# nLab prefibered category

A category $\pi:E\to B$ over $B$ is a prefibered category if for every morphism $f:b\to b'$ there is a weakly cartesian morphism $u:e\to e'$ with $\pi(e) = f$.

This differs from a fibered category by not requiring that a composition of weakly cartesian morphisms is weakly cartesian, or equivalently that strongly cartesian liftings exist.

Just as a fibred category corresponds to a pseudofunctor $B^{op}\to Cat$, a prefibred category corresponds to a normal lax functor $B^{op}\to Cat$.

Prefibered (prefibred) category (Rus. предрасслоённая категория, Fr. catégorie préfibrée)