Categorifying the algebra of indexing systems

Abstract

The homotopy category of $N_{\infty }$ operads is equivalent to a finite lattice, and as the ambient group 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 derived operadic coproducts and products, and we show that the image constructions correspond to derived operadic induction, restriction, and coinduction, at least when taken along an injective homomorphism. We also prove that a derived variant of the Boardman-Vogt tensor product lifts the join. Our result does not resolve Blumberg and Hill's conjecture on the usual tensor product, but it does imply that every $N_{\infty }$ ring spectrum can be replaced with an equivalent spectrum, which is equipped with a self-interchanging operad action.

35 pages, comments welcome. Significant portions of this paper were based on work from the author's dissertation, and from an early version of arXiv:1903.08723

References in corpus (2)

• Equivariant symmetric monoidal structures
• Characterizations of equivariant Steiner and linear isometries operads