Vector Bundles#

class sage.categories.vector_bundles.VectorBundles(base_space, base_field, name=None)[source]#

Bases: Category_over_base_ring

The category of vector bundles over any base space and base field.

See also

TopologicalVectorBundle

EXAMPLES:

Sage

sage: M = Manifold(2, 'M', structure='top')
sage: from sage.categories.vector_bundles import VectorBundles
sage: C = VectorBundles(M, RR); C
Category of vector bundles over Real Field with 53 bits of precision
 with base space 2-dimensional topological manifold M
sage: C.super_categories()
[Category of topological spaces]

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> from sage.categories.vector_bundles import VectorBundles
>>> C = VectorBundles(M, RR); C
Category of vector bundles over Real Field with 53 bits of precision
 with base space 2-dimensional topological manifold M
>>> C.super_categories()
[Category of topological spaces]

class Differentiable(base_category)[source]#

Bases: CategoryWithAxiom_over_base_ring

The category of differentiable vector bundles.

A differentiable vector bundle is a differentiable manifold with differentiable surjective projection on a differentiable base space.

class Smooth(base_category)[source]#

Bases: CategoryWithAxiom_over_base_ring

The category of smooth vector bundles.

A smooth vector bundle is a smooth manifold with smooth surjective projection on a smooth base space.

class SubcategoryMethods[source]#

Bases: object

Differentiable()[source]#

Return the subcategory of the differentiable objects of self.

EXAMPLES:

Sage

sage: M = Manifold(2, 'M')
sage: from sage.categories.vector_bundles import VectorBundles
sage: VectorBundles(M, RR).Differentiable()
Category of differentiable vector bundles over Real Field with
 53 bits of precision with base space 2-dimensional
 differentiable manifold M

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> from sage.categories.vector_bundles import VectorBundles
>>> VectorBundles(M, RR).Differentiable()
Category of differentiable vector bundles over Real Field with
 53 bits of precision with base space 2-dimensional
 differentiable manifold M

Smooth()[source]#

Return the subcategory of the smooth objects of self.

EXAMPLES:

Sage

sage: M = Manifold(2, 'M')
sage: from sage.categories.vector_bundles import VectorBundles
sage: VectorBundles(M, RR).Smooth()
Category of smooth vector bundles over Real Field with 53 bits
 of precision with base space 2-dimensional differentiable
 manifold M

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> from sage.categories.vector_bundles import VectorBundles
>>> VectorBundles(M, RR).Smooth()
Category of smooth vector bundles over Real Field with 53 bits
 of precision with base space 2-dimensional differentiable
 manifold M

base_space()[source]#

Return the base space of this category.

EXAMPLES:

Sage

sage: M = Manifold(2, 'M', structure='top')
sage: from sage.categories.vector_bundles import VectorBundles
sage: VectorBundles(M, RR).base_space()
2-dimensional topological manifold M

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> from sage.categories.vector_bundles import VectorBundles
>>> VectorBundles(M, RR).base_space()
2-dimensional topological manifold M

super_categories()[source]#

EXAMPLES:

Sage

sage: M = Manifold(2, 'M')
sage: from sage.categories.vector_bundles import VectorBundles
sage: VectorBundles(M, RR).super_categories()
[Category of topological spaces]

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> from sage.categories.vector_bundles import VectorBundles
>>> VectorBundles(M, RR).super_categories()
[Category of topological spaces]