Representability (category theory)

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Representability is a term from the mathematical branch of category theory . It describes the fact that there are "classifying objects" for certain constructions.

definition

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 following forget function can be represented:
from to represented by
Abelian groups amounts
Vector spaces over a body amounts
unitary rings amounts
Topological spaces amounts (a single point space)
  • 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
is pictured. In general, there are descriptive spaces for the functors for any Abelian group and natural numbers . They are called Eilenberg-MacLane rooms .

See also

The representations of the form presented above also occur in the Yoneda lemma .