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

TL;DR

This paper studies representations of various combinatorial categories, providing bounds on their homological degrees, characterizing finitely presented representations, and describing sheaves over a specific site with irreducibles linked to primitive roots.

Contribution

It offers explicit bounds for homological degrees, characterizes abelian subcategories of finitely presented representations, and describes the structure of sheaves over a ringed atomic site.

Findings

Finitely presented representations form abelian subcategories.

Explicit bounds for homological degrees are established.

Irreducible sheaves are parameterized by primitive roots of unity.

Abstract

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