Abacus bicomodule configurations and the Bergner-Osorno-Ozornova-Rovelli-Scheimbauer equivalence

A theorem of Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer states an equivalence between 2-Segal spaces and certain augmented stable double Segal spaces. In this paper we establish more general equivalences, involving simplicial maps of 2-Segal spaces and abacus bicomodule configurations, extending results of Carlier. The BOORS equivalence is recovered from the special case of the identity map. One main ingredient is an analysis of the relationship between the BOORS and Carlier notions of augmentation, hitherto considered unrelated.