Realizing compatible pairs of transfer systems by combinatorial $N_\infty$-operads

TL;DR

This paper explores the relationship between pairings of operads and compatible pairs of indexing systems within the framework of $N_ fty$-operads, establishing conditions under which these structures correspond.

Contribution

It demonstrates that pairings of operads induce pairings of indexing systems and that compatible indexing systems can often be realized by pairings of $N_ inf$-operads.

Findings

Pairings of operads induce pairings on indexing systems.

Compatible indexing systems can be realized by $N_ inf$-operad pairings.

Established correspondence between operad pairings and indexing system pairings.

Abstract

We investigate how the notions of pairings of operads of May and compatible pairs of indexing systems of Blumberg--Hill relate via the correspondence between indexing systems and $N_{\infty}$-operads. We show that a pairing of operads induces a pairing on the associated indexing systems. Conversely, we show that in many cases, compatible pairs of indexing systems can be realized by a pairing of $N_{\infty}$-operads.