Re: gr-stacks in the literature

Re: gr-stacks in the literature

[Ettore Aldrovandi]

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)

Re: gr-stacks in the literature

[David Roberts]

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)

Re: gr-stacks in the literature

[jim stasheff]

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.
>
>

gr-stacks in the literature

[David Roberts]

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