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This entry is lacking a good textbook reference. It used to list, on the one extreme, an expository webpage, and on the other end a bunch of references which are citeable but way too over-specialized for any reader who just needs to look up what a torsor is.
I have now added pointer to
which comes closer, but still buries the simple idea under faithfully flat verbiage.
Do we maybe have a discrete group theory textbook that states the definition in a way useful for those readers who don’t already know it?
I don’t know about a textbook reference, but this paper by Breen cites the definition on page 2 here, and it doesn’t look too over-specialized for a quick look-up.
Okay, I have added a line saying “see also at principal bundle”.
But what I have in mind here is discussion of less than that: torsors over the point, preferably in discrete sets. Such as the default meaning when people speak of $\mathbb{Z}$-torsors. Such as in John Baez’s expository note but inside a citable textbook. Probably John wrote that note because such textbooks are not common.
Do such textbooks exist? I’ve just checked Artin, Lang, Dummit–Foote, Aluffi, and none of them mention torsors.
Thanks for checking! Maybe not.
Our Symmetry book mentions torsors (see current Sec. 4.6+4.7), and is meant to be an undergraduate textbook. But of course, the book is still far from finished, and we take as the main definition of a torsor that it’s any G-set merely equal to the principal G-torsor. (I don’t remember if we prove the equivalence with the standard definition yet, but it’ll be there in time—Ch. 4 is not very polished yet.)
Thanks, that sounds promising! But I don’t see a book behind that link. Do you have a pointer to a pdf for me?
Here you go — but as I said, it’s still quite preliminary.
In a model theoretic context of definable sets, principal homogeneous spaces are studied in
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