As I'm sure someone else will say in more detail than I'm qualified to, your question is currently of considerable interest to n-category theorists (Ross Street and Mike Johnson in particular), topologists (Ronnie Brown, Richard Steiner, F. Al-Agl), and at least one computer scientist, namely myself in connection with modeling concurrent computation. A (the?) central question is whether omega-categories (Brown's terminology) (based on either simplexes or cubes) are equivalent to infinity-categories (based on n-cells) or just a special case. For myself, I'm very interested in applying homotopy *monoids* to improving on our the existing notion of computation as a path, generalizing it to a homotopy class. My paper "Modeling Concurrency with Geometry" (available by anonymous ftp as pub/cg.{tex,dvi} from boole.stanford.edu, see pub/README for abstracts of related work) defines a concurrent automaton with n concurrent processes to be an n-category, and (elsewhere in the paper) suggests the notion of monoidal homotopy without giving a knock-down definition of it, and contrasts it with group homotopy. Since then Rob van Glabbeek has encouraged me to look at neither simplicial sets nor n-categories but rather cubical sets. For now these seem somewhat more tractable than n-categories for modeling concurrency, and better capture the essence of concurrent automata. n-categories seem just a bit too general. However I'm still quite open-minded about this since I don't see really persuasive arguments for either. Vaughan Pratt =========================