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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 4th 2010
• (edited Feb 4th 2010)

a bit, added hyperlinks here and there, in particular linked to homotopy group of an infinity-stack -- also referenced the chapter in Johnstone's book.

It would be good if we could highlight what exactly the theorem described there actually says. Currently it is easy to miss for the reader what the punchline is. But I have to do something else right now...

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeFeb 4th 2010
• (edited Feb 4th 2010)

I added ref. J. P. Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group, Tata Inst. of Fund. Res. Lectures on Mathematics 40, Bombay, 1967. iv+176+iv pp. and references of Joyal Tierney and Borceux Janelidze. By the way I think that the axioms in Dubuc are just literally taken from Murre.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeFeb 4th 2010

New entry categorical Galois theory. Just the main reference to start.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 4th 2010

Thanks! Wasn't aware of that.

Can you give me a hint on how that relates to the discussion in Johnstone's book chapter 8?

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeFeb 4th 2010

I must go to the bus immediately. BUt categorical Glaois theory is discovered later...see works of Janelidze. I might have sent you the book long time ago, look at old emails. if not ask me tomorrow. Running now...

• CommentRowNumber6.
• CommentAuthorTim_Porter
• CommentTimeFeb 5th 2010

Urs, The main reference to Galois Theories in category theory is as Zoran says: F. Borceux and G. Janelidze, 2001, Galois theories , volume 72 of Cambridge Studies in Advanced Mathematics , Cambridge University Press.

That book has a very good treatment of all the earlier stuff. It introduced Janelidze's theory as well but one really has to look at the original papers (or do the exercises in the book) to get the full picture. There is also the earlier AMS Memoir by Joyal and Tierney.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 5th 2010
• (edited Feb 5th 2010)

The main reference to Galois Theories in category theory is as Zoran says:

I do believe you two on that. But can you give me a shortcut by just sayingh roughly how what is discussed in that book relates to what Jonstone discusses in chapter 8 of Topos theory . Is one of these a special case of the other. If not, what's the difference?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeFeb 5th 2010
• (edited Feb 5th 2010)

Is this in reply to me?

shows how to reconstruct the category of G-sets from a fibre functor ?,

Yes, I know, and in Johnstone's chapter 8 this is done in great generality over any object in any topos.

All I want to know is if this is already what people call "categorical Galois theory" or if it's something else.

In other words, all I want to know is how the first sentence of the entry categorical Galois theory, once that sentence is written, will distinguish it from the material that we already have at Grothendieck's Galois theory and at fundamental group of a topos.

Who of you has read the reference

F. Borceux, G. Janelidze, Galois theories, Cambridge Studies in Adv. Math. 72, 2001. xiv+341 pp.

given at categorical Galois theory? If you have read it, write two sentences into that entry saying what it is about and how it is different from the material at the other entries.

• CommentRowNumber9.
• CommentAuthorTim_Porter
• CommentTimeFeb 5th 2010

I wrote a review of it for the Edinburgh Math Soc. I will look at that and then create a new entry based on it. (Copyright means I cannot use it as it.) Needl;ess to say I have read it from end to end!

1. I guess jack34 is spam (the post is a cut from the wikipedia entry on Grothendieck's Galois theory, with the insertion of a link to a "search engine marketing" (?) ).
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeFeb 5th 2010
This comment is invalid XML; displaying source. <blockquote> <p>I guess jack34 is spam</p> </blockquote> <p>Oh, I see. Stupid me. <blush> I had been wondering about that link...</p> 
• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeFeb 5th 2010

No, categorical Galois theory is not a special case of Grothendieck Galois theory. It is a theory which among the rest does generalize the Grothendieck version of Galois correspondence but does not generalize the general framework of Grothendieck Galois theory. So in a sense neither is generalization of another. Categorical Galois theory is cleaner as it requires some data which are just categorical, actually less, it requires precategories.

• CommentRowNumber13.
• CommentAuthorzskoda
• CommentTimeFeb 5th 2010

But can you give me a shortcut by just sayingh roughly how what is discussed in that book relates to what Jonstone discusses in chapter 8 of Topos theory .

Urs, I can't. The book is about Categorical Galois Theory. It is too abstract for me, and few attempts I made to read it failed. It was 3-4 years ago, maybe I could do it now, but it would take me a lot of time. For guys who are categorically unclined like you may be a better reading than for a noncommutative geometer like me (I was pretty unhappy that the book skips the general case which can cover the Hopf Galois descent in noncommutative case, though it claims it can do it; but that example was THE example of my interest).

• CommentRowNumber14.
• CommentAuthorzskoda
• CommentTimeFeb 5th 2010

It seems Tim will be able to say something more meaningful.

• CommentRowNumber15.
• CommentAuthorzskoda
• CommentTimeFeb 5th 2010

Urs, I have extracted the statement of the Grothendieck's Galois theorem for the classical case of fields as a new section in Grothendieck's Galois theory. It would be a pity having an entry of a theory, without its main theorem.

• CommentRowNumber16.
• CommentAuthorzskoda
• CommentTimeFeb 5th 2010

I added a Galois theorem for Grothendieck topoi.

• CommentRowNumber17.
• CommentAuthorTim_Porter
• CommentTimeFeb 5th 2010

I have created an entry SGA1 to anchor the original references.

• CommentRowNumber18.
• CommentAuthorTim_Porter
• CommentTimeFeb 5th 2010

I found some floating terms I needed to refer to Profinite spaces, they weren't there, so I created them and linked to Stone spaces, .... oh dear,they weren't there either!! They are now!

• CommentRowNumber19.
• CommentAuthorzskoda
• CommentTimeFeb 5th 2010
• (edited Feb 5th 2010)

Entry category over a category cites SGA I with arXiv number and link, what is useful online, so I corrected the entry with a more useful multiple citation. To note that we also have entry EGA which should be the top entry from which all entries to EGA, SGA and FGA volumes should stem.

• CommentRowNumber20.
• CommentAuthorTim_Porter
• CommentTimeFeb 5th 2010

That is good. Perhaps we should have a SGA page with a brief summary of the contents (not a detailed list every time) of each SGA and the links to those that are available in NUMDAM or elsewhere. If we keep it brief that would not be dic=fficult to do but might prove useful for other researchers (and oursevles!!!!)

• CommentRowNumber21.
• CommentAuthorTobyBartels
• CommentTimeFeb 5th 2010

I guess jack34 is spam

I deleted that comment (#8) to reduce the risk that somebody might click on the link (which puts referer headers in their logs indicating that the ad is successful, encouraging more spam … I guess). If anyone wants to read it, it was (as domenico pointed out) from Wikipedia's article.

• CommentRowNumber22.
• CommentAuthorzskoda
• CommentTimeFeb 5th 2010

Great, Toby, thanks.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeFeb 5th 2010

Thanks, Zoran. I added some hyperlinks to The classical case of fields

• CommentRowNumber24.
• CommentAuthorzskoda
• CommentTimeFeb 7th 2010

The Galois correspondence there is sort of descent "along torsors" theorem. Like equivariant sheavs on total space vs usual sheavs over base space. In categorical Galois theory also some case of an adjunction and some case of descent play a role; plus generalizations of profinite business. I am not getting it yet.

• CommentRowNumber25.
• CommentAuthorTim_Porter
• CommentTimeAug 15th 2011

I just came upon a neat master’s thesis from IST Lisboa that gives a nice (classical) treatment of the Galois theory side of SGA1 . The link is here. It does nothing that is really needed for us but it may be useful in introducing the ideas to some students that you have.

2. Removing the redirect for Galois category, as I will create a more focused page on this notion. Not altering the content of the page for the moment until the Galois category page is in a good state.