Completeness of the induced cotorsion pairs in functor categories

TL;DR

This paper investigates the completeness of induced cotorsion pairs in functor categories of quiver representations, removing previous hereditary assumptions and establishing new conditions for their properties.

Contribution

It extends prior work by removing the hereditary condition on cotorsion pairs, broadening the applicability of completeness results in functor categories.

Findings

Proves completeness of cotorsion pairs without hereditary assumption.

Shows subcategories can be extended to functor categories as precovering or preenveloping.

Provides conditions under which properties are preserved in representation categories.

Abstract

This paper focuses on a question raised by Holm and J{\o}rgensen, who asked if the induced cotorsion pairs $(\Phi({\sf X}),\Phi({\sf X})^{\perp})$ and $(^{\perp}\Psi({\sf Y}),\Psi({\sf Y}))$ in $\mathrm{Rep}(Q,{\sf{A}})$ -- 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. Recently, Odaba\c{s}{\i} gave an affirmative answer if the quiver $Q$ is rooted and the cotorsion pair $(\sf X,\sf Y)$ is further hereditary. In this paper, we improve Odaba\c{s}{\i}'s work by removing the hereditary assumption on the cotorsion pair. 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 $\mathrm{Rep}(Q,{\sf{A}})$.