Categories of quiver representations and relative cotorsion pairs

Abstract

We study the category $\operatorname{Rep}(Q,\mathcal{C})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{C}$. For this purpose we introduce the mesh and the cone-shape cardinal numbers associated to the quiver $Q$ and we use them to impose conditions on $\mathcal{C}$ that allow us to prove interesting homological properties of $\operatorname{Rep} (Q,\mathcal{C})$ that can be constructed from $\mathcal{C}.$ For example, we compute the global dimension of $\operatorname{Rep} (Q,\mathcal{C})$ in terms of the global one of $\mathcal{C}.$ We also review a result of H. Holm and P. Jørgensen which states that (under certain conditions on $\mathcal{C}$) every hereditary complete cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{C}$ induces the hereditary complete cotorsion pairs $(\operatorname{Rep}(Q,\mathcal{A}),\operatorname{Rep}(Q,\mathcal{A})^{\bot_{1}})$ and $(^{\bot_{1}}Ψ(\mathcal{B}),Ψ(\mathcal{B}))$ in $\operatorname{Rep}(Q,\mathcal{C})$, and then we obtain a strengthened version of this and others related results. Finally, we will apply the above developed theory to study the following full abelian subcategories of $\operatorname{Rep}(Q,\mathcal{C}),$ finite-support, finite-bottom-support and finite-top-support representations. We show that the above mentioned cotorsion pairs in $\operatorname{Rep}(Q,\mathcal{C})$ can be restricted nicely on the aforementioned subcategories and under mild conditions we also get hereditary complete cotorsion pairs.