Adjoint functors: Encyclopedia II - Adjoint functors - Motivation

Adjoint functors - Motivation

Adjoint functors - 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

in the category of abelian groups, where F was the functor –⊗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 <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 Mac Lane, any idea such as adjoint functors that occurs widely enough in mathematics should be studied for its own sake.

Adjoint functors - 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 - 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 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 a discussion of 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.

Adjoint functors - 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 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.

Adapted from the Wikipedia article "Motivation", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki