Inmathematics, abundle is a generalization of afiber bundle dropping the condition of alocal product structure. The requirement of a local product structure rests on the bundle having atopology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π:E →B withE andBsets. It is no longer true that thepreimages${\displaystyle {\pi }^{-1}(x)}$ must all look alike, unlike fiber bundles, where the fibers must all beisomorphic (in the case ofvector bundles) andhomeomorphic.

A bundle is a triple(E,p,B) whereE,B are sets andp :E →B is a map.^[1]

• E is called thetotal space
• B is thebase space of the bundle
• p is theprojection

This definition of a bundle is quite unrestrictive. For instance, theempty function defines a bundle. Nonetheless it serves well to introduce the basic terminology, and every type of bundle has the basic ingredients of above with restrictions onE,p,B and usually there is additional structure.

For eachb ∈B,p⁻¹(b) is thefibre orfiber of the bundle overb.

A bundle(E*,p*,B*) is asubbundle of(E,p,B) ifB* ⊂B,E* ⊂E andp* =p|_E*.

Across section is a maps :B →E such thatp(s(b)) =b for eachb ∈B, that is,s(b) ∈p⁻¹(b).

• IfE andB aresmooth manifolds andp is smooth,surjective and in addition asubmersion, then the bundle is afibered manifold. Here and in the following examples, the smoothness condition may be weakened to continuous or sharpened to analytic, or it could be anything reasonable, like continuously differentiable (C¹), in between.
• If for each two pointsb₁ andb₂ in the base, the corresponding fibersp⁻¹(b₁) andp⁻¹(b₂) arehomotopy equivalent, then the bundle is afibration.
• If for each two pointsb₁ andb₂ in the base, the corresponding fibersp⁻¹(b₁) andp⁻¹(b₂) arehomeomorphic, and in addition the bundle satisfies certain conditions oflocal triviality outlined in the pertaining linked articles, then the bundle is afiber bundle. Usually there is additional structure, e.g. agroup structure or avector space structure, on the fibers besides a topology. Then is required that the homeomorphism is an isomorphism with respect to that structure, and the conditions of local triviality are sharpened accordingly.
• Aprincipal bundle is a fiber bundle endowed with a rightgroup action with certain properties. One example of a principal bundle is theframe bundle.
• If for each two pointsb₁ andb₂ in the base, the corresponding fibersp⁻¹(b₁) andp⁻¹(b₂) arevector spaces of the same dimension, then the bundle is avector bundle if the appropriate conditions of local triviality are satisfied. Thetangent bundle is an example of a vector bundle.

More generally, bundles orbundle objects can be defined in anycategory: in a categoryC, a bundle is simply anepimorphism π:E →B. If the category is notconcrete, then the notion of a preimage of the map is not necessarily available. Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do; for instance, for a category withpullbacks and aterminal object 1 the points ofB can be identified with morphismsp:1→B and the fiber ofp is obtained as the pullback ofp and π. The category of bundles overB is a subcategory of theslice category (C↓B) of objects overB, while the category of bundles without fixed base object is a subcategory of thecomma category (C↓C) which is also thefunctor categoryC², the category ofmorphisms inC.

The category of smooth vector bundles is a bundle object over the category of smooth manifolds inCat, thecategory of small categories. Thefunctor taking each manifold to itstangent bundle is an example of a section of this bundle object.

• Fiber bundle
• Fibration
• Fibered manifold

• Category theory
• Fiber bundles

• Articles with short description
• Short description is different from Wikidata