Representations of ${\mathbb{N}}^{\infty }$-type combinatorial categories

Abstract

In this paper we consider representations of certain combinatorial categories, including the poset $\D$ of positive integers and division, the Young lattice $\mathcal{Y}$ of partitions of finite sets, the opposite category of the orbit category $\mathcal{Z}$ of $(\mathbb{Z},+)$ with respect to nontrivial subgroups, and the category $\mathcal{CI}$ 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 $(\mathcal{Z},J_{at},\underline{\mathbb{C}})$, and show that irreducible sheaves are parameterized by primitive roots of the unit.

A revised version with some small changes