Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
  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×PP×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 \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

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)