Representations of generalized linear Reedy categories and abelian model structures

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.