Abstract
If is a reduced operad in a symmetric monoidal category of spectra (–modules), an –algebra 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 admits a canonical (and homotopically meaningful) decreasing –algebra filtration 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 a commutative –algebra, an –bimodule defines an endofunctor of the category of –algebras in –modules by sending such an –algebra to . We explore the use of the bar construction as a derived version of this. Letting run through a decreasing –bimodule filtration of 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 , fitting nicely with previously studied structure.
As a formal consequence, an –algebra map induces compatible maps for all . 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.
Citation
Nicholas Kuhn. Luís Pereira. "Operad bimodules and composition products on André–Quillen filtrations of algebras." Algebr. Geom. Topol. 17 (2) 1105 - 1130, 2017. https://doi.org/10.2140/agt.2017.17.1105
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