nLab VectBund




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



Paths and cylinders

Homotopy groups

Basic facts




This entry is concerned with categories VectBundVectBund 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 VectBundVectBund are vector bundles over any base space BB in the ambient category, which we shall denote like this:

    [E B] \left[ \array{ E \\ \big\downarrow \\ B } \right]
  • the morphisms of VectBundVectBund are commuting squares (in the ambient category of spaces) of the form

    E ϕ E B f B \array{ E &\overset{\phi}{\longrightarrow}& E' \\ \big\downarrow && \big\downarrow \\ B &\underset{f}{\longrightarrow}& B' }

    which are fiber-wise linear maps.

    Notice here that given the base map ff, such a diagram is equivalent to a homomorphism of vector bundles over BB from EE into the pullback vector bundle f *Ef^\ast E', which we may denote by the same symbol ϕ\phi:

    E ϕ f *E B \array{ E && \overset{\phi}{\longrightarrow} && f^\ast E' \\ & \searrow && \swarrow \\ && B }

Accordingly, for each base space BB there is a full subcategory-inclusion

VectBund(B)VectBund VectBund(B) \xhookrightarrow{\phantom{--}} VectBund

of the category VectBund(B) of vector bundles over BB,

Closed monoidal structures

For a fixed base space BB, the monoidal category-structures on VectBund B VectBund_B are well known, given fiber-wise by the respective structures on VectorSpaces:

But analogous – if subtly different – monoidal structures exist on the total category VectBundVectBund, 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 VectBundVectBund 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

B×B pr B pr B B B \array{ && B \times B' \\ & \mathllap{ {}^{pr_{B}} }\swarrow && \searrow \mathrlap{ {}^{ pr_{B'} } } \\ B && && B' }

and then forming the direct sum of vector bundles there:

(1)[E B]×[E B][(pr B) *E(pr B) *E B×B] \left[ \begin{array}{c} E \\ \big\downarrow \\ B \end{array} \right] \times \left[ \begin{array}{c} E' \\ \big\downarrow \\ B' \end{array} \right] \;\;\simeq\;\; \left[ \begin{array}{c} (pr_{B})^\ast E \oplus (pr_{B'})^\ast {E'} \\ \big\downarrow \\ B \times B' \end{array} \right]

From this one finds that the cartesian mapping space (internal hom)

Maps(,):VectBund op×VectBundVectBund Maps(-,-) \;\colon\; VectBund^{op} \times VectBund \longrightarrow VectBund

is given by forming the vector bundle whose base is the space of vector bundle homomorphisms ϕ\phi covering maps of base spaces ff, and whose fiber over (f,ϕ)(f,\phi) is (independent of ϕ\phi and) given by the vector-space of sections Γ B()\Gamma_B(-) of the pullback vector bundle f *Ef^\ast E' of the codomain bundle EE' along ff:

Maps([E B],[E B])[(f,ϕ)Γ B(f *E) {f:BB, ϕ:Elin/Bf *E}] Maps \left( \left[ \begin{array}{c} E \\ \big\downarrow \\ B \end{array} \right] \;,\;\; \left[ \begin{array}{c} E' \\ \big\downarrow \\ B' \end{array} \right] \right) \;\;\simeq\;\; \left[ \begin{array}{c} (f,\phi) \mapsto \Gamma_B\big(f^\ast E'\big) \\ \big\downarrow \\ \left\{ \array{ f \colon B \to B', \\ \phi \colon E \underset{lin/B}{\to} f^\ast E' } \;\; \right\} \end{array} \right]

Namely, one checks the defining hom-isomorphism

Hom(,Map(,))Hom(×,) Hom \big( - ,\, Map(-,-) \big) \;\;\simeq\;\; Hom \big( - \times - ,\, - \big)

by testing it on vector bundles over the point space (ignoring the topology for the moment) and using (1) on the right:

Hom([V *],Maps([E B],[E B]))(v((f,ϕ),σ v))(f,(ϕ+σ ()))Hom([EV B],[E B]) Hom \left( \left[ \begin{array}{c} V \\ \big\downarrow \\ \ast \end{array} \right] \;,\;\; Maps \left( \left[ \begin{array}{c} E \\ \big\downarrow \\ B \end{array} \right] \;,\;\; \left[ \begin{array}{c} E' \\ \big\downarrow \\ B' \end{array} \right] \right) \right) \;\; \underoverset {\sim} { \Big( v \mapsto \big( (f,\phi) , \sigma_v \big) \Big) \;\mapsto\; \big(f, (\phi + \sigma_{(-)}) \big) }{\longrightarrow} \;\; Hom \left( \left[ \begin{array}{c} E \oplus V \\ \big\downarrow \\ B \end{array} \right] \;,\;\; \left[ \begin{array}{c} E' \\ \big\downarrow \\ B' \end{array} \right] \right)

Tensor monoidal structure

Consider now the symmetric monoidal structure on VectBundVectBund given by the external tensor product

:VectBund×VectBundVectBund \otimes \;\colon\; VectBund \times VectBund \longrightarrow VectBund

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)[E B][E B][(pr B) *E(pr B) *E B×B] \left[ \begin{array}{c} E \\ \big\downarrow \\ B \end{array} \right] \otimes \left[ \begin{array}{c} E' \\ \big\downarrow \\ B' \end{array} \right] \;\;\coloneqq\;\; \left[ \begin{array}{c} (pr_{B})^\ast E \otimes (pr_{B'})^\ast {E'} \\ \big\downarrow \\ B \times B' \end{array} \right]

The corresponding internal hom

LMaps:VectBund op×VectBundVectBund LMaps \;\colon\; VectBund^{op} \times VectBund \longrightarrow VectBund

forms the vector bundle whose base space is the mapping space of base spaces, and whose fiber over f:BBf \colon B \to B' is the vector space [E,f *E] B[E, f^\ast E']_B of vector bundle homomorphisms Ef *EE \to f^\ast E' over BB:

LMaps([E B],[E B])[f[E,f *E] B {f:BB}] LMaps \left( \left[ \begin{array}{c} E \\ \big\downarrow \\ B \end{array} \right] \;,\;\; \left[ \begin{array}{c} E' \\ \big\downarrow \\ B' \end{array} \right] \right) \;\;\coloneqq\;\; \left[ \begin{array}{c} f \mapsto \big[E, f^\ast E'\big]_B \\ \big\downarrow \\ \left\{ \array{ f \colon B \to B' } \;\; \right\} \end{array} \right]

To see this, one checks the required hom-isomorphism

Hom(,LMap(,))Hom(,) Hom \big( - ,\, LMap(-,-) \big) \;\;\simeq\;\; Hom \big( - \otimes - ,\, - \big)

over the point, now using (2) on the right:

Hom([V *],LMaps([E B],[E B]))(v(f,σ v))(f,σ ())Hom([EV B],[E B]) Hom \left( \left[ \begin{array}{c} V \\ \big\downarrow \\ \ast \end{array} \right] \;,\;\; LMaps \left( \left[ \begin{array}{c} E \\ \big\downarrow \\ B \end{array} \right] \;,\;\; \left[ \begin{array}{c} E' \\ \big\downarrow \\ B' \end{array} \right] \right) \right) \;\; \underoverset {\sim} { \Big( v \mapsto \big(f, \sigma_v\big) \Big) \;\mapsto\; \big(f, \sigma_{(-)} \big) }{\longrightarrow} \;\; Hom \left( \left[ \begin{array}{c} E \otimes V \\ \big\downarrow \\ B \end{array} \right] \;,\;\; \left[ \begin{array}{c} E' \\ \big\downarrow \\ B' \end{array} \right] \right)

Created on November 21, 2022 at 17:56:45. See the history of this page for a list of all contributions to it.