Representations of N^-type combinatorial categories

In this paper we consider representations of certain combinatorial categories, including the poset of positive integers and division, the Young lattice Y of partitions of finite sets, the opposite category of the orbit category Z of (Z, +) with respect to nontrivial subgroups, and the category 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 (Z, \, J₀ₓ, \, C), and show that irreducible sheaves are parameterized by primitive roots of the unit.