Model structures on the category of complexes of quiver representations

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 $\Mod R$.

We have decided to divide the paper into two parts. The first part, the present version, has been published by Payam Behiraei in the Bulletin of the Iranian Mathematical Society. In the second part, where we delve into the examination of morphism categories, we have separated it as we aim to achieve better and stronger results. Currently, we are working on this part