Re: lax crossed modules

["Urs Schreiber" <Urs.Schreiber@uni-essen.de>]

Ronald Brown wrote, in response to David Roberts:


> I have a gut feeling that these strengthened sesquicategories (with a
> *measure* of the failure of the interchange law) will crop up in a variety
> of situations, e.g. in rewriting, 2-dimensional holonomy, ...., since the
> interchange law makes things too abelian, sometimes.



One can have a 2-holonomy for nonabelian gerbes if a funny condition holds,
called the "fake flatness condition", which is a differential version of the
exchange law, appearing when one realizes a 2-holonomy in a gerbe as a
2-functor from 2-paths to 2-group 2-torsors.

Some people working on bundle gerbes feel that this constraint, which is
derived in the context of strict 2-groups (crossed modules) is "too strong".
While there are straightforward ways to relax conditions in the formalism,
for instance by passing to weak (coherent) structure 2-groups (I guess these
are essentially "the same" as lax crossed modules?) this does not seem to
really address these people's concerns, because after weakening one no
longer deals with Lie groups and Lie algebras, which is what they do.

Hence I'd be extremely interested if somebody came up with a nice weakened
version of crossed modules that would allow to realize 2-holonomy in
non-fake flat gerbes.

Best regards,
Urs Schreiber