Dear Jim I don't see the big deal about not having a unit in the monoid when looking at (pseudo-)actions; throw one in - we know how it should act. If you agree to this then what you are looking at is a pseudofunctor (or homomorphism of bicategories) R : M --> Cat where M is the monoid regarded as a one-object category. Indeed, M can be any category. There is an equivalence between the 2-category Hom(M,Cat) of homomorphisms from M to Cat and the 2-category Fib/M of opfibrations over M. The study of fibrations has gone off in many directions: for example, it provides the appropriate way of dealing with categories of all sizes (not just small) when working in a topos. As to higher homotopies, Cat doesn't really have enough dimensions for them. But there are trihomomorphisms M --> Bicat. More generally, there will be higher homomorphisms M --> WOC where WOC is the weak omega-category of weak omega-categories - someday - for now we have several fairly good definitions of the objects and arrows of WOC but that's as far as it goes. Higher fibrations is another interesting topic: Claudio Hermida knows about the 2-category case which is relevant to braids since 2-opfibrations over a 2-category M correspond to homomorphisms M --> 2-Cat where 2-Cat is self-enriched via the internal hom for the Gray tensor product of 2-categories (and it is in proving the coherence for this tensor product where braid groups first seriously entered category theory). Also, consider any braided monoidal bicategory B and let t be the n-th tensor power of some object of B. Then there is an action of the kind you describe of the n-string braid group on the hom-category B(t,t). But this is part of a longer story. Some References: John Gray, "Fibred and cofibred categories" Proc Conf Cat Alg, La Jolla 1965 (Springer 1966) [See references to Grothendieck's work in the Gray paper] John Gray, Formal Category Theory SLNM 391 (1974) John Gray, Coherence for the tensor product of 2-categories, and braid groups "Algebra, Topology, and Category Theory" (Academic Press 1976) 63-76 Jean Benabou, Introduction to bicategories SLNM 47 (1967) Jean Benabou, Fibrations petites et localement petites CR Acad Sci Paris A 281 (1975) 897-900 Benabou-Roubaud, Monades et descente CR Acad Sc Paris 270 (1970) 96-98 Gordon-Power-Street, Coherence for tricategories, Memoirs AMS #558 (Sept 1995) Day-Street, Monoidal bicategories and Hopf algebroids, Advances in Math (to appear; galley proofs returned) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ross Street email: street@mpce.mq.edu.au Mathematics Department phone: +612 9850 8921 Macquarie University fax: +612 9850 8114 Sydney, NSW 2109 Australia Internet: http://www.mpce.mq.edu.au/~street/ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~