nLab (infinity,1)-module bundle

Idea

The notion of an $(\infty,1)$-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 $k$ 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 $k Mod$ is the category Vect of vector spaces. There is the subcategory $k Line \hookrightarrow k Mod$ of 1-dimensional $k$-vector bundles, and morphisms that factor as $\alpha : X \to k Line \hookrightarrow k Mod$ 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 $(\infty,1)$-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 $(\infty,n)$-module over an E-∞-ring $K$ is an object of the (∞,n)-category $(\cdots (K Mod) Mod ) \cdots Mod$, where we are iteratively forming module $(\infty,k)$-categories over the monoidal $(\infty,k-1)$-category of $(\infty,k-1)$-modules, $n$ times.

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

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

• $K$ – an E-∞ ring

• $A$ – a $K$-algebra,

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

Denote

• $A Mod$ – 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 $\infty$-bundles over $X$ is the (∞,1)-functor (∞,1)-category

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

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

$$\alpha : \Pi(X) \to A Mod \,.$$

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 $X = B G$ 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 $A Mod$.

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

Definition

Write

$$A Line \hookrightarrow A Mod$$

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

This appears as (ABG def. 3.12), (ABGHR 08, 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 $(\pi_0 \Omega^\infty A)^\times$ for its group of units.

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

$$\array{ GL_1(A) &\to& \Omega^\infty A \\ \downarrow && \downarrow \\ (\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A } \,.$$

Proposition

There is an equivalence of ∞-groups

$$GL_1(A) \simeq Aut_{A Line}(A)$$

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:

$$A Line \simeq B GL_1(A) \simeq B Aut(A)$$

that identifies $A Line$ 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 $A$-line $\infty$-bundle is canonically associated to a $GL_1(A)$-principal ∞-bundle over $X$ which is modulated by a map $X \to B GL_1(A)$.

Definition

A $GL_1(A)$-principal ∞-bundle on $X$ is also called a twist – or better: a local coefficient ∞-bundle – 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 $(\infty,1)$-module bundle $f : X \to A Mod$ is the (∞,1)-colimit over this functor

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

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

$$\Gamma(f) := Mod_A(X^f, A) \,.$$

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 08, def. 7.27, remark 7.28) the Thom A-module of $f$ and written $M f$.

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

Proposition

For $K = S$ the sphere spectrum, $f : X \to K Line = S Line$ 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 $X \stackrel{f}{\to} S Mod \to A Mod$ is the classical Thom spectrum of $f$ tensored with $A$:

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

This is (ABGHR 08, theorem 4.5).

Trivializations and orientations

Definition

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

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

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

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

Corollary

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

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

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

$$\pi_\bullet \Gamma(f) \simeq H_\bullet(X,A) \,.$$

This appears as (ABGHR 08, 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 $f : X \to A Line$ an $A$-line $\infty$-bundle, the $f$-twisted A-homology of $A$ is

$$H_\bullet^f(X, A) := \pi_\bullet(\Gamma(f)) := \pi_\bullet(M f) \,.$$

The $f$-twisted A-cohomology is

$$H^\bullet_f(X,A) := \pi_0 A Mod(M f, \Sigma^\bullet A) \,.$$

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

We discuss now $(\infty,1)$-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 $\infty$-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. this implies that the space of states of the $\sigma$-model is the $A$-homology spectrum $\Gamma(P) \simeq X \edge A$ of $X$, and that for every suitable surface $\Sigma$ with incoming and outgoing boundary components $\partial_{in} \Gamma \stackrel{in}{\to} \Gamma \stackrel{out}{\leftarrow} \partial_{out} \Gamma$ the mapping space span

  $$X^{\partial_{in} \Gamma} \stackrel{X^{in}}{\leftarrow} X^{\Gamma} \stackrel{X^{out}}{\rightarrow} X^{\partial_{out} \Gamma}$$

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

  $$(X^{out})_* (X^{in})^! : H_\bullet(X^{\partial_{in} \Gamma},A) \to H_\bullet(X^{\partial_{out} \Gamma}, A) \,.$$

Related concepts

• ∞-representation

• twisted cohomology

• Picard ∞-stack

References

A systematic discussion of discrete $(\infty,1)$-module bundles has a precursor in

• Matthew Ando, Michael Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)

(discussing the string orientation of tmf) and is then discussed in more detail 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, American Mathematical Society (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

A streamlined version of (ABGHR 08) appears as

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology (arXiv:1403.4325)

Lecture notes on these articles are in

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

Interpretation of the algebraic K-theory $K(R)$ of a ring spectrum $R$ (see at iterated algebraic K-theory) as the Grothendieck group of (∞,1)-module bundles over $R$:

• John Lind, Bundles of spectra and algebraic K-theory, Pacific J. Math. 285 (2016) 427-452 (arXiv:1304.5676, doi:10.2140/pjm.2016.285.427)