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

TL;DR

This paper generalizes the Bergner-Osorno-Ozornova-Rovelli-Scheimbauer equivalence by establishing broader relationships between 2-Segal spaces, abacus bicomodule configurations, and simplicial maps, extending prior results.

Contribution

It introduces a more general framework connecting 2-Segal spaces and abacus bicomodule configurations, extending the BOORS equivalence and analyzing augmentation notions.

Findings

Established new equivalences involving simplicial maps and bicomodule configurations.

Extended the BOORS equivalence to a broader context.

Clarified the relationship between different augmentation concepts.

Abstract

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.