The universal Hall bialgebra of a double 2-Segal space

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.