nLab prefibered category

Definition

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 $B^{op}\to Cat$. For constrast, every functor into $B$ is a normal lax functor $B^{op} \to Prof$, see at displayed category.

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

See also

• foliated category
• fibered category
• displayed category

References

• A. Grothendieck, M. Raynaud et al. Revêtements étales et groupe fondamental (SGA1), Lecture Notes in Mathematics 224, Springer 1971 (retyped as math.AG/0206203; published version Documents Mathématiques 3, Société Mathématique de France, Paris 2003)