group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
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.
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.
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
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
In this form this appears as (ABG def. 3.7). Compare this to the analogous definition at principal ∞-bundle.
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
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 $A Mod$.
Write
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).
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
There is an equivalence of ∞-groups
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:
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).
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 $X \to B GL_1(A)$.
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.
The $A$-module of (dual) sections of an $(\infty,1)$-module bundle $f : X \to A Mod$ is the (∞,1)-colimit over this functor
The corresponding spectrum of sections is the $A$-dual
This is (ABG, def. 4.1) and (ABG, p. 15), (ABG11, remark 10.16).
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 $f$:
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 $A$:
This is (ABGHR 08, theorem 4.5).
For $f : X \to A Line$ an $A$-line $\infty$-bundle, its ∞-groupoid of trivializations is the $\infty$-groupoid of lifts
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).
Every trivialization/orientation of an $A$-line $\infty$-bundle $f : X \to A Line$ induces an equivalence
of the $A$-module of sections of $f$ / the Thom A-module of $f$ with the generalized A-homology-spectrum of $X$:
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:
For $f : X \to A Line$ an $A$-line $\infty$-bundle, the $f$-twisted A-homology of $A$ is
The $f$-twisted A-cohomology is
We discuss now $(\infty,1)$-vector bundles in more general (∞,1)-toposes.
(…)
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. 1 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
acts by path integral as a pull-push transform on these spaces of states
A systematic discussion of discrete $(\infty,1)$-module bundles has a precursor in
(discussing the string orientation of tmf) and is then discussed in more detail in the triple of articles
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
A streamlined version of (ABGHR 08) appears as
Lecture notes on these articles are in