Categories of quiver representations and relative cotorsion pairs

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