The following arxiv.org submission may be of interest to subscribers of this list: http://arxiv.org/abs/1005.3979 It is in a similar vein to the paper: C. Balteanu, Z. Fiedorowicz, R. Schwanzl, and R. M. Vogt, Iterated monoidal categories, Adv. in Math. 176 (2003), 277-349 where a hierarchy of commutativity conditions on categories were analyzed that correspond to those on iterated loop spaces upon passage to nerves. In this paper we analyze a hierarchy of associativity conditions on monoidal structures on categories which correspond to Stasheff's A_n hierarchy on H-spaces upon passage to nerves. Zig Fiedorowicz [For admin and other information see: http://www.mta.ca/~cat-dist/ ]