definition. category of elements [leinster2016basic, 6.2.16] [tt-004C]

✍️source

Let \({\cal C}\) be a category and \(\mathscr {X} : {\cal C}^{op} \to \mathbf {Set}\) a presheaf on \({\cal C}\). The category of elements \({\cal E}(\mathscr {X})\) of \(\mathscr {X}\) is the category in which:

There is a projection functor \(\mathscr {P}: \mathbf {E}(\mathscr {X}) \rightarrow {\cal C}\) defined by \(\mathscr {P}(X, x) = X\) and \(\mathscr {P}(f) = f\).

The Yoneda lemma implies that for a presheaf \(\mathscr {X}\), the generalized elements of \(\mathscr {X}\) of representable shape correspond to objects of the category of elements.