Contents

# Contents

## Definition

Let Top be a category of topological spaces and $B$ an object in $Top$ (the ‘base’ space). The slice category $Top/B$ is called the category of (topological) spaces over $B$ (or sometimes simply bundles).

An étale space (or étale map, sometimes étalé space) 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 stalk of $p$ over $x$.

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

## Definition for arbitrary toposes

As long as $C$ has a small set of topological generators (i.e., as long as $Sh(C,J)$ isn’t too large to be a topos), there always exists a certain version of an étale space: If $F$ is a sheaf on $(C,J)$, the slice topos $Sh(C,J)/F$ has a canonical étale projection $\pi_F:Sh(C,J)/F \to Sh(C,J)$. This map is a local homeomorphism of topoi. This topos with this local homeomorphism is the étale space of $F$. Indeed, we may make this construction for each object $c \in C$, call it $U(c) \coloneqq Sh(C,J)/y(c)$, where $y(c)$ is the (possibly sheafified) Yoneda embedded object. Then, “sections of $\pi_F$ over $U(c) \to Sh(C,J)$” are in bijection with elements of $F(c)$. If the Grothendieck site $(C,J)$ happened to be the canonical site of a topological space, then each slice $Sh(C,J)/F$ is equivalent to sheaves on the étale space of that sheaf, and the projection corresponds to the usual one. In particular, $U(c) \to Sh(C,J)$ corresponds to the inclusion of an open subset. So, this is reduces to the usual construction for spaces. Another example is, if $(C,J)$ were the small étale site of some scheme $S$, then each $Sh(C,J)/F$ is the small étale site of some algebraic space (with no separation conditions) étale over $S$, with $\pi_F$ corresponding to the étale map from this algebraic space to $S$.

## Properties

### Relation to sheaves

Let $p:E\to B$ be in $Top/B$. The (local) 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 étale map $p$, 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.

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 étale 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 étale space $L(P) = (E(P)\to B)$ the space of the sheaf $P$, having in mind the adjoint equivalence above. (This is also called the sheaf space or the display space (alias étale space, or in French: espace étalé); 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 $Shv_B$.

### Relation to covering spaces

Every covering space (even in the more general sense not requiring any connectedness axiom) is étale but not vice versa:

• 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 étale 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.

• even if the the stalks of the étale space are finite, it need not be locally trivial. For instance the disjoint union $\coprod_i U_i$ of a collection of open subsets of a topological space $X$ with the obvious projection $(\coprod_i U_i) \to X$ is étale, 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.

## Grammar note

In French, the verb ‘étaler’ means, roughly, to spread out; ‘-er’ becomes ‘-é’ to make a past participle. So an ‘espace étalé’ is a space that has been spread out over $B$. On the other hand, ‘étale’ is a (relatively obscure, distantly related) nautical adjective that can be translated as ‘calm’ or ‘slack’.

To quote from the Wiktionnaire français:

‘étale’ qualifie la mer qui ne monte ni ne descend à la fin du flot ou du jusant (‘flot’ = ‘flow’ and ‘jusant’ = ‘ebb’).

There is an interesting stanza from a song of Léo Ferré:

Et que les globules figurent

Une mathématique bleue,

Sur cette mer jamais étale

D’où me remonte peu à peu

Cette mémoire des étoiles.

— (Léo Ferré, La mémoire et la mer)

He also mentions geometry and ‘théorème’ elsewhere in the song.

For generalizations to étale spaces of stacks in groupoids see

• David Carchedi, An étalé space construction for stacks, Algebr. Geom. Topol. 13 (2013), no. 2, 831–903.