Category of elements
Encyclopedia
In category theory
if C is a category and
a set-valued functor the category of elements of F 
(also denoted by ∫CF) is the category defined as follows:
A more concise way to state this is that the category of elements of F is the comma category
where 
is a one-point set. The category of elements of F comes with a natural projection 
that sends an object (A,a) to A and an arrow 
to its underlying arrow in C.
is a presheaf the category of elements of P (again denoted by 
or to make the distinction to the above definition clear ∫C P) is the category defined as follows:
As one sees the direction of the arrows is reversed and in fact one can once again state this definition in a more concise manner: the category we just defined is nothing but
. Consequentially, in the spirit of adding a "co" in front of the name for a construction to denote its dual, one should rather call this category the category of coelements of P.
For C small, this construction can be extended into a functor ∫C from
to 
, the category of small categories
. In fact, using the Yoneda lemma
one can show that ∫CP
, where 
is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to 
.
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...
if C is a category and
a set-valued functor the category of elements of F (also denoted by ∫CF) is the category defined as follows:
- Objects are pairs
where 
and 
. - An arrow
is an arrow 
in C such that 
.
A more concise way to state this is that the category of elements of F is the comma category
Comma category
In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W...
where is a one-point set. The category of elements of F comes with a natural projection that sends an object (A,a) to A and an arrow to its underlying arrow in C.The Category of Elements of a Presheaf
Somewhat confusingly in some texts (e.g. Mac Lane, Moerdijk) the category of elements for a presheaf is defined differently. If is a presheaf the category of elements of P (again denoted by or to make the distinction to the above definition clear ∫C P) is the category defined as follows:
- Objects are pairs
where 
and 
. - An arrow
is an arrow 
in C such that 
.
As one sees the direction of the arrows is reversed and in fact one can once again state this definition in a more concise manner: the category we just defined is nothing but
. Consequentially, in the spirit of adding a "co" in front of the name for a construction to denote its dual, one should rather call this category the category of coelements of P.For C small, this construction can be extended into a functor ∫C from
to , the category of small categoriesCategory of small categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories...
. In fact, using the Yoneda lemma
Yoneda lemma
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
one can show that ∫CP
, where is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to .External links
- http://ncatlab.org/nlab/show/category+of+elements, definition and elementary properties of the category of elements in the nLab