Operad bimodules and composition products on Andre-Quillen filtrations of algebras

If O is a reduced operad in a symmetric monoidal category of spectra ( S-modules), an O-algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. From the literature on topological Andre-Quillen homology, one can see that such an I admits a canonical ( and homotopically meaningful) decreasing O-algebra filtration I (<-) over tilde I-1 <- I-2 <- I-3 <- ... satisfying various nice properties analogous to powers of an ideal in a ring. We more fully develop such constructions in a manner allowing for more flexibility and revealing new structure. With R a commutative S-algebra, an O-bimodule M defines an endofunctor of the category of O-algebras in R-modules by sending such an O-algebra I to M circle O I. We explore the use of the bar construction as a derived version of this. Letting M run through a decreasing O-bimodule filtration of O itself then yields the augmentation ideal filtration as above. The composition structure of the operad then induces pairings among these bimodules, which in turn induce natural transformations. (I-i)(j) -> I-ij, fitting nicely with previously studied structure. As a formal consequence, an O-algebra map I -> J(d) induces compatible maps I-n -> J(dn) for all n. This is an essential tool in the first author's study of Hurewicz maps for infinite loop spaces, and its utility is illustrated here with a lifting theorem.