nLab representable fibered category

Definition

Given an object $X$ in a category $B$, the domain functor $(Y \overset{f} \longrightarrow X)\mapsto Y$, from the slice category $B/X$ to $B$, is a fibered category (a Grothendieck fibration).

Any fibered category equivalent to this $dom \colon B/X \longrightarrow B$ is said to be representable.

This is because under the Grothendieck construction representable fibered categories correspond precisely to representable functors $B^{op} \longrightarrow Set \hookrightarrow Cat$: the category $B/X$ is the category of elements of the representable functor $B(-,X)$.

References

• Angelo Vistoli; Def. 3.43 in: Notes on Grothendieck topologies, fibered categories and descent theory [math.AG/0412512, pdf, MR2223406] in: Fantechi et al. (eds.): Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) 1–104 [ISBN:978-0-8218-4245-4, MR2007f:14001]

• The Stacks Project: Representable categories fibred in groupoids [tag:0046]