Category of elements

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In category theory, if C is a category and F: C \to \mathbf{Set} is a set-valued functor, the category of elements of F \mathop{\rm el}(F) (also denoted by ∫CF) is the category defined as follows:

  • Objects are pairs (A,a) where A \in \mathop{\rm Ob}(C) and a \in FA.
  • An arrow (A,a) \to (B,b) is an arrow f: A \to B in C such that (Ff)a = b.

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[edit]

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:

  • Objects are pairs (A,a) where A \in \mathop{\rm Ob}(C) and a\in P(A).
  • An arrow (A,a)\to (B,b) is an arrow f:A\to B in C such that (Pf)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 (\ast\downarrow P)^{\rm 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 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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