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 where and . Morphisms are arrows of such that . 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 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 . 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 where and . An arrow is an arrow in such that See also Grothendieck construction References External links Representable functors