Fibred category

Summary

A fibred category is a functor $p \colon \mathcal{E} \to \mathcal{B}$ along which objects can be pulled back. For every object $y$ of $\mathcal{E}$ and every morphism $f$ into $p(y)$, there is a universal, or cartesian, morphism over $f$ ending at $y$. Fibred categories describe families of categories indexed by the objects of $\mathcal{B}$, such as sheaves or vector bundles over varying spaces. They are the setting of Grothendieck's descent theory and of stacks.