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

TL;DR

This paper explicitly constructs and analyzes filtrations of O-algebras in symmetric spectra, exploring their composition structures and applications to homotopical algebra, particularly in the context of Andre-Quillen homology.

Contribution

It provides explicit constructions of filtrations of O-algebras and studies their composition products, linking operad bimodule structures to Andre-Quillen filtrations.

Findings

Established a formal framework for the augmentation ideal filtration in O-algebras.

Connected operad composition structures to algebra maps between powers of ideals.

Applied the theory to prove a lifting theorem for Hurewicz maps in infinite loop spaces.

Abstract

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.