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Hall algebras via 2-Segal spaces
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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.
Forward citations
Cited by 2 Pith papers
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2-Segal sets and pseudomonoids in the bicategory of spans
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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Combinatorial examples and applications of 2-Segal sets
An expository survey introducing 2-Segal sets through graph and tree examples, with worked Hall algebra and discrete Waldhausen S-construction applications.
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