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\begin{document}

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\section*{(infinity,1)-module bundle}

[[!redirects (infinity,1)-vector bundle]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{discrete_vector_bundles}{Discrete $(\infty,1)$-vector bundles}\dotfill \pageref*{discrete_vector_bundles} \linebreak
\noindent\hyperlink{modules_and_module_bundles}{$\infty$-Modules and $\infty$-Module bundles}\dotfill \pageref*{modules_and_module_bundles} \linebreak
\noindent\hyperlink{lines_and_line_bundles}{$\infty$-Lines and $\infty$-line bundles}\dotfill \pageref*{lines_and_line_bundles} \linebreak
\noindent\hyperlink{SectionsAndTwistedCohomology}{Sections and twisted cohomology}\dotfill \pageref*{SectionsAndTwistedCohomology} \linebreak
\noindent\hyperlink{trivializations_and_orientations}{Trivializations and orientations}\dotfill \pageref*{trivializations_and_orientations} \linebreak
\noindent\hyperlink{Structured}{Structured $(\infty,1)$-vector bundles}\dotfill \pageref*{Structured} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of an \emph{$(\infty,1)$-module bundle} is a [[categorification]]/[[homotopy theory|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 \emph{discrete} or \emph{flat} vector bundles $k Mod$ is the [[category]] [[Vect]] of vector spaces. There is the subcategory $k Line \hookrightarrow k Mod$ of 1-[[dimension]]al $k$-vector bundles, and morphisms that factor as $\alpha : X \to k Line \hookrightarrow k Mod$ are $k$-[[line bundle]]s. In the discrete case the vector space of [[section]]s of the vector bundle classified by $\alpha$ is the [[colimit]] $\lim_\to \alpha$.

These statements [[categorification|categorify]] in a straightforward manner to the case where $k$ is generalized to a commutative [[∞-ring]]: an \emph{[[E-∞ ring]]} or \emph{[[ring spectrum]]} . Modules are replaced by [[module spectrum|module spectra]] and colimits by [[homotopy colimit]]s.

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

Further generalization of the concept leads to [[(∞,n)-vector bundle]]s: 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.

\hypertarget{discrete_vector_bundles}{}\subsection*{{Discrete $(\infty,1)$-vector bundles}}\label{discrete_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 \emph{[[discrete ∞-groupoids]]}.

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

\hypertarget{modules_and_module_bundles}{}\subsubsection*{{$\infty$-Modules and $\infty$-Module bundles}}\label{modules_and_module_bundles}

Assume in the following choices

\begin{itemize}%
\item $K$ -- an [[E-∞ ring]]


\item $A$ -- a $K$-[[algebra spectrum|algebra]],

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



\end{itemize}
Denote

\begin{itemize}%
\item $A Mod$ -- the [[(∞,1)-category]] of $A$-[[module spectrum|module spectra]].

\end{itemize}
\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[discrete ∞-groupoid]] (often presented as a [[topological space]]), the [[(∞,1)-category]] of \textbf{$A$-module $\infty$-bundles} over $X$ is the [[(∞,1)-functor (∞,1)-category]]

\begin{displaymath}
A Mod(X) := Func(X, A Mod)
  \,.
\end{displaymath}
\end{defn}
In this form this appears as (\hyperlink{ABG}{ABG def. 3.7}). Compare this to the analogous definition at \emph{[[principal ∞-bundle]]}.

\begin{remark}
\label{}\hypertarget{}{}
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

\begin{displaymath}
\alpha : \Pi(X) \to A Mod
  \,.
\end{displaymath}
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 [[discrete ∞-groupoid|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 $A Mod$.

\end{remark}
\hypertarget{lines_and_line_bundles}{}\subsubsection*{{$\infty$-Lines and $\infty$-line bundles}}\label{lines_and_line_bundles}

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
A Line \hookrightarrow A Mod
\end{displaymath}
for the [[full sub-(∞,1)-category]] on the \emph{$A$-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 \textbf{$A$-line $\infty$-bundles}.

\end{defn}
This appears as (\hyperlink{ABG}{ABG def. 3.12}), (\hyperlink{ABGHR08}{ABGHR 08, 7.5}).

\begin{defn}
\label{}\hypertarget{}{}
Let $A$ be an [[A-∞ algebra|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

\begin{displaymath}
\itexarray{
    GL_1(A) &\to& \Omega^\infty A
    \\
    \downarrow && \downarrow
    \\
    (\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A
  }
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
There is an [[equivalence in an (∞,1)-category|equivalence]] of [[∞-groups]]

\begin{displaymath}
GL_1(A) \simeq Aut_{A Line}(A)
\end{displaymath}
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]]:

\begin{displaymath}
A Line \simeq B GL_1(A) \simeq B Aut(A)
\end{displaymath}
that identifies $A Line$ as the [[delooping]] [[∞-groupoid]] of either of these two [[∞-groups]].

\end{prop}
This appears in (\hyperlink{ABG}{ABG, 3.6}) (\href{http://arxiv.org/pdf/1002.3004v2.pdf#page=10}{p. 10}). See also (\hyperlink{ABGHR08}{ABGHR 08, section 6}).

\begin{remark}
\label{}\hypertarget{}{}
This means that every $A$-line $\infty$-bundle is canonically [[associated ∞-bundle|associated]] to a $GL_1(A)$-[[principal ∞-bundle]] over $X$ which is [[moduli space|modulated]] by a map $X \to B GL_1(A)$.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A $GL_1(A)$-[[principal ∞-bundle]] on $X$ is also called a \textbf{[[twisted cohomology|twist]]} -- or better: a \textbf{[[local coefficient ∞-bundle]]} -- for $A$-[[generalized (Eilenberg-Steenrod) cohomology|cohomology]] on $X$.

\end{defn}
For the moment see \emph{[[twisted cohomology]]} for more on this.

\hypertarget{SectionsAndTwistedCohomology}{}\subsubsection*{{Sections and twisted cohomology}}\label{SectionsAndTwistedCohomology}

\begin{defn}
\label{}\hypertarget{}{}
The $A$-module of (dual) [[sections]] of an $(\infty,1)$-module bundle $f : X \to A Mod$ is the [[(∞,1)-colimit]] over this functor

\begin{displaymath}
X^f := \lim_\to (X \stackrel{\alpha}{\to} A Mod)
  \,.
\end{displaymath}
The corresponding \emph{spectrum of sections} is the $A$-dual

\begin{displaymath}
\Gamma(f) := Mod_A(X^f, A)
  \,.
\end{displaymath}
\end{defn}
This is (\hyperlink{ABG}{ABG, def. 4.1}) and (\hyperlink{ABG}{ABG, p. 15}), (\hyperlink{ABGII}{ABG11, remark 10.16}).

\begin{remark}
\label{}\hypertarget{}{}
For $f$ an $A$-line bundle $\Gamma(f)$ is called in (\hyperlink{ABGHR08}{ABGHR 08, def. 7.27, remark 7.28}) the \textbf{[[Thom spectrum|Thom A-module]]} of $f$ and written $M f$.

\end{remark}
Because for $A = S$ the [[sphere spectrum]], $M f$ is indeed the classical [[Thom spectrum]] of the spherical fibration given by $f$:

\begin{prop}
\label{}\hypertarget{}{}
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 $A$:

\begin{displaymath}
\Gamma(X \stackrel{f}{\to} S Line \to A Line \to A Mod)
  \simeq
   X^f \wedge_S A
  \,.
\end{displaymath}
\end{prop}
This is (\hyperlink{ABGHR08}{ABGHR 08, theorem 4.5}).

\hypertarget{trivializations_and_orientations}{}\subsubsection*{{Trivializations and orientations}}\label{trivializations_and_orientations}

\begin{defn}
\label{}\hypertarget{}{}
For $f : X \to A Line$ an $A$-line $\infty$-bundle, its [[∞-groupoid]] of \textbf{trivializations} is the $\infty$-groupoid of lifts

\begin{displaymath}
\itexarray{  
     && *
     \\
     & \nearrow & \downarrow
    \\
    X &\stackrel{f}{\to}& A Line
  }
  \,.
\end{displaymath}
For $K \to A$ the canonical inclusion and $f : X \to K Line$ a $K$-line bundle, we say that an \textbf{$A$-orientation} of $f$ is a trivialization of the associated $A$-line bundle $X \stackrel{f}{\to} K Line \to A Line$.

\end{defn}
That this encodes the notion of [[orientation in generalized cohomology|orientation in A-cohomology]] is around (\hyperlink{ABGHR08}{ABGHR 08, 7.32}).

\begin{cor}
\label{ThomModuleInOrientedCase}\hypertarget{ThomModuleInOrientedCase}{}
Every trivialization/orientation of an $A$-line $\infty$-bundle $f : X \to A Line$ induces an equivalence

\begin{displaymath}
\Gamma(f) \simeq (\Sigma^\infty X )\wedge A
\end{displaymath}
of the $A$-module of sections of $f$ / the [[Thom spectrum|Thom A-module]] of $f$ with the [[homology|generalized A-homology]]-spectrum of $X$:

\begin{displaymath}
\pi_\bullet \Gamma(f) \simeq H_\bullet(X,A)   
  \,.
\end{displaymath}
\end{cor}
This appears as (\hyperlink{ABGHR08}{ABGHR 08, cor. 7.34}).

Therefore if $f$ is \emph{not} trivializable, we may regard its $A$-module of sections as encoding $f$-[[twisted cohomology|twisted A-cohomology]]:

\begin{defn}
\label{}\hypertarget{}{}
For $f : X \to A Line$ an $A$-line $\infty$-bundle, the $f$-\textbf{[[twisted cohomology|twisted A-homology]] of $A$ is}

\begin{displaymath}
H_\bullet^f(X, A) := \pi_\bullet(\Gamma(f)) := \pi_\bullet(M f)
  \,.
\end{displaymath}
The \textbf{$f$-[[twisted cohomology|twisted A-cohomology]] is}

\begin{displaymath}
H^\bullet_f(X,A) := \pi_0 A Mod(M f, \Sigma^\bullet A)
  \,.
\end{displaymath}
\end{defn}
\hypertarget{Structured}{}\subsection*{{Structured $(\infty,1)$-vector bundles}}\label{Structured}

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

(\ldots{})

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item 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 spectrum|Thom spaces]] induced from the [[tangent bundle]] if the manifold is [[oriented]] in the ordinary sense).

By prop. \ref{ThomModuleInOrientedCase} this implies that the space of [[state]]s 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]]

\begin{displaymath}
X^{\partial_{in} \Gamma}
   \stackrel{X^{in}}{\leftarrow}
  X^{\Gamma}
   \stackrel{X^{out}}{\rightarrow}
  X^{\partial_{out} \Gamma}
\end{displaymath}
acts by [[path integral as a pull-push transform]] on these spaces of states

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


\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[twisted cohomology]]


\item [[Picard ∞-stack]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

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

\begin{itemize}%
\item [[Matthew Ando]], [[Michael Hopkins]], [[Charles Rezk]], \emph{Multiplicative orientations of KO-theory and the spectrum of topological modular forms}, 2010 (\href{http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf}{pdf})

\end{itemize}
(discussing the [[string orientation of tmf]]) and is then discussed in more detail in the triple of articles

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], \emph{Units of ring spectra and Thom spectra} (\href{http://arxiv.org/abs/0810.4535}{arXiv:0810.4535})


\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Twists of K-theory and TMF}, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], \emph{Superstrings, Geometry, Topology, and $C^*$-algebras}, Proceedings of Symposia in Pure Mathematics \href{http://www.ams.org/bookstore-getitem/item=PSPUM-81}{vol 81}, American Mathematical Society (\href{http://arxiv.org/abs/1002.3004}{arXiv:1002.3004})


\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map} (\href{http://arxiv.org/abs/1112.2203}{arXiv:1112.2203})



\end{itemize}
The last of these explains the relation to

\begin{itemize}%
\item [[Peter May|May]], Sigurdsson, \emph{[[Parametrized Homotopy Theory]]}

\end{itemize}
A streamlined version of (\hyperlink{ABGHR08}{ABGHR 08}) appears as

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], \emph{An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology} (\href{http://arxiv.org/abs/1403.4325}{arXiv:1403.4325})

\end{itemize}
Lecture notes on these articles are in

\begin{itemize}%
\item Ben Knudsen, Scott Slinker, Paul VanKoughnett, Brian Williams, and [[Dylan Wilson]], \emph{Thom spectra reading course} (\href{http://www.math.northwestern.edu/~bwill/thom/}{web})

\end{itemize}
[[!redirects (∞,1)-vector bundle]]

[[!redirects (infinity,1)-vector bundles]] [[!redirects (∞,1)-vector bundles]]

[[!redirects (∞,1)-module bundle]] [[!redirects (∞,1)-module bundles]] [[!redirects (infinity,1)-module bundle]] [[!redirects (infinity,1)-module bundles]]

[[!redirects (∞,1)-line]] [[!redirects (∞,1)-lines]] [[!redirects (infinity,1)-line]] [[!redirects (infinity,1)-lines]]

[[!redirects (∞,1)-vector space]] [[!redirects (∞,1)-vector spaces]] [[!redirects (infinity,1)-vector space]] [[!redirects (infinity,1)-vector spaces]]

[[!redirects (∞,1)-line bundle]] [[!redirects (∞,1)-line bundles]] [[!redirects (infinity,1)-line bundle]] [[!redirects (infinity,1)-line bundles]]

[[!redirects ∞-line bundle]] [[!redirects ∞-line bundles]] [[!redirects infinity-line bundle]] [[!redirects infinity-line bundles]]



\end{document}