Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers

This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs (Φ(X), Φ(X)^⊥) and (^⊥Ψ(Y), Ψ(Y)) in Rep(Q, A)—the category of all A-valued representations of a quiver Q—are complete whenever (X, Y) is a complete cotorsion pair in an abelian category A satisfying some mild conditions. We give an affirmative answer if the quiver Q is rooted. As an application, we show under certain mild conditions that if a subcategory L, which is not necessarily closed under direct summands, of A is special precovering (resp., preenveloping), then Φ(L)(resp., Ψ(L)) is special precovering (resp., preenveloping) in Rep(Q, A).