as the full subcategory of C consisiting of those objects X of C for which η_X is an isomorphism, and define D₁ as the full subcategory of D consisting of those objects Y of D for which ε_Y is an isomorphism. Then F and G can be restricted to C₁ and D₁ 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 1_D) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses.

General existence theorem

Not every functor G : D → C admits a left adjoint. If D is complete, then the functors with left adjoints can be characterized by the Freyd Adjoint Functor Theorem: G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object X of C there exists a family of morphisms f_i : X → G(Y_i) (where the indices i come from a set I, not a proper class -- this is the whole point), such that every morphism h : X → G(Y) can be written as h = G(t) o f_i for some i in I and some morphism t : Y_i → Y in D.

An analogous statement characterizes those functors with a right adjoint.