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\section*{étalé space}

[[!redirects etale space]]

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

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{tale_morphisms}{}\paragraph*{{\'E{}tale morphisms}}\label{tale_morphisms}

[[!include etale morphisms - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToSheaves}{Relation to sheaves}\dotfill \pageref*{RelationToSheaves} \linebreak
\noindent\hyperlink{relation_to_covering_spaces}{Relation to covering spaces}\dotfill \pageref*{relation_to_covering_spaces} \linebreak
\noindent\hyperlink{grammar}{Grammar note}\dotfill \pageref*{grammar} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Let [[Top]] be a category of [[topological space]]s and $B$ an object in $Top$ (the `base' space). The [[over category|slice category]] $Top/B$ is called the category of (topological) spaces over $B$ (or sometimes simply bundles).

An \textbf{\'e{}tal\'e{} space} (or \emph{[[étale map]]}) over $B$ is an object $p:E\to B$ in $Top/B$ such that $p$ is a [[local homeomorphism]]: that is, for every $e\in E$, there is an open set $U \ni e$ such that the [[image]] $p(U)$ is open in $B$ and the restriction of $p$ to $U$ is a [[homeomorphism]] $p|_U: U \to p(U)$.

The set $E_x = p^{-1}(x)$ where $x\in B$ is called the \textbf{[[stalk]]} of $p$ over $x$.

The underlying set of the \emph{total space} $E$ is the union of its stalks (notice that we do not say fiber!). $p$ is sometimes refered to as the \emph{projection}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToSheaves}{}\subsubsection*{{Relation to sheaves}}\label{RelationToSheaves}

Let $p:E\to B$ be in $Top/B$. The (local) \textbf{[[sections]]} of $p$ over an open set $U\subseteq B$ are the [[continuous maps]] $s:U\to E$ such that $p\circ s = \mathrm{id}_U$. It is an elementary but central fact that for an \'e{}tale map $p$, \emph{the images of local sections form a base for the topology} of the total space $E$. The topology of $E$ is then typically non-[[Hausdorff space|Hausdorff]].

The set of sections of $p$ over $U$ is denoted by $\Gamma_U p = (\Gamma p)(U) = \Gamma_U E = (\Gamma E)(U)$ and may be shown to extend to a [[functor]] $\Gamma : Top/B\to PShv_B$ where $PShv_B$ is the [[category of presheaves]] over $B$. The functor $\Gamma$ has a [[left adjoint]] $L : PShv_B\to Top/B$, whose [[essential image]] is the [[full subcategory]] $Et/B$ of \'e{}tal\'e{} spaces over $B$. The [[essential image]] of the functor $\Gamma$ is the [[category of sheaves]] $Shv_B$ over $B$, and this [[adjunction]] restricts to an [[equivalence of categories]] between $Et/B$ and $Shv_B$ (that is, it is an [[idempotent adjunction]]).

If $P:Open(B)^{op}\to Set$ is a [[sheaf]], then one sometimes calls the total space $E(P)$ of the \'e{}tal\'e{} space $L(P) = (E(P)\to B)$ the \textbf{space of the sheaf} $P$, having in mind the adjoint equivalence above. (This is also called the \textbf{sheaf space} or the \textbf{display space}; compare also a [[display morphism]] of [[contexts]].) The associated sheaf functor $a:PShv_B\to Shv_B\hookrightarrow PShv_B$ decomposes as $a = \Gamma\circ L$, and $a$ may be considered as an endofunctor part of an [[idempotent monad]] in $PShv_B$ whose corresponding [[reflective subcategory]] is $Shv_B$.

(e.g. \hyperlink{MacLaneMoerdijk}{MacLane-Moerdijk, section II.5, II.6})

\hypertarget{relation_to_covering_spaces}{}\subsubsection*{{Relation to covering spaces}}\label{relation_to_covering_spaces}

Every [[covering space]] (even in the more general sense not requiring any connectedness axiom) is \'e{}tal\'e{} but not vice versa:

\begin{itemize}%
\item for a covering space the [[inverse image]] of some [[open subset]] in the base $B$ needs to be, by the definition, a [[disjoint union]] of homeomorphic open sets in $E$; however the `size' of the [[open neighborhoods]] over various $e$ in the same [[stalk]] required in the definition of \'e{}tal\'e{} space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.


\item even if the the stalks of the \'e{}tal\'e{} space are finite, it need not be locally trivial. For instance the [[disjoint union]] $\coprod_i U_i$ of a collecton of [[open subsets]] of a topological space $X$ with the obvious projection $(\coprod_i U_i) \to X$ is \'e{}tale, but does not have a typical fiber: the fiber over a given point has [[cardinality]] the number of open sets $U_i$ that contain this particular point.



\end{itemize}
\hypertarget{grammar}{}\subsection*{{Grammar note}}\label{grammar}

In French, the verb `\'e{}taler' means, roughly, to spread out; `-er' becomes `-\'e{}' to make a past participle. So an `espace \'e{}tal\'e{}' is a space that has been spread out over $B$. On the other hand, `\'e{}tale' is a (relatively obscure, distantly related) nautical adjective that can be translated as `calm' or `slack'.

To quote from the \href{http://fr.wiktionary.org/wiki/%C3%A9tale}{Wiktionnaire française}:

\begin{quote}%
`\'e{}tale' \emph{qualifie la mer qui ne monte ni ne descend \`a{} la fin du flot ou du jusant} (`flot' = `flow' and `jusant' = `ebb').


\end{quote}
There is an interesting stanza from a song of L\'e{}o Ferr\'e{}:

\begin{quote}%
Et que les globules figurent

Une math\'e{}matique bleue,

Sur cette mer jamais \'e{}tale

D'o\`u{} me remonte peu \`a{} peu

Cette m\'e{}moire des \'e{}toiles.

--- (L\'e{}o Ferr\'e{}, La m\'e{}moire et la mer)


\end{quote}
He also mentions geometry and `th\'e{}or\`e{}me' elsewhere in the song.

\begin{itemize}%
\item \href{http://secure.wikimedia.org/wikipedia/en/wiki/Wikipedia_talk:WikiProject_Mathematics/Archive11#french_spelling}{Further reference}

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

\begin{itemize}%
\item [[étale groupoid]], [[étale infinity-groupoid]]

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

\begin{itemize}%
\item [[Saunders Mac Lane]], [[Ieke Moerdijk]], sections II.5 and II.6 of \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
[[!redirects etale space]] [[!redirects etale spaces]] [[!redirects étalé space]] [[!redirects étalé spaces]] [[!redirects étale space]] [[!redirects étale spaces]] [[!redirects etalé spaces]] [[!redirects etalé space]] [[!redirects espace étalé]] [[!redirects espaces étalés]] [[!redirects sheaf space]] [[!redirects sheaf spaces]] [[!redirects display space]] [[!redirects display spaces]]

[[!redirects espaces etales]]



\end{document}