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nLab > Latest Changes: torsor

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  1.  

  1. fix wrong definition of free group action

    Alexey Muranov

    diff, v32, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 9th 2019
    • (edited Jan 9th 2019)

    Interpreted the freeness and transitivity in terms of the map ⟨ρ,π 2⟩:G×P→P×P\langle \rho, \pi_2 \rangle: G \times P \to P \times P.

    diff, v33, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2021
    • (edited Jan 6th 2021)

    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?

    diff, v36, current

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 6th 2021

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2021

    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 ℤ\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.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 7th 2021

    Do such textbooks exist? I’ve just checked Artin, Lang, Dummit–Foote, Aluffi, and none of them mention torsors.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2021

    Thanks for checking! Maybe not.

    • CommentRowNumber8.
    • CommentAuthorUlrik
    • CommentTimeJan 8th 2021

    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.)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2021

    Thanks, that sounds promising! But I don’t see a book behind that link. Do you have a pointer to a pdf for me?

    • CommentRowNumber10.
    • CommentAuthorUlrik
    • CommentTimeJan 8th 2021

    Here you go — but as I said, it’s still quite preliminary.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2021

    Okay, thanks. I have added the pointer to the list of references (here)

    diff, v38, current

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeJan 22nd 2021

    In a model theoretic context of definable sets, principal homogeneous spaces are studied in

    • Anand Pillay, Remarks on Galois cohomology and definability, The Journal of Symbolic Logic 62:2 (1997) 487-492 doi

    diff, v39, current

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