nLab
(infinity,1)-vector bundle

Idea

The notion of an $(\mathrm{\infty },1)$-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 $k\mathrm{Line}\hookrightarrow k\mathrm{Mod}$ of 1-dimensional $k$-vector bundles, and morphisms that factor as $\alpha :X\to k\mathrm{Line}\hookrightarrow k\mathrm{Mod}$ are $k$-line bundles. In the discrete case the vector space of sections of the vector bundle classified by $\alpha$ is the colimit ${\lim }_{\to }\alpha$.

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 $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },n)$-module over an E-∞-ring $K$ is an object of the (∞,n)-category $(\cdots (K\mathrm{Mod})\mathrm{Mod})\cdots \mathrm{Mod}$, where we are iteratively forming module $(\mathrm{\infty },k)$-categories over the monoidal $(\mathrm{\infty },k-1)$-category of $(\mathrm{\infty },k-1)$-modules, $n$ times.

Discrete $(\mathrm{\infty },1)$-vector bundles

We discuss $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-topos of discrete ∞-groupoids.

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

$\mathrm{\infty }$-Modules and $\mathrm{\infty }$-Module bundles

Assume in the following choices

• $K$ – an E-∞ ring

• $A$ – a $K$-algebra,

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

Denote

• $A\mathrm{Mod}$ – the (∞,1)-category of $A$-module spectra.

Definition

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

$$A\mathrm{Mod}(X):=\mathrm{Func}(X,A\mathrm{Mod}).$$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

$$\alpha :\Pi (X)\to A\mathrm{Mod}.$$\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 $\Pi (X)$. Notaby if $X=BG$ is the classifying space of a discrete group or discrete ∞-group, a $K$-module $\mathrm{\infty }$-bundle over $X$ is the same as an ∞-representation of $G$ on $A\mathrm{Mod}$.

$\mathrm{\infty }$-Lines and $\mathrm{\infty }$-line bundles

Definition

Write

$$A\mathrm{Line}\hookrightarrow A\mathrm{Mod}$$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 $A\mathrm{Mod}(X)$ on morphisms factoring through this inclusion we call the $(\mathrm{\infty },1)$-catgeory of $A$-line $\mathrm{\infty }$-bundles.

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

Definition

Let $A$ be an A-∞ ring spectrum.

For ${\Omega }^{\mathrm{\infty }}A$ the underlying A-∞ space and ${\pi }_0{\Omega }^{\mathrm{\infty }}A$ the ordinary ring of connected components, writ $({\pi }_0{\Omega }^{\mathrm{\infty }}A{)}^{\times }$ for its group of units.

Then the ∞-group of units of $A$ is the (∞,1)-pullback ${\mathrm{GL}}_1(A)$ in

$$\begin{array}{ccc}{\mathrm{GL}}_1(A)& \to & {\Omega }^{\mathrm{\infty }}A\\ \downarrow & & \downarrow \\ ({\pi }_0{\Omega }^{\mathrm{\infty }}A{)}^{\times }& \to & {\pi }_0{\Omega }^{\mathrm{\infty }}A\end{array}.$$\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

$${\mathrm{GL}}_1(A)\simeq {\mathrm{Aut}}_{A\mathrm{Line}}(A)$$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:

$$A\mathrm{Line}\simeq B{\mathrm{GL}}_1(A)\simeq B\mathrm{Aut}(A)$$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 $\mathrm{\infty }$-bundle is canonically associated to a ${\mathrm{GL}}_1(A)$-principal ∞-bundle $X\to B{\mathrm{GL}}_1(A)$.

Definition

A ${\mathrm{GL}}_1(A)$-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 $(\mathrm{\infty },1)$-module bundle $f:X\to A\mathrm{Mod}$ is the (∞,1)-colimit over this functor

$$X^f:=\underrightarrow{\lim }(X\stackrel{\alpha }{\to }A\mathrm{Mod}).$$X^f := \lim_\to (X \stackrel{\alpha}{\to} A Mod) \,.

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

$$\Gamma (f):={\mathrm{Mod}}_A(X^f,A).$$\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 $\Gamma (f)$ 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 $\mathrm{\infty }$-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 $A$:

$$\Gamma (X\stackrel{f}{\to }S\mathrm{Line}\to A\mathrm{Line}\to A\mathrm{Mod})\simeq X^f{\wedge }_{S}A.$$\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 $\mathrm{\infty }$-bundle, its ∞-groupoid of trivializations is the $\mathrm{\infty }$-groupoid of lifts

$$\begin{array}{ccc}& & *\\ & \nearrow & \downarrow \\ X& \stackrel{f}{\to }& A\mathrm{Line}\end{array}.$$\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 $\mathrm{\infty }$-bundle $f:X\to A\mathrm{Line}$ induces an equivalence

$$\Gamma (f)\simeq ({\Sigma }^{\mathrm{\infty }}X)\wedge A$$\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 $X$:

$${\pi }_{•}\Gamma (f)\simeq H_{•}(X,A).$$\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 $\mathrm{\infty }$-bundle, the $f$-twisted A-homology of $A$ is

$$H_{•}^f(X,A):={\pi }_{•}(\Gamma (f)):={\pi }_{•}(Mf).$$H_\bullet^f(X, A) := \pi_\bullet(\Gamma(f)) := \pi_\bullet(M f) \,.

The $f$-twisted A-cohomology is

$$H_f^{•}(X,A):={\pi }_{0}A\mathrm{Mod}(Mf,{\Sigma }^{•}A).$$H^\bullet_f(X,A) := \pi_0 A Mod(M f, \Sigma^\bullet A) \,.

Structured $(\mathrm{\infty },1)$-vector bundles

We discuss now $(\mathrm{\infty },1)$-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 $\mathrm{\infty }$-bundle $(P,\nabla )$ which is $A$-oriented over $X$, hence trivializabe over $X$ (for instance for $A=H\mathbb{Q}$ 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 $\Gamma (P)\simeq XedgeA$ of $X$, and that for every suitable surface $\Sigma$ with incoming and outgoing boundary components ${\partial }_{\mathrm{in}}\Gamma \stackrel{\mathrm{in}}{\to }\Gamma \stackrel{\mathrm{out}}{\leftarrow }{\partial }_{\mathrm{out}}\Gamma$ the mapping space span

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

  $$(X^{\mathrm{out}}{)}_{*}(X^{\mathrm{in}}{)}^{!}:H_{•}(X^{{\partial }_{\mathrm{in}}\Gamma },A)\to H_{•}(X^{{\partial }_{\mathrm{out}}\Gamma },A).$$(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 $(\mathrm{\infty },1)$-module bundles is in the triple of articles

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)

• Matthew Ando, Andrew Blumberg, David Gepner, Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and $C^{*}$-algebras, Proceedings of Symposia in Pure Mathematics vol 81 (arXiv:1002.3004)

• Matthew Ando, Andrew Blumberg, David Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map (arXiv:1112.2203)

The last of these explains the relation to

• May, Sigurdsson, Parametrized Homotopy Theory