Representations of ℕ^∞-type combinatorial categories

2024/04/23 by Zhenxing Di, Liping Li, Di, Zhenxing +3
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2404.15125

openalex publication_date 2024/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider representations of certain combinatorial categories, including the poset \D of positive integers and division, the Young lattice \mathscrY of partitions of finite sets, the opposite category of the orbit category \mathscrZ of (ℤ, +) with respect to nontrivial subgroups, and the category \mathscrCI of finite cyclic groups and injective homomorphisms. We describe explicit upper bounds for homological degrees of their representations, and deduce that finitely presented representations (resp., representations presented in finite degrees) over a field form abelian subcategories of the representation categories. We also give an explicit description for the category of sheaves over the ringed atomic site (\mathscrZ, J_at, \underlineℂ), and show that irreducible sheaves are parameterized by primitive roots of the unit.

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