An addition to Ross Street's answer. Wilst the notion of weak \omega-functor is not yet worked out adequately the corresponding topological theory of A_{\infty} and E_{\infty} maps between A_{\infty} and E_{\infty}-spaces has been constructed by Boardman,Vogt, May, Segal and others. The theory of natural transformations up to ALL higher homotopies between simplicial functors has been developed by Dawer,Kan, Cordier, Porter, Bourn, Batanin, Heller and others. Simplicial A_{\infty}-categories and A_{\infty}-functors were defined and studied by Batanin and topological version of it by Schwanzl and Vogt (they call them \Delta-categories and use the idea related to the Segal delooping mashine.) Some (not all) references: 1. Batanin M.A., Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers Topologie et Geom. Diff., vol.XXXIV-4, pp.279-304, 1993. 2. Batanin M.A., Homotopy coherent category theory and A_{\infty}-structures in monoidal categories, to appear, dvi file available at http://www-math.mpce.mq.edu.au/~mbatanin/papers.html 3. Boardman J.M., R.M.Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Math., vol. 347, Springer-Verlag, Berlin, Heidelberg, New York, 1973. 4. Cordier J.-M., Porter T., Vogt's Theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Phil. Soc., vol. 100, pp 65-90, 1986. 5. Cordier J.-M., Porter T., Maps between homotopy coherent diagrams, Topology and its Appl., 28, pp.255-275, 1988. 6. Cordier J.-M., Porter T., Homotopy coherent category theory, to appear in Transactions of the AMS, 7. Dwyer W.G., Kan D.M., Realizing diagrams in the homotopy category by means of diagrams of simplicial sets, Proc. Amer. Math. Soc., 91, pp.456-460, 1984. 8.Heller A., Homotopy in Functor Categories, Transactions AMS., v.272, pp.185-202, 1982. 10. Schw\"{a}nzl R., Vogt R., Homotopy homomorphisms and the Hammock localization, Boletin de la Soc. Mat. Mexicana, 37, 1-2, pp.431-449, 1992.