End (category theory)
Encyclopedia
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
, where e is an object of X and
is a dinatural transformation from the constant functor whose value is e on every object and
on every morphism, 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
) and is written
If X is complete
, 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 
.
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 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
Dually, if X is cocomplete, then the coend can be described as the coequalizer in the diagram
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 F to G. Intuitively, a natural transformation from F to G is a morphism from F(c) to G(c) for every c in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.
Let T be a simplicial set. That is, T 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 [n] of 
to the standard n simplex inside 
. Finally there is a functor 
which takes the product of two topological spaces. Define S to be the composition of this product functor with 
. The coend of S is the geometric realization of T.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, an end of a functor
is a universal dinatural transformation from an object e of X to S.More explicitly, this is a pair
, where e is an object of X andis a dinatural transformation from the constant functor whose value is e on every object and
on every morphism, such that for every dinatural transformationthere 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 writtenIf X is complete
Complete category
In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C. Dually, a cocomplete category is one in which all small colimits exist...
, 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 andis 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
Dually, if X is cocomplete, then the coend can be described as the coequalizer in the diagram
Examples
Suppose we have functors then . In this case, the category of sets is complete, so we need only form the equalizer and in this casethe natural transformations from F to G. Intuitively, a natural transformation from F to G is a morphism from F(c) to G(c) for every c in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.
Let T be a simplicial set. That is, T 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 [n] of to the standard n simplex inside . Finally there is a functor which takes the product of two topological spaces. Define S to be the composition of this product functor with . The coend of S is the geometric realization of T.