Urs Schreiber
20050414 10:42:49 UTC
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Please note, the kernel of the idea here comes from a comment in an
email from B Jurco to my supervisor, Michael Murray.
In Baez and Schreiber's paper `2connections on 2bundles', they talk
about the automorphism 2group AUT(H) corresponding to the crossed
module t:H Aut(H). I've been looking at nonabelian bundle gerbes
(NABG) and one way to define them is to look at an Hbitorsor (or
principal Hbibundle) P defined on the fibre product Y^[2] of a
submersion Y \to M. In that case there is a local automorphism (comes
from a section, and here i indexes a cover of Y^[2] = \coprod U_i)
u_i : U_i \to Aut(H)
(called in AschieriCantiniJurco (hepth/0312154) \varphi_e  I've
`locally trivialised' my bitorsor) which obeys
h.s_i = s_i.u_i(h).
However, an automorphism is just a selfdiffeomorphism respecting the
group structure of the Lie group H. When we look at the contracted
product of two Hbitorsors
However, for a general diffeomorphism f:H \to H (here's the point,
finally), this is not so.
And this is what we would _like_, considering that for general
bitorsors (thinking more alg. geom here),
As far as I am aware this idea arose in a discussion when I was visitingemail from B Jurco to my supervisor, Michael Murray.
In Baez and Schreiber's paper `2connections on 2bundles', they talk
about the automorphism 2group AUT(H) corresponding to the crossed
module t:H Aut(H). I've been looking at nonabelian bundle gerbes
(NABG) and one way to define them is to look at an Hbitorsor (or
principal Hbibundle) P defined on the fibre product Y^[2] of a
submersion Y \to M. In that case there is a local automorphism (comes
from a section, and here i indexes a cover of Y^[2] = \coprod U_i)
u_i : U_i \to Aut(H)
(called in AschieriCantiniJurco (hepth/0312154) \varphi_e  I've
`locally trivialised' my bitorsor) which obeys
h.s_i = s_i.u_i(h).
However, an automorphism is just a selfdiffeomorphism respecting the
group structure of the Lie group H. When we look at the contracted
product of two Hbitorsors
However, for a general diffeomorphism f:H \to H (here's the point,
finally), this is not so.
And this is what we would _like_, considering that for general
bitorsors (thinking more alg. geom here),
Branislav Jurco and Paolo Aschieri in Torino/Italy last year. I was
mentioning how it seemed to me that 2bundles with weak coherent structure
2groups (as opposed to strict 2groups), whose product is not associative
on the nose, would capture the idea of a basespacedependent group product
in some sense and hence account for the 'algebra bundle'freedom that Andrew
Neitzke identified as a plausible candidate for the n^3scaling behaviour on
5branes:
http://golem.ph.utexas.edu/string/archives/000461.html
Branislav Jurco and Paolo Aschieri noted that this idea might correspond to
the "weak automorphisms" in a NABG that you are discussing in your post.
the product (where defined,
if we want HGbitorsors) is _not_ associative. Also, dealing with a
\in Aut(H) which satisfies
a(h_1).a(h_2) = a(h_1.h_2)
^


bad!
seems very uncategorylike.
Right, and the way to do it is to go to coherent 2groups instead. Butif we want HGbitorsors) is _not_ associative. Also, dealing with a
\in Aut(H) which satisfies
a(h_1).a(h_2) = a(h_1.h_2)
^


bad!
seems very uncategorylike.
coherent 2groups are much less well understood than strict ones. Since we
know that a strict one is just a crossed module, we would want to know which
weak form of a crossed modules describes a coherent 2group. I have once
started working that out
(http://golem.ph.utexas.edu/string/archives/000471.html), but it's not
really finished yet.
One `snag'  H \to Diff(H) won't give us a Lie crossed module, but
something weaker. This is probably where the coherent
Lie2group/?something like a crossed module? correspondence comes
in.
Yes, that's what I am talking about above.something weaker. This is probably where the coherent
Lie2group/?something like a crossed module? correspondence comes
in.
Can this concept be made a bit less handwavy?
There is a precise way to define a coherent 2group, a 2bundle with acoherent 2group as structure 2group as well as what connection and curving
on such a 2bundle would be. There is also a known way how to get a
nonabelian gerbe from a strict 2bundle.
What is hard is to fill the definition of a 2connection for a coherent
2group with life by given concrete realizations in terms of local data. One
possible approach I am discussing here:
http://golem.ph.utexas.edu/string/archives/000542.html. There is also a
complementary approach using connections on path space which is more
directly related with Hofman's ideas
I thought I'd have lots of time to work this out more completely. But now
that Adelaide is also working on this... :)
If these 2bundle ideas help you to work out the construction of a weakened
nonabelian bundle gerbe I don't know. But since all this is really just
different ways to look at the same thing it should really be related.