category of elements

In category theory, if is a category and is a set-valued functor, the category of elements of (also denoted ) is the following category:
* Objects are pairs $(A,a)$ where $A \in \mathop(C)$ and $a \in FA$.
* Morphisms $(A,a) \to (B,b)$ are arrows $f: A \to B$ of $C$ such that $(Ff)a = b$.

A more concise way to state this is that the category of elements of is the comma category , where is a singleton (a set with one element). The category of elements of comes with a natural projection that sends an object to , and an arrow to its underlying arrow in .

The category of elements of a presheaf

In some texts (e.g. Mac Lane, Moerdijk) the category of elements is used for presheaves. We state it explicitly for completeness. If is a presheaf, the category of elements of (again denoted by , or, to make the distinction to the above definition clear, ) is the following category:
* Objects are pairs $(A,a)$ where $A \in \mathop(C)$ and $a \in P(A)$.
* Morphisms $(A,a) \to (B,b)$ are arrows $f:A \to B$ of $C$ such that $(Pf)b = a$.

As one sees, the direction of the arrows is reversed. One can, once again, state this definition in a more concise manner: the category just defined is nothing but . Consequentially, in the spirit of adding a "co" in front of the name for a construction to denote its opposite, one should rather call this category the category of coelements of .

For small , this construction can be extended into a functor from to , the category of small categories. In fact, using the Yoneda lemma one can show that , where is the Yoneda embedding. This isomorphism is natural in and thus the functor is naturally isomorphic to .

The category of elements of an operad algebra

Given a (colored) operad and a functor, also called an algebra, , one obtains a new operad, called the ''category of elements'' and denoted , generalizing the above story for categories. It has the following description:
* Objects are pairs $(o,x)$ where $o\in \mathrm(O)$ and $x\in A(o)$.
* An arrow $(o_1,x_1)\to (o_2,x_2)$ is an arrow $f\colon o_1\to o_2$ in $O$ such that $A(f)(x_1)=x_2.$

See also

* Grothendieck construction

References

*
*

External links

* {{nlab, id=category+of+elements, title=Category of elements

Representable functors