Operad bimodules, and composition products on Andre-Quillen filtrations   of algebras
  If O is a reduced operad in symmetric spectra, an O-algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. Implicit in the literature on Topological Andre-Quillen homology is that such an I admits a canonical (and homotopically meaningful) decreasing O-algebra filtration I > I^2 > I^3 > ... satisfying various nice properties analogous to powers of an ideal in a ring.   In this paper, we are explicit about these constructions. With R a commutative S-algebra, we study derived versions of the circle product M o_O I, where M is an O-bimodule, and I is an O-algebra in R-modules. 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 induces algebra maps from (I^i)^j to I^{ij}, fitting nicely with previously studied structure.   As a formal consequence, an O-algebra map from I to J^d induces compatible maps from I^n to 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. 
Nicholas J. Kuhn and Luis A. Pereira
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