Representations of N∞-type combinatorial categories

In this paper we consider representations of certain combinatorial categories, including the poset $D$ 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 $\mathrm{CI}$ 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 $(Z,J_{at},\underset{\_}{C})$, and show that irreducible sheaves are parameterized by primitive roots of the unit.