I have not seen Fukaya's paper. If the problem is to formalise higher homotopies for, say, topological spaces or chain complexes, together with their operations, there is a simple solution based on the path endofunctor (or, dually, the cylinder endofunctor; or their adjunction) and its powers. The path endofunctor P is constructed so that a homotopy a: f -> g: X -> Y amounts to a map a: X -> PY; the maps f, g are recovered by means of the two faces d-, d+: PY -> Y. (For Top, PY is obviously the space Y^[0, 1] of paths in Y, with the compact-open topology; for chain complexes (PY)_n = Y_n + Y_(n+1) + Y_n, with suitable differential.) P comes equipped with various natural transformations (faces, degeneracy, connections, symmetries, concatenation...), satisfying "algebraic" coherence axioms (a sort of "cubical comonad" with additional structure). These transformations represent the basic structure of lower order homotopies. But now you have, practically for free: - n-tuple homotopies, represented by the power endofunctor P^n, - n-homotopies, represented by a subfunctor P_n of P^n, - their operations, via the usual algebra of natural transformations, - the deduced coherence relations of the latter. Eg, a double homotopy, represented by a map X -> P^2(Y), has four faces given by the four natural transformations d-P, d+P, Pd-, Pd+: P^2 -> P; if k is the concatenation of homotopies, kP and Pk are the vertical and horizontal concatenation of double homotopies, and so on; if k is associative, as is the case for chain complexes but not for spaces, so are all higher concatenations. A double homotopy is said to be a 2-homotopy when its "vertical" faces (say) are trivial, i.e. factor through the degeneracy 1 -> P. (For Top, P^2(Y) is the space of maps from the standard square [0, 1]^2 to Y; P_2(Y) is the subspace of those maps which are constant on the vertical faces of the square.) This way of deducing higher homotopies and their operations from the lower ones can be found in: M. Grandis, Categorically algebraic foundations for homotopical algebra, Appl. Categ. Structures 5 (1997), 363-413. M. Grandis, On the homotopy structure of strongly homotopy associative algebras, J. Pure Appl. Algebra, 134 / 1 (1999 ?), 15-81. Regards, Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it http://pitagora.dima.unige.it/webdima/STAFF/GRANDIS/ ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/