Vector Bundles - Category Framework

Skip to content
Back to top
View this page
Edit this page

# Vector Bundles

class sage.categories.vector_bundles.VectorBundles(base_space, base_field, name=None)

  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)

    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)

    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

    Bases: `object`

    Differentiable()

      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()

      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()

    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()

    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]