Home > Representable functor

1 Definition

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

• 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 cofunctors. Let be an arbitrary category. For each object

• 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

by the consistent family of bijections , such that

for all in .

Note again that .

2 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 .

3 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 functionsThis article is about the characteristic function in set theory. For characteristic function in probability theory see characteristic function In mathematical subfield of set theory, the indicator function or characteristic function is a function defined.

• Let , the category of ringsIn ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. History See Ring theory Definition and notation A ring is an a and let be the forgetful functorA forgetful functor is a type of functor in mathematics. The nomenclature is suggestive of such a functor's behaviour: given some algebraic object as input, some or all of the object's structure is 'forgotten' in the output. For an algebraic structure of. 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 ringIn abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a commutative ring. More precisely, let R be a commutative ring. The polynomial ring in n variables X . X is the set of all polynomials in those in one variableIn computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. In mathematics, a variable often represents an unknown quantity; in computer science, it represents a place where a quantity can be stored. Variabl with integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which st coefficientIn mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable, a basis vector, a basis function and so on. Usually, the objects and the coefficients are indexed in the same way, leading to expressions such ass, and . Then any ring homomorphism in is uniquely determined by , where any in can be used.