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