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Stream: learning: questions

Topic: large-representable elementary fibrations

James Deikun (May 19 2025 at 16:51):

An elementary (i.e. faithful and conservative, i.e. fibered in posetal groupoids) fibration $P : \mathcal{E \to B}$ is representable iff its total category has a terminal object. Then the fibration $P$ is equivalent to the slice projection $\pi : \mathcal B/P(I) \to \mathcal B$. So far so good.

But there are some fibrations that seem to represent a large "representable" presheaf, for example the projection from the category of arrows (as objects) and pullback squares (as arrows) in $\mathcal B$, which represents the "slice over the true universe". Is there some known characterization of these elementary fibrations?