nLab
étale space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

étale morphisms

Contents

Definition

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

An étale space (or étale map, sometimes étalé space) over BB is an object p:EBp:E\to B in Top/BTop/B such that pp is a local homeomorphism: that is, for every eEe\in E, there is an open set UeU \ni e such that the image p(U)p(U) is open in BB and the restriction of pp to UU is a homeomorphism p| U:Up(U)p|_U: U \to p(U).

The set E x=p 1(x)E_x = p^{-1}(x) where xBx\in B is called the stalk of pp over xx.

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

Definition for arbitrary toposes

The following is adapted from a MathOverflow answer by David Carchedi:

As long as CC has a small set of topological generators (i.e., as long as Sh(C,J)Sh(C,J) isn’t too large to be a topos), there always exists a certain version of an étale space: If FF is a sheaf on (C,J)(C,J), the slice topos Sh(C,J)/FSh(C,J)/F has a canonical étale projection π F:Sh(C,J)/FSh(C,J)\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 FF. Indeed, we may make this construction for each object cCc \in C, call it U(c)Sh(C,J)/y(c)U(c) \coloneqq Sh(C,J)/y(c), where y(c)y(c) is the (possibly sheafified) Yoneda embedded object. Then, “sections of π F\pi_F over U(c)Sh(C,J)U(c) \to Sh(C,J)” are in bijection with elements of F(c)F(c). If the Grothendieck site (C,J)(C,J) happened to be the canonical site of a topological space, then each slice Sh(C,J)/FSh(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)Sh(C,J)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)(C,J) were the small étale site of some scheme SS, then each Sh(C,J)/FSh(C,J)/F is the small étale site of some algebraic space (with no separation conditions) étale over SS, with π F\pi_F corresponding to the étale map from this algebraic space to SS.

Properties

Relation to sheaves

Let p:EBp:E\to B be in Top/BTop/B. The (local) sections of pp over an open set UBU\subseteq B are the continuous maps s:UEs:U\to E such that ps=id Up\circ s = \mathrm{id}_U. It is an elementary but central fact that for an étale map pp, the images of local sections form a base for the topology of the total space EE. The topology of EE is then typically non-Hausdorff.

The set of sections of pp over UU is denoted by Γ Up=(Γp)(U)=Γ UE=(ΓE)(U)\Gamma_U p = (\Gamma p)(U) = \Gamma_U E = (\Gamma E)(U) and may be shown to extend to a functor Γ:Top/BPShv B\Gamma : Top/B\to PShv_B where PShv BPShv_B is the category of presheaves over BB. The functor Γ\Gamma has a left adjoint L:PShv BTop/BL : PShv_B\to Top/B, whose essential image is the full subcategory Et/BEt/B of étale spaces over BB. The essential image of the functor Γ\Gamma is the category of sheaves Shv BShv_B over BB, and this adjunction restricts to an equivalence of categories between Et/BEt/B and Shv BShv_B (that is, it is an idempotent adjunction).

If P:Open(B) opSetP:Open(B)^{op}\to Set is a sheaf, then one sometimes calls the total space E(P)E(P) of the étale space L(P)=(E(P)B)L(P) = (E(P)\to B) the space of the sheaf PP, 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 BShv BPShv Ba:PShv_B\to Shv_B\hookrightarrow PShv_B decomposes as a=ΓLa = \Gamma\circ L, and aa may be considered as an endofunctor part of an idempotent monad in PShv BPShv_B whose corresponding reflective subcategory is Shv BShv_B.

(e.g. MacLane-Moerdijk, section II.5, II.6)

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 BB needs to be, by the definition, a disjoint union of homeomorphic open sets in EE; however the ‘size’ of the open neighborhoods over various ee 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 iU i\coprod_i U_i of a collection of open subsets of a topological space XX with the obvious projection ( iU i)X(\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 iU_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 BB. 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 française:

‘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.

References

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.

Last revised on May 26, 2020 at 23:54:00. See the history of this page for a list of all contributions to it.