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\begin{document}

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\section*{locally constant sheaf}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{pattern}{Pattern}\dotfill \pageref*{pattern} \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 locally constant [[sheaf]] $A$ over a [[topological space]] is a sheaf of [[section]]s of a [[covering space]] of $X$: there is a [[cover]] of $X$ by open subsets $\{U_i\}$ such that the [[inverse image|restrictions]] $A|_{U_i}$ are [[constant sheaf|constant sheaves]].

More elegantly said: locally constant sheaves are the sections of [[constant stack]]s:

Let $C = Core(FinSet)\in$ [[Grpd]] be the [[core]] of the [[category]] [[FinSet]] of finite set, let $const_C : Op(X)^{op} \to Grpd$ the [[constant presheaf|presheaf constant]] on $C$, i.e. the [[functor]] on the [[opposite category]] of the [[category of open subsets]] of $X$ that sends everything to (the identity on) $C$. Then the [[constant stack]] on $C$ is the [[stackification]] $\bar const_C : Op(X)^{op} \to Grpd$.

Write then $X$ for the space $X$ regarded as a sheaf or trivial [[covering space]] over itself, i.e. the [[terminal object]] $X$ in sheaves and hence in stacks over $X$. Then by definition of stackification morphisms

\begin{displaymath}
X \to \bar const_C
\end{displaymath}
are represented by

\begin{itemize}%
\item an open cover $\{U_i\}$ of $X$;


\item over each $U_i$ a choice $F_i \in C$ of object in $C$, hence a finite set in $C$;


\item over each double overlap $U_{i j} = U_i \cap U_j$ an morphism $g_{i j} : F_i|_{I_{i j}} \stackrel{\simeq}{\to} F_j|_{U_{i j}}$, hence a bijection of finite sets;


\item such that on triple overlaps we have $g_{i k}|_{U_{i j k}}= g_{j k}|_{U_{i j k}}\circ g_{i j}|_{U_{i j k}}$.



\end{itemize}
Such data clearly is the local data for a [[covering space]] over $X$ with typical fiber any of the $F_i$.

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

Let $(\Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \mathcal{S}$ be the [[global section]] [[geometric morphism]] of a [[topos]] $\mathcal{E}$ over base $\mathcal{S}$.

Without further assumption on $\mathcal{E}$ we have the following definition.

\begin{defn}
\label{}\hypertarget{}{}
For $U \to *$ an [[epimorphism]] in $\mathcal{E}$, an [[object]] $E \in \mathcal{E}$ is called \textbf{locally constant and split by $U$} if in the [[over category]] $\mathcal{E}/U$ we have an [[isomorphism]]

\begin{displaymath}
E \times U \simeq (\Delta S) \times U
  \,,
\end{displaymath}
for some $S \in$ [[Set]].

An object which is locally constant and $U$-split for \emph{some} $U$ is called \textbf{locally constant}.

A locally constant object $E$ which is in addition an $\Delta Aut(X)$-[[principal bundle]] is called a \textbf{[[Galois object]]} .

\end{defn}
If $\mathcal{E}$ is a [[locally connected topos]] there is another characterization of locally constant sheaves.

\begin{defn}
\label{}\hypertarget{}{}
For $C$ and $C$ [[cartesian closed categories]], a [[functor]] $F : C \to D$ that preserves [[product]]s is called a \textbf{[[cartesian closed functor]]} if the canonical [[natural transformation]]

\begin{displaymath}
F(B^A) \to (F(B))^{F(A)}
\end{displaymath}
(which is the [[adjunct]] of $F(A) \times F(B^A) \simeq F(A \times B^A) \to F(B)$) is an [[isomorphism]].

\end{defn}
From the discussion at [[locally connected topos]] we have that

\begin{prop}
\label{}\hypertarget{}{}
The [[constant sheaf]]-functor $\Delta : \mathcal{S} \to \mathcal{E}$ is a [[cartesian closed functor]] precisely if $\mathcal{E}$ is a [[locally connected topos]].

\end{prop}
In this case the above definition is equivalent to the following one.

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathcal{E} = Sh(C)$ be a [[locally connected topos]]. Let $p : core(Set^\kappa_*) \to core(Set^\kappa)$ be the [[core]] of the [[generalized universal bundle]] for sets of [[cardinality]] less than some $\kappa$.

A \textbf{locally constant $\kappa$-bounded object} in $\mathcal{E}$ is the [[pullback]] of $\Delta(p)$ along a morphism $* \to core(Set^\kappa)$ in the [[2-topos|(2,1)-topos]] $Sh_{(2,1)}(C)$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This says that locally constant sheaves are the sections of the [[constant stack]] on the [[groupoid]] $core(Set^\kappa)$.

Notice that

\begin{displaymath}
core(Set \kappa) \simeq \coprod_i \mathbf{B}Aut(F_i) 
  \,,
\end{displaymath}
where the [[coproduct]] is over all [[cardinal]]s smaller than $\kappa$ and where $\mathbf{B}Aut(F_i)$ denotes the [[delooping]] [[groupoid]] of the [[automorphism group]] of the set $F_i$: the [[symmetric group]] on $F_i$.

This means that on each connected component of $\mathcal{E}$ a locally constant sheaf is the $\Delta \rho$-[[associated bundle]] to an $\Delta Aut(F)$-[[principal bundle]] induced by the canonical [[permutation representation]] $\rho : \mathbf{B} Aut(F) \to Set$ of the [[automorphism group]] $Aut(F)$ on $F$.

Specifically for $g : * \to \Delta \mathbf{B} Aut(F) \simeq \mathbf{B} \Delta Aut(F) \hookrightarrow \Delta core(set)$ the classifying morphism of a locally constant sheaf and for $U \to *$ an [[epimorphism]] on which it trivializes, we have a pasting diagram of [[pullback]]s

\begin{displaymath}
\itexarray{
    U \times \Delta F 
    &\to&
    P \times_{\Delta Aut(F) (\Delta (F // Aut(F)))}
    &\to&
    \Delta(F // Aut(F))
    &\to&
    \Delta Set^\kappa
    \\
    \downarrow &&
    \downarrow &&
    \downarrow &&
    \downarrow
    \\
    U &\to& * &\stackrel{g}{\to}& \mathbf{B} \Delta Aut(F)
    &\hookrightarrow&
    \Delta core(Set^\kappa)
  }
  \,,
\end{displaymath}
where $F//Aut(F)$ is the [[action groupoid]], the [[2-colimit]] of $\rho \mathbf{B}Aut(F) \to Grpd$.

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

\begin{itemize}%
\item Locally constant sheaves are sheaves of sections of [[covering space]]s.


\item When used as coefficient objects in [[cohomology]] they are also called [[local system]]s.


\item The [[action]] of the [[fundamental groupoid]] $\Pi_1(X)$ on the fibers of a local system give rise to the notion of [[monodromy]].


\item This may be used to define [[homotopy group (of an infinity-stack)|homotopy groups of]] general objects in a [[topos]], and the [[fundamental group of a topos]].


\item This is the content of [[Galois theory]].



\end{itemize}
\hypertarget{pattern}{}\subsection*{{Pattern}}\label{pattern}

In sufficiently highly locally connected cases, we have:

\begin{itemize}%
\item A [[locally constant function]] is a section of a [[constant sheaf]];


\item a \textbf{locally constant sheaf} is a section of a [[constant stack]];


\item a [[locally constant stack]] is a section of (\ldots{} and so on\ldots{})


\item a [[locally constant ∞-stack]] is a section of a [[constant ∞-stack]].



\end{itemize}
A locally constant sheaf / $\infty$-stack is also called a [[local system]].

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

\begin{itemize}%
\item [[constructible sheaf]]

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

The definition of \emph{locally constant sheaf} originates in the notion of \emph{covering projection}

\begin{itemize}%
\item [[SGA]] 4, Expos\'e{} IX, 2.0 .

\end{itemize}
Lecture notes are in

\begin{itemize}%
\item [[James Milne]], section 6 of \emph{[[Lectures on Étale Cohomology]]}

\end{itemize}
The topos-theoretic definition is reproduced in the context of a discussion of the notion of [[Galois topos]] as definition 5.1.1 in

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{On the representation theory of Galois and Atomic Topoi} (\href{http://arxiv.org/abs/math/0208222}{arXiv:0208222})

\end{itemize}
and definition 2.2 in

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{The fundamental progroupoid of a general topos} (\href{http://arxiv.org/abs/0706.1771}{arXiv:0706.1771})

\end{itemize}
or as definition 1 in

\begin{itemize}%
\item [[Michael Barr]], [[Radu Diaconescu]], \emph{On locally simply connected toposes and their fundamental groups} (\href{http://www.numdam.org/item?id=CTGDC_1981__22_3_301_0}{NUMDAM})

\end{itemize}
Discussion of the notions of locally constant sheaves is at

\begin{itemize}%
\item [[Michael Shulman]], \emph{Locally constant sheaves} ()

\end{itemize}
[[!redirects locally constant sheaves]]

[[!redirects locally constant object]] [[!redirects locally constant objects]]



\end{document}