Representable functor

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

Definition

A contravariant functor from one category to the category of sets is called representable if there is a pair consisting of an object and of an element such that

Is bijective for all objects. You then simply write

A covariant functor is called representable if there is an analogous pair, so that

Is bijective.

Other names:

• For an element of ie, the corresponding morphism also classifying morphism.
• Means representative object, even if by itself the natural equivalence

• Is often called universal because every element of for any object picture from under a suitable morphism

Properties

• If a contravariant functor as above, on the one hand through, but on the other hand, also represented by, so there is exactly one isomorphism applies. He is the classifying morphism of respect.
• Displayable functors are left exact, that is,

Examples

• The formation of the power set of a set can be regarded as a contravariant functor: for a mapping of sets is the induced map the archetype of subsets:.

• Forget The following functors are representable:

• An example of the commutative algebra form the Kähler differentials with the universal derivation.
• The fundamental group of a punctured topological space is by definition a representable functor on the category of topological spaces dotted with the homotopy classes of punctured illustrations as morphisms:

• Cohomology group with the first coefficient in the integers is a contravariant functor that of the 1- sphere together with one of the two generators