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Hall algebras via 2-Segal spaces

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arxiv 2409.19384 v1 pith:NSO2TYCD submitted 2024-09-28 math.CT math.ATmath.RT

Benjamin Cooper , Matthew B. Young This is my paper · ORCID
classification math.CTmath.ATmath.RT
keywords hallsegalalgebrasspacesconstructionexplainperspectivealgebra
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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.

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Cited by 1 Pith paper

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  1. 2-Segal sets and pseudomonoids in the bicategory of spans

    math.CT 2025-05 conditional novelty 4.0 of 10

    2-Segal sets are shown to correspond one-to-one, up to isomorphism, with pseudomonoids in the bicategory of spans, using a graphical proof that avoids higher category theory.

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