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 $B$,

Properties

Bireflective inclusion of zero-bundles

The construction which to any base space $B$ assigns the zero-vector bundle over $B$ constitutes a bireflective subcategory-inclusion (see there) of the category of base spaces into that over vector bundles over these base spaces.

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 of vector bundles

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

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:

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

Distributive monoidal structure

With respect to the coproduct and the external tensor product of vector bundles, $VectBund$ is a distributive monoidal category. We spell this out in detail for the case of discrete base spaces (sets):

Definition

For any ground field, write $Vect_{Set}$ for the category of indexed sets of vector spaces.

Remark

We may and will present objects $\mathcal{V}$ of $Vect_{Set}$ as pairs consisting of a set $S$ and a function $\mathcal{V}_{(-)}$ (really a functor on the discrete category on $S$) to Vect:

$$\left( \array{ S &\longrightarrow& Vect \\ s &\mapsto& \mathcal{V}_s } \right) \;\; \in \;\; Vect_{Set} \mathrlap{\,.}$$

Definition

The “external” tensor product on $Vect_{Set}$ is the functor

$$\boxtimes \,\colon\, Vect_{Set} \times Vect_{Set} \longrightarrow Vect_{Set}$$

given by

$$\big( \mathcal{V}_{(-)} \,\colon\, S \to \Vect \big) \,\boxtimes\, \big( \mathcal{V}'_{(-)} \,\colon\, S' \to \Vect \big) \;\; \coloneqq \;\; \left( \array{ S \times S' &\longrightarrow& \Vect \\ (s, s') &\mapsto& \mathcal{V}_s \otimes \mathcal{V}'_{s'} } \right) \mathrlap{\,.}$$

Proposition

The coproduct in $Vect_{Set}$ is given by disjoint union of bundles:

$$\big( \mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to \Vect \big) \; \sqcup \; \big( \mathcal{V}^{(1)}_{(-)} \,\colon\, S_2 \to \Vect \big) \;\; \simeq \;\; \left( \array{ S_1 \sqcup S_2 &\longrightarrow& \Vect \\ s_i &\mapsto& \mathcal{V}^{(i)}_{s_i} } \right)$$

Proof

It is immediate to check the universal property characterizing the coproduct.

Proposition

The external tensor product $\boxtimes$ (Def. ) distributes over the coproduct (Prop. ):

$$\mathcal{E} \boxtimes ( \mathcal{V}^{(1)} \sqcup \mathcal{V}^{(2)}) \;\simeq\; \big( \mathcal{E} \boxtimes \mathcal{V}^{(1)} \big) \,\sqcup\, \big( \mathcal{E} \boxtimes \mathcal{V}^{(2)} \big)$$

and hence gives a distributive monoidal category:

$$\big( Vect_{Set} , \sqcup , \boxtimes \big) \;\in\; DistMonCat \mathrlap{\,.}$$

Proof

Unwinding the above definitions and using that Set is a distributive category, we have the following sequence of natural isomorphisms:

$$\begin{array}{l} \mathcal{E} \,\boxtimes\, \big( \mathcal{V}^{(1)} \,\sqcup\, \mathcal{V}^{(2)} \big) \\ \;\equiv\; \big( \mathcal{E}_{(-)} \,\colon\, S_E \to Vect \big) \,\boxtimes\, \Big( \big( \mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to Vect \big) \,\sqcup\, \big( \mathcal{V}^{(2)}_{(-)} \,\colon\, S_2 \to Vect \big) \Big) \\ \;\simeq\; \big( \mathcal{E}_{(-)} \,\colon\, S_E \to Vect \big) \,\boxtimes\, \left( \array{ S_1 \sqcup S_2 &\longrightarrow& Vect \\ s_i &\mapsto& \mathcal{V}^{(i)}_{s_i} } \right) \\ \;\simeq\; \left( \array{ S_E \times (S_1 \sqcup S_2) &\longrightarrow& Vect \\ (s_E , s_i) &\mapsto& \mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i} } \right) \\ \;\simeq\; \left( \array{ (S_E \times S_1) \,\sqcup\, (S_2 \times S_2) &\longrightarrow& Vect \\ (s_E , s_i) &\mapsto& \mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i} } \right) \\ \;\simeq\; \left( \array{ S_E \times S_1 &\longrightarrow& Vect \\ (s_E , s_1) &\mapsto& \mathcal{E}_{s_E} \otimes \mathcal{V}^{(1)}_{s_1} } \right) \,\sqcup\, \left( \array{ S_E \times S_2 &\longrightarrow& Vect \\ (s_E , s_2) &\mapsto& \mathcal{E}_{s_E} \otimes \mathcal{V}^{(2)}_{s_2} } \right) \\ \;\equiv\; \mathcal{E} \boxtimes \mathcal{V}^{(1)} \,\sqcup\, \mathcal{E} \boxtimes \mathcal{V}^{(2)} \end{array}$$

Amalgamation of monoidal and parameter structures

under construction

We want to point out a way in which $VectBund$ with its external tensor product-structure is the canonical “amalgamation” of

1. vector spaces equipped with their tensor product,

2. vector spaces equipped with parametrization, namely vector bundles,

at least in the case that the base space is discrete.

The argument generalizes immediately to flat vector bundles over arbitrary base spaces (by enhancing the following parameter sets to fundamental groupoids), and in fact to $(\infty,1)$-vector bundles with flat $\infty$-connections (by enhancing the fundamental groupoids further to fundamental $\infty$-groupoids and the category of vector spaces to the (infinity,1)-category of chain complexes).

In order to formalize this idea, we need to work inside a 2-category which suitably subsumes all of the following very large (2,1)-categories:

• $Cat$ of categories

• $MonCat$ of monoidal categories;

• $CoCartCart$ of cocartesian monoidal categories;

• $DistMonCat$ of distributive monoidal categories (with underlying cocartesian monoidal categories).

Here we understand that

• the 1-morphisms in these 2-categories are functors which respect all given tensor products up to isomorphism, i.e. strong monoidal functors

• the 2-morphisms are natural isomorphisms between these functors.

The system of forgetful 2-functors between these (2,1)-categories forms a square diagram which we may regard as the image of a corresponding contravariant pseudofunctor from the commuting square-diagram category to 2Cat:

$$\mathbf{C} \;\; \colon \;\; \array{ (1,0) &\longrightarrow& (1,1) \\ \big\uparrow && \big\uparrow \\ (0,0) &\longrightarrow& (1,0) } \;\;\;\; \mapsto \;\;\;\; \array{ MonCat &\longleftarrow& DistMonCat \\ \big\downarrow && \big\downarrow \\ Cat &\longleftarrow& CoCartCat }$$

The 2-categorical context which we are after is then the (2,1)-Grothendieck construction $\int \mathbf{C}$ on this pseudofunctor: Here 1-morphisms are functors which strictly preserve the tensor products present on their domain category, but whose codomain may be equipped with further tensor products (though not with fewer tensor products).

Claim. In $\int \mathbf{C}$ the following diagram is a (homotopy) pushout

Proof

We check the universal property. To this end, observe that any cocone under the diagram must have as tip a distributive monoidal category, since only these can receive morphisms in $\int \mathbf{C}$ both from a monoidal and from a cocartesian category.

This means that a general cocone looks like the following solid diagram:

We need to see that this uniquely admits the dashed arrow. Unwinding the definitions, the existence of the dashed arrow means that any functor $F \,\colon\, Vect_{Set} \to \mathcal{C}$ which preserves both the coproduct of vector bundles as well as the tensor product on fibers already respects the external tensor product. But this follows by the distributivity and using that any set is the coproduct of its elements:

$$\begin{array}{l} F\big( \mathcal{E} \boxtimes \mathcal{E}' \big) \\ \;\simeq\; F \Bigg( \underset{s \in S}{\coprod} \left( \array{ \{s\} &\longrightarrow& Vect \\ s &\mapsto& \mathcal{E}_s } \right) \;\; \boxtimes \;\; \underset{s' \in S'}{\coprod} \left( \array{ \{s'\} &\longrightarrow& Vect \\ s' &\mapsto& \mathcal{E}'_{s'} } \right) \Bigg) \\ \;\simeq\; F \Bigg( \underset{(s, s') \in S \times S'}{\coprod} \Big( \big( (s \colon \{s\}) \mapsto \mathcal{E}_s \big) \boxtimes \big( (s' \colon \{s'\}) \mapsto \mathcal{E}_{s'} \big) \Big) \Bigg) \\ \;\simeq\; \underset{(s, s') \in S \times S'}{\coprod} \bigg( \Big( F \big( (s \colon \{s\}) \mapsto \mathcal{E}_s \big) \Big) \otimes \Big( F \big( (s' \colon \{s'\}) \mapsto \mathcal{E}_{s'} \big) \Big) \bigg) \\ \;\simeq\; \Big( F \big( \underset{s \in S}{\coprod} (s \colon \{s\}) \mapsto \mathcal{E}_s \big) \Big) \otimes \Big( F \big( \underset{s' \in S'}{\coprod} (s' \colon \{s'\}) \mapsto \mathcal{E}_{s'} \big) \Big) \\ \;\simeq\; F(\mathcal{E}) \otimes F(\mathcal{E}') \,. \end{array}$$

Related concepts

• Mod

• VectBund(B)

• tangent $\infty$-category

• parameterized spectra

• doubly closed monoidal category