You should at most ask that the morphisms out of the slices are open immersions.

]]>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 $\alpha$?

]]>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”.

]]>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.

]]>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.

]]>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).

]]>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.

]]>@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.

]]>@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.

]]>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!

]]>Yup. That’s the point of the generalization. For further discussion, see above.

]]>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.

]]>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.

]]>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).

]]>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.

]]>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.

]]>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?

]]>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.

]]>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

]]>And how about the canonical site on the category of etale covers (or any of the flat topologies)?

]]>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.

]]>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”?

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.

]]>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.