Representable functor Guide, Meaning , Facts, Information and Description

In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functionss) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

Table of contents

1 Definition 2 Uniqueness 3 Examples 4 Representability and Adjoints 5 See also

Definition

Let be an arbitrary category and the be the category of sets. For each object in we define a functor as follows:

• maps each object in to the set of morphisms
• maps each morphism to the function given by .

An arbitrary functor is said to be 'represented by a pair', , where is an object of and is in , if there is a natural isomorphism , given by the consistent family of bijections , such that

for all in .

It is also common in this case to say that is 'representable'. Note that .

A dual set of definitions and statements apply to cofunctorss. Let be an arbitrary category. For each object

in  we define a cofunctor  as follows:

maps each object in to the set of morphisms

maps each morphism to the function given by .

An arbitrary cofunctor is said to be represented by a pair, , where is an object in

and  is in , if there is a natural isomorphism , given

by the consistent family of bijections , such that

for all in .

Note again that .

Uniqueness

The representing pair is unique in the following sense. If and represent the same functor, then there exists one and only one isomorphism from to so that in maps to in . This is because we have the isomorphisms and and so we have an isomorphism . By the Yoneda lemma, is isomorphic to via the isomorphism determined by and , and this maps to . Uniqueness follows as everything is determined by and .

Examples

• Let and consider the cofunctor given by the power set of , and if is a map of sets, the morphism is the map that sends every subset to its inverse image, , in . To represent this functor we need a set, , and a in , that is, in some subset of , so that is isomorphic to via .

  Now, we know that is the map that sends a subset, , of to its inverse image, , a subset of . So, is the inverse image of our chosen . Take and . Then subsets of are exactly of the form for the various in , which are thus characteristic functions.

• Let , the category of rings and let be the forgetful functor. To represent this we need a ring, , and an element in , so that for all rings , is isomorphic to via

in in .

Take , the polynomial ring in one variable with integer coefficients, and . Then any ring homomorphism in is uniquely determined by , where any in can be used.

Representability and Adjoints

The following result shows the relationship between representability of a functor and adjointness.

Proposition: A functor, , has a left adjoint if and only if, for every in , the functor from to mapping to is representable. If represents this functor then is the object part of a left-adjoint of for which the isomorphism is functorial in and yields the adjointness.

See also

• Yoneda lemma

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