More fancies on lax crossed modules and cubical ideas

More fancies on lax crossed modules and cubical ideas

[Ronald Brown]

To pursue some ideas suggested by David Robert's queries on lax crossed
modules and Vaughan Pratt's interest:

As said in the previous contribution: crossed modules (over groupoids) are
equivalent to (edge symmetric) double groupoids with connections or thin
structures, and the latter generalise easily to all dimensions. Why use
crossed modules? For the work with Higgins, the aim was  (a) to relate to
classical invariants (relative homotopy groups) , and (b) for calculations.
One thinks of calculation as serial, so the `linear' crossed modules are
appropriate. But for theory, one wants the clear 2-dimensional compositions,
particularly to get `algebraic inverses to subdivisions' for applications to
`local-to-global problems'.

Now multiple groupoids arose from considering the structure held by the
singular cubical complex of a space, SC(X), which in dimension n consists of
maps I^n --> X.  So SC(X) has some claim to be a model of a weak
omega-groupoid. Now for a filtered space X_* one can consider also R(X_*)
which in dimension n is filtered maps I^n --> X_*. This again is a weak
omega-groupoid, at least as much as SC(X) is.

But an advantage is there is Kan fibration p:R(X_*) --> \rho(X_*) where the
latter is a strict omega-groupoid.

32.  (with P.J. HIGGINS), ``Colimit theorems for relative homotopy
groups'', {\em J. Pure Appl. Algebra} 22 (1981) 11-41.
(except that now we would modify the definition to take homotopies rel
vertices of the cubes, and avoid the J_0-condition, and the theorems still
work).

So this suggests that a `controlled lax omega-groupoid'  R should come
with a Kan fibration R->G where G is a strict omega-groupoid and where R has
lots of lax multiple compositions, [a_{(r)}],  as considered in [32].  (why
not?) This would allow for liftings of multiple compositions from G to R
(loc cit), which should be helpful.  An advantage of cubical over globular
or simplicial is the ease of formulating multiple compositions.

Notice that SC(X) has strict interchange for a 2 x 2 composition, and the
connections have strict transport laws (2 x 2 again) but lax cancellation of
\Gamma^-_i with \Gamma^+_i (in the terms of Al-Agl/Brown/Steiner).

To backtrack a bit:
\mu : M \to P is a crossed module (of groups!) if and only if there is a
pointed fibration F \to E \to B such that \mu is \pi_1 F \to \pi_1 E.
(Loday) The lax version of this is that given such a fibration, then \Omega
F \to \Omega E may be given the structure of lax crossed module, where I
cheat by saying lax means the structure that this has --  `crossed module up
to homotopy'. (I am not sure if this has been written down somewhere!) This
is related to an old paper of Philip R. Heath on, if I remember correctly,
`Groupoid Operations and fibre homotopy equivalences'. Presumably there is
also a recognition principle involved.

This should answer Vaughan's question on the geometry (here topology)
related to lax crossed modules.

Ronnie
www.bangor.ac.uk/r.brown
r.brown@bangor.ac.uk