End (category theory)


In category theory, an end of a functor is a universal extranatural transformation from an object e of X to S.
More explicitly, this is a pair, where e is an object of X and is an extranatural transformation such that for every extranatural transformation there exists a unique morphism
of X with
for every object a of C.
By abuse of language the object e is often called the end of the functor S and is written
Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram
where the first morphism being equalized is induced by and the second is induced by.

Coend

The definition of the coend of a functor is the dual of the definition of an end.
Thus, a coend of S consists of a pair, where d is an object of X and
is an extranatural transformation, such that for every extranatural transformation there exists a unique morphism
of X with for every object a of C.
The coend d of the functor S is written
Characterization as colimit: Dually, if X is cocomplete and C is small, then the coend can be described as the coequalizer in the diagram

Examples

Suppose we have functors then
In this case, the category of sets is complete, so we need only form the equalizer and in this case
the natural transformations from to. Intuitively, a natural transformation from to is a morphism from to for every in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.
Let be a simplicial set. That is, is a functor. The discrete topology gives a functor, where is the category of topological spaces. Moreover, there is a map sending the object of to the standard -simplex inside. Finally there is a functor that takes the product of two topological spaces.
Define to be the composition of this product functor with. The coend of is the geometric realization of.