End (category theory)

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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 (forgetting ) and is written

Characterization as limit: If X is complete and C is small, the end can be described as the equaliser in the diagram

where the first morphism is induced by and the second morphism is induced by .

Coend[edit]

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[edit]

  • 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.

  • Geometric realizations:

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 which sends the object of to the standard simplex inside . Finally there is a functor which 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[edit]