reply to here Vaughan, David An interesting question! It raises several possible red herrings. 1) What is a lax action of a group (or groupoid) G on a group (or groupoid) A? There is a paper by Brylinski in Cahier on this, with applications to K-theory, if I remember rightly. Another interpretation of this seems to be as a Schreier cocycle (factor set). A relevant paper is 97. (with T. PORTER), ``On the Schreier theory of non-abelian extensions: generalisations and computations''. {\em Proceedings Royal Irish Academy} 96A (1996) 213-227. It is a useful exercise (which I meant to write down, but ...) to translate Brylinski into the terms of a map of a free crossed resolution, and so put this into nonabelian cohomology terms, and potentially allow for calculation using a small free crossed resoution when possible .... This suggests what might be a lax action, but does not complete in an obvious way into a lax crossed module. 2) Another way is to go to 2-crossed modules (Daniel Conduche), which brings in relations with simplicial groups (Tim Porter) and higher Peiffer elements. See also the relations with braided crossed modules and other things in 59. (with N.D. GILBERT), ``Algebraic models of 3-types and automorphism structures for crossed modules'', {\em Proc. London Math. Soc.} (3) 59 (1989) 51-73. 3) There are equivalences of categories crossed modules of groupoids <--> 2-groupoids <--> double groupoids with connections <--> double groupoids with thin elements. I have long found the cubical approach easier to follow and to use than the globular, but it turns out one needs also the globular to define commutative cubes in cubical omega-categories with connections (see a recent paper by Philip Higgins in TAC). This raises the question of "lax cubical omega-categories with connections". What do you laxify, to get an equivalence with one or other notion of weak globular (or other?) omega-category??!! Quite an amusing step, and more do-able, would be to generalise Gray categories to: cubical omega-categories C with an algebra structure C \otimes C \to C, generalising Brown-Gilbert, and using the monoidal closed structure given in 116. (with F.A. AL-AGL and R. STEINER), `Multiple categories: the equivalence between a globular and cubical approach', Advances in Mathematics, 170 (2002) 71-118 A nice point about such algebra structures is that they allow for a failure of the interchange law, with a measure of that failure, similar to the way 2-crossed modules give a measure of the failure of the Peiffer law for a crossed module by using a map { , }: P_1 \times P_1 \to P_2. Is this related to Sjoerd Crans' teisi? 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. This brings in automorphism structures for crossed modules, I guess (Brown-Gilbert again, and of course derived from JHC Whitehead, who first studied such automorphisms). Another thought: the non abelian tensor product of groups derives from properties of the commutator map on groups. Why not develop the corresponding theory for the Peiffer commutator map? Hope that helps Ronnie ------------------------------------------------- *Date:* Mon, 19 Sep 2005 09:41:44 -0700 *From:* Vaughan Pratt <pratt@cs.stanford.edu> *To:* Ronnie Brown <mas010@bangor.ac.uk> *Reply-to:* pratt@cs.stanford.edu *Subject:* [Fwd: categories: Question re lax crossed modules] I'd be interested in knowing this too, in particular what the geometric significance of laxness is. Presumably laxness only enters in the passage from pre-crossed to crossed. Vaughan -------- Original Message -------- Subject: categories: Question re lax crossed modules Date: Mon, 19 Sep 2005 12:44:30 +0930 From: David Roberts <droberts@maths.adelaide.edu.au> To: categories@mta.ca I have been looking at categorical groups a little and was wondering what a lax crossed module is. A search through various databases has turned up nothing. It would seem that they should be like crossed modules but only satisfy a weakened equivariance property. Any pointers toward a definition would be great. ------------------------------------------------------------------------ -- David Roberts School of Mathematical Sciences University of Adelaide SA 5005 ------------------------------------------------------------------------ -- droberts@maths.adelaide.edu.au www.maths.adelaide.edu.au/~droberts