Category of elements

In category theory if C is a category and F: C \to \mathbf{Set} a set-valued functor the category of elements of F \mathop{\rm el}(F) (also denoted by ∫CF) 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 \ast\downarrow F where \ast is a one-point set. The category of elements of F comes with a natural projection \mathop{\rm el}(F) \to C that sends an object (A,a) to A and an arrow (A,a) \to (B,b) to its underlying arrow in C.

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 P \in\hat C�:= \mathbf{Set}^{C^{op}} is a presheaf the category of elements of P (again denoted by \mathop{\rm el}(P) or to make the distinction to the above definition clear ∫C P) is the category defined as follows:

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 (\ast\downarrow P)^{\rm op}. 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 \hat C to \mathbf{Cat}, the category of small categories. In fact, using the Yoneda lemma one can show that ∫CP \cong \mathop{\textbf{y}}\downarrow P, where \mathop{\textbf{y}}: C \to \hat{C} is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to \mathop{\textbf{y}}\downarrow-: \hat C \to \textbf{Cat}.

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