# End (category theory)

In category theory, an **end** of a functor is a universal extranatural transformation from an object *e* of **X** to *S*.[1]

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* (forgetting ) 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

- Natural transformations:

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 .

## References

- end in
*nLab* - Loregian, Fosco (2015). "This is the (co)end, my only (co)friend". arXiv:1501.02503 [math.CT].

- Mac Lane, Saunders (2013).
*Categories for the working mathematician*. Springer Science & Business Media. pp. 222–226.