Ronnie Brown wrote; Has anyone formulated a cubical (square?) version of bicategory or higher?
Dominic Verity invented "double bicategories", which are the fully weakened version of double categories. Unfortunately, his thesis can only be obtained by special shipment from Australia. Someday someone will scan it and make it more widely available! Luckily, you can see the definition in this paper: Jeffrey Morton Double bicategories and double cospans http://arxiv.org/abs/math/0611930 For higher dimensions, try: Marco Grandis HIGHER COSPANS AND WEAK CUBICAL CATEGORIES (COSPANS IN ALGEBRAIC TOPOLOGY, I) http://tac.mta.ca/tac/volumes/18/12/18-12abs.html and also parts II and III of this series. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]