Modulated categories and their representations via higher categories

2025/06/25 by Fei Xu, Xu, Fei, Minghao Zhang +1
Computer Science · Decision Sciences · #16B50 #16G10 #18N10 #Advanced Algebra and Logic #FOS: Mathematics #Fuzzy and Soft Set Theory #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2506.20426

openalex publication_date 2025/06/25 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28

Abstract

We consider the 3-category 2\mathfrakCat 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 C be a small 1-category and \mathfrakBim_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 C into \mathfrakBim_k a modulation (resp. comodulation) on 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\mathfrakCat, and thus it exhibits an interesting application of higher category theory to representation theory.

Citations

• On cohomological characterizations of endotrivial modules
• Sheaves of modules on atomic sites and discrete representations of topological groups
• 2-Dimensional Categories
• Gluing derived equivalences together
• Hochschild cohomology of abelian categories and ringed spaces

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