\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{local system}

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

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{sheaftheoretic_case}{Sheaf-theoretic case}\dotfill \pageref*{sheaftheoretic_case} \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}

A \emph{local system} -- which is short for \emph{local system of [[coefficients]] for [[cohomology]]} -- is a system of coefficients for [[twisted cohomology]].

Often this is presented or taken to be presented by a [[locally constant sheaf]]. Then cohomology with coefficients in a local system is the corresponding [[sheaf cohomology]].

More generally, we say a \emph{local system} is a [[locally constant stack]], \ldots{} and eventually a [[locally constant ∞-stack]].

Under suitable conditions (if we have [[Galois theory]]) local systems on $X$ correspond to [[functor]]s out of the [[fundamental groupoid]] of $X$, or more generally to [[(∞,1)-functor]]s out of the [[fundamental ∞-groupoid]]. These in turn are equivalently [[flat connections]] (this relation is known as the \emph{[[Riemann-Hilbert correspondence]]}) or generally [[flat ∞-connections]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A notion of [[cohomology]] exists intrinsically within any [[(∞,1)-topos]]. We discuss local systems first in this generality and then look at special cases, such as local systems as ordinary [[sheaves]].

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

For $\mathbf{H}$ an [[(∞,1)-sheaf (∞,1)-topos]], write

\begin{displaymath}
(LConst \dashv \Gamma)
  : 
  \mathbf{H}
   \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
\end{displaymath}
for the [[terminal object|terminal]] [[(∞,1)-geometric morphism]], where $\Gamma$ is the [[global section]] [[(∞,1)-functor]] and $LConst$ the [[constant ∞-stack]]-functor.

Write $\mathcal{S} := core(Fin \infty Grpd) \in$ [[∞Grpd]] for the [[core]] [[∞-groupoid]] of the [[(∞,1)-category]] of finite $\infty$-groupoids. (We can drop the finiteness condition by making use of a higher [[universe]].) This is canonically a [[pointed object]] $* \to \mathcal{S}$, with points the terminal groupoid.

\begin{defn}
\label{}\hypertarget{}{}
For $X \in \mathbf{H}$ an [[object]], a \textbf{local system} or \emph{[[locally constant ∞-stack]]} on $X$ is a morphism

\begin{displaymath}
\tilde \nabla \colon X \longrightarrow LConst \mathcal{S}
\end{displaymath}
in $\mathbf{H}$ or equivalently the object in the [[over-(∞,1)-topos]]

\begin{displaymath}
(P \to X) \in \mathbf{H}/X
\end{displaymath}
that is classified by $\tilde \nabla$ under the [[(∞,1)-Grothendieck construction]]

\begin{displaymath}
\itexarray{
    P &\to& LConst \mathcal{Z}
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\tilde \nabla}{\to}&  LConst \mathcal{S}
  }
\end{displaymath}
In other words, local systems are [[locally constant ∞-stacks]] or equivalently their classifying [[cocycles]] for [[cohomology with constant coefficients]].

\end{defn}
(See [[principal ∞-bundle]] for discussion of how [[cocycle]]s $\tilde \nabla : X \to LConst \mathcal{S}$ classify morphisms $P \to X$.)

\hypertarget{remark}{}\paragraph*{{Remark}}\label{remark}

If $\mathbf{H}$ happens to be a [[locally ∞-connected (∞,1)-topos]] in that there is the further [[left adjoint|left]] [[adjoint (∞,1)-functor]] $\Pi$

\begin{displaymath}
(\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd
\end{displaymath}
we call $\Pi(X)$ the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]. In this case, by the adjunction hom-equivalence we have

\begin{displaymath}
\mathbf{H}(X, LConst \mathcal{S})
  \simeq
  Func(\Pi(X), \mathcal{S})
  \,.
\end{displaymath}
This means that local systems are naturally identified with [[representations]] ($\infty$-[[permutation representation]]s, as it were) of the [[fundamental ∞-groupoid]] $\Pi(X)$:

\begin{displaymath}
Maps(X, LConst \mathcal{S}) \simeq Maps(\Pi(X), \mathcal{S})
  \,.
\end{displaymath}
This is essentially the basic statement around which [[Galois theory]] revolves.

The [[(∞,1)-sheaf (∞,1)-topos]] over a [[locally contractible space]] is locally $\infty$-connected, and many authors identify local systems on such a topological space with representations of its [[fundamental groupoid]].

\begin{defn}
\label{}\hypertarget{}{}
Given a local system $\tilde \nabla : X \to LConst \mathcal{S}$, the cohomology of $X$ with this \textbf{local system of coefficients} is the intrinsic [[cohomology]] of the [[over-(∞,1)-topos]] $\mathbf{H}/X$:

\begin{displaymath}
H(X,\tilde \nabla)
    :=
    \mathbf{H}_{/X}(X, P_{\tilde \nabla})
    \,,
\end{displaymath}
where $P_{\tilde\nabla}$ is the [[homotopy fiber]] of $\tilde \nabla$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Unwinding the definitions and using the universality of the [[(∞,1)-pullback]], one sees that a [[cocycle]] $c \in \mathbf{H}(X,\tilde \nabla)$ is a [[diagram]]

\begin{displaymath}
\itexarray{
    X &&\stackrel{c}{\to}&& *
    \\
    & \searrow &\swArrow& \swarrow
    \\
    && LConst \mathcal{S}
  }
\end{displaymath}
in $\mathbf{H}$. This is precisely a [[section]] of the [[locally constant ∞-stack]] $\tilde \nabla$.

\end{remark}
\hypertarget{sheaftheoretic_case}{}\subsubsection*{{Sheaf-theoretic case}}\label{sheaftheoretic_case}

Local systems can also be considered in abelian contexts. One finds the following version of a local system

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{linear local system} is a [[locally constant sheaf]] on a [[topological space]] $X$ (or manifold, analytic manifold, or algebraic variety) whose stalk is a finite-dimensional [[vector space]].

\end{defn}
Regarded as a sheaf $F$ with values in [[abelian group]]s, such a linear local system serves as the coefficient for [[abelian sheaf cohomology]]. As discussed there, this is in degree $n$ nothing but the intrinsic cohomology of the $\infty$-topos with coefficients in the [[Eilenberg-MacLane object]] $\mathbf{B}^n F$.

\begin{lemma}
\label{}\hypertarget{}{}
On a connected topological space this is the same as a sheaf of sections of a finite-dimensional [[vector bundle]] equipped with flat [[connection on a bundle]]; and it also corresponds to the [[representations]] of the [[fundamental group]] $\pi_1(X,x_0)$ in the typical stalk. On an analytic manifold or a variety, there is an equivalence between the category of non-singular coherent $D_X$-[[D-module|modules]] and local systems on $X$.

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

\begin{itemize}%
\item [[flat vector bundle]]


\item [[simplicial local system]]: within Sullivan's (1977) theory of \emph{Infinitesimal computations in topology}, he refers to `local systems' several times. This seems to be simplicial in nature. [[simplicial local system|This]] entry explores some of the uses of that notion based on Halperin's lecture notes on minimal models

\begin{itemize}%
\item D. Sullivan, \emph{Infinitesimal computations in topology} (\href{http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1977__47_/PMIHES_1977__47__269_0/PMIHES_1977__47__269_0.pdf}{pdf})

\end{itemize}

\item [[twisted cohomology]], [[local coefficient bundle]], [[twisted infinity-bundle]]



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

An early version of the definition of local system appears in

\begin{itemize}%
\item [[Norman Steenrod]]: \emph{Homology with local coefficients}, Annals 44 (1943) pp. 610 - 627,

\end{itemize}
This is before the formal notion of [[sheaf]] was published by [[Jean Leray]]. (Wikipedia's entry on \href{http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29#History}{Sheaf theory} is interesting for its historical perspective on this.)

A definition appears as an exercise in

\begin{itemize}%
\item [[Edwin Spanier]], 1966, Algebraic Topology , McGraw Hill. (republished by Springer, 1982).

\end{itemize}
on page 58 :

\begin{quote}%
\emph{A local system on a space $X$ is a covariant functor from the fundamental groupoid of $X$ to some category.}


\end{quote}
A blog exposition of some aspects of linear local system is developed here:

\begin{itemize}%
\item [[David Speyer]], \emph{Three ways of looking at a local system}

\begin{itemize}%
\item \href{http://sbseminar.wordpress.com/2009/04/20/three-ways-of-looking-at-a-local-system-introduction-and-connection-to-cohomology-theories/}{Introduction and connection to cohomology theories}


\item \href{http://sbseminar.wordpress.com/2009/04/21/local-systems-the-path-groupoid-approach/}{the path groupoid approach}


\item \href{http://sbseminar.wordpress.com/2009/04/30/three-ways-of-looking-at-a-local-system-the-infinitesimal-perspective/}{the infinitesimal perspective}


\item \href{http://sbseminar.wordpress.com/2009/05/06/the-infinitesimal-site/}{the infinitesimal site}



\end{itemize}


\end{itemize}
A clear-sighted description of locally constant $(n-1)$-stacks / $n$-local systems as sections of constant $n$-stacks is in

\begin{itemize}%
\item [[Pietro Polesello]], [[Ingo Waschkies]], Higher monodromy, Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150;\href{http://arxiv.org/abs/math/0407507}{arXiv:0407507}

\end{itemize}
for [[locally constant stack]]s on [[topological space]]s. The above formulation is pretty much the evident generalization of this to general [[(∞,1)-topos]]es.

Discussion of [[Galois representations]] as encoding local systems in [[arithmetic geometry]] includes

\begin{itemize}%
\item Tom Lovering, \emph{\'E{}tale cohomology and Galois Representations}, 2012 (\href{http://tlovering.files.wordpress.com/2012/06/essay-body1.pdf}{pdf})

\end{itemize}
See also at \emph{[[function field analogy]]}.

[[!redirects local systems]]

[[!redirects local system of coefficients]] [[!redirects local systems of coefficients]]

[[!redirects cohomology with a local system of coefficients]]



\end{document}