Concentration structures on categories and horizontal categorification
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918157084524544 |
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| author | Luo, Yangxiao Wan, Shunyu |
| author_facet | Luo, Yangxiao Wan, Shunyu |
| contents | We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\textit{concentration monoid}$, which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that every group arises as the concentration monoid of a trivial category, up to category equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07553 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration structures on categories and horizontal categorification Luo, Yangxiao Wan, Shunyu Category Theory Algebraic Topology Geometric Topology We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\textit{concentration monoid}$, which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that every group arises as the concentration monoid of a trivial category, up to category equivalence. |
| title | Concentration structures on categories and horizontal categorification |
| topic | Category Theory Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2510.07553 |