nLab VectBund

Idea

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

  $$\left[ \array{ E \\ \big\downarrow \\ B } \right]$$

• the morphisms of $VectBund$ are commuting squares (in the ambient category of spaces) of the form

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

  which are fiber-wise linear maps.

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

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

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

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

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

Closed monoidal structures

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

• the cartesian monoidal structure on $VectBund(B)$ is given by direct sum of vector bundles, which in turn is fiber-wise the direct sum of vector spaces;

• the other symmetric monoidal structure on $VectBund(B)$ is given by tensor product of vector bundles, which in turn is fiber-wise the tensor product of vector spaces.

But analogous – if subtly different – monoidal structures exist on the total category $VectBund$, discussed in the following:

• Cartesian monoidal structure on VectBund

• Tensor monoidal structure on VectBund.

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 $VectBund$ 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

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

and then forming the direct sum of vector bundles there:

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

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

$$Maps(-,-) \;\colon\; VectBund^{op} \times VectBund \longrightarrow VectBund$$

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

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

Namely, one checks the defining hom-isomorphism

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

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

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

Tensor monoidal structure

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

$$\otimes \;\colon\; VectBund \times VectBund \longrightarrow VectBund$$

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

The corresponding internal hom

$$LMaps \;\colon\; VectBund^{op} \times VectBund \longrightarrow VectBund$$

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

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

To see this, one checks the required hom-isomorphism

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

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

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

Related concepts

• Mod

• VectBund(B)

• tangent $\infty$-category