#### Volume 17, issue 2 (2017)

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Operad bimodules and composition products on André–Quillen filtrations of algebras

### Nicholas J Kuhn and Luís Alexandre Pereira

Algebraic & Geometric Topology 17 (2017) 1105–1130
##### Abstract

If $\mathsc{O}$ is a reduced operad in a symmetric monoidal category of spectra ($S$–modules), an $\mathsc{O}$–algebra $I$ can be viewed as analogous to the augmentation ideal of an augmented algebra. From the literature on topological André–Quillen homology, one can see that such an $I$ admits a canonical (and homotopically meaningful) decreasing $\mathsc{O}$–algebra filtration $I\stackrel{\sim }{←}{I}^{1}←{I}^{2}←{I}^{3}←\cdots \phantom{\rule{0.3em}{0ex}}$ 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 $\mathsc{O}$–bimodule $M$ defines an endofunctor of the category of $\mathsc{O}$–algebras in $R$–modules by sending such an $\mathsc{O}$–algebra $I$ to $M{\circ }_{\mathsc{O}}I$. We explore the use of the bar construction as a derived version of this. Letting $M$ run through a decreasing $\mathsc{O}$–bimodule filtration of $\mathsc{O}$ 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 ${\left({I}^{i}\right)}^{j}\to {I}^{ij}$, fitting nicely with previously studied structure.

As a formal consequence, an $\mathsc{O}$–algebra map $I\to {J}^{d}$ induces compatible maps ${I}^{n}\to {J}^{dn}$ 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.