Operadic lifts of the algebra of indexing systems

Abstract

Abstract For a given finite group G , the homotopy category of N ∞ G -operads is equivalent to a finite lattice , and G varies, 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 ∞ coproducts and products, and we show that the image constructions correspond to N ∞ induction, restriction, and coinduction constructions, at least when taken along an injective homomorphism. We also prove that a N ∞ 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 ∞ operads models the join, but it does imply a closely related result. If O and P are operads, then an algebra over the Boardman-Vogt tensor product O ⊗ P is equipped with a pair of interchanging O and P -actions. We prove that under mild hypotheses on a N ∞ operad O , every orthogonal O -ring spectrum is weakly equivalent to a spectrum over an operad O ′ ≃ O that interchanges with itself.