gr-stacks in the literature
Dear all, after a bit of searching, I cannot find much in the literature about gr- stacks, more specifically, charts and presentations thereof reflecting (in a non-technical sense) the group-like structure. Also, aside from self equivalences of gerbes and quotients of groups (G/H for non-normal H), I cannot dream up other "interesting" examples - and these are the opposite ends of the spectrum I want to consider. All pointers welcome. -- David Roberts Pure Mathematics University of Adelaide South Australia, 5005
Greetings: I take this chance to post my first message to this list... On Sat, Apr 29, 2006 at 01:27:20PM +0930, David Roberts wrote:
Dear all,
after a bit of searching, I cannot find much in the literature about gr- stacks, more specifically, charts and presentations thereof reflecting (in a non-technical sense) the group-like structure. Also, aside from self equivalences of gerbes and quotients of groups (G/H for non-normal H), I cannot dream up other "interesting" examples - and these are the opposite ends of the spectrum I want to consider.
Another example would be stack associated to the presheaf of groupoids defined by a crossed module d: G --> H. You end up with the stack of G-torsors equipped with a trivialization of the associated H-torsor via the homomorphism d. But because you have a crossed module, the trivialization can be used to induce a G-bitorsor structure compatible with the section giving the trivialization above. The bitorsor product will give the gr-stack structure. This is spelled out in some detail in L. Breen's "Bitorseurs et cohomologie non ab\'elienne," in The Grothendieck Festschrift, Vol. I. Also, Aut(x) for x an object in a 2-gerbe gives an example. Hope this helps a bit, -- Ettore Aldrovandi Department of Mathematics http://www.math.fsu.edu/~ealdrov Florida State University aldrovandi at math.fsu.edu Tallahassee, FL 32306-4510, USA +1 (850) 644-9717 (FAX: 4053)
You've seen Larry Breen's Asterisque vol and others of his papers? David Roberts wrote:
Dear all,
after a bit of searching, I cannot find much in the literature about gr- stacks, more specifically, charts and presentations thereof reflecting (in a non-technical sense) the group-like structure. Also, aside from self equivalences of gerbes and quotients of groups (G/H for non-normal H), I cannot dream up other "interesting" examples - and these are the opposite ends of the spectrum I want to consider.
All pointers welcome.
Thanks for replies. I am familiar with Breen's work, and the correspondence between crossed modules and gr-stacks. I ws thinking more along the lines of "a smooth gr- chart K (or whatever it is called) for a differentiable gr-stack G is a representable surjection K --> G satisfying the properties..." (with some 2- cartesian squares following). Such a thing might be in Champs Algebriques , in the category of schemes, but I haven't got it at the library. -- David Roberts Pure Mathematics University of Adelaide South Australia, 5005 You know we all became mathematicians for the same reason: we were lazy. -Max Rosenlicht(1949)
Thanks for replies. I am familiar with Breen's work, and the correspondence between crossed modules and gr-stacks. I ws thinking more along the lines of "a smooth gr- chart K (or whatever it is called) for a differentiable gr-stack G is a representable surjection K --> G satisfying the properties..." (with some 2- cartesian squares following). Such a thing might be in Champs Algebriques , in the category of schemes, but I haven't got it at the library. -- David Roberts Pure Mathematics University of Adelaide South Australia, 5005 You know we all became mathematicians for the same reason: we were lazy. -Max Rosenlicht(1949)
participants (3)
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David Roberts -
Ettore Aldrovandi -
jim stasheff