# nLab (infinity,1)-vector bundle

bundles

## Examples and Applications

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The notion of an $\left(\infty ,1\right)$-vector bundle is a categorification of the notion of a vector bundle, where fields and rings are replaced by ∞-rings.

Recall that for $k$ a field, a vector space is a $k$-module, and a vector bundle over a space $X$ is classified by a morphism $\alpha :X\to k$Mod with $k$Mod regarded as an object in the relevant topos. For instance for discrete or flat vector bundles $k\mathrm{Mod}$ is the category Vect of vector spaces. There is the subcategory $k\mathrm{Line}↪k\mathrm{Mod}$ of 1-dimensional $k$-vector bundles, and morphisms that factor as $\alpha :X\to k\mathrm{Line}↪k\mathrm{Mod}$ are $k$-line bundles. In the discrete case the vector space of sections of the vector bundle classified by $\alpha$ is the colimit ${\mathrm{lim}}_{\to }\alpha$.

These statements categorify in a straightforward manner to the case where $k$ is generalized to a commutative ∞-ring: an E-∞ ring or ring spectrum . Modules are replaced by module spectra and colimits by homotopy colimits.

The resulting notion of $\left(\infty ,1\right)$-vector bundles plays a central role in many constructions in orientation in generalized cohomology, twisted cohomology and Thom isomorphisms.

Further generalization of the concept leads to (∞,n)-vector bundles: an $\left(\infty ,n\right)$-module over an E-∞-ring $K$ is an object of the (∞,n)-category $\left(\cdots \left(K\mathrm{Mod}\right)\mathrm{Mod}\right)\cdots \mathrm{Mod}$, where we are iteratively forming module $\left(\infty ,k\right)$-categories over the monoidal $\left(\infty ,k-1\right)$-category of $\left(\infty ,k-1\right)$-modules, $n$ times.

## Discrete $\left(\infty ,1\right)$-vector bundles

We discuss $\left(\infty ,1\right)$-vector bundles internal to the (∞,1)-topos ∞Grpd $\simeq$ Top. Since we are discussing objects with geometric interpretation, we are to think of this as the $\left(\infty ,1\right)$-topos of discrete ∞-groupoids.

Discussion of $\infty$-vector bundles internal to structured (non-discrete) $\infty$-groupoids is below.

### $\infty$-Modules and $\infty$-Module bundles

Assume in the following choices

• $K$ – an E-∞ ring

• $A$ – a $K$-algebra,

hence an A-∞ algebra in Spec equipped with a $\infty$-algebra homomorphism $K\to A$.

Denote

###### Definition

For $X$ a discrete ∞-groupoid (often presented as a topological space), the (∞,1)-category of $A$-module $\infty$-bundles over $X$ is the (∞,1)-functor (∞,1)-category

$A\mathrm{Mod}\left(X\right):=\mathrm{Func}\left(X,A\mathrm{Mod}\right)\phantom{\rule{thinmathspace}{0ex}}.$A Mod(X) := Func(X, A Mod) \,.

In this form this appears as (ABG def. 3.7). Compare this to the analogous definition at principal ∞-bundle.

###### Remark

If $X$ is regarded as a topological space then the corresponding discrete ∞-groupoid is $\Pi X$, the fundamental ∞-groupoid of $X$ and the morphism encoding an $K$-module bundle over $X$ is reads

$\alpha :\Pi \left(X\right)\to A\mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$\alpha : \Pi(X) \to A Mod \,.

This assignment of $A$-modules to points in $X$, of $A$-module morphism to paths in $X$ etc. may be regarded as the higher parallel transport of the (unique and flat, due to discreteness) connection on an ∞-bundle on $\alpha$.

Equivalently, this morphism may be regarded as an ∞-representation of $\Pi \left(X\right)$. Notaby if $X=BG$ is the classifying space of a discrete group or discrete ∞-group, a $K$-module $\infty$-bundle over $X$ is the same as an ∞-representation of $G$ on $A\mathrm{Mod}$.

### $\infty$-Lines and $\infty$-line bundles

###### Definition

Write

$A\mathrm{Line}↪A\mathrm{Mod}$A Line \hookrightarrow A Mod

for the full sub-(∞,1)-category on the $A$-lines : on those $A$-modules that are equivalent to $A$ as an $A$-module. The full subcatgeory of $A\mathrm{Mod}\left(X\right)$ on morphisms factoring through this inclusion we call the $\left(\infty ,1\right)$-catgeory of $A$-line $\infty$-bundles.

This appears as (ABG def. 3.12), (ABGHR, 7.5).

###### Definition

Let $A$ be an A-∞ ring spectrum.

For ${\Omega }^{\infty }A$ the underlying A-∞ space and ${\pi }_{0}{\Omega }^{\infty }A$ the ordinary ring of connected components, writ $\left({\pi }_{0}{\Omega }^{\infty }A{\right)}^{×}$ for its group of units.

Then the ∞-group of units of $A$ is the (∞,1)-pullback ${\mathrm{GL}}_{1}\left(A\right)$ in

$\begin{array}{ccc}{\mathrm{GL}}_{1}\left(A\right)& \to & {\Omega }^{\infty }A\\ ↓& & ↓\\ \left({\pi }_{0}{\Omega }^{\infty }A{\right)}^{×}& \to & {\pi }_{0}{\Omega }^{\infty }A\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ GL_1(A) &\to& \Omega^\infty A \\ \downarrow && \downarrow \\ (\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A } \,.
###### Proposition

There is an equivalence of ∞-groups

${\mathrm{GL}}_{1}\left(A\right)\simeq {\mathrm{Aut}}_{A\mathrm{Line}}\left(A\right)$GL_1(A) \simeq Aut_{A Line}(A)

of the ∞-group of units of $A$ with the automorphism ∞-group of $A$, regarded canonically as a module over itself.

Since every $A$-line is by definition equivalent to $A$ as an $A$-module, there is accordingly, an equivalence of (∞,1)-categories, in fact of ∞-groupoids:

$A\mathrm{Line}\simeq B{\mathrm{GL}}_{1}\left(A\right)\simeq B\mathrm{Aut}\left(A\right)$A Line \simeq B GL_1(A) \simeq B Aut(A)

that identifies $A\mathrm{Line}$ as the delooping ∞-groupoid of either of these two ∞-groups.

This appears in (ABG, 3.6) (p. 10). See also (ABGHR, section 6.

###### Remark

This means that every $A$-line $\infty$-bundle is canonically associated to a ${\mathrm{GL}}_{1}\left(A\right)$-principal ∞-bundle $X\to B{\mathrm{GL}}_{1}\left(A\right)$.

###### Definition

A ${\mathrm{GL}}_{1}\left(A\right)$-principal ∞-bundle on $X$ is also called a twist for $A$-cohomology on $X$.

For the moment see twisted cohomology for more on this.

### Sections and twisted cohomology

###### Definition

The $A$-module of (dual) sections of an $\left(\infty ,1\right)$-module bundle $f:X\to A\mathrm{Mod}$ is the (∞,1)-colimit over this functor

${X}^{f}:=\underset{\to }{\mathrm{lim}}\left(X\stackrel{\alpha }{\to }A\mathrm{Mod}\right)\phantom{\rule{thinmathspace}{0ex}}.$X^f := \lim_\to (X \stackrel{\alpha}{\to} A Mod) \,.

The corresponding spectrum of sections is the $A$-dual

$\Gamma \left(f\right):={\mathrm{Mod}}_{A}\left({X}^{f},A\right)\phantom{\rule{thinmathspace}{0ex}}.$\Gamma(f) := Mod_A(X^f, A) \,.

This is (ABG, def. 4.1) and (ABG, p. 15), (ABG11, remark 10.16).

###### Remark

For $f$ an $A$-line bundle $\Gamma \left(f\right)$ is called in (ABGHR, def. 7.27, remark 7.28) the Thom $A$-module of $f$ and written $Mf$.

Because for $A=S$ the sphere spectrum, $Mf$ is indeed the classical Thom spectrum of the spherical fibration given by $f$:

###### Proposition

For $K=S$ the sphere spectrum, $f:X\to K\mathrm{Line}=S\mathrm{Line}$ an $S$-line bundle – hence a spherical fibration, and $A$ any other $\infty$-ring with canonical inclusion $S\to A$, the Thom $A$-module of the composite $X\stackrel{f}{\to }S\mathrm{Mod}\to A\mathrm{Mod}$ is the classical Thom spectrum of $f$ tensored with $A$:

$\Gamma \left(X\stackrel{f}{\to }S\mathrm{Line}\to A\mathrm{Line}\to A\mathrm{Mod}\right)\simeq {X}^{f}{\wedge }_{S}A\phantom{\rule{thinmathspace}{0ex}}.$\Gamma(X \stackrel{f}{\to} S Line \to A Line \to A Mod) \simeq X^f \wedge_S A \,.

This is (ABGHR, theorem 4.5).

### Trivializations and orientations

###### Definition

For $f:X\to A\mathrm{Line}$ an $A$-line $\infty$-bundle, its ∞-groupoid of trivializations is the $\infty$-groupoid of lifts

$\begin{array}{ccc}& & *\\ & ↗& ↓\\ X& \stackrel{f}{\to }& A\mathrm{Line}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && * \\ & \nearrow & \downarrow \\ X &\stackrel{f}{\to}& A Line } \,.

For $K\to A$ the canonical inclusion and $f:X\to K\mathrm{Line}$ a $K$-line bundle, we say that an $A$-orientation of $f$ is a trivialization of the associated $A$-line bundle $X\stackrel{f}{\to }K\mathrm{Line}\to A\mathrm{Line}$.

That this encodes the notion of orientation in A-cohomology is around (ABGHR, 7.32).

###### Corollary

Every trivialization/orientation of an $A$-line $\infty$-bundle $f:X\to A\mathrm{Line}$ induces an equivalence

$\Gamma \left(f\right)\simeq \left({\Sigma }^{\infty }X\right)\wedge A$\Gamma(f) \simeq (\Sigma^\infty X )\wedge A

of the $A$-module of sections of $f$ / the Thom $A$-module of $f$ with the generalized A-homology-spectrum of $X$:

${\pi }_{•}\Gamma \left(f\right)\simeq {H}_{•}\left(X,A\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_\bullet \Gamma(f) \simeq H_\bullet(X,A) \,.

This appears as (ABGHR, cor. 7.34).

Therefore if $f$ is not trivializable, we may regard its $A$-module of sections as encoding $f$-twisted A-cohomology:

###### Definition

For $f:X\to A\mathrm{Line}$ an $A$-line $\infty$-bundle, the $f$-twisted A-homology of $A$ is

${H}_{•}^{f}\left(X,A\right):={\pi }_{•}\left(\Gamma \left(f\right)\right):={\pi }_{•}\left(Mf\right)\phantom{\rule{thinmathspace}{0ex}}.$H_\bullet^f(X, A) := \pi_\bullet(\Gamma(f)) := \pi_\bullet(M f) \,.

The $f$-twisted A-cohomology is

${H}_{f}^{•}\left(X,A\right):={\pi }_{0}A\mathrm{Mod}\left(Mf,{\Sigma }^{•}A\right)\phantom{\rule{thinmathspace}{0ex}}.$H^\bullet_f(X,A) := \pi_0 A Mod(M f, \Sigma^\bullet A) \,.

## Structured $\left(\infty ,1\right)$-vector bundles

We discuss now $\left(\infty ,1\right)$-vector bundles in more general (∞,1)-toposes.

(…)

## Applications

• The string topology operations on a compact smooth manifold $X$ may be understood as arising from a sigma-model quantum field theory with target space $X$ whose background gauge field is a flat $A$-line $\infty$-bundle $\left(P,\nabla \right)$ which is $A$-oriented over $X$, hence trivializabe over $X$ (for instance for $A=Hℚ$ the Eilenberg-MacLane spectrum this may be the sphereical fibration of Thom spaces induced from the tangent bundle if the manifold is oriented in the ordinary sense).

By prop. 1 this implies that the space of states of the $\sigma$-model is the $A$-homology spectrum $\Gamma \left(P\right)\simeq XedgeA$ of $X$, and that for every suitable surface $\Sigma$ with incoming and outgoing boundary components ${\partial }_{\mathrm{in}}\Gamma \stackrel{\mathrm{in}}{\to }\Gamma \stackrel{\mathrm{out}}{←}{\partial }_{\mathrm{out}}\Gamma$ the mapping space span

${X}^{{\partial }_{\mathrm{in}}\Gamma }\stackrel{{X}^{\mathrm{in}}}{←}{X}^{\Gamma }\stackrel{{X}^{\mathrm{out}}}{\to }{X}^{{\partial }_{\mathrm{out}}\Gamma }$X^{\partial_{in} \Gamma} \stackrel{X^{in}}{\leftarrow} X^{\Gamma} \stackrel{X^{out}}{\rightarrow} X^{\partial_{out} \Gamma}

acts by path integral as a pull-push transform on these spaces of states

$\left({X}^{\mathrm{out}}{\right)}_{*}\left({X}^{\mathrm{in}}{\right)}^{!}:{H}_{•}\left({X}^{{\partial }_{\mathrm{in}}\Gamma },A\right)\to {H}_{•}\left({X}^{{\partial }_{\mathrm{out}}\Gamma },A\right)\phantom{\rule{thinmathspace}{0ex}}.$(X^{out})_* (X^{in})^! : H_\bullet(X^{\partial_{in} \Gamma},A) \to H_\bullet(X^{\partial_{out} \Gamma}, A) \,.

## References

A systematic discussion of discrete $\left(\infty ,1\right)$-module bundles is in the triple of articles

The last of these explains the relation to

Revised on June 25, 2012 22:15:37 by Urs Schreiber (89.204.139.149)