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