Category of elements


In category theory, if C is a category and is a set-valued functor, the category of elements of F 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 to A, and an arrow to its underlying arrow in C.

The category of elements of a presheaf

In some texts the category of elements is used for presheaves. We state it explicitly for completeness. If is a presheaf, the category of elements of P is the category defined as follows:
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 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.

The category of elements of an operad algebra

Given a 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: