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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

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.

operads, Andre–Quillen homology
Mathematical Subject Classification 2010
Primary: 55P43
Secondary: 18D50
Received: 21 February 2016
Revised: 20 June 2016
Accepted: 1 August 2016
Published: 14 March 2017
Nicholas J Kuhn
Department of Mathematics
University of Virginia
Charlottesville, VA 22904
United States
Luís Alexandre Pereira
Department of Mathematics
University of Virginia
Charlottesville, VA 22904
United States