Representability is a term from the mathematical branch of category theory . It describes the fact that there are "classifying objects" for certain constructions.
A contravariant functor from a category into the category of sets is said to be representable if there is a pair consisting of an object of and an element such that
for all objects of bijective is. Then you just write
A covariant functor is said to be representable if there is an analog pair such that
is bijective.
Further designations:
For an element of , the corresponding morphism is also called a classifying morphism .
is called the performing object , even if by itself the natural equivalence
or.
is not yet established.
is often called universal because each element of for some object is an image of below with a suitable morphism
is. (The same applies in the case of covariant functors.)
properties
If a contravariant functor is represented by , on the one hand , and, on the other hand, as above , then there is exactly one isomorphism for which applies. He is the classifying morphism of respect .
Representable functors are left exact; H.
or .
Examples
The formation of the power set of a set can be counterproductive functor are considered: an illustration of sets is the induced map the archetype of subsets: .
This functor is represented by the pair , because if an object, i.e. a set, is bijective. The classifying mapping of a subset is therefore the characteristic function of , because .
The first cohomology group with coefficients in the integers is a contravariant functor that passes through the 1- sphere together with one of the two generators of