Operad Bimodule Characterization of Enrichment. V2

In a recent talk at CT06 http://faculty.tnstate.edu/sforcey/ct06.htm and in a research proposal at http://faculty.tnstate.edu/sforcey/class_home/research.htm a definition of weak enrichment over a strict monoidal n-category is introduced. We would like to generalize the definition of enrichment in a way which fits naturally into the world of weakened category theory, where multiplication and composition are unbiased and parameterized rather than being strictly binary and associative. The basic idea of classical enrichment is to allow a general binary product in a some category to reprise the role which the cartesian product of sets usually plays in describing binary composition of morphisms. This role is that of forming the domain for composition. It seems that in the new world we should expand the idea of enrichment to allow an unbiased and weakly associative product to form the domain for composition, while simultaneously allowing that composition itself to be weakened. The structure which appears to accomplish this is that of a bimodule. Broadly speaking a bimodule is an object upon which two other objects may act, from two different “directions.” The actors are monoids: they are each assumed to have an associative, unital, self action of their own. That is, they each possess a multiplication which in turn must be respected by their action on the bimodule they share.