Contents
Context
Bundles
Linear algebra
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Contents
Idea
This entry is concerned with categories of vector bundles whose morphisms are allowed to cover non-trivial maps between their base spaces.
This means that after choosing a ground field and an ambient category of spaces (e.g. Sets TopologicalSpaces, SmoothManifolds, etc.):
-
the objects of are vector bundles over any base space in the ambient category, which we shall denote like this:
-
the morphisms of are commuting squares (in the ambient category of spaces) of the form
which are fiber-wise linear maps.
Notice here that given the base map , such a diagram is equivalent to a homomorphism of vector bundles over from into the pullback vector bundle , which we may denote by the same symbol :
Accordingly, for each base space there is a full subcategory-inclusion
of the category VectBund(B) of vector bundles over ,
Closed monoidal structures
For a fixed base space , the monoidal category-structures on are well known, given fiber-wise by the respective structures on VectorSpaces:
But analogous – if subtly different – monoidal structures exist on the total category , discussed in the following:
In discussing this now, we (have to and want to) assume that the ambient category of TopologicalSpaces is itself cartesian closed (a “convenient category of topological spaces”), such as that of compactly generated topological spaces or of D-topological spaces.
Cartesian monoidal structure
The cartesian product in is readily checked to be the “external direct sum”, namely the result of pulling back the two vector bundles to the product space of their base spaces along the projection maps
and then forming the direct sum of vector bundles there:
(1)
From this one finds that the cartesian mapping space (internal hom)
is given by forming the vector bundle whose base is the space of vector bundle homomorphisms covering maps of base spaces , and whose fiber over is (independent of and) given by the vector-space of sections of the pullback vector bundle of the codomain bundle along :
Namely, one checks the defining hom-isomorphism
by testing it on vector bundles over the point space (ignoring the topology for the moment) and using (1) on the right:
Tensor monoidal structure
Consider now the symmetric monoidal structure on given by the external tensor product
which itself is constructed by first pulling back both bundles to the product space of their base spaces, and then forming their tensor product of vector bundles there:
(2)
The corresponding internal hom
forms the vector bundle whose base space is the mapping space of base spaces, and whose fiber over is the vector space of vector bundle homomorphisms over :
To see this, one checks the required hom-isomorphism
over the point, now using (2) on the right: