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(infinity,1)-module bundle

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cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

The notion of an (,1)(\infty,1)-module bundle is a categorification/homotopification of the notion of a module bundle/vector bundle, where fields and rings are replaced by ∞-rings and modules by ∞-modules; a central notion in parameterized stable homotopy theory.

Recall that for kk a field, a vector space is a kk-module, and a vector bundle over a space XX is classified by a morphism α:Xk\alpha : X \to kMod with kkMod regarded as an object in the relevant topos. For instance for discrete or flat vector bundles kModk Mod is the category Vect of vector spaces. There is the subcategory kLinekModk Line \hookrightarrow k Mod of 1-dimensional kk-vector bundles, and morphisms that factor as α:XkLinekMod\alpha : X \to k Line \hookrightarrow k Mod are kk-line bundles. In the discrete case the vector space of sections of the vector bundle classified by α\alpha is the colimit lim α\lim_\to \alpha.

These statements categorify in a straightforward manner to the case where kk 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 (,1)(\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 (,n)(\infty,n)-module over an E-∞-ring KK is an object of the (∞,n)-category ((KMod)Mod)Mod(\cdots (K Mod) Mod ) \cdots Mod, where we are iteratively forming module (,k)(\infty,k)-categories over the monoidal (,k1)(\infty,k-1)-category of (,k1)(\infty,k-1)-modules, nn times.

Discrete (,1)(\infty,1)-vector bundles

We discuss (,1)(\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 (,1)(\infty,1)-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

Denote

Definition

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

AMod(X):=Func(X,AMod). 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 XX is regarded as a topological space then the corresponding discrete ∞-groupoid is ΠX\Pi X, the fundamental ∞-groupoid of XX and the morphism encoding an KK-module bundle over XX is reads

α:Π(X)AMod. \alpha : \Pi(X) \to A Mod \,.

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

Equivalently, this morphism may be regarded as an ∞-representation of Π(X)\Pi(X). Notaby if X=BGX = B G is the classifying space of a discrete group or discrete ∞-group, a KK-module \infty-bundle over XX is the same as an ∞-representation of GG on AModA Mod.

\infty-Lines and \infty-line bundles

Definition

Write

ALineAMod A Line \hookrightarrow A Mod

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

This appears as (ABG def. 3.12), (ABGHR 08, 7.5).

Definition

Let AA be an A-∞ ring spectrum.

For Ω A\Omega^\infty A the underlying A-∞ space and π 0Ω A\pi_0 \Omega^\infty A the ordinary ring of connected components, writ (π 0Ω A) ×(\pi_0 \Omega^\infty A)^\times for its group of units.

Then the ∞-group of units of AA is the (∞,1)-pullback GL 1(A)GL_1(A) in

GL 1(A) Ω A (π 0Ω A) × π 0Ω A. \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

GL 1(A)Aut ALine(A) GL_1(A) \simeq Aut_{A Line}(A)

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

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

ALineBGL 1(A)BAut(A) A Line \simeq B GL_1(A) \simeq B Aut(A)

that identifies ALineA Line as the delooping ∞-groupoid of either of these two ∞-groups.

This appears in (ABG, 3.6) (p. 10). See also (ABGHR 08, section 6).

Remark

This means that every AA-line \infty-bundle is canonically associated to a GL 1(A)GL_1(A)-principal ∞-bundle over XX which is modulated by a map XBGL 1(A)X \to B GL_1(A).

Definition

A GL 1(A)GL_1(A)-principal ∞-bundle on XX is also called a twist – or better: a local coefficient ∞-bundle – for AA-cohomology on XX.

For the moment see twisted cohomology for more on this.

Sections and twisted cohomology

Definition

The AA-module of (dual) sections of an (,1)(\infty,1)-module bundle f:XAModf : X \to A Mod is the (∞,1)-colimit over this functor

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

The corresponding spectrum of sections is the AA-dual

Γ(f):=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 ff an AA-line bundle Γ(f)\Gamma(f) is called in (ABGHR 08, def. 7.27, remark 7.28) the Thom A-module of ff and written MfM f.

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

Proposition

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

Γ(XfSLineALineAMod)X f SA. \Gamma(X \stackrel{f}{\to} S Line \to A Line \to A Mod) \simeq X^f \wedge_S A \,.

This is (ABGHR 08, theorem 4.5).

Trivializations and orientations

Definition

For f:XALinef : X \to A Line an AA-line \infty-bundle, its ∞-groupoid of trivializations is the \infty-groupoid of lifts

* X f ALine. \array{ && * \\ & \nearrow & \downarrow \\ X &\stackrel{f}{\to}& A Line } \,.

For KAK \to A the canonical inclusion and f:XKLinef : X \to K Line a KK-line bundle, we say that an AA-orientation of ff is a trivialization of the associated AA-line bundle XfKLineALineX \stackrel{f}{\to} K Line \to A Line.

That this encodes the notion of orientation in A-cohomology is around (ABGHR 08, 7.32).

Corollary

Every trivialization/orientation of an AA-line \infty-bundle f:XALinef : X \to A Line induces an equivalence

Γ(f)(Σ X)A \Gamma(f) \simeq (\Sigma^\infty X )\wedge A

of the AA-module of sections of ff / the Thom A-module of ff with the generalized A-homology-spectrum of XX:

π Γ(f)H (X,A). \pi_\bullet \Gamma(f) \simeq H_\bullet(X,A) \,.

This appears as (ABGHR 08, cor. 7.34).

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

Definition

For f:XALinef : X \to A Line an AA-line \infty-bundle, the ff-twisted A-homology of AA is

H f(X,A):=π (Γ(f)):=π (Mf). H_\bullet^f(X, A) := \pi_\bullet(\Gamma(f)) := \pi_\bullet(M f) \,.

The ff-twisted A-cohomology is

H f (X,A):=π 0AMod(Mf,Σ A). H^\bullet_f(X,A) := \pi_0 A Mod(M f, \Sigma^\bullet A) \,.

Structured (,1)(\infty,1)-vector bundles

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

(…)

Applications

  • The string topology operations on a compact smooth manifold XX may be understood as arising from a sigma-model quantum field theory with target space XX whose background gauge field is a flat AA-line \infty-bundle (P,)(P,\nabla) which is AA-oriented over XX, hence trivializabe over XX (for instance for A=HA = 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 σ\sigma-model is the AA-homology spectrum Γ(P)XedgeA\Gamma(P) \simeq X \edge A of XX, and that for every suitable surface Σ\Sigma with incoming and outgoing boundary components inΓinΓout outΓ\partial_{in} \Gamma \stackrel{in}{\to} \Gamma \stackrel{out}{\leftarrow} \partial_{out} \Gamma the mapping space span

    X inΓX inX ΓX outX outΓ 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 out) *(X in) !:H (X inΓ,A)H (X outΓ,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 (,1)(\infty,1)-module bundles has a precursor in

(discussing the string orientation of tmf) and is then discussed in more detail in the triple of articles

The last of these explains the relation to

A streamlined version of (ABGHR 08) appears as

Lecture notes on these articles are in

  • Ben Knudsen, Scott Slinker, Paul VanKoughnett, Brian Williams, and Dylan Wilson, Thom spectra reading course (web)

Revised on May 21, 2014 13:18:58 by Urs Schreiber (82.136.246.44)