Hall algebras via 2-Segal spaces

TL;DR

This paper introduces a new perspective on Hall algebras using 2-Segal spaces, connecting them to decomposition spaces and functoriality, and explaining how this approach recovers and extends known Hall algebra constructions.

Contribution

It provides a novel 2-Segal space framework for understanding Hall algebras, including their functoriality and representation theory, unifying various known constructions.

Findings

Recovering Hall algebras from 2-Segal spaces via Waldhausen's S-construction

Establishing functoriality of Hall algebra constructions within the 2-Segal framework

Introducing relative 2-Segal spaces leading to Hall algebra representations

Abstract

This is an introduction to Hall algebras from the perspective of $2$-Segal spaces or decomposition spaces, as introduced by Dyckerhoff and Kapranov and G\'{a}lvez-Carrillo, Kock and Tonks, respectively. We explain how linearizations of the $2$-Segal space arising as the Waldhausen $\mathcal{S}_{\bullet}$-construction of a proto-exact category recover various previously known Hall algebras. We use the $2$-Segal perspective to study functoriality of the Hall algebra construction and explain how relative variants of $2$-Segal spaces lead naturally to representations of Hall algebras.