Non-abelianess of the category of modules over a sum-id bipresheaf of rings

TL;DR

This paper introduces bipresheaves of rings and their module categories, showing that modules over a sum-id bipresheaf form a non-abelian category, extending the understanding of categorical structures in algebraic topology.

Contribution

It defines bipresheaves of rings and their modules, and characterizes the module category as bipresheaves on a Grothendieck construction, proving it is non-abelian.

Findings

Category of modules over a sum-id bipresheaf of rings is non-abelian.

Modules over bipresheaves can be characterized via bipresheaves on a Grothendieck construction.

The framework generalizes bisheaves of abelian groups in algebraic topology.

Abstract

Let $\mathcal{C}$ be a small category, motivated by the definition of bisheaves of abelian groups of MacPherson and Patel (see the Definition 5.1 of the paper: R. MacPherson and A. Patel. Persistent local systems. Adv. in Math. 386: 107795, 2021), we first introduce the notions of bipresheaves of rings $\mathfrak{R}$ on $\mathcal{C}$ and their module categories $\mbox{Mod-} \mathfrak{R}$. Then the linear Grothendieck construction $Gr(\mathfrak{R})$ of $\mathfrak{R}$ is defined. With this linear Grothendieck construction, we show that the category of bipresheaves of modules over a sum-id bipresheaf of rings $\mathfrak{R}$ can be characterized as the category of bipresheaves of abelian groups on $Gr(\mathfrak{R})$. It follows that the category $\mbox{Mod-} \mathfrak{R}$ of modules over a sum-id bipresheaf of rings $\mathfrak{R}$ is non-abelian.