nLab prefibered category

A category π:EB\pi:E\to B over BB is a prefibered category if for every morphism f:bbf:b\to b' there is a weakly cartesian morphism u:eeu:e\to e' with π(e)=f\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 opCatB^{op}\to Cat, a prefibred category corresponds to a normal lax functor B opCatB^{op}\to Cat.

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

See also foliated category, fibered category

  • 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)

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