Bundle (mathematics)
Encyclopedia
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 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 sets. It is no longer true that the preimages π − 1(x) must all look alike, unlike fiber bundles where the fibers must all be isomorphic (in the case of vector bundle
s) and homeomorphic.
More generally, bundles or bundle objects can be defined in any category
: in a category C, a bundle is simply an epimorphism
π: E → B. If the category is not 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 pullbacks
and a 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. Just as fiber bundles have section
s, a bundle can have a (global) section, which is a morphism s:B → E such that πs=idB. The category of bundles over B is therefore 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
C², 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a bundle is a generalization of a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E→ B with E and B sets. It is no longer true that the preimages π − 1(x) must all look alike, unlike fiber bundles where the fibers must all be isomorphic (in the case of vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s) and homeomorphic.
More generally, bundles or bundle objects can be defined in any category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
: in a category C, a bundle is simply an epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
π: E → B. If the category is not concrete
Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...
, 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 pullbacks
Pullback (category theory)
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...
and a terminal object
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
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. Just as fiber bundles have section
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
s, a bundle can have a (global) section, which is a morphism s:B → E such that πs=idB. The category of bundles over B is therefore 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
Comma category
In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W...
(C↓C) which is also the functor category
Functor category
In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...
C², the category of morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
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
Category of small categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories...
. The functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
taking each manifold to its tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
is an example of a section of this bundle object.