Zhen Xing DI, Li Ping LI, Li LIANG, Fei XU
This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs $(\Phi({\sf X}),\Phi({\sf X})^{\perp})$ and $(^{\perp}\Psi({\sf Y}),\Psi({\sf Y}))$ in Rep $({Q},{\sf{A}})$-the category of all $\sf A$-valued representations of a quiver $Q$-are complete whenever $(\sf X,\sf Y)$ is a complete cotorsion pair in an abelian category $\sf{A}$ 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 $\sf L$, which is not necessarily closed under direct summands, of $\sf A$ is special precovering (resp., preenveloping), then $\Phi(\sf L)$ (resp., $\Psi(\sf L)$) is special precovering (resp., preenveloping) in Rep$({Q},{\sf{A}})$.
