Pseudoskew category algebras and modules over representations of small categories

TL;DR

This paper introduces pseudoskew category algebras and characterizes modules over small category representations, providing new insights into their structure and torsion pairs.

Contribution

It defines pseudoskew category algebras, characterizes module categories over small category representations, and classifies hereditary torsion pairs.

Findings

Characterization of Mod-R as functors on Gr(R)

Introduction of pseudoskew category algebras R[C]

Classification of hereditary torsion pairs

Abstract

Let $\mathcal {C}$ be a small category and let $R$ be a representation of the category $\mathcal {C}$, that is, a pseudofunctor from a small category to the category of small preadditive categories. In this paper, we mainly study the category $\mbox{Mod-} R$ of right modules over $R$. We characterize it both as a category of the Abelian group valued functors on $Gr(R)$ and as a category of modules over a new family of algebras: the pseudoskew category algebras $R[{\mathcal C}]$, where $Gr(R)$ is the linear Grothendieck construction of $R$. Moreover, we also classify the hereditary torsion pairs in $\mbox{Mod-} R$ and reprove Theorem 3.18 of the paper (S. Estrada and S. Virili. Cartesian modules over representations of small categories. Adv. in Math. 310: 557-609, 2017) of Estrada and Virili.