Representations of generalized linear Reedy categories and abelian model structures

TL;DR

This paper develops a framework for representations of generalized linear Reedy categories, constructs abelian model structures on their module categories, and explores applications in representation theory and model category theory.

Contribution

It introduces a new perspective viewing these categories as infinite analogues of stratified algebras and constructs abelian model structures using cotorsion pairs.

Findings

Parameterization of irreducible representations

Equivalence conditions to product of local module categories

Construction of abelian model structures on generalized categories

Abstract

In this paper we consider representations of generalized $k$-linear Reedy categories $\underline{\mathscr{C}}$, a common generalization of $k$-linear Reedy categories introduced by Georgiois-\v{S}t'ov\'{\i}\v{c}ek and $k$-linearizations of generalized Reedy categories introduced by Berger-Moerdijk, and construct abelian model structures on $\underline{\mathscr{C}} \text{-}\mathrm{Mod}$. In the first part, we show that $\underline{\mathscr{C}}$ can be viewed as an infinite categorical analogue of standardly stratified algebras. Explicitly, we give a parameterization of irreducible representations of $\underline{\mathscr{C}} \text{-}\mathrm{Mod}$, provide several sufficient criteria such that $\underline{\mathscr{C}} \text{-}\mathrm{Mod}$ is equivalent to the Cartesian product of module categories over the ``local" endomorphism algebras of $\underline{\mathscr{C}}$, and describe applications of these results to representation theory of some interesting combinatorial categories including categories of spans and the category of finite dimensional vector spaces over a finite field and linear maps. In the second part, using the technique of Grothendieck bifibrations, we glue a family of complete cotorsion pairs in the module categories of these ``local" endomorphism algebras to a complete cotorsion pair in $\underline{\mathscr{C}} \text{-}\mathrm{Mod}$, and deduce that under certain mild conditions a family of abelian model structures on these ``local" module categories can be glued to an abelian model structure on $\underline{\mathscr{C}} \text{-}\mathrm{Mod}$. As applications, we obtain a few abelian model structures on generalized $k$-linear direct or inverse categories.