Operadic lifts of the algebra of indexing systems

[摘要] For a given finite group G, the homotopy category of N-infinity G-operads is equivalent to a finite lattice, and Gvaries, there are various image constructions between these lattices. In this paper, we explain how to lift this algebraic structure back to the operad level. We show that lattice joins and meets correspond to N-infinity coproducts and products, and we show that the image constructions correspond to N-infinity induction, restriction, and coinduction constructions, at least when taken along an injective homomorphism. We also prove that a N-infinity variant of the Boardman-Vogt tensor product lifts the join. Our result does not resolve Blumberg and Hill's conjecture that the ordinary tensor product of suitably cofibrant N-infinity operads models the join, but it does imply a closely related result. If O and Pare operads, then an algebra over the Boardman-Vogt tensor product O circle times P is equipped with a pair of interchanging O and P-actions. We prove that under mild hypotheses on a N-infinity operad O, every orthogonal O-ring spectrum is weakly equivalent to a spectrum over an operad O' similar or equal to O that interchanges with itself. (c) 2021 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

[发布日期] 2021-12-01