In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, an end of a functor
is a universal
extranatural transformation (dually co-wedges and co-ends), by setting F (dually G) constant.
Extranatural transformations can be defined in terms of dinatural transformations, of which they are a special case.
See also
* Dinatural transformation
External links
* {{n ...
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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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.
- Geometric realizations:
Let be a
simplicial set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
. That is, is a functor . The discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
gives a functor , where is the category of topological spaces. Moreover, there is a map sending the object
Notes
References
*
*
External links
*
{{Category theory
Functors