In category theory, if C is a category and F : C → S e t is a set-valued functor, the category of elements of F e l ( F ) (also denoted by ∫CF) is the category defined as follows:
Objects are pairs ( A , a ) where A ∈ O b ( C ) and a ∈ F A .An arrow ( A , a ) → ( B , b ) is an arrow f : A → B in C such that ( F f ) a = b .A more concise way to state this is that the category of elements of F is the comma category ∗ ↓ F , where ∗ is a one-point set. The category of elements of F comes with a natural projection e l ( F ) → C that sends an object (A,a) to A, and an arrow ( A , a ) → ( B , b ) to its underlying arrow in C.
Somewhat confusingly in some texts (e.g. Mac Lane, Moerdijk), the category of elements for a presheaf is defined differently. If P ∈ C ^ := S e t C o p is a presheaf, the category of elements of P (again denoted by e l ( P ) , or, to make the distinction to the above definition clear, ∫C P) is the category defined as follows:
Objects are pairs ( A , a ) where A ∈ O b ( C ) and a ∈ P ( A ) .An arrow ( A , a ) → ( B , b ) is an arrow f : A → B in C such that ( P f ) 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 ( ∗ ↓ P ) o p . 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 C ^ to C a t , the category of small categories. In fact, using the Yoneda lemma one can show that ∫CP ≅ y ↓ P , where y : C → C ^ is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to y ↓ − : C ^ → Cat .
