Adjoint functor

A concept expressing the universality and naturalness of many important mathematical constructions, such as a free universal algebra, various completions, and direct and inverse limits.

Let be a covariant functor in one argument from a category into a category . induces a functor

where is the category dual to , is the category of sets, and is the basic set-valued functor. The functor is contravariant in the first argument and covariant in the second. Similarly, any covariant functor induces a functor

which is also contravariant in the first argument and covariant in the second. The functors and are adjoint, or form an adjoint pair, if and are isomorphic, that is, if there is a natural transformation that establishes a one-to-one correspondence between the sets of morphisms and for all objects and . The transformation is called the adjunction of with , is called the left adjoint of and the right adjoint of (this is written , or simply ). The transformation is called the co-adjunction.

Let . For all and , let

The families of morphisms and define natural transformations and , called the unit and co-unit of the adjunction . They satisfy the following equations:

In general, a pair of natural transformations and leads to an adjoint pair (or adjunction) if the following equations hold:

for all objects and . A natural transformation is the unit of some adjunction if and only if for any morphism in there is a unique morphism in such that . This property expresses the fact that is a free object over with respect to the functor in the sense of the following definition. An object together with a morphism is free over an object if every morphism can be written uniquely in the form for some morphism . A functor has a left adjoint if and only if for every there is an object that is free over with respect to .

Examples of adjoint functors.

1) If , where is the category of sets, then has a left adjoint only if it is representable. A representable functor has a left adjoint if and only if all co-products exist in , where and for all .

2) In the category of sets, for any set the basic functor is the right adjoint of the functor .

3) In the category of Abelian groups, the functor is the right adjoint of the functor of tensor multiplication by , and the imbedding functor of the full subcategory of torsion groups is the left adjoint of the functor of taking the torsion part of any Abelian group.

4) Let be the forgetful functor from an arbitrary variety of universal algebras into the category of sets. The functor has a left adjoint , which assigns to every set the free algebra of the variety with as set of free generators.

5) The imbedding functor of an arbitrary reflective subcategory of a category is the right adjoint of the -reflector (cf. also Reflexive subcategor). In particular, the imbedding functor of the category of Abelian groups in the category of groups has a left adjoint, which assigns to every group its quotient group by the commutator subgroup.

Properties of adjoint functors.

The left adjoint functor of a given functor is uniquely determined up to isomorphism of functors. Left adjoints commute with co-limits (e.g. co-products) and send null objects and null morphism into null objects and null morphisms, respectively.

Let and be categories that are complete on the left and locally small on the left. A functor has a left adjoint if and only if the following conditions hold: a) commutes with limits; b) for every , at least one of the sets , , is non-empty; and c) for every , there is a set such that every morphism is representable in the form , where , , .

By passing to dual categories, one may establish a duality between the concepts of a "left adjoint functor" and a "right adjoint functor" ; this enables one to deduce the properties of right adjoints from those of left adjoints.

The concept of an adjoint functor is directly connected with the concept of a triple (or monad) in a category.

References

[1]	M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)
[2]	S. Maclane, "Categories for the working mathematician" , Springer (1971)

Comments

A category is called complete on the left if small diagrams have limits. A category is called locally small on the left if it has small hom-sets. The statement that a functor has a left adjoint if and only if a), b) and c) above holds, is called the Freyd adjoint functor theorem.