# nLab representable fibered category

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

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

Revised on April 27, 2011 13:03:29 by Urs Schreiber (131.211.239.144)