Related papers: Categories of quiver representations and relative …

We study the category $\mathrm{Rep}(Q,\mathcal{M})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{M}$. Under certain assumptions, we show that every cotorsion pair $(\mathcal{A},\mathcal{B})$ in…

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}),…

In this paper we introduce a special kind of relative (co)resolutions associated to a pair of classes of objects in an abelian category $\mathcal{C}.$ We will see that, by studying these relative (co)resolutions, we get a possible…

In this paper, we study the category $C(Rep(\mathcal{Q}, \mathcal{A}))$ of complexes of representations of quiver $\mathcal{Q}$ with values in an abelian category $\mathcal{A}$. We develop a method for constructing some model structures on…

The aim of this paper is to construct exact model structures from so called extendable cotorsion pairs. Given a hereditary Hovey triple $(\mathcal{C}, \mathcal{W}, \mathcal{F})$ in a weakly idempotent complete exact category with enough…

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…

Let $\mathcal{M}$ be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in $\mathcal{M}$. This is based on…

Let $(\mathcal A, \mathcal B, \mathcal C)$ be a recollement of extriangulated categories. In this paper, we provide bounds on the coresolution dimensions of the subcategories involved in $\mathcal A$, $\mathcal B$ and $\mathcal C$. We show…

We show that a complete hereditary cotorsion pair $(\C,\C^\bot)$ in an exact category $\E$, together with a subcategory $\Z\subseteq\E$ containing $\C^\bot$, determines a Waldhausen category structure on the exact category $\C$, in which…

Given a hereditary complete cotorsion pair $(\mathsf A,\mathsf B)$ generated by a set of objects in a Grothendieck category $\mathsf K$, we construct a natural equivalence between the Becker coderived category of the left-hand class…

We study the category $\textrm{Rep}(Q,\mathbb{F}_1)$ of representations of a quiver $Q$ over "the field with one element", denoted by $\mathbb{F}_1$, and the Hall algebra of $\textrm{Rep}(Q,\mathbb{F}_1)$. Representations of $Q$ over…

Let $\mathcal{B}$ be an abelian category with enough projective objects and enough injective objects and let $\mathcal{A}=\mathcal{B}\ltimes_\eta\mathsf{F}$ be an $\eta$-extension of $\mathcal{B}$. Given a cotorsion pair…

Let B be an extriangulated category with enough projectives and enough injectives. Let C be a fully rigid subcategory of B which admits a twin cotorsion pair ((C,K),(K,D)). The quotient category B/K is abelian, we assume that it is…

We introduce the notion of (twisted) quiver representations in abelian categories and study the category of such representations. We construct standard resolutions and coresolutions of quiver representations and study basic homological…

Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver…

Let Q be a strongly locally finite quiver and denote by rep(Q) the category of locally finite dimensional representations of Q over some fixed field k. The main purpose of this paper is to get a better understanding of rep(Q) by means of…

Given a complete hereditary cotorsion pair $(\mathcal{A},\mathcal{B})$ in a Grothendieck category $\mathcal{G}$, the derived category $\mathcal{D}(\mathcal{B})$ of the exact category $\mathcal{B}$ is defined as the quotient of the category…

Let $\mathcal B$ be an extriangulated category with enough projectives and enough injectives. We define a proper $m$-term subcategory $\mathcal G$ on $\mathcal B$, which is an extriangulated subcategory. Then we give a correspondence…

Let $(\mathscr{X}$, $\mathscr{Y})$ be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to $(\mathscr{X}$, $\mathscr{Y})$, and then give some equivalent characterizations when a relative…

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander-Solberg relative tilting modules on Artin algebras to…