## Algebraic & Geometric Topology

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

#### Abstract

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 André–Quillen homology, one can see that such an $I$ admits a canonical (and homotopically meaningful) decreasing $O$–algebra filtration $I← ∼I1 ← I2 ← I3 ←⋯$ 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 $M ∘OI$. 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 $(Ii)j → Iij$, fitting nicely with previously studied structure.

As a formal consequence, an $O$–algebra map $I → Jd$ induces compatible maps $In → Jdn$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 17, Number 2 (2017), 1105-1130.

Dates
Revised: 20 June 2016
Accepted: 1 August 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.agt/1508431455

Digital Object Identifier
doi:10.2140/agt.2017.17.1105

Mathematical Reviews number (MathSciNet)
MR3623683

Zentralblatt MATH identifier
1362.55008

Keywords

#### Citation

Kuhn, Nicholas; Pereira, Luís. Operad bimodules and composition products on André–Quillen filtrations of algebras. Algebr. Geom. Topol. 17 (2017), no. 2, 1105--1130. doi:10.2140/agt.2017.17.1105. https://projecteuclid.org/euclid.agt/1508431455

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