Talk:Principal homogeneous space

message: :Header added. - Dusa McDuff (Dusam (talk) 14:27, 24 October 2010 (UTC)
thread_title:  Notation

message: :Header added. —Nils von Barth (nbarth) (talk) 18:43, 2 April 2009 (UTC)
thread_title:  Definition 

message: I'm curious about the name “torsor”. Is this statement accurate?:
: This “torque”-related word comes from the action of rotation? That is, if I apply a 20° rotation, it twists everything by 20°—there is a zero rotation, but there needn't be a zero orientation.
Is that about right? Also, does this relate to other mathematical uses of the word “torsion”? —Ben FrantzDale 18:53, 4 January 2007 (UTC)
thread_title: “Torsor”

message: As far as I understand, the definition given here, i.e. a free action of G on E, so that the given map von E x G to E x E is bijective, implies transitivity. It also implies that the action is free. So I don't see the difference to the definition given above.
Florianhe 21:28, 11 January 2007 (UTC)
replies: text: You're correct, assuming that G is an ordinary group, and E is an object whose isomorphisms are determined by bijections on an underlying set.  But when G and E are sheaves on some topological space, for instance, with G being group-valued and E being set-valued, the requirement that the map E x G to E x E be an isomorphism makes sense, whereas requiring that G act freely and transitively on E no longer has any meaning beyond that.  The point is, since E x G --> E x E being an isomorphism is a category-theoretic requirement, that definition will work in a much larger range of categories.  QBobWatson 04:40, 25 September 2007 (UTC)
thread_title:  Section "Other usage" 

message: This is partially in response to the anonymous comment at the top of the page -- I agree that a principal homogeneous space should be nonempty; more generally (as in the "Other Usage" section), a torsor should be locally trivial on the base.  This is certainly required if one is going to classify torsors using cohomology groups.  I'm going to go ahead and update the page to reflect this. QBobWatson 23:42, 23 October 2007 (UTC)
thread_title:  Local Triviality 

message: The English is poor and the sentence structure nonsensical.
Unfortunately, my understanding is insufficient to attempt to edit for style without the possibility of introducing mathematical errors.  — Preceding unsigned comment added by 50.0.193.12 (talk) 14:31, 3 January 2013 (UTC)
replies: text: This is readable and might be worth abstracting some examples from. 67.117.146.66 (talk) 18:05, 8 January 2013 (UTC)
thread_title:  "Applications" section needs a rewrite 

message: I've found a webpage that explains torsors far more clearly than this article: http://math.ucr.edu/home/baez/torsors.html
I don't consider myself up to the task of matching that clarity without direct copying, but I propose that webpage as a model for how this article might be improved. --Brilliand (talk) 01:31, 22 January 2014 (UTC)
thread_title:  Better explanation 

message: I think the following paragraph has an error:
In mathematics, a principal homogeneous space,[1] or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively (meaning that, for any x, y in X, there exists a unique g in G such that x·g = y, where · denotes the (right) action of G on X).
The error is that every point having trivial stabilizer is equivalent to being free, not to being free AND transitive. The correct definition (I think) should require both free AND transitive.
-Sam W
2607:9880:1A18:10A:4D16:B0A6:FC95:5616 (talk) 07:48, 6 December 2020 (UTC)
replies: text: It’s fine since we are taking about a homogeneous space; i.e., the action is transitive. It might be better to say "transitive" more explicitly in the article, though. —- Taku (talk) 00:58, 10 December 2020 (UTC)
thread_title:  I think I spotted an error