End (category theory)
From Wikipedia, the free encyclopedia
-
- This page is not about the use of End to represent (categories of) endomorphisms.
In category theory, an end of a functor 
is a universal dinatural transformation from an object e of X to S.
More explicitly, this is a pair (e,ω), where e is an object of X and
is a dinatural transformation, such that for every dinatural 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 ω) and is written
or just 
.
[edit] Coend
The definition of the coend of a functor is the dual of definition of an end.
Thus, a coend of S consists of a pair (d,ζ), where d is an object of X and
is a dinatural transformation, such that for every dinatural transformation
there exists a unique morphism
of X with
for every object a of C.
The coend d of the functor S is written
or 
.
