Higher categorical aspects of Hall Algebras

TL;DR

This paper explores the higher categorical structures of Hall algebras, emphasizing their relation to Waldhausen's S-construction, and extends classical theory to broader categorical contexts.

Contribution

It introduces higher categorical perspectives on Hall algebras, connecting them with Waldhausen's S-construction, and broadens their applicability beyond classical abelian categories.

Findings

Establishes links between Hall algebras and higher category theory.

Extends classical Hall algebra constructions to more general categorical settings.

Highlights the role of Waldhausen's S-construction in understanding Hall algebras.

Abstract

These are extended notes for a series of lectures on Hall algebras given at the CRM Barcelona in February 2015. The basic idea of the theory of Hall algebras is that the collection of flags in an exact category encodes an associative multiplication law. While introduced by Steinitz and Hall for the category of abelian p-groups, it has since become clear that the original construction can be applied in much greater generality and admits numerous useful variations. These notes focus on higher categorical aspects based on the relation between Hall algebras and Waldhausen's S-construction.