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Created small site and large site of an object in a site, as a spin off from discussion around petit topos. The latter is so named because large site is taken for sites that happen to be large. The content of this page, however, looks as though it could go somewhere discussing sheaves.
Moved stuff on small sites from petit topos to small site, but it leaves the former looking a bit bare. May have to play around to massage the entries into shape.
I would kind of rather keep large site for what it is (a site which happens to be large), although we definitely need the other one too. What about big site for a slice category with its induced topology? I’ve definitely heard people say things like “the big Zariski site”.
Could we call them big and little sites rather than big and small?
I like big and little site (forgot about people calling it the big site!). I can’t edit at the moment, so if a friendly lab elf (or someone else) was to rename the pages and fix the links, it would be nice :)
We could also say big topos and small topos. But nobody does, for some reason.
Well, “small topos” might be misleading. Little topos?
The antonym pairs in English are:
big - little
large - small
tall - short
I renamed the pages to big site and little site.
I like the idea of using “big” and “little” for the toposes as well (of course with the proper redirects). This way big and little become consistent descriptions in topos theory rather than just words that were chosen without too much thought.
You mean, renaming gros topos and petit topos to big topos and little topos respectively? I wouldn’t be against that. It’s funny that usually mathematical words get translated, but every so often there are words that people almost universally fail to translate.
You mean, renaming gros topos and petit topos to big topos and little topos respectively?
Yes. Few weeks back in Oxford Peter Johnstone was thinking aloud about how that would really be the natural thig to do.
For once, I agree with Johnstone (blast you, “covering”!).
espace etale (sorry for lack of accents) is another one.
The espace étalé is called that because if not, it would be confused with the notion of an étale morphism, to which it is only tangentially related. (All étale morphisms induce local isomorphisms locally in the étale topology, but this is essentialy a tautology when one looks at the definition of the étale topology). The espace étalé is the topological space defined as something like the disjoint union of the fibers topologized so that all sections are continuous.
I remember Zoran had convinced me at one point that they were related, but he was using a nonstandard definition of the espace étalé.
Ah, nerd-sniping :D (Sorry, Harry)
…And the espace étalé is a local isomorphism in the open cover topology. I guess I just dislike saying ’local homeomorphism’/’local diffeomorphism’/etc as it pins down the category I’m working in too much, and ’local isomorphism’, while correct, seems a bit strange to my ears.
Nope, the espace étalé is not that. The espace étalé is the bundle functorially associated with a sheaf on a topological space.
You are talking about étale morphisms, and your definition is nonstandard (now that I know a lot more about étale morphisms). This is because étale morphisms (of schemes, algebraic spaces, stacks, etc.) are those morphisms that are étale-local local isomorphisms. This means that in the étale topology, étale maps can be covered by affine étale maps. This is, as I said, a foregone conclusion, essentially by the definition! They are not local isomorphisms in the Zariski topology, which is the only one of the topologies for schemes that is an honest topology (and for which the notion of the éspace étalé makes sense).
Can you cite your definition of étale for me from a source that isn’t the nLab?
Is there an accent over the ’e’ in ’espace’?
espace étalé - “spread out space”
vs.
morphisme étale - “slack morphism”
The way you can see the words are not the same is that étalé is the past participle of étaler, to spread out, while étale is an honest adjective coming from the noun for slack water or slack tide.
Thanks for the “lesson” :-D, but you misread what I wrote. Try again.
Hmm.. I guess not. My professor wrote it like that, but I guess he got carried away with the accents (at least I remember him writing it like that! It could also be my mistake.)
I went back and fixed all of the errors.
I stand by this:
the espace étalé is a local isomorphism in the open cover topology [Edit: on Top!]
because it is true by construction (=disjoint union of fibres such that sections of the sheaf are continuous). Every etale map of spaces (or local iso in Top for open cover topology, if you insist) is an espace etale for some sheaf, namely the sheaf of sections. I’m not saying the espace etale is defined as a local isomorphism, I’m just saying it is one.
Can you point me to a definition of etale-local for Top? What is an affine in Top?
Can we say J-etale? (I jest :-) because a J-sheaf on Top for some pretopology J (or more generally some concrete site with enough colimits) should give rise to a J-local isomorphism.
Not that I like citing wikipedia, but here it discusses the relation of etale to local diffeomorphisms for smooth complex varieties, namely they are the same. The extension to topological spaces seems to be via the link to manifolds, not the algebra.
I’m telling you that an étale map has no meaning in the topological category. An étale map is explicitly a map of (commutative rings, schemes, algebraic spaces, algebraic stacks) satisfying the unique nilpotent lifting property (nilpotent thickening property, (not sure about the one for algebraic stacks and algebraic spaces, but it’s an appropriate extension of the unique nilpotent lifting property)). Here’s the definition for commutative rings, which is the one I’m most comfortable with:
A map R->S is étale if it is finitely presented (i.e. $S\cong \frac{R[X_1,...,X_n]}{I}$ where $I$ is finitely generated as an $R$-module) and given any commutative diagram
$\begin{matrix} R&\to &T\\ \downarrow&&\downarrow\\ S&\to& T/J\end{matrix}$of commutative rings such that the ideal $J^2=0$, there exists a unique lift $S\to T$ making the whole diagram commute.
By use of the module of relative Kähler differentials, we can see that yes, such maps induce isomorphisms on the tangent spaces, but this is the sense in which they are the same. There is probably a deeper homological connection (probably about how étale cohomology gives us a better cohomology closer to singular cohomology (related to the smooth case by deRham’s theorem)) that I am not qualified to talk about, but conflating the espace étalé and étale morphisms seems like it’s taking the analogy too far if the only justification is the one you just gave. This is why I wanted a real citation from a book or published paper, because it sounds completely outlandish to me.
Note also that the espace étalé of the structure sheaf on a scheme does not give an étale map onto the original scheme (in particular, there is no reason to believe that the éspace étalé is even a scheme at all).
The espace étalé is a construction for sheaves on a topological space and does not make sense in general (sheaves on general sites). The espace étalé is in reality a topological presentation of the discrete fibration that we get from a sheaf on a topological space. This is why we topologize it with the weak topology induced by the sections. The fact that it is an “étale bundle” to use the (rather unfortunate) terminology of Mac Lane and Moerdijk is simply an expression of the fact that it satisfies descent (if we rephrase our definition of descent in terms of fibered categories). The special factor for all of this is that our categories are fibered in sets, so the corresponding fibered category is a set. But since we’re taking fibers over points and fibers over open sets, we can encode this in a very convenient way.
I can explain why it works: The fiber over an open set is the union of the fibers over its points. So not only can we take the discrete fibration over the open sets, but we can also take the discrete fibration over the points, and these fit together in a compatible way (that is, the functor sending open sets of the espace étalé is compatible with the function projecting stalks down onto their points). So it’s almost natural to guess that it’s going to be a set equipped with some structure. People with better topological intuition than I do realized that we can just encode everything as a continuous map, which sends points to points and pulls open sets back to open sets. Further, we want to refine the topology on the total space so its projection sends the right open sets back down to the right open sets, but this is what we did by giving it the weak topology.
This is the situation that Lurie abstracts when he considers in addition to a structure sheaf also a sheaf of points!
such maps induce isomorphisms on the tangent spaces, but this is the sense in which they are the same.
and this can be extended to arbitrary manifolds, not just algebraic ones (a tangent space at a point is diffeomorphic to a chart around the point). This gives a way to define an ’etale map’ of manifolds as a local diffeomorphism.
I’m telling you that an étale map has no meaning in the topological category.
Well, I shall have to be careful and refer to ’local isomorphisms for the open cover topology’ then :) But sometimes my revisionist nature will shine through and extend the definition of etale to the topological category, despite my best efforts.
Actually the construction of the espace etale is a sort of representability argument: for Sh(Open/X) the petit topos we get that on passing into Sh(Top/X) (the gros topos) any object in the petit topos is representable. It would be nice to abstract this to other petit/gros topoi Sh(J/a), Sh(C/a) (but note that Sh(C) will need to have enough points, as Sh(Top) does)
and this can be extended to arbitrary manifolds, not just algebraic ones (a tangent space at a point is diffeomorphic to a chart around the point).
If you read the wikipedia page, it says that the inverse function theorem (this is the result you’re citing) fails in the algebraic case. I don’t see how any nice characterization filters down to general topological spaces (although it may work for certain kinds of manifolds).
Anyway, I haven’t read through anything about the cotangent complex, but étale maps have a characterization in terms of it (the Illusie-Quillen cotangent complex c.f. HAG II Toen-Vezzosi). They are not as simple as you’d expect.
Actually the construction of the espace etale is a sort of representability argument
Exactly. It doesn’t really hold in general in any naive way. We could state it as follows: Let $F:Shv((O(X)\downarrow X))\to Shv((Top\downarrow X))$ be the geometric morphism given by the canonical map of sites $(O(X)\downarrow X)\to (Top\downarrow X)$. Then for any object $S\in Shv((O(X)\downarrow X))$, $F(S)$ is representable.
This fails for schemes, since it says that every sheaf in the small étale topos of S is representable by an affine scheme over S, which is palpably false. I think that if we tried it out for differentiable or smooth manifolds, it would fail as well. I suspect that it has to do with the fact that the Grothendieck topology on $Top$ is somehow very special (it seems like it’s very close to canonical.)
To quote wikipedia (link above in 21)
More precisely, a morphism between smooth varieties is étale at a point iff the differential between the corresponding tangent spaces is an isomorphism. This is in turn precisely the condition needed to ensure that a map between manifolds is a local diffeomorphism, i.e. for any point y \in Y, there is an open neighborhood U of x such that the restriction of f to U is a diffeomorphism.
So we have the following chain of reasoning for a smooth variety: etale at a point $\leftrightarrow$ diffeo of tangent spaces at a point $\Rightarrow$ local diffeo about that point (using open cover pretopology, not Zariski covers). At which point we can state this for arbitrary manifolds (=local diffeomorphisms, in the open cover pretopology), and then state this for topological spaces (=local homeomorphisms in the open cover pretopology) and thence to general sites (perhaps with enough points, or concrete or something).If the arrow $\Rightarrow$ is a two-way implication, then we can recover etale maps between smooth varieties from the notion of local isomorphism, but I’m not sure if it is.
Actually, come to think of it, I haven’t seen the definition of a local isomorphism in a general site before, so here goes:
Definition: Let (C,J) be a site (J a pretopology). A map $f:a \to b$ is a J-local isomorphism if there are covering families $(v_i \to b)$ and $(u_j \to a)$ such that for each $u_j$ the restriction $f|u_j$ is an isomorphism onto some $v_i$.
I’ll pop this into a page tomorrow, together with any rewrites or suggestions people come up with.
Well, but étale morphisms carry more information than that. The tangent space isn’t always nice (unless the scheme is a smooth variety). The problem I have with the whole chain of reasoning is that it assumes that schemes are fundamentally locally ringed spaces, but this is not accurate.
Anyway, as you can see, it’s really a big stretch to call such maps étale, and it’s nonstandard, so could we just call it a local isomorphism?
Also, I think the condition for a local J-isomorphism is slightly more involved. In particular, the map on the target should be any map from a representable, and we require that the pullback is representable such that the induced map u->v represents an element of a cover of v.
My argument is only intended to work for smooth varieties, indeed smooth complex varieties.
BTW here’s that reference you wanted: Moerdijk and Mrcun, “Introduction to foliations and Lie groupoids”, page 134 (try Google books, search for “etale map”) in which they call a local diffeomorphism (in the category of smooth manifolds) an etale map. It’s quite an influential book in the “differentiable stack world”. It’s where I learned a chunk of groupoid theory, so perhaps it is where I picked it up (and how can you argue with Urs’ boss?)
To answer some of your points from #27:
the map on the target..
which map is this?
..should be any map from a representable
I’m not talking about sheaves here: I’m still only in the site. And I’m not sure that it is right that $u\to v$ should just be an element of a covering family (modulo worries about site or sheaf category): where then is the ’isomorphism’ in ’local isomorphism’?
Calling a local diffeomorphism étale is something I could live with. Calling a local homeomorphism étale is not.
Cool. I guess calling a morphism étale should be based on its lifting behavior w/r/t infinitesimal thickenings or based on its effect on cohomology. I suspect that étale morphisms and local diffeomorphisms are extremely similar in one of those two regards. I’m sure if Urs took a look at the algebraic case for a moment or two, he could tell us precisely how they’re related.
I’m jumping in late, but I don’t really understand the objection to calling a local homeomorphism étale. Unique lifting of nilpotents seems to me to be a way of saying that the map induces an isomorphism on the “infinitesimal structure” of the spaces involved. Isn’t this what a local homeomorphism also does, if you interpret “infinitesimal structure” in the only possible way in the topological category?
One way to make this precise is found in the paper “Local homeomorphisms via ultrafilter convergence” by Maria Manuel Clementino, Dirk Hofmann, and George Janelidze, where they prove that a continuous map $f\colon X\to Y$ is a local homeomorphism if and only if for any point $x\in X$ and any ultrafilter $\Phi$ in $Y$ converging to $f(x)$ (i.e. any “ideal point infinitesimally close to $f(x)$”) there exists a unique ultrafilter $\Psi$ in $X$ converging to $x$ such that $f(\Psi)=\Phi$, and moreover any pullback of $f$ also has this property.
Another way to make this precise is in the language of nonstandard analysis, where a continuous map of standard spaces is a local homeomorphism if and only if it induces a bijection on the halo of every point. Again, a unique lifting of infinitesimal neighborhoods.
@Mike: There’s definitely some other complicated stuff going on over and above the Grothendieck topology in the smooth and algebraic cases. The definition of an etale map between sheaves on Aff involves a representability condition (representable by an algebraic space), and the definition of an algebraic space (it has an atlas of affines satisfying certain conditions) directly involves the admissibility structure. This means that we can’t just extend the definitions automatically. The case of Top is a completely different animal. It’s nothing like the topological manifold case, the smooth case, or the algebraic case.
Well, sure, but I think that when the nLab entry about all of this is written, care should be taken to explain the nuances, since it’s not perfectly straightforward.
By the way, have I convinced you about the espace étalé and étale morphisms being unrelated? That’s the more important point.
If I call a local homeomorphism an ’etale map’, then by definition the espace etale comes equipped with an etale map to the base space. If one doesn’t accept this definition of etale maps in Top, then there is not much relating ’espace etale’ and ’etale map’. And since the espace etale construction doesn’t work in Sch (or even for smooth varieties, I bet) the notion of etale map there has nothing to be compared to.
And as far as the nlab page goes, I’m happy with saying local isomorphism, since if we try to define a local isomorphism in CRing there are all sorts of clashes of terminology. I’d just put in a paragraph explaining that some people use etale for local isomorphism in Top or Diff, and why (via smooth varieties etc)
BTW here’s a characterisation of local homeomorphism that may bring it closer to etale (it just occured to me, I haven’t sat down and proved it yet): a map $X \to Y$ is a local homeo if it has local sections through every point in $X$ and the fibres are discrete.
If I call a local homeomorphism an ’etale map’, then by definition the espace etale comes equipped with an etale map to the base space. If one doesn’t accept this definition of etale maps in Top, then there is not much relating ’espace etale’ and ’etale map’. And since the espace etale construction doesn’t work in Sch (or even for smooth varieties, I bet) the notion of etale map there has nothing to be compared to.
Yes, my point is that if you want to have a general definition of an étale map that works for various categories, the analogy between the espace étalé and étale maps should generalize, which it clearly doesn’t.
I agree with David #35: when you start passing to sheaves or stacks, then of course there will be representability conditions, but for objects of the categories themselves (topological spaces, or rings, or whatever) that shouldn’t be an issue.
Harry, I don’t understand how your comments #36 and #38 fit together. If “espace etale” and “etale map” are unrelated, then shouldn’t it be okay to have a general notion of etale map that works for various categories, because it is unrelated to the espace etale in topological spaces? And conversely, if the analogy between espace etales and etale maps is important enough to prevent us from generalizing the notion of etale map, then surely the two notions aren’t unrelated.
By the way, the notion of “espace etale” of a sheaf can be generalized beyond topological spaces. It works perfectly well for locales: local homeomorphisms over a given locale X can be identified with sheaves on X. (Moreover, this equivalence sits inside the equivalence between arbitrary locales over X and internal locales in Sh(X), which is more than you can say for spaces.) And you can even make it work for arbitrary toposes, if you define a “local homeomorphism” of toposes to be a geometric morphism equivalent to one of the form $\mathcal{E}/A \to \mathcal{E}$ for some object $A\in\mathcal{E}$. Admittedly this is taking the theorem as a definition, but at least it generalizes the situation for locales and spaces, and there might be a more intrinsic characterization of local homeomorphisms that I don’t know.
There is a different general notion of “etale map” given here.
Also perhaps of interest is the abstract notion of “a class of etale maps” proposed by Joyal+Moerdijk in their paper “A completeness theorem for open maps”?
the analogy between the espace étalé and étale maps should generalize, which it clearly doesn’t.
I don’t see it as an analogy, only a happy coincidence: the projection map from the espace etale is etale and in Top all etale maps are (isomorphic to) projection from an espace etale. If the espace etale cannot be defined, then there is no problem - it’s not as if the two concepts suddenly become completely divergent. Perhaps it’s related to equation (4) at Kleene equality: one side is always defined, the other side only sometimes, but when they are both defined they coincide.
In top all etale maps are (isomorphic to) projection from an espace etale.
This isn’t true though. The espace étalé projection map is always surjective.
The espace étalé projection map is always surjective.
Surely not for, say, the espace étalé associated to the initial sheaf? (-:
The espace étalé construction is always an equivalence of categories from the category Sh(X) of sheaves on X to the category LH/X of local homeomorphisms over X.
I was under the impression that Mac Lane and Moerdijk prove that the espace étalé has to be a surjective bundle.
I meant to say this before, but I don’t like calling an local homeomorphism/etale map a bundle, unless one insists that arbitrary maps can be called bundles. At the most one could call it a fibre map/fibration (being a bit old-fashioned there). For a locally constant sheaf the espace etale is a covering space, but it doesn’t preclude there being empty fibres over some components of the base. If one calls a covering space a bundle (where fibres are not required to be all the same - this can only occur if the base is not (path) connected), then I grant you, it is a natural extension to call any espace etale a bundle. But it is certainly not locally trivial, and even in the covering space case, not surjective.
Checking Moerdijk+MacLane, I see that what they call a bundle on X is just an object of Top/X. (see e.g. page 89). Also on page 88, they define a map in Top (in their terminology, a bundle) to be etale if it is a local homeomorphism. So they do call arbitrary maps ’bundles’ over their codomain, it’s just the category they live in is a slice cat.
Ah, that’s not good terminology at all.
Yeah, let’s not use “bundle.” And it’s definitely not surjective in general.
Anyway, I really thought that the definition of étale using factorization systems was really very close to the right definition of a correct generalization.
14 said: Zoran had convinced me at one point that they were related, but he was using a nonstandard definition of the espace étalé.
No, I did not use anything nonstandard. I use the same and standard notion of espace étalé since 1987.
I meant your definition of the generalization. You gave a definition for sheaves on a topological space, then extended it into more generality in a way that I don’t think works. In particular, the generalization does not make sense in any context past Top (and probably Locales) because, for example, the object corresponding to the espace étalé of a sheaf on a scheme is not a scheme, and hence, the map induced is not a map of schemes either.
No. I used the etale space in the usual sense. The question was if the etale morphisms have anything to do with it. There is a generalized PROPERTY (open local isomorphism when pulled back to sufficiently fine cover) which in topological case have only etale projection; while in etale topology it is shared by etale morphisms as well (this is not a tautology). So the connection is established in a generalization, but contrary to your accusations, I do not use name etale space for that generalization. Projections of etale spaces are etale in the sense that they are open local homeomorphisms where local is in the sense of their own topology. Etale morphisms can be viewed as projections which are open etale-local isomorphisms. As a third example, in smooth category, etale in diffeo sense maps are open local isomorphisms.
Thus I am not using nonstandard definitions. I just go down to the definition that we refer to open local isomorphisms. If I recall the argument right (I am not as concentrated on the issue as when we had the discussion, so I am talking from memory).
It is a tautology that étale morphisms are étale local isomorphisms. It’s pretty much by the construction of the topology. We could pick any topology we want and this is still true (with a suitable definition of what it means to be $\tau$-local).
The way that étale morphisms are similar to local diffeomorphisms is not the “local isomorphism” property, but rather the cohomological behavior. This may also have to do with the factorization system explanation that Mike linked to.
Apart from ramification locus it is true already in the usual (Zariski) topology but this is not good argument enough. Now you are right that one can make the original morphism artificially part of a bigger cover in any topology and pull back there. I should think of more precise argument; I mean if one can get down to the germs almost everywhere and have a regular extension for every branch it determines it uniquely, just like in the classical case. This will not happen in flat case as one can not go down to usual topology on the Zariski open dense subset.
Cohomologically it is as you say close to complex analytic behaviour.
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