Adjoint functors
In category theory, sometimes used as a general theory to discuss the structure concept in mathematics as a whole, the existence of many pairs of adjoint functors is one of the major observations. (For structure see also algebraic structure, structure (category theory).) It is also on an abstract level somewhat beyond everyday usage. This article starts with a number of introductory sections. There are several points of view.
The idea of adjoint functors in general took shape during the 1950s. Like many of the concepts in category theory, it probably arose from the needs of homological algebra, which was at heart devoted to computations. Presentations of the subject would have noticed relations like
Hom (TB, C) = Hom (B, UC)
in the category of abelian groups, where T was the functor 'take the tensor product with A', and U the functor Hom(A, .). Here Hom (X,Y) means 'all homomorphisms of abelian groups'; and the use of the equals sign is an abuse of notation. That is, those two groups aren't identical: what is meant is that 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.
Such relations turn out to be everywhere in abstract algebra if one looks for them; and elsewhere too. Based on the Hilbert space idea of adjoint operators T, U with
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.
By itself, the generality of the adjoint functor concept isn't a recommendation to most mathematicians. Concepts are judged according to their use in problem-solving, at least as much as for theory-building - a tension that, at the time in question during the 1950s, was perhaps as burning a question as it ever has been. 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 it was inherent in the whole approach he took to recognise the role of adjunctions. 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.
One good way to motivate adjoint functors is this: explain what problem it is that 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, be said that the concept pinned down by adjoint functors is the best structure of a type one is interested in constructing. For example in ring theory an elementary question is how to add a multiplicative identity to a ring that doesn't have one (the Wikipedia definition actually assumes one: see ring (mathematics). The best way is to add 1, add nothing extra you don't need (you will need to have r+1 for r in the ring, clearly), and add no equations s=t in the new ring that aren't forced by axioms. This is rather vague, though suggestive. There are several ways to make it precise, adjoint functors only being one.
The way that is a popular alternative is that of the universal property, which is 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 as like the attainment of a supremum. To do it is something of a knack: take the given ring R, and make a category C of ring homomorphisms R -> S, with S a ring having multiplicative identity as objects. 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 is to look at two categories C0 and C1, of rings without assumption of 1, and of rings with 1. 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'. Now this is honest in a way that category theory has no intention of being.
The answer about the way to get a (unitary) ring from one thatis not not unitary 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.
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. (Provided, that is, it is contravariant: that may require a cosmetic change to one of the orders, to its opposite). See that article for a number of examples: the case of Galois theory of course is a leading one. As for Galois groups, the real interest is is in refining a correspondence to a duality (i.e. order isomorphism with the opposite). 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:
In category theory, 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 equivalence consisting of bijective functions
We 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 IdC to GF consisting of morphisms
Free objects. If F : Set → Group is the functor assiging 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, free modules etc. 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 (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 unitary This example was discussed in section 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 construct a left adjoint to the functor taking a ring to the underlying non-unitary 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) = HomZ(A, M) for every abelian group A, is a right adjoint to F.
From monoids and groups to rings The monoid ring and group ring constructions are left adjoints of functors on the category of rings. The two cases are, respectively, take the underlying multiplicative monoid, or the multiplicative group of units in it. Those adjoints will give the integral versions Z[M], and Z[M]. For a more general coefficient ring one can for example start with a field K and the category of K-algebras over it instead, to get the monoid and group rings over K.
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: See induced representation - this 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.
Let G be the functor on topological spaces (X, &t;) taking the underlying set X (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 topology with open sets Y and the empty set only.
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.
The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is a right adjoint) is continuous; every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (see limit (category theory)).
Not every functor G : D → C admits a left adjoint. If D is complete (see limit (category theory)), 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 for every object x of C there exists a family of morphisms fi : x → G(di) (where the indices i come from a set (not a proper class) I), such that every morphism h : x → G(d) can be written as h = G(t) o fi for some i in I and some morphism t : di → d in D.Adjoint functors - ubiquity
Adjoint functors - when existence is deep
Adjoint functors - optimal talk
Adjoint functors - partial order case
Together these observations provide explanatory value all over mathematics. Formal definitions
for all objects X in C and Y in D. All such pairs of adjoint functors arise from universal constructions.
for every X in C. ηX is defined as φX,F(X) (idF(X)). Analogously, one may define a co-unit ε, a natural transformation consisting of morphisms
for every Y in D.Examples
Functors having a left and a right adjoint
Adjoints preserve certain limits
General existence theorem
