- Coend
- Examples
- References
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 equalizer in the diagram
where the first morphism is induced by 
and the second morphism 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.
- 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
- {{nlab|id=end}}
- {{cite arxiv |last1=Loregian |first1=Fosco |title=This is the (co)end, my only (co)friend |eprint=1501.02503|class=math.CT |year=2015 }}
- {{cite book |last1=MacLane |first1=Saunders |title=Categories for the working mathematician |date=2013 |publisher=Springer Science & Business Media |pages=222–226}}
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