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I added to covering space a section In terms of homotopy fibers that explains the universal covering space as the homotopy fiber/principal oo-bundle classified by the cocycle that is the constant path inclusion of topological groupoids.
To fit this into the entry, I added some new sections and restructured slightly. Todd and David should please have a look.
What I just added is essentially what David Roberts says in various query boxes, notably in what is currently the last query box. Back then we talked about the "Roberts-Schreiber construction" or whatnot, but really what this is is just the standard way to compute homotopy fibers in the oo-category of oo-groupoids.
I suspect that Todd's bar construction described there can similarly be understood as being nothing but another way to compute the more abstractly defined homotopy pullback in concrete terms. I'll have to think about this, though. But probably Tim Porter or Mike Shulman will immediately recognize this as the relevant bar construction of homotopy pullbacks in homotopy coherent category theory.
Yes, the sheaf of its sections. As we just discussed in the other thread: the stack of covering spaces is the stackification of the one constant on . A section of this is a locally constant sheaf, the locally constant sheaf of sections of the covering space.
I am about to add more on this to the Lab. But currently I can't quite decide where it should go!
So does that mean that the category of covering spaces is the category of etale bundles (as in Mac Lane and Moerdijk's sense)?
Etale spaces are more general. Every sheaf (on a topological space) X is the sheaf of sections of some etale space over X. Locally constant sheaves are those that are sections of covering spaces.
I am about to add more on this to the Lab. But currently I can't quite decide where it should go!
I started now locally constant sheaf and constant sheaf
I think what makes finite covering spaces special etale spaces is that they have a more restrictive condition on what thy look like locally: they need to be what is called evenly covering instead of just being local homeomorphisms.
there is a remark on this at etale space, which I now also copied over to covering space:
Every covering space (even in the more general sense not requiring any connectedness axiom) is an etale space but not vice versa: for a covering space the inverse image of some open set in the base B needs to be, by the definition, a disjoint union of homeomorphic open sets in E; however the ‘size’ of the neighborhoods over various e in the same stalk required in the definition of étalé space may differ, hence the intersection of their projections does not need to be an open set.
Possibly there is room to say this in a bit more detail.
Oh, I am now happy that I wrote that remark at etale space some time ago :)
Yes very good!
But maybe we could add a comment about finiteness, then: what if we have etale spaces with finite stalks. How different can they be from covering spaces? I guess now Harry's original idea comes into play: in that case it is only connectedness that distinguishes covering spaces from general finite etale spaces. I guess.
Well intersection of finitely many opens downstairs is open.
Connectedness is not always asked for in covering spaces. This is more special definition in the theory, in order to relate it to Galois theory/fundamental group. Spanier has for example no restrictions in the very definition of covering space. If one wants to have the connection to the fundamental group, then and deck transformations one needs more conditions, including that the base space is linearly connected and locally linearly connected and that the covering space is connected as well. We should treat all these levels of generality, including the definition of a regular covering. In general I like the approach in Spanier (which I reviewed just few weeks ago).
Well intersection of finitely many opens downstairs is open.
Yes, so that's why I am saying with finite fibers, etale spaces and covering spaces must be pretty much the same
I'll agree that that map's étale, but the obvious question is: What about a connected Hausdorff étalé space with finite fibres?
Thanks David, of course. I added that remark to etale space and covering space.
(Let me all remind you: it is forbidden to say anything useful here without making sure a copy of it survives on the nLab, in some form! :-))
Urs: As far as surviving, I have the feeling that many good things from café which appeared after the nlab did not get into nlab. I am not complaining as all of us are too busy, but suggesting what I did before, I mean if somebody can suggest to Jacques or somebody, if it is not too hard, to get "source" button for indvidual posts at café available. Namely, some discussions, especially those most technical, hence frequently most valuable ones, have lots of formulas. In answering to those posts, inquoting them in nlab, and finally for pedagogical purpose of learning the script, it woudl be very useful to be able to see the source code of chosen individual comment or post in cafe. If the source code is preserved at all (I do not know how it works there, as I am not insider or author there).
Right, so as I may have said before: drop me a note when you want source code to be copied. I can access it.
That may be a solution for me, and I remember your kind offer, but still if there is any lite effort in improving the cafe system in sight, this issue is my preferred choice :)
@ Zoran
It is preserved. If the poster PGP-signs their work, then you can even view it.
Espace etale or etale space is the one which by definition corresponds to etale map in continuous world. In algebraic geometry, a morphism is etale if it is smooth of relative dimension 0. Equivalently flat and unramified, or smooth and unramified. So in the category of schemes and flat morphisms of schemes. Thus etale morphisms (word used in algebraic geometry) is a precise analogue of the notion of etale map from usual topological setup. Harry, why do you object ? See etale morphism entry distinct from etale map entry (I think that I initially wrote both entries).
You described for rings in some detail the infinitesimal lifting property for formally smooth morphisms. I do not distinguish rings and affine schemes in terminology, and this is not an issue.
The projection of an etale space is etale map in topological category. Period.
Etale means local homeomorphism there, the terminology being standard.
This is totally parallel to the situations in smooth category and in the category of algebraic schemes.
So far I do not get which mistake you alude ? So far it looks to me a strange claim.
To add, espace etale is just the total space of etale map in topology, by definition. I am not sure if you object the variant of the standard terminology or the sufficiently formal semantic parallelism in algebraic, differential and topological category ?
No, we don't. We have the definitions of etale maps in at least 4 contexts, which appear to be the category of topological spaces and continuous maps, category of schemes (or analytic spaces) and smooth morphisms of schemes (or of analytic spaces) and category of smooth manifolds and smooth morphisms. The category of affine schemes is just a subcategory of all schemes, hence your special case. A smooth map of schemes is etale iff there is a cover of the total scheme by Zariski-open subschemes which project by an isomorphism of schemes onto open subschemes of the base. In particular it induces the topological etale maps of the underlying topological spaces, but local isomorphism includes the sheaf pat as well.
A smooth map of schemes is etale iff there is a cover of the total scheme by Zariski-open subschemes which project by an isomorphism of schemes onto open subschemes of the base. In particular it induces the topological etale maps of the underlying topological spaces, but local isomorphism includes the sheaf pat as well.
This should be added to etale morphism.
We could take a cover U of the scheme/space E above and a cover V of the scheme/space M below, and an isomorphism iso : U to V forming a commutative square U to E to M equals U to V to M. This characterizes etale maps in all 4 categories mentioned, equipped with corresponding topologies (Zariski, not etale in the case of schemes).
I entered the claim suggested by Urs.
For Harry: I don't know, now we are getting to terminological issue closer to Toby's expertise.
By the way, Michael Artin has written a fairly long paper in 1994 where one of the main issues is the study of etale morphisms of noncommutative rings. This is unpublished (I have the file somewhere). There are alterantive approaches nowdays, but Misha's treatment is elementary and very concrete, I liked it (it was long since i looked at it, and forgot it, but remember the feeling).
Wait, sorry, something is wrong in one of my newest claims above, namely it is OK but it should be in etale topology, not Zariski (this makes it circular for the definition, but appropriately consistent with the picture); Zariski opens are too big for this to work in any example. The disjointness is exactly the thing which is achieved by etale covers (this is very essential and makes it really parallel to classical situation).
On the other hand, passage from the condition from affine schemes to the general case is not difficult. For example the relative dimension is defined in each fiber and formal smoothness is clearly Zariski local.
Smooth morphisms of schemes are finitely presented so the difference between covering maps and etale maps in topology does not play a role here. This makes it easier to see that taking pullbacks of etale morphism by appropriate etale cover leads to morphism which is etale at the level of locally ringed spaces (unlike the original map which is not).
Edit: when one mentions locally ringed space it must be Zariski, etale makes no sense as usual topology here. But etale cover makes the situation at the pullback disjoint enough that by then the usual topology suffices.
for schemes local is in terms of the opposite of maps called 'etale' in CRing
Right David, but not only that etale are used to define the locality needed here, the etale morphisms in CRing are themselves (opposite to) etale maps of corresponding affine schemes.
Zoran (#32) refers above to etale map as distinct from etale morphism, but the former does not actually exist. (Possibly Zoran was thinking of etale space, where the term is mentioned.) So I started it, with a note that as to how it appears in Zoran's four examples.
Does a universally strict epimorphism have anything to do with a strict epimorphism? The latter makes sense in any category whatsoever.
Right Toby, the entry etale map from my memory is actually the entry etale space.
Sorry, Harry, I'm not sure what your response is to.
Edit: As for strict epimorphisms, I just searched and found that they are defined in Grothendieck's Seminaire Bourbaki No 190 (definition 2.2) - collected in FGA. I should add this to the page strict epimorphism for reference.
I have streamlined the statement of the “fundamental theorem” at covering space a little.
Eventually I’ll want this page to be ready for public consumption, usable as lecture notes. Presently it’s a bit of a mess. There are a handful of ancient query box comments sitting there, various from David (Roberts), some from Todd. Eventually I will be wishing to and working on removing these such as to make the entry text be finalized and polished.
Anything that I wrote there can be removed. If I get the urge to think about such topics again I can go back to the history of the page.
I have added the statement and the detailed elementary proof of the path lifting property of covering spaces: here.
I have also added statement and proof of “the lifting theorem” for covering spaces (here)
Presently the very last step in the proof is hard to read. I’ll see if I can improve this tomorrow.
Okay, I have improved (I hope) the phrasing of the last bit of the proof of “the lifting theorem” (in the section Properties – Lifting properties). In the course I added some further basics, such as the statement and proof that a covering projection is an open map (in the section Properties – Basic properties).
I have splitt off an entry
from the entry covering space and moved to it Todd’s category-theoretic discussion In terms of a coend. Then I added also a description of the elementary construction.
(I am working towards a detailed expositional discussion of the fundamental theorem of covering spaces and am finding it helpful to organize the material a bit more, for readability. There is still other stuff at covering space that might usefully be moved to elsewhere.)
I noticed. I would probably like to weigh in at some point on organizational matters, but I’m glad you’re thinking about this. One thing I might mention is that the whole “monodromy” discussion provides an explicit adjoint equivalence and obviates any need to bring up the axiom of choice (or at least this aspect should be brought out better), i.e., the fundamental theorem of covering space theory should make this adjoint equivalence explicit and hopefully not just say a certain functor is essentially surjective and fully faithful.
Thanks, Todd, for some patience. I might be about to change my mind and move it all back. But in either case the entry deserves some tidying-up.
I have tried to spell out the proof of the “fundamental theorem” now here.
It’s all very tautological, of course, but so I need to beware not falling into traps. For instance, I seem to think that the assumption on the base space needed is, besides semi-local simply-connectedness, that it be locally path-connected. Elsewhere I see instead local connectedness assumed. (?)
It’s totally fine, Urs. I agree that the article was in need of tidying up, and please take your time.
It may be tautological, but this is the sort of “trivially trivial” material that category theory is good at rendering, and besides it’s hard to get the kind of quality of exposition in textbooks that I think we are ultimately after here. (Most textbooks still futz around with basepoints and stick to groups, not groupoids. It should be done right after all.)
the assumption on the base space needed is, semi-local simply-connectedness, that it be locally path-connected
Correct.
For completeness I have
split off an entry groupoid representation from representation
created an entry category of covering spaces.
I have been editing the section Lifting properties.
Now the homotopy lifting property is stated in the generality that deserves this name (here) and the corollary “lift of homotopy between paths relative starting point for given lift of paths” is now an example afterwards (here).
after the homotopy lifting lemma for covering spaces, I added a remark here saying that
this means that covering spaces are Serre fibrations,
there are counterexamples to them being always Hurewicz fibrations,
but the latter holds if base and total space admit CW-structure.
Hm, now I got myself mixed up. In the proof of the homotopy lifting lemma for covering spaces here, does it really need local (path-)connectedness?
Ah, that counterexample (here) is for “generalized covering spaces”.
Okay, so I have added to Hurewicz fibration a mentioning of covering spaces as examples here.
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