Hall algebras via 2-Segal spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929520029728768 |
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| author | Cooper, Benjamin Young, Matthew B. |
| author_facet | Cooper, Benjamin Young, Matthew B. |
| contents | This is an introduction to Hall algebras from the perspective of $2$-Segal spaces or decomposition spaces, as introduced by Dyckerhoff and Kapranov and Gálvez-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19384 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hall algebras via 2-Segal spaces Cooper, Benjamin Young, Matthew B. Category Theory Algebraic Topology Representation Theory Primary 54C40, 14E20, Secondary 46E25, 20C20 This is an introduction to Hall algebras from the perspective of $2$-Segal spaces or decomposition spaces, as introduced by Dyckerhoff and Kapranov and Gálvez-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. |
| title | Hall algebras via 2-Segal spaces |
| topic | Category Theory Algebraic Topology Representation Theory Primary 54C40, 14E20, Secondary 46E25, 20C20 |
| url | https://arxiv.org/abs/2409.19384 |