\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*{simplicial local system}

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

\hypertarget{differential_graded_objects}{}\paragraph*{{Differential Graded Objects}}\label{differential_graded_objects}

[[!include differential graded objects - contents]]

\hypertarget{simplicial_local_system}{}\section*{{Simplicial local system}}\label{simplicial_local_system}

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{simplicial local system} on a [[simplicial set]] $X$ with coefficients in a category $C$ (like [[abelian groups]], for example) is a [[functor]]

\begin{displaymath}
\pi_{\le1}(X)\to C,
\end{displaymath}
where $\pi_{\le1}(X)$ is (one of the models for) the [[simplicial fundamental groupoid]] of $X$.

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

\begin{itemize}%
\item NB. There is an entry at [[local system|local systems]] together with a blog link to 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}

\end{itemize}


\end{itemize}
Here we will concentrate on the combinatorial and simplicial version of local systems.

\hypertarget{local_systems_in_a_simplicial_context}{}\subsection*{{Local Systems in a simplicial context}}\label{local_systems_in_a_simplicial_context}

By the category of \textbf{$n$-graded spaces}, we mean the category whose objects are the $n$-graded vector spaces

\begin{displaymath}
V = \sum_{p_1,\ldots,p_n\geq0}V^{p_1,\ldots,p_n}
\end{displaymath}
and whose morphisms are the linear maps, homogeneous of multidegree zero.

The category of $n$-graded differential vector spaces has for objects pairs $(V,d)$, where $V$ is an $n$-graded vector space, $d$ is a linear map of \emph{total} degree 1, and $d^2 = 0$. The morphisms are the linear maps, homogeneous of multidegree zero, which commute with $d$.

We will denote by $\mathcal{C}$ one of the following categories:

\begin{itemize}%
\item $n$-graded vector spaces.


\item The category of $n$-graded algebras,


\item The subcategory of commutative $n$-graded algebras,


\item $n$-graded differential vector spaces,


\item The subcategory of $n$-graded differential algebras,


\item The subcategory of commutative $n$-graded differential algebras.



\end{itemize}
[[Urs Schreiber|Urs]]: How does the $n$-grading affect the nature of the following definition? It seems that chain homotopies are not used in the following, just the 1-categorical structure?

In the `differential' examples, the differential will usually be denoted $d$. Almost always we will be restricting ourselves to the case $n = 1$. Extensions of any results or definitions to the general case are usually routine.

Let $K$ be a simplicial set. A \textbf{local system} $F$ on $K$ with values in $\mathcal{C}$ is:

\begin{enumerate}%
\item a family of objects $F_\sigma =\sum_{p\geq 0} F^p_\sigma$ in $\mathcal{C}$ indexed by the simplices $\sigma$ of $K$;


\item a family of morphisms (called the \emph{face} and \emph{degeneracy} operators)



\end{enumerate}
\begin{displaymath}
d_i :F_\sigma \to F_{d_i\sigma} \quad and\quad s_i : F_\sigma \to F_{s_i\sigma}
\end{displaymath}
satisfying the simplicial identities.

\hypertarget{remarks}{}\subsubsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item Here we will often just refer to `local system' rather than the fuller `simplicial local system', if no confusion will be likely to result.


\item There is an obvious way of assigning a small category to a simplicial set in which the simplices are the objects and the face and degeneracy maps generate the morphisms:



\end{itemize}
regarding the [[simplicial set]] as a [[functor]]

\begin{displaymath}
K : \Delta^{op} \to Set
\end{displaymath}
on the [[simplex category]], its category of cells is the [[comma category]]

\begin{displaymath}
(Y, const_K) =
  \left\{
    \itexarray{
       Y(\Delta^n) &&\stackrel{}{\to}&& Y(\Delta^{n'})
       \\
       & {}_c\searrow && \swarrow_{c'}
       \\
       && K
    }
  \right\}
\end{displaymath}
where $Y : \Delta \to [\Delta^{op}, Set]$ is the [[Yoneda embedding]] for which $Y(\Delta^n)$ is the standard simplicial $n$-[[simplex]], so that $c : Y(\Delta^n) \to K$ is an $n$-simplex $c \in K_n$ of the simplicial set $n$.

A simplicial local system is then just a functor

\begin{displaymath}
F : (Y,const_K) \to \mathcal{C}
\end{displaymath}
from that category to $\mathcal{C}$.

[[Urs Schreiber|Urs]]: Here it says ``a local system''. I suppose ``simplicial local system'' is meant? We should have a discussion about how this notion of simplicial local system relates to the functors from fundamental groupoids discussed at [[local system]].

[[Tim Porter|Tim]]: That has been amended! Halperin just calls them `local systems', so in the notes that were the basis for this so did I. I copied and pasted from them, so this slip may occur elsewhere as well.

\hypertarget{back_to_discussion}{}\subsubsection*{{Back to discussion}}\label{back_to_discussion}

Let $\varphi : L \to K$ be a simplicial map and $F$ a local system over $K$. The \emph{pullback} of $F$ to $L$ (or along $\varphi$) is the local system $\varphi^*F$ over $L$ given by

\begin{displaymath}
(\varphi^*F)_\sigma = F_{\varphi\sigma} ; \quad d_i = d_i ; \quad s_i = s_i.
\end{displaymath}
If $\varphi$ is an inclusion of a simplicial subset then we may say that $\varphi^*F$ is the \emph{restriction} of $F$ to $L$.

Now let $F$ be a local system on $K$ with values in $\mathcal{C}$. Define a graded space $F(K)$ as follows : an element $\Phi$ of $F^p(K)$ is a function which assigns to each simplex $\sigma$ of $K$ an element $\Phi_\sigma \in F^p_\sigma$ such that for all $\sigma$

\begin{displaymath}
\Phi_{d_i\sigma} = d_i(\Phi_\sigma) \quad  and  \quad \Phi_{s_i\sigma}  = s_i(\Phi_\sigma).
\end{displaymath}
[[Urs Schreiber|Urs]]: Do I understand correctly that when the simplicial local system is expressed as a functor, then $F(K)$ is the space of natural transformations from the simplicial local system constant on the [[generator]] (if any) of $\mathcal{C}$ (for instance the tensor unit if $\mathcal{C}$ is graded vector spaces).

For ordinary [[local system]]s this gives the flat sections.

[[Tim Porter|Tim]]: I'm not sure.

The linear structure is the obvious one, defined `componentwise' and if $\mathcal{C}$ is one of the algebra (resp. differential) variants of the generic receiving category then the multiplication (resp. the differential) is defined componentwise as well. In this way $F(K)$ becomes an object of $\mathcal{C}$, called the \textbf{object of global sections} of $F$.

[[Tim Porter|Tim]]: This construction also has (I think) a neat categorical description, that will be worth investigating. It would seem to be the analogue of the Grothendieck construction / homotopy colimit (at least partially) in this context. (enlightenment sought!!!)

If $\varphi : L \to K$ is a simplicial map, it determines a morphism $F(\varphi) : (\varphi^*F)(L)\to F(K)$ given by

\begin{displaymath}
(F(\varphi)\Phi)_\sigma = \Phi_{\varphi\sigma}.
\end{displaymath}
If $\varphi$ is an inclusion of $L$ into $K$, then we denote $(\varphi^*F)(L)$ simply by $F(L)$ and call the morphism $F(K)\to F(L)$ \emph{restriction}.

Now suppose $F$ is a local system over $K$. Assume $M_n \subset K_n$ are subsets ($n \geq 0$) such that $d_i : M_n \to M_{n-1}$ This family $\{M_n\}$ generates a subsimplicial set $L\subset K$ and if $s_i\sigma \in M_{n+1}$ then $\sigma = d_i s_i\sigma \in M_n$.

[[Urs Schreiber|Urs]]: So what are simplicial local systems used for? Is there a good motivating example? Relating it to the other definition of [[local system]], maybe?

[[Tim Porter|Tim]]: Aha! All will be revealed in the next entry `Differential forms on a simplicial set' \ldots{} when I get to putting it in! There is some more to go here as well, describing special properties, but it was getting late last night.

\begin{ulemma}
Suppose $\Phi_\sigma \in F^p_\sigma$ ( $\sigma \in M_n$), $n \geq 0$, satisfy $\Phi_{d_i\sigma} = d_i\Phi_\sigma$ and $\Phi_{s_i\sigma} = s_i\Phi_\sigma$ (this is with $s_i\sigma\in M_n$, and $n\geq 0$). Then there is a unique element $\Phi\in F^p(L)$ extending $\Phi_\sigma$.

\end{ulemma}
The proof is by induction and can be found in Halperin's notes if required.

For any simplicial set $K$, any $n$-simplex $\sigma \in K_n$ determines a unique simplicial map, which we will also write as $\sigma$ from $\Delta[n]$ to $K$ that sends the unique non-degenerate $n$-simplex of the standard $n$-simplex $\Delta[n]$ to the element $\sigma$. In particular, if $F$ is a local system over $K$, then we can form $\sigma^*F$ over $\Delta[n]$. We will say that $F$ is \emph{extendable} if for each $\sigma$ the restriction

\begin{displaymath}
\sigma^*(F)(\Delta[n]) \to \sigma^*(F)(\partial\Delta[n])
\end{displaymath}
is surjective, where $\partial\Delta[n]$ is the boundary of the $n$-simplex.

\begin{uprop}
Suppose $\varphi :L \to K$ is a simplicial map and $F$ is an extendable system over $K$, then $\varphi^*F$ is an extendable local system over $L$.

\end{uprop}
The proof is easy.

\begin{uprop}
Suppose that $L\subset K$ is a subsimplicial set and $F$ is an extendable local system over $K$. Then the restriction morphism $F(K)\to F(L)$ is surjective.

\end{uprop}
The proof is again by induction up the skeleta of $K$ and $L$, for details see Halperin, p.XII 10.

If $F$ is an extendable local system over $K$ and $L\subset K$, we denote the kernel of $F(K)\to F(L)$ by $F(K,L)$ and call it the \emph{space of relative global sections}. (A description of $F(K,L)$ is given in detail in Halperin, p.XII-12.)

It may be useful to have some more of the terminology of local systems available. A local system $F$ over $K$ is \textbf{constant} if for some $F_0 \in \mathcal{C}$, each $F_\sigma = F_0$ and each $d_i$ and $s_j$ is the identity map on $F_0$. We say $F$ is \textbf{constant by dimension} if for some sequence $F_n\in \mathcal{C}$ ($n \geq 0$), $F_\sigma = F_n$, for $\sigma \in K_n$ and $d_i$, $s_j$ depend only on $\dim \sigma$.

A local system $F$ over $K$ is a \textbf{local system of coefficients} if for each $\sigma$ and each $i$,

\begin{displaymath}
d_i : F_\sigma \to F_{d_i\sigma} \quad  and s_i : F_\sigma \to F_{s_i \sigma}
\end{displaymath}
are isomorphisms. Finally $F$ is a \textbf{local system of differential coefficients} if $\mathcal{C}$ is one of the categories with differentials above, and for each $\sigma$, and $i$

\begin{displaymath}
d_i^* : H(F_\sigma) \to H(F_{d_i\sigma}) \quad  and  s_i^* : H(F_\sigma) \to H(F_{s_i\sigma})
\end{displaymath}
are isomorphisms, in other words if the corresponding cohomology is a local system of coefficients.

\begin{utheorem}
Let $F$ and $G$ be extendable local systems of differential coefficients over $K$. Assume we are given morphisms

\begin{displaymath}
\varphi_\sigma : F_\sigma \to G_\sigma, \quad \sigma \in K,
\end{displaymath}
compatible with the face and degeneracy operators. Then a morphism $\varphi : F(K)\to G(K)$ is given by $(\varphi\Phi)\sigma = \varphi_\sigma (\Phi_\sigma)$, and

\begin{displaymath}
\varphi^* : H(F(K))\to H(G(K))
\end{displaymath}
is an isomorphism.

\end{utheorem}
\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[local system]]


\item [[twisted cohomology]]



\end{itemize}
\hypertarget{references_2}{}\subsection*{{References}}\label{references_2}

\begin{itemize}%
\item S. Halperin, \emph{Lectures on minimal models}, M\'e{}moires de la S. M. F. 2e s\'e{}rie, tome 9-10 (1983), p. 1-261 (\href{http://eudml.org/doc/94833}{eduml}, \href{http://archive.numdam.org/article/MSMF_1983_2_9-10__1_0.pdf}{pdf})

\end{itemize}


\end{document}