Category of elements

In , if C is a and F:C→Set is a set-valued functor, the category el(F) of elements of F (also denoted ∫CF) is the following category: are pairs where and . Morphisms are arrows of such that . A more concise way to state this is that the category of elements of F is the ∗↓F, where ∗ is a singleton (a set with one element). The category of elements of F comes with a natural projection el(F)→C that sends an object (A, a) to A, and an arrow (A,a)→(B,b) to its underlying arrow in C. In some texts (e.g. Mac Lane, Moerdijk) the category of elements is used for presheaves. We state it explicitly for completeness. If P∈Ĉ:=SetCop is a , the category of elements of P (again denoted by el(P), or, to make the distinction to the above definition clear, ∫C P = ∫CopP) is the following category: Objects are pairs where and . Morphisms are arrows of such that . 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 (∗↓P)op. 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, this construction can be extended into a functor ∫C from Ĉ to Cat, the . In fact, using the Yoneda lemma one can show that ∫C P≅y↓P, where y:C→Ĉ is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to y↓–:Ĉ→Cat. Given a O and a functor, also called an algebra, A:O→Set, one obtains a new operad, called the category of elements and denoted ∫OA, generalizing the above story for categories. It has the following description: Objects are pairs where and .