Completeness of the induced cotorsion pairs in categories of quiver representations

TL;DR

This paper investigates the conditions under which induced cotorsion pairs in categories of quiver representations are complete, extending known results to infinite line quivers and providing new completeness conditions.

Contribution

It establishes the completeness of induced cotorsion pairs in quiver representation categories under various conditions, including for infinite line quivers.

Findings

Cotorsion pairs are complete when the quiver is left or right rooted.

Completeness is also achieved for the infinite line quiver $A_{ ext{infinity}}^{ ext{infinity}}$.

Results extend the understanding of cotorsion pairs in more general quiver categories.

Abstract

Given a complete hereditary cotorsion pair $(\mathcal{A}, \mathcal{B})$ in an abelian category $\mathcal{C}$ satisfying certain conditions, we study the completeness of the induced cotorsion pairs $(\Phi(\mathcal{A}), \Phi(\mathcal{A})^{\perp})$ and $(^{\perp}\Psi(\mathcal{B}), \Psi(\mathcal{B}) )$ in the category $\mbox{Rep}(Q, \mathcal{C})$ of $\mathcal{C}$-valued representations of a given quiver $Q$. We show that if $Q$ is left rooted, then the cotorsion pair $(\Phi(\mathcal{A}), \Phi(\mathcal{A})^{\perp})$ is complete, and if $Q$ is right rooted, then the cotorsion pair $(^{\perp}\Psi(\mathcal{B}), \Psi(\mathcal{B}) )$ is complete. Besides, we work on the infinite line quiver $A_{\infty}^{\infty}$, which is neither left rooted nor right rooted. We prove that these cotorsion pairs in $\mbox{Rep}(A_{\infty}^{\infty}, R)$ are complete, as well.