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created etale geometric morphism
(david R.: can we count this as a belated reply to your recent question, which I can’t find anymore?)
Objection!
Well, I think it’s bad terminology since an étale morphism of schemes is definitely unrelated to this, unless we’re in the étale topology. This was one of the reasons I was against calling them the same thing, because this does not generalize the concept of an étale morphism.
To illustrate exactly why, I give you the following: an étale morphism of schemes $X\to Y$ only induces an étale geometric morphism between the two slice toposes
$Sh(Aff_{et})/X \to Sh(Aff_{et})/Y$or the two petit étale toposes:
$Sh(Et/X)\to Sh(Et/Y).$We note that this is totally independent of the actual concept of an étale morphism! It depends only on the Grothendieck topology! This is what I was talking about. It isn’t a true generalization of the notion of an étale morphism. Therefore it fails the test of the principle of reduction (generalizations canonically restrict to the original concept).
Étale morphisms in the sense of algebraic geometry should be generalized as follows: Étale morphisms should induce equivalences (in some useful sense) on infinitesimal data.
I move for the term to be changed to:
If this bad terminology is here to stay, and I am unable to convince you otherwise, can I at least propose a minor change that might lessen its negative effects?
Call it one of the following:
(This is a compromise, since étalé, the nasty word from which all of this comes, is used to describe the space, not the morphism. It comes from the fact that the espace étalé is the space of “spread out” sheaf data. When we’re talking about morphisms, the obviously correct terminology for the concept in question is local. I also don’t like how this sounds in English, since étalé, a french adjectivization of a verb, should follow the word, since in English, almost no adjectives (and never adjectivizations of verbs) end in the sound é (I use the qualifier almost because although I’ve never heard of one, there might be one that exists, so I’m just covering my bases). This somewhat more correct terminology sounds much better in French: “morphisme géométrique étalé”, which translates properly to a “spread out geometric morphism”.)
or
By the way, the definition on the nLab page is incorrect. In particular, we want the left map of the factorization to be an equivalence (categorical equivalence in the oo,1 case), and we want the righthand map to be right adjoint to the inverse image (pullback, the guy with the upper star) of the projection.
Well, I think it’s bad terminology since an étale morphism of schemes is definitely unrelated to this, unless we’re in the étale topology.
Er? So this is the point of having a topos-theoretic abstraction of a notion: depending on which topos exactly you look at, one abstract notion gives you several concrete notions, which are thus unified conceptually. Here we have unification of the notion of etale space over a topological space with etale morphisms of schemes, depending on just which topos we choose to work with.
I have added to etale geometric morphism a little bit more to reflect this. See the detailed discussion in Structured Spaces on how it all relates to the etale morphisms that you are thinking of.
By the way, the definition on the nLab page is incorrect. In particular, we want the left map of the factorization to be an equivalence (categorical equivalence in the oo,1 case), and we want the righthand map to be right adjoint to the inverse image (pullback, the guy with the upper star) of the projection.
Well. Okay, I fixed the missing equivalence sign. For the rest, it is supposed to be understood that $\pi : \mathbf{H}/X \to \mathbf{H}$ is the geometric morphism discussed just two lines earlier.
Feel free to add clarifications to the entry, if you feel need for them.
Well, I think it’s bad terminology since an étale morphism of schemes is definitely unrelated to this, unless we’re in the étale topology.
Er? So this is the point of having a topos-theoretic abstraction of a notion: depending on which topos exactly you look at, one abstract notion gives you several concrete notions, which are thus unified conceptually. Here we have unification of the notion of etale space over a topological space with etale morphisms of schemes, depending on just which topos we choose to work with.
No, this is circular. I’m saying that an étalé geometric morphism of Zariski slice toposes is not an étale map of schemes. It is not an abstraction at all, as I’ve been saying. If you check out the notion of an actual étale morphism of CRings in Structured Spaces (2.6), you can see the problems a bit more.
I’m saying that an étalé geometric morphism of Zariski slice toposes is not an étale map of schemes.
Sure. And not just you are saying this. Another example: an étale geometric morhism of localic toposes is a topological étalé space.
The notion depends on which topos you choose. That’s the point of it.
If you check out the notion of an actual étale morphism in Structured Spaces,
This is a funny phrase after I just pointed you to that. I am not sure what to make of your messages. Maybe it would be good if we leave it at that for the moment and you come back to it when you feel a bit more relaxed.
Yes, here’s the problem. We define a topology called the étale topology whose covers consist of étale morphisms. In particular, this means that the notion of étale should be independent of the topology.
Here’s what is awful about this terminology:
The étale topology is the topology where étale morphisms become étale.
A meaningless sentence, but this terminology makes it true. Rather, here’s what it should be:
The étale topology is the topology where étale morphisms become étalé.
Or more properly:
The étale topology is the topology where étale morphisms become local isomorphisms.
The Elephant calls this a “local homeomorphism” of toposes. Perhaps that would make Harry happier, although it has exactly the same issue but for topological spaces rather than schemes – only for the usual local-homeomorphism topology on petit sheaf toposes do local homeomorphisms of topological spaces induce local homeomorphisms of toposes. I don’t really have a problem with either “local homeomorphism” or “étale,” I think. It seems to me that what’s going on is that we have “intrinsic” notions of “étale morphism” for (1) topological spaces, (2) schemes, and (3) toposes, that we can choose many different Grothendieck topologies on Top or Sch, resulting in many different functors $Top \to Topoi$ and $Sch \to Topoi$, but that only for the “étale topology” does the induced functor preserve étale morphisms, i.e. take étale morphisms of spaces/schemes to étale morphisms of toposes.
By the way, a “local geometric morphism” is something quite different, so one should avoid clashing with that.
I’m 100% behind calling a it local homeomorphism of toposes. This restricts to the correct definition on topological spaces, while étale does not restrict to the correct notion on schemes. The reason why it’s different is that a topological space has a canonical site (namely $\mathcal{O}(X)$).
This is not true for schemes or algebraic spaces, which are typically considered with the étale, Zariski, fppf, fpqc, or Nisnevich topologies, the corresponding local homeomorphisms of their toposes emphatically not all being étale.
There’s another reason too, involving T-open covers on the underlying site (C,T), but I can’t quite figure out how to formulate it. There’s a sense in which you can find a T-open cover of an object such that the restriction on each “open set” is an isomorphism onto its image, at least when we’re looking at local homeomorphisms between slice toposes
a topological space has a canonical site
says who? What’s canonical about this over and above the etale site (or any other) on a scheme? And what about big sites associated to a space/scheme? I claim that the numerable site on a space is actually where classical algebraic topology happens, and for paracompact spaces this and $O(X)$ coincide. There are a whole bunch other other little sites one can consider on a space (and even more if the space is a manifold). Perhaps for some (most) applications particular sites are more appropriate, but why is the Zariski site not considered canonical? Is it merely because it is not so useful?
One needs to separate the existence of different categories of ’covers’ and the topologies on them. What about the category of finite covers $\coprod_{i=1}^N U_i \to X$ of a topological space $X$? This is sort of useless except for compact spaces, obviously, but it is what is actually used in that case. There are obvious generalisations to countable covers, locally finite covers, $\sigma$-finite covers and so on, for the classes of spaces and applications on/for which they are sensible.
Anyway, my claim is that different sorts of spaces have different sites that are used for them, and only in nice cases do these coincide. When for nasty spaces these bifurcate, which is the ’canonical’ site? It depends on applications, just as for schemes.
Are you really trying to argue that there is no canonical site on a topological space? I can guarantee to you that the majority of work done with sheaves is done on this site for a topological space X.
Anyway, the term local homeomorphism is a much more accurate description than the mistranslation étale. In addition, there’s the thing in my last paragraph, which we discussed last time (namely that we actually have a “local isomorphism of sites” whenever we have a “local homeomorphism” of toposes.
I’m trying to argue that there are many natural sites on topological spaces, and they coincide when the space is nice. This is also application driven. I’m also trying to argue that saying there is no canonical site for schemes is a bit strong: it depends on what you want to do. The difference between them is just a matter of degree.
I don’t know if I agree calling a map of topoi etale, because of the potential for conflict. I personally see etale and local homeomorphism/diffeomorphism as synonyms in Top/Diff resp. as there is no cause for confusion. There is a saving of 6 syllables in such a practice. I would then talk about local homeomorphisms of topoi, to avoid clashes with maps of topoi of sheaves on little etale sites of schemes.
Of course, a scheme is a particular sort of topological space, so if a topological space has a canonical site, then so does a scheme qua topological space, namely its Zariski site. Any chance that it would be accurate to say that the notion of “étale morphism” as distinguished from “local homeomorphism” depends on the fact that a scheme is not just a topological space but a ringed one? And that therefore we should talk about “local homeomorphisms of toposes” but “étale morphisms of ringed toposes”?
BTW, the usual topology on (the open-set lattice of) a topological space (or locale) is definitely “canonical” in the precise sense of the canonical topology.
And how about the canonical site on the category of etale covers (or any of the flat topologies)?
Actually, since etale cohomology is a Weil cohomology, that makes the little etale site on a scheme pretty close to being canonical in my eyes
Of course, a scheme is a particular sort of topological space, so if a topological space has a canonical site, then so does a scheme qua topological space, namely its Zariski site.
If you, in your heart of hearts, really want to call local homeomorphisms “zariski morphisms”, I won’t object…
And how about the canonical site on the category of etale covers (or any of the flat topologies)?
This is something of a straw man. The étale topology on a scheme is not part of the definition of a scheme.
Regardless, we can either go with an english word that is technically closer to the truth, or we can use a french word, which should be written étalé.
Well, the lattice of open sets doesn’t have to be part of the definition of a topological space. (-:
Harry, my question in #16 was mostly addressed at you; any thoughts?
I think this distinction in #16 is a good one: “local homeomorphism” for the general notion of “étale geometric morphism” for the refined definition of structured toposes.
I’ll implement that now at etale geometric morphism.
The Elephant calls this a “local homeomorphism” of toposes.
Which page is that? I tried to find the term related to topos morphisms there, but failed. But then, I am looking at the djVu, browsing throw which is a bit awkward.
Of course, a scheme is a particular sort of topological space, so if a topological space has a canonical site, then so does a scheme qua topological space, namely its Zariski site. Any chance that it would be accurate to say that the notion of “étale morphism” as distinguished from “local homeomorphism” depends on the fact that a scheme is not just a topological space but a ringed one? And that therefore we should talk about “local homeomorphisms of toposes” but “étale morphisms of ringed toposes”?
@Mike #16: The fact that étale morphisms are “like” local diffeomorphisms is very deep and relies on Artin’s approximation theorem and a whole bunch of cohomological properties. There is no sense in which étale and étalé are related aside from the observation that they coincide in the étale topology (a nearly vacuous observation).
There is no sense in which étale and étalé are related
There is, that’s what we are trying to convey all along:
after passing to suitable toposes both are examples of topos geometric morphisms projecting out of an overcategory topos.
That is true for any class of morphisms defining a site. That is, take any class of morphisms stable under pullback and composition (may need a little more here, but it’s unimportant), call it H, and just take the covering sieves on objects to be finite collections of H-morphisms with the same target. Then all H-morphisms in the site induce local homeomorphisms on the resulting toposes.
Yup. That’s the point of the generalization. For further discussion, see above.
Then don’t call them étale! If you come up with a really compelling reason other than “other people have done it in the past, and étale can fill the space of H”, I would be happy to concede, but we have perfectly acceptable terminology from Johnstone that makes a whole lot of sense!
@Harry #24: I don’t understand how that answers the question; can you clarify? I didn’t use the words local diffeomorphisms or étalé at all.
@Mike: Sorry.
Re: #16
I don’t think it works either (restricting only to locally ringed toposes). The notion of étale is very specific and doesn’t a priori make sense in an arbitrary context (although there are definite parallels we can draw between étale morphisms and local diffeomorphisms). The fact that local diffeomorphisms and étale morphisms describe something very similar is a happy coincidence, and the properties that they share are deep. The properties they share are not the properties being abstracted by this definition (which has to do with properties regarding coverings in their respective sites).
I’m not saying that there is no proper abstraction of an étale morphism (there probably is!), just that this is not the right one.
I guess I wasn’t clear – I wasn’t arguing in favor of the definition in question being called “étale.” And I still don’t understand where the local diffeomorphisms appeared from. What I was saying was, if we call the notion in question a “local homeomorphism of toposes,” could the definition of “étale morphism of schemes” be separately extended to a different notion of “étale morphism of ringed toposes”? If so, I think giving that definition would be a good way to help explain why local homeomorphisms of toposes shouldn’t be called “étale.”
Are you talking about local diffeomorphisms because manifolds can be regarded as ringed spaces and therefore as ringed toposes, so that any such definition of “étale morphism of ringed toposes” could be specialized to them? That’s true, but I wouldn’t necessarily expect that such a definition would specialize to be a local diffeomorphism in that case. Fortunately the words “étale” and “local diffeomorphism” are also different.
I was saying that the proper generalization of an étale morphism should include étale morphisms of schemes as well as local diffeomorphisms/local biholomorphisms as special cases (based on cohomological properties). If I remember correctly, Toen/Vezzosi give the right definition (it involves the cotangent complex in the sense of André/Quillen).
By the way, here’s the page in the elephant that explains why étale is wrong.
However, I was unable to find the definition equivalent to Lurie’s. He gives an identical definition when the topos is localic, though.
Your link is to a bunch of pages; which one do you mean? The definition of “local homeomorphism” for general topoi is on page 651.
Crud, I hit the wrong “link” button on google books. There’s a page where Johnstone notes exactly as I have that “étale morphism” is a mistranslation of “espace étalé”, which means “display space”.
Is there a “local” characterization for local homeomorphisms of toposes, maybe something like the following?
A geometric morphism $X \to Y$ is a local homeomorphism iff there exists a family of objects $(x_\alpha)$ in $X$ such that the morphism to the terminal object $\coprod x_\alpha \to e_X$ is an epimorphism, and $X/x_\alpha \to Y$ is an equivalence for each $\alpha$?
You should at most ask that the morphisms out of the slices are open immersions.
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