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I created Galois topos following Dubuc’s article.
But I must be missing something about the notation: does it really mean to say that $A$ is an $\Delta Aut(A)$-torsor, as opposed to saying that it is associated to an $\Delta Aut(A)$-torsor?
Thanks. It sure seems to me like it ought to be “associated to.”
I do not see what the objection is to the current wording. Any object $X$ naturally has a $\Delta Aut(A)$ action. (One might want to say that that was a different type of object, but is that really necessary. Perhaps it does need verifying and spelling out.) Then Dubuc’s use of words just says that $X$ satisfies the usual conditions of a torsor for that action (i.e. the iso). The wording may not be ’optimal’ but was and still is current.
I am not sure what ’associated to’ means without a specification of how it is associated to whatever.
I do not see what the objection is to the current wording.
The problem is that there does not seem to be a reason why we would want to demand that there is an isomorphism between a fiber $F$ and its automorphism group $Aut(F)$.
In the topos $Set$ the only objects $A$ that are $Aut(A)$-torsors are the singleton and the two-element set, because this are the only two solutions to $n = n!$.
There is no good reason to declare that only sets of cardinality lower than three would count as locally constant objects.
edit: hm, maybe the point is that nevertheless these two sets form a set of generators for $Set$?
I am not sure what ’associated to’ means without a specification of how it is associated to whatever.
There is a canonical notion of associated bundle if the structure group is given as an automorphism group: for $P$ an $Aut(F)$-bundle, the canonical associated bundle is $P \times_{Aut(F)} F$.
Looking at Dubuc’s paper, I think that he intended exactly what he seems to say. The point that you, Urs, make is exactly right and is what makes it work. This identifies the analogues of those solutions to $n = n!$. I have not got Moerdijk’s article in front of me so cannot compare with his notion of Galois object. Tonini in his notes defines them differently but is nearer the classical notion.
I found this, which uses Debuc’s definition.
Yes, I looked at that, too. he just seems to copy Dubuc’s definition, though. Or does he shed light on the above issue?
What are the intended examples of “Galois objects”?
Dubuc does not seem to give any, whilst Tonini gives some reasonable ones but is using a different definition.
Hmm… if $X$ is a connected and locally simply connected topological space, then in the category of covering spaces of $X$ (which is a topos, since it is equivalent to the category of $\pi_1(X)$-sets), the universal cover $\tilde{X}$ of $X$ is, in fact, a torsor for $Aut(\tilde{X})=\pi_1(X)$.
This all feels so wrong to me because it is so focused on the locally connected case! (-: Surely things will become clearer when we have good definitions that work in generality.
Tim,
after three “Debuc”s maybe it’s not just your fingers: I think the name is Dubuc ;-)
OOPS ! Eduardo would not be pleased.:-(
Sorry, i should have said this by email. But you can easily edit all your comments and fix the typo (just hit “edit” on the top of each comment.) And then we can blank these last three comments here.
It does not really bother me and see no real reason to cover up a slip, since it was made. (That looks like papering over cracks in the plaster. It may look better but ….) I was not checking what I was typing and my fingers were on automatic pilot. It is like when I type ’categroy’. I have known Eduardo since …. if you see what I mean, so the slip is very silly on my part.
It is like when I type ’categroy’.
I always find myself typing “catgegory”. I can’t stop my fingers doing this. Not sure why that happens.
I always find it heartening when I mistype a search term in Google getting something ridiculous and then find that it gives me some hits, and that they are relevant! :-)
It is then interesting to note that I might not have found them if I had not made the slip.
For some reason my typo is always “catetgory”.
Why do we all type THAT word incorrectly?
Could somebody upload the Moerdijk’s proceedings article cited in the entry ? The proceedings are hard to find.
@zskoda: it’s on ScienceDirect: http://www.sciencedirect.com/science/article/pii/S1385725889800307
Great, I have the access to this!
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