The universal Hall bialgebra of a double 2-Segal space

TL;DR

This paper constructs a universal Hall bialgebra from double 2-Segal spaces within an $( abla,2)$-category framework, unifying various Hall algebra structures and their coproducts, with applications to Waldhausen's $S$-construction.

Contribution

It introduces the universal Hall bialgebra of a double 2-Segal space as a lax bialgebra in the $( abla,2)$-category, linking double 2-Segal spaces to Waldhausen's $S$-construction.

Findings

Constructed the universal Hall bialgebra as a lax bialgebra in bispans.

Connected double 2-Segal spaces with Waldhausen's $S$-construction.

Provided a unifying framework for Hall algebras and their coproducts.

Abstract

Hall algebras and related constructions have had diverse applications in mathematics and physics, ranging from representation theory and quantum groups to Donaldson-Thomas theory and the algebra of BPS states. The theory of $2$-Segal spaces was introduced independently by Dyckerhoff-Kapranov and G\'alvez-Carrillo-Kock-Tonks as a unifying framework for Hall algebras: every $2$-Space defines an algebra in the $\infty$-category of spans, and different Hall algebras correspond to different linearisations of this universal Hall algebra. A recurring theme is that Hall algebras can often be equipped with a coproduct which makes them a bialgebra, possibly up to a `twist'. In this paper will explain the appearance of these bialgebraic structures using the theory of $2$-Segal spaces: We construct the universal Hall bialgebra of a double $2$-Segal space, which is a lax bialgebra in the $(\infty,2)$-category of bispans. Moreover, we show how examples of double $2$-Segal spaces arise from Waldhausen's $S$-construction.