Modulated categories and their representations via higher categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912526688583680 |
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| author | Xu, Fei Zhang, Maoyin |
| author_facet | Xu, Fei Zhang, Maoyin |
| contents | We consider the 3-category $2\mathfrak{C}at$ whose objects are 2-categories, 1-morphisms are lax functors, 2-morphisms are lax transformations and 3-morphisms are modifications. The aim is to show that it carries interesting representation-theoretic information.
Let $\mathcal{C}$ be a small 1-category and $\mathfrak{B}im_k$ be the 2-category of bimodules over $k$-algebras, where $k$ is a commutative ring with identity. We call a covariant (resp. contravariant) pseudofunctor from $\mathcal{C}$ into $\mathfrak{B}im_k$ a modulation (resp. comodulation) on $\mathcal{C}$, define and study its representations. This framework provides a unified approach to investigate 2-representations of finite groups, modulated quivers and their representations, as well as presheaves of $k$-algebras and their modules. Moreover, several key constructions are natural ingredients in $2\mathfrak{C}at$, and thus it exhibits an interesting application of higher category theory to representation theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20426 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modulated categories and their representations via higher categories Xu, Fei Zhang, Maoyin Representation Theory 16B50, 16G10, 18N10 We consider the 3-category $2\mathfrak{C}at$ whose objects are 2-categories, 1-morphisms are lax functors, 2-morphisms are lax transformations and 3-morphisms are modifications. The aim is to show that it carries interesting representation-theoretic information. Let $\mathcal{C}$ be a small 1-category and $\mathfrak{B}im_k$ be the 2-category of bimodules over $k$-algebras, where $k$ is a commutative ring with identity. We call a covariant (resp. contravariant) pseudofunctor from $\mathcal{C}$ into $\mathfrak{B}im_k$ a modulation (resp. comodulation) on $\mathcal{C}$, define and study its representations. This framework provides a unified approach to investigate 2-representations of finite groups, modulated quivers and their representations, as well as presheaves of $k$-algebras and their modules. Moreover, several key constructions are natural ingredients in $2\mathfrak{C}at$, and thus it exhibits an interesting application of higher category theory to representation theory. |
| title | Modulated categories and their representations via higher categories |
| topic | Representation Theory 16B50, 16G10, 18N10 |
| url | https://arxiv.org/abs/2506.20426 |