representable fibered category

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

Any fibered category isomorphic to the dom:B/XBdom: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 opSetCatB^{op} \to Set \hookrightarrow Cat: the category B/XB/X is the category of elements of the representable functor B(,X)B(-,X).

Last revised on April 27, 2011 at 13:03:29. See the history of this page for a list of all contributions to it.