Editing Adjoint functors (section)

== Introduction and motivation ==

{{Quote|The slogan is "Adjoint functors arise everywhere".|Saunders Mac Lane, ''[[Categories for the Working Mathematician]]''}}

Common mathematical constructions are very often adjoint functors.  Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve [[Limit (category theory)|colimits/limits]] (which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

=== Solutions to optimization problems ===
In a sense, an adjoint functor is a way of giving the ''most efficient'' solution to some problem via a method that is ''formulaic''.  For example, an elementary problem in [[ring theory]] is how to turn a [[Rng (algebra)|rng]] (which is like a ring that might not have a multiplicative identity) into a [[ring (mathematics)|ring]].  The ''most efficient'' way is to adjoin an element '1' to the rng, adjoin all (and only) the elements that are necessary for satisfying the ring axioms (e.g. ''r''+1 for each ''r'' in the ring), and impose no relations in the newly formed ring that are not forced by axioms.  Moreover, this construction is ''formulaic'' in the sense that it works in essentially the same way for any rng.

This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is ''most efficient'' if it satisfies a [[universal property]], and is ''formulaic'' if it defines a [[functor]].  Universal properties come in two types: initial properties and terminal properties.  Since these are [[dual (category theory)|dual]] notions, it is only necessary to discuss one of them.

The idea of using an initial property is to set up the problem in terms of some auxiliary category ''E'', so that the problem at hand corresponds to finding an [[initial object]] of ''E''.  This has an advantage that the ''optimization''—the sense that the process finds the ''most efficient'' solution—means something rigorous and recognisable, rather like the attainment of a [[supremum]].  The category ''E'' is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.

Back to our example: take the given rng ''R'', and make a category ''E'' whose ''objects'' are rng homomorphisms {{math|''R'' → ''S''}}, with ''S'' a ring having a multiplicative identity. The ''morphisms'' in ''E'' between {{math|''R'' → ''S''<sub>1</sub>}} and {{math|''R'' → ''S''<sub>2</sub>}} are [[commutative diagram|commutative triangles]] of the form ({{math|''R'' → ''S''<sub>1</sub>, ''R'' → ''S''<sub>2</sub>, ''S''<sub>1</sub> → ''S''<sub>2</sub>}}) where {{math|S<sub>1</sub> → S<sub>2</sub>}} is a ring map (which preserves the identity). (Note that this is precisely the definition of the [[comma category]] of ''R'' over the inclusion of unitary rings into rng.) The existence of a morphism between {{math|''R'' → ''S''<sub>1</sub>}} and {{math|''R'' → ''S''<sub>2</sub>}} implies that ''S''<sub>1</sub> is at least as efficient a solution as ''S''<sub>2</sub> to our problem: ''S''<sub>2</sub> can have more adjoined elements and/or more relations not imposed by axioms than ''S''<sub>1</sub>.
Therefore, the assertion that an object {{math|''R'' → ''R''{{sup|&lowast;}}}} is initial in ''E'', that is, that there is a morphism from it to any other element of ''E'', means that the ring ''R''* is a ''most efficient'' solution to our problem.

The two facts that this method of turning rngs into rings is ''most efficient'' and ''formulaic'' can be expressed simultaneously by saying that it defines an ''adjoint functor''. More explicitly:  Let ''F'' denote the above process of adjoining an identity to a rng, so ''F''(''R'')=''R''{{sup|&lowast;}}.  Let ''G'' denote the process of "forgetting" whether a ring ''S'' has an identity and considering it simply as a rng, so essentially ''G''(''S'')=''S''. Then ''F'' is the ''left adjoint functor'' of ''G''.

Note however that we haven't actually constructed ''R''{{sup|&lowast;}} yet; it is an important and not altogether trivial algebraic fact that such a left adjoint functor {{math|''R'' → ''R''{{sup|&lowast;}}}} actually exists.

===Symmetry of optimization problems===

It is also possible to ''start'' with the functor ''F'', and pose the following (vague) question: is there a problem to which ''F'' is the most efficient solution?

The notion that ''F'' is the ''most efficient solution'' to the problem posed by ''G'' is, in a certain rigorous sense, equivalent to the notion that ''G'' poses the ''most difficult problem'' that ''F'' solves.

This gives the intuition behind the fact that adjoint functors occur in pairs: if ''F'' is left adjoint to ''G'', then ''G'' is right adjoint to ''F''.