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

Kuhn, Nicholas¹ ; Pereira, Luís¹

¹Department of Mathematics, University of Virginia, Charlottesville, VA 22904, United States

Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 1105-1130

Résumé

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.

Détail

DOI : 10.2140/agt.2017.17.1105

Classification : 55P43, 18D50
Keywords: operads, Andre–Quillen homology

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