Linear Reedy categories, quasi-hereditary algebras and model structures

TL;DR

This paper explores linear Reedy categories and their connection to quasi-hereditary algebras, establishing new model structures and highest weight categories in the context of finite dimensional algebras.

Contribution

It introduces the concept of linear Reedy categories over a field and demonstrates their relation to quasi-hereditary algebras and abelian model structures.

Findings

Category of left modules admits a highest weight structure

Finite linear Reedy categories correspond to quasi-hereditary algebras

Lifted cotorsion pairs and model structures in functor categories

Abstract

We study linear versions of Reedy categories in relation with finite dimensional algebras and abelian model structures. We prove that, for a linear Reedy category $\mathcal{C}$ over a field, the category of left $\mathcal{C}$--modules admits a highest weight structure, which in case $\mathcal{C}$ is finite corresponds to a quasi-hereditary algebra with an exact Borel subalgebra. We also lift complete cotorsion pairs and abelian model structures to certain categories of additive functors indexed by linear Reedy categories, generalizing analogous results from the hereditary case.