Hall algebras via 2-Segal spaces

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.