Editing Bundle (mathematics)

{{distinguish|Bundle (geometry)}}

In [[mathematics]], a '''bundle''' is a generalization of a [[fiber bundle]] dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a [[topological space|topology]]. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: ''E'' → ''B'' with ''E'' and ''B'' [[Set (mathematics)|sets]]. It is no longer true that the [[preimage]]s <math>\pi^{-1}(x)</math> must all look alike, unlike fiber bundles, where the fibers must all be [[isomorphic]] (in the case of [[vector bundle]]s) and [[homeomorphic]].

== Definition ==
A bundle is a triple {{math|(''E'', ''p'', ''B'')}} where {{math|''E'', ''B''}} are sets and {{math|''p'' : ''E'' → ''B''}} is a map.<ref>{{harvnb|Husemoller|1994}} p 11.</ref>
*{{math|''E''}} is called the '''total space'''
*{{math|''B''}} is the '''base space''' of the bundle
*{{math|''p''}} is the '''projection'''

This definition of a bundle is quite unrestrictive. For instance, the [[empty 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 on {{math|''E'', ''p'', ''B''}} and usually there is additional structure.

For each {{math|''b'' ∈ ''B'', ''p''<sup>−1</sup>(''b'')}} is the '''fibre''' or '''fiber''' of the bundle over {{math|''b''}}.

A bundle {{math|(''E*'', ''p*'', ''B*'')}} is a '''subbundle''' of {{math|(''E'', ''p'', ''B'')}} if {{math|''B*'' ⊂ ''B'', ''E*'' ⊂ ''E''}} and {{math|''p*'' {{=}} ''p''{{!}}<sub>''E*''</sub>}}.

A [[Section (fibre bundle)|cross section]] is a map {{math|''s'' : ''B'' → ''E''}} such that {{math|''p''(''s''(''b'')) {{=}} ''b''}} for each {{math|''b'' ∈ ''B''}}, that is, {{math|''s''(''b'') ∈ ''p''<sup>−1</sup>(''b'')}}.

== Examples ==
*If {{math|''E''}} and {{math|''B''}} are [[smooth manifold]]s and {{math|''p''}} is smooth, [[Surjective function|surjective]] and in addition a [[Submersion (mathematics)|submersion]], then the bundle is a [[fibered 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 ({{math|''C''<sup>1</sup>}}), in between.
*If for each two points {{math|''b''<sub>1</sub>}} and {{math|''b''<sub>2</sub>}} in the base, the corresponding fibers {{math|''p''<sup>−1</sup>(''b''<sub>1</sub>)}} and {{math|''p''<sup>−1</sup>(''b''<sub>2</sub>)}} are [[homotopy equivalent]], then the bundle is a [[fibration]].
*If for each two points {{math|''b''<sub>1</sub>}} and {{math|''b''<sub>2</sub>}} in the base, the corresponding fibers {{math|''p''<sup>−1</sup>(''b''<sub>1</sub>)}} and {{math|''p''<sup>−1</sup>(''b''<sub>2</sub>)}} are [[homeomorphic]], and in addition the bundle satisfies certain conditions of ''local triviality'' outlined in the pertaining linked articles, then the bundle is a [[fiber bundle]]. Usually there is additional structure, e.g. a [[Group (mathematics)|group structure]] or a [[Vector space#Definition|vector 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.
*A [[principal bundle]] is a fiber bundle endowed with a right [[Group action (mathematics)|group action]] with certain properties. One example of a principal bundle is the [[frame bundle]].
*If for each two points {{math|''b''<sub>1</sub>}} and {{math|''b''<sub>2</sub>}} in the base, the corresponding fibers {{math|''p''<sup>−1</sup>(''b''<sub>1</sub>)}} and {{math|''p''<sup>−1</sup>(''b''<sub>2</sub>)}} are [[vector space]]s of the same dimension, then the bundle is a [[vector bundle]] if the appropriate conditions of local triviality are satisfied. The [[tangent bundle]] is an example of a vector bundle.

==Bundle objects==
More generally, bundles or '''bundle objects''' can be defined in any [[category (mathematics)|category]]: in a category '''C''', a bundle is simply an [[epimorphism]] π: ''E'' → ''B''.  If the category is not [[concrete category|concrete]], 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 with [[pullback (category theory)|pullbacks]] and a [[initial object|terminal object]] 1 the points of ''B'' can be identified with morphisms ''p'':1→''B'' and the fiber of ''p'' is obtained as the pullback of ''p'' and π. The category of bundles over ''B'' is a subcategory of the [[slice category]] ('''C'''↓''B'') of objects over ''B'', while the category of bundles without fixed base object is a subcategory of the [[comma category]] (''C''↓''C'') which is also the [[functor category]] '''C'''², the category of [[morphism]]s in '''C'''.

The category of smooth vector bundles is a bundle object over the category of smooth manifolds in '''Cat''', the [[category of small categories]].  The [[functor]] taking each manifold to its [[tangent bundle]] is an example of a section of this bundle object.

==See also==
* [[Fiber bundle]]
* [[Fibration]]
* [[Fibered manifold]]

==Notes==
{{reflist}}

==References==
{{refbegin}}
*{{cite book
|last=Goldblatt
|first=Robert 
|title=Topoi, the Categorial Analysis of Logic
|url=http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=gold010
|accessdate=2009-11-02
|origyear=1984
|year=2006
|publisher=Dover Publications
|isbn=978-0-486-45026-1
}}
*{{citation|last=Husemoller|first=Dale|authorlink=Dale Husemoller |title=Fibre bundles|year=1994|publisher=Springer|origyear=1966|isbn=0-387-94087-1|series=Graduate Texts in Mathematics|volume=20}}
*{{citation|last=Vassiliev|first=Victor|title=Introduction to Topology|year=2001|publisher=Amer Mathematical Society|origyear=2001|isbn=0821821628|series=Student Mathematical Library}}
{{refend}}

{{Topology}}

{{DEFAULTSORT:Bundle (Mathematics)}}
[[Category:Category theory]]
[[Category:Fiber bundles]]