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

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.

v2: minor update with focused revisions in section 7. v1: 37 pages, 1 figure