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Category of elements

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In category theory, for every presheaf P \in\hat C := \mathbf{Set}^{C^{op}} the category of elements \mathbf{El}(P) of P is the category defined as follows:

  • its objects are pairs (A,a) where A is an object of C and a\in P(A),
  • its morphisms (A,a)\to (B,b) are the morphisms f:A\to B of C such that a = P(f)(b).

This construction can be extended into a functor \mathbf{El} from \hat C to \mathbf{Cat}, the category of small categories.

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