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I made étale homotopy and étale homotopy theory be redirects to the entry geometric homotopy groups in an (infinity,1)-topos
but étale homotopy does not just handle the homotopy groups so this may be too drastic! It describes the THEORY as well so the title is now inaccurate. The subject as put forward by Artin and Mazur really centers on the profinite completion functor and other related things. there need to be links here but your change may not be optimal.
Like many things on the $n$Lab, this is not optimal, but for the moment I think it is better to have the redirect than to not have it.
But notice that the entry is also not just about homotopy groups. The main restriction is rather that the entry talks about the locally $\infty$-connected case (which is also the reason why the need for pro-completion disappears).
Somebody should write the generalization to the locally connected case that is usually considered in the context of étale homotopy.
(It keeps making me wonder, though: étale homotopy of topological spaces comes out wrong if the space is not locally $\infty$-connected. This makes me feel that assuming local $\infty$-connectedness generally is “good”. But maybe that is too naive.)
What is ‘wrong’? Is it not just shape theory? and that corresponds to nice things for C^*-algebras.
Strictly étale homotopy should not make that much sense for ordinary spaces as it refers to the homotopy theory relative to the étale topology and that is not a topological notion in its origin.
But notice that the entry is also not just about homotopy groups. The main restriction is rather that the entry talks about the locally ∞-connected case
But the title refers to homotopy groups not homotopy types! The restriction is then too strong as the non-locally $\infty$-connected case is also important.
What is ‘wrong’?
The homotopy groups of a topological space are those computed by its étale homotopy only if the space is locally contractible. This is the original result of Artin-Mazur. The entry geometric homotopy groups in an (infinity,1)-topos also explains “why” this is so: only the in the locally $\infty$-connected case is the corresponding $\infty$-topos locally $\infty$-connected and then the central statement of étale homotopy / higher Galois theory simply follows from the $(\Pi \dashv Disc)$-adjunction.
Strictly étale homotopy should not make that much sense for ordinary spaces
It makes sense for every locally connected site and every locally connected topos. See
for a review (or, for that matter, the above $n$Lab entry).
But the title refers to homotopy groups not homotopy types!
Sure, things could be written in a better way. But for the moment I neverthess think that “étale homotopy” should redirect to that entry, because that entry discusses étale homotopy.
Eventually somebody will find the time to beautify the corresponding $n$Lab entries further. There are lots of points that would deserve improvement.
I’m afraid I continue to disagree. You say ’that entry discusses étale homotopy’ but it only discusses étale homotopy in a very special case, not in general. It does discuss the étale homotopy of locally $\infty$-connected sites, but the subject of étale homotopy is not restricted to such sites. so this should be clear from the entry.
I’m afraid I continue to disagree.
No, I agree with that! :-) As I said in #3.
When I have some time I will see if I can come up with some suggestions for redistribution of material so that various other bits of work (e.g. work of Isaksen and Quick) will fit in… but it can wait!! Happy New Year.
@Tim: would it be better if étale homotopy redirected to shape of an (infinity,1)-topos?
Probably eventually, étale homotopy theory should be its own page. Maybe it would be worth making it a stub right now.
Right, I should have done that right away. I tried to save myself the time, but that strategy didn’t work out ;-).
Here are some first words: étale homotopy.
To be expanded (by whoever feels like expanding!).
No. 9 I think the étale homotopy should have its own page with a link to shape etc. I have some material that can be adapted, but not for today. :-(
No. 10 Urs: that looks a good start.
On the other hand, the new sentence you added at nerve theorem sort of does not make sense. I have suggested an amednment there , so check that you are ok with it.
Hi Tim,
I can live with your version.
But I am not sure why the previous version “sort of does not make sense”.
The point that I think is important here is that for paracompact spaces the limit over hypercovers that one has to take in the explicitl definition of etale homotopy type localizes at the Cech nerve of any good cover (since these always exist and are already cofibrant in the projective model structure!).
In this sense the nerve theorem is an avatar of the full definition in a simple special case.
I found it did not work ’linguistically’, but I also think that the nerve theory is not central to étale homotopy. That grew out of the corresponding cohomology theory and includes profinite completion, links with Galois theory, use of pro-objects etc., and the study of objects for which the nerve theorem does not apply as they do not have good covers. They are not even topological spaces. The sentence therefore ended up being too restrictive and might have put some people off (e.g. Algebraic geometers).
It is interesting to Google on étale homotopy and see what areas it is being applied to, e.g. a recent paper (ETALE HOMOTOPY AND SUMS-OF-SQUARES FORMULAS, DANIEL DUGGER AND DANIEL C. ISAKSEN ). There is also work on étale homotopy type of Voevodsky spaces about 10 years ago, Geirion QUick’s work on profinite homotopy and étale realisations, and Jon Pridhams (http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.0928v4.pdf), and that is just picking at random. But the effect of your sentence was almost to suggest that étale homotopy theory really was about classical homotopy types. You may not have intended that, or you may feel that by going to the $\infty$-topos context they all become somehow comparable but I think the Artin-Mazur approach suitably modernised deserves a better ’press’ in the nLab than it has had so far…. hence my change.
I also think that the nerve theory is not central to étale homotopy. […] and the study of objects for which the nerve theorem does not apply as they do not have good covers. They are not even topological spaces. The sentence therefore ended up being too restrictive and might have put some people off (e.g. Algebraic geometers).
I don’t understand why you think any of this is in contradiction with a sentence that briefly asserted “that we have here a special case” of étale homotopy.
Notice also that your modification of the sentence still says that. Also this very statement is currently the lead-in at etale homotopy itself, which you called a “good start”.
So I don’t understand what you say is wrong. But since we seem to agree now on what is in the entries, it’s not so important. Let’s leave it at that.
Fine by me.
Added the reference
Thanks. That looks useful.
added two kinds of examples related to étale (non-)contractibiity.
I have added some more of the fundamental references with brief comments.
In the course of this I also added a remark to analytification on Relation to etale homotopy type. (But that whole entry on analytification is in need of attenion.)
Have added a pointer to
at étale homotopy.
I admit that I am not sure what #23 is after.
For a moment I thought it’s asking why schemes, in general, are not locally contractible, like manifolds are, which is the reason why smooth $\infty$-groupoids are cohesive, but not so algebraic $\infty$-groupoids.
But on re-reading #23 I am not sure if that’s what it’s about, or what it’s about.
Could you maybe clarify?
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