Completeness of the induced cotorsion pairs in functor categories

Abstract

This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs $(\Phi (\mathsf{X}),\Phi (\mathsf{X}{)}^{\perp })$ and ${(}^{\perp }\Psi (\mathsf{Y}),\Psi (\mathsf{Y}))$ in $\mathrm{Rep}(Q,\mathsf{A})$ -- the category of all $\mathsf{A}$-valued representations of a quiver $Q$ -- are complete whenever $(\mathsf{X},\mathsf{Y})$ is a complete cotorsion pair in an abelian category $\mathsf{A}$ satisfying some mild conditions. Recently, Odabaşı gave an affirmative answer if the quiver $Q$ is rooted and the cotorsion pair $(\mathsf{X},\mathsf{Y})$ is further hereditary. In this paper, we improve Odabaşı's work by removing the hereditary assumption on the cotorsion pair. As an application, we show under certain mild conditions that if a subcategory $\mathsf{L}$, which is not necessarily closed under direct summands, of $\mathsf{A}$ is special precovering (resp., preenveloping), then $\Phi (\mathsf{L})$ (resp., $\Psi (\mathsf{L})$) is special precovering (resp., preenveloping) in $\mathrm{Rep}(Q,\mathsf{A})$.