The universal Hall bialgebra of a double 2-Segal space

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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 Gálvez-Carrillo-Kock-Tonks as a unifying framework for Hall algebras: every 2-Space defines an algebra in the ∞-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 (∞,2)-category of bispans. Moreover, we show how examples of double 2-Segal spaces arise from Waldhausen's S-construction.