Construction of Grothendieck categories with enough compressible objects using colored quivers

TL;DR

This paper presents a novel method to construct Grothendieck categories from colored quivers, enabling the realization of finite partially ordered sets as atom spectra with specific properties.

Contribution

It introduces a new construction technique for Grothendieck categories with enough compressible objects based on colored quivers, extending previous methods.

Findings

Constructed Grothendieck categories with desired atom spectra

Proved existence of locally noetherian Grothendieck categories with compressible objects

Demonstrated realization of finite posets as atom spectra

Abstract

We introduce a new method to construct a Grothendieck category from a given colored quiver. This is a variant of the construction used to prove that every partially ordered set arises as the atom spectrum of a Grothendieck category. Using the new method, we prove that for every finite partially ordered set, there exists a locally noetherian Grothendieck category such that every nonzero object contains a compressible subobject and its atom spectrum is isomorphic to the given partially ordered set.