Model structures on the category of complexes of quiver representations

TL;DR

This paper develops model structures on the category of complexes of quiver representations in an abelian category, showing they are monoidal and applying them to describe derived categories of such representations.

Contribution

It introduces a componentwise construction of model structures on complexes of quiver representations and proves their monoidality, advancing the homotopical understanding of these categories.

Findings

Constructed model structures on complexes of quiver representations.

Proved these model structures are monoidal.

Applied the model structures to describe derived categories.

Abstract

In this paper, we study the category $C(Rep(\mathcal{Q}, \mathcal{A}))$ of complexes of representations of quiver $\mathcal{Q}$ with values in an abelian category $\mathcal{A}$. We develop a method for constructing some model structures on $C(Rep(\mathcal{Q}, \mathcal{A}))$ based on componentwise notion. Moreover, we also show that these model structures are monoidal. As an application of these model structures, we introduce some descriptions of the derived category of complexes of representations of $\mathcal{Q}$ in $\Mod R$.