The result of Maltsiniotis referred to by Jonathan is very welcome. But I wonder if there is still a problem with cubical sets with connection: the geometric realisation of a simplicial group is, in a convenient category, a topological group, because of the homeomorphism f: |K \times Y| \to |K| \times |Y| . However in the case of cubical sets with connections this map f is a homotopy equivalence but it seems is not a homeomorphism (?). As Grothendieck wrote: `homotopically speaking' that is not a problem! For homotopies and higher homotopies cubes are nice and easy because of the basic formula I^m \times I^n = I^{m+n}. This leads to monoidal closed structures on strict cubical higher categories and groupoids. For a basic discussion of other issues such as algebraic inverses to subdivision and commutative cubes I refer to my 2009 Liverpool seminar on`What is and what should be `Higher dimensional group theory'?' http://pages.bangor.ac.uk/~mas010/pdffiles/liverpool-beamer-handout.pdf Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]