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