Re: lax crossed modules

["John Baez" <baez@math.ucr.edu>]

Urs Schreiber wrote:

> [...] weak (coherent) structure 2-groups (I guess these
> are essentially "the same" as lax crossed modules?) [...]

Since not everyone will understand this remark by my esteemed
coauthor, let me elaborate.

There's a nice way to weaken the concept of crossed module.
A crossed module is just another way of looking at a group
object in Cat - otherwise known as a "categorical group" or
"strict 2-group".

But, starting with the concept of group object in Cat, one can
then weaken the usual group axioms to natural isomorphisms
and impose suitable coherence laws, obtaining the notion of
"gr-category" or "coherent 2-group".

One could then backtrack and formulate this concept so that it
resembles the concept of crossed module as closely as possible.
I guess this would deserve to be called a "weak crossed module"
or something like that.

All this stuff except the last paragraph is well-known and
summarized here:

John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-Groups,
Theory and Applications of Categories 12 (2004), 423-491.
http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html

One might also seek a "lax" version of the concept of crossed
module, where "lax" is taken in the Australian sense of replacing
equations by morphisms rather than isomorphisms - "lax" as opposed
to "pseudo".  If I were forced to do this, I'd try to do it by
laxifying the concept of group object in Cat.  But, I don't see
which way all the 2-arrows should point.

Best,
jb