adjunction n : an act of joining or adjoining things [syn: junction]
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. The general notion of adjoint functor is studied in a branch of mathematics known as category theory.
Specifically, functors F : C → D and G : D → C between categories C and D form an adjoint pair if there is a family of bijections
- \mathrm_(FX,Y) \cong \mathrm_(X,GY)
- F\dashv G.
Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some of these.
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(F(X), Y) = Hom(X, G(Y))
in the category of abelian groups, where F was the functor - \otimes A (i.e. take the tensor product with A), and G was the functor Hom(A,–).
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 X × A to Y. 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 = , 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. Note that G may have itself a right adjoint that is quite different from F (see below for an example). The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts http://www.arxiv.org/abs/q-alg/9609018.
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.
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 definition in this encyclopedia 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 provides 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 a 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 H. 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'.
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 a discussion of how to formulate the question.
The major argument in favour of the use 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 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 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 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.
There are a variety of equivalent ways of defining adjoint functors. We give three such definitions here. The whole picture, and the relationships between these definitions, will be given in the next section.
Hom set definition
An adjunction between two categories C and D consists of two functors F : C → D and G : D → C and a natural isomorphism
- Φ : HomD (F–, –) → HomC (–, G–)
- ΦX,Y : HomD (F(X), Y) → HomC (X, G(Y))
In order to interpret Φ as a natural isomorphism, one must recognize HomD(F–, –) and HomC(–, G–) as functors. In fact, they are both bifunctors from Cop × D 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 : X ′ → X in C and all morphisms g : Y → Y ′ in D the following diagram commutes:
The horizontal arrows in this diagram are those induced by f and g.
Universal morphism definition
An adjunction between two categories C and D is given by a functor G : D → C together with a universal morphism (FX, ηX) from X to G for each X in C. The assignment X\mapsto FX defines a functor F : C → D called the left adjoint of G. The morphisms ηX form the components of a natural transformation η : 1C → GF called the unit of the adjunction.
Dually, an adjunction between two categories C and D is given by a functor F : C → D together with a universal morphism (GY, εY) from F to Y for each Y in D. The assignment Y\mapsto GY defines a functor G : D → C called the right adjoint of F. The morphisms εY form the components of a natural transformation ε : FG → 1D called the counit of the adjunction.
Unit and counit definition
An adjunction between two categories C and D consists of two functors F : C → D and G : D → C and two natural transformations
There are numerous functors and natural transformations associated with every adjunction, and there exists an intricate web of relationships among these objects. These relationships allow one to determine all of the data based on knowledge of only some of the pieces.
An adjunction between categories C and D consists of
The web of relationships between these functors and transformations can be neatly summarized by a family of commutative diagrams. Let X be an object in C and Y an object in D. Then for every morphism f : X → G(Y) there is a unique morphism g : F(X) → Y such that the following two diagrams commute: Likewise, for every g : F(X) → Y there is a unique f : X → G(Y) such that the above diagrams commute. Each adjunction, therefore, gives rise to a family of universal morphisms:
- For each object X in C, the pair (F(X), ηX) is a universal morphism from X to G.
- For each object Y in D, the pair (G(Y), εY) is a universal morphism from F to Y.
The morphisms f and g in the above diagrams are related by the natural isomorphism Φ, which for each X in C and Y in D gives a bijection
- \Phi_:\mathrm_(F(X),Y) \cong \mathrm_(X,G(Y))
- \beginf &= \Phi_(g) = G(g)\circ \eta_X\\
We consider two special cases of the above diagrams:
The unitIf Y = F(X) then HomD(F(X),Y) contains the identity morphism. The corresponding element of HomC(X,G(Y)) is just the unit ηX. That is, if Y = F(X) and g = idF(X), then f = ηX so that
- \eta_X = \Phi_\left(\mathrm_\right),\qquad
The counitIf X = G(Y) then HomC(X,G(Y)) contains the identity morphism. The corresponding element of HomD(F(X),Y) is just the counit εY. That is, if X = G(Y) and f = idG(Y), then g = εY so that
- \mathrm_ = G(\varepsilon_Y)\circ\eta_,\qquad\varepsilon_Y = \Phi_^\left(\mathrm_\right).
- Free objects and forgetful functors. If F : Set → Grp is the functor assigning to each set X the free group over X, and if G : Grp → 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.
- Products. Let Δ : Grp → Grp2 be the diagonal functor which assigns to every group X the pair (X, X) in the product category Grp2, and Π : Grp2 → Grp the functor which assigns to each pair (Y1, Y2) the product group Y1×Y2. The universal property of the product group shows that Π is right-adjoint to Δ. The co-unit gives the natural projections from the product to the 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.
- 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 (gA, gB) of morphisms such that gBf1 = f2gA. 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-unital 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.
- 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) = 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.
- 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 −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).
- 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.
- 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 spatial 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.
- The functor π0 which assigns to a category its sets 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 antidiscrete category on that set.
Uniqueness of adjoints
If the functor F : C → D has two right adjoints G1 and G2, then G1 and G2 are naturally isomorphic. The same is true for left adjoints.
Conversely, if F is left adjoint to G1, and G1 is naturally isomorphic to G2 then F is also left adjoint to G2. More generally, if 〈F, G, η, ε〉 is an adjunction and
- σ : F → F′
- τ : G → G′
Adjunctions can be composed in a natural fashion. Specifically, if 〈F1, G1, η1, ε1〉 is an adjunction between C and D and 〈F2, G2, η2, ε2〉 is an adjunction between D and E then the functor
- F_2 \circ F_1 : \mathcal \to \mathcal
- G_1 \circ G_2 : \mathcal \to \mathcal.
One can then form a category whose objects are all small categories and whose morphisms are adjunctions.
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:
If C and D are preadditive categories and F : C → D is an additive functor with a right adjoint G : D → C, then G is also an additive functor and the Hom-set bijections
- \Phi_ : \mathrm_(FX,Y) \cong \mathrm_(X,GY)
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.
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 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 X of C there exists a family of morphisms
- fi : X → G(Yi)
- h : X → G(Y)
can be written as
- h = G(t) o fi
for some i in I and some morphism
- t : Yi → Y in D.
An analogous statement characterizes those functors with a right adjoint.
Relationship to other categorical concepts
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 : D → C from every object of C, 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 C (equivalently, every object of D).
Equivalences of categories
Every equivalence of categories defines an adjunction. If F : C → D and G : D → C are functors with natural isomorphisms η : 1C → GF and ε : FG → 1D then (F, G) form an adjoint pair with unit η and counit ε. Conversely, an adjunction 〈F, G, η, ε〉 defines an equivalence of categories if and only if, the unit and counit are natural isomorphisms (and not just natural transformations).
If (F, G) define an equivalence of categories, then F is not only a left adjoint of G but a right adjoint as well. Explicitly, if 〈F, G, η, ε〉 is an adjoint equivalence then so is 〈G, F, ε−1, η−1〉.
Every adjunction 〈F, G, η, ε〉 extends an equivalence of certain subcategories. Define C1 as the full subcategory of C consisting of those objects X of C for which ηX is an isomorphism, and define D1 as the full subcategory of D consisting of those objects Y of D for which εY 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 1D) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses.
Every adjunction 〈F, G, η, ε〉 gives rise to an associated monad 〈T, η, μ〉 in the category C. The functor
- T : \mathcal \to \mathcal
- \eta : 1_ \to T
- \mu : T^2 \to T\,
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.
- Abstract and Concrete Categories
- Categories for the Working Mathematician
- Adjunctions Seven short lectures on adjunctions.
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