lax crossed modules

lax crossed modules

[Ronald Brown]

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

Re: lax crossed modules

[jim stasheff]

Ronnie,

     I don't see these words below but `lax functor' is what came to mind.
As a monoid is a category with one object,
what is the many object version of an ordinary crossed module?

jim

Re: lax crossed modules

[Urs Schreiber]

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

Re: lax crossed modules

[John Baez]

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