fibred's Usage Examples:

Unlike cleavages, not all fibred categories admit splittings.

Fibre bundles: Fibre products exist in the category Top of topological spaces and thus by the previous example A(Top) is fibred over Top.

The choice of a (normalised) cleavage for a fibred E-category F specifies, for each morphism f: T → S in E, a functor f*: FS → FT: on objects f* is simply the inverse image by the corresponding transport morphism, and on morphisms it is defined in a natural manner by the defining universal property of cartesian morphisms.

manifold Connection (affine bundle) Connection (composite bundle) Connection (fibred manifold) Connection (principal bundle), gives the derivative of a section.

pullbacks; fibred categories then make a good framework to discuss the possibility of such gluing.

If E has a terminal object e and if F is fibred over E, then the functor ε from cartesian sections to Fe defined at the end of the previous section is an equivalence of categories and moreover surjective on objects.

The two preceding constructions of split categories are used in a critical way in the construction of the stack associated to a fibred category (and in particular stack associated to a pre-stack).

This Top-category is also fibred, and the inverse image functors are the ordinary pull-back functors for vector bundles.

is a fibred manifold X → R {\displaystyle X\to \mathbb {R} } .

Ni, Yi (2007), "Knot Floer homology detects fibred knots", Inventiones Mathematicae, 170 (3): 577–608, arXiv:math/0607156,.

Any indexed category has an associated Grothendieck construction, which gives rise to a fibred category.

Another way to view this is as a fibred category [ U / R ] → ( S c h / S ) {\displaystyle [U/R]\to (\mathrm {Sch}.

manifold and π : E → M a surjective submersion, so that E is a manifold fibred over M.