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

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.