Category of Elements - The Category of Elements of A Presheaf

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 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 .