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

Abstract

If $\mathcal{O}$ is a reduced operad in a symmetric monoidal category of spectra ($S$–modules), an $\mathcal{O}$–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 $\mathcal{O}$–algebra filtration $I\stackrel{\sim }{\leftarrow }{I}^1\leftarrow I^2\leftarrow I^3\leftarrow \cdots$ 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 $\mathcal{O}$–bimodule $M$ defines an endofunctor of the category of $\mathcal{O}$–algebras in $R$–modules by sending such an $\mathcal{O}$–algebra $I$ to $M{\circ }_{\mathcal{O}}I$. We explore the use of the bar construction as a derived version of this. Letting $M$ run through a decreasing $\mathcal{O}$–bimodule filtration of $\mathcal{O}$ 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 $\mathcal{O}$–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.