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Model structures on the category of complexes of quiver representations

Payam Bahiraei

Abstract

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 $C(Rep(\mathcal{Q}, \mathcal{A}))$ based on componentwise notion. Moreover, we also show that these model structures are monoidal. As an application of these model structures, we introduce some descriptions of the derived category of complexes of representations of $\mathcal{Q}$ in $\Mod R$.

Model structures on the category of complexes of quiver representations

Abstract

In this paper, we study the category of complexes of representations of quiver with values in an abelian category . We develop a method for constructing some model structures on based on componentwise notion. Moreover, we also show that these model structures are monoidal. As an application of these model structures, we introduce some descriptions of the derived category of complexes of representations of in .

Paper Structure

This paper contains 8 sections, 9 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The homotopy category of complexes
  4. The category of representation of quiver:
  5. Model structures and Hovey pairs.
  6. Homotopy category of model category:
  7. Componentwise monoidal model structures
  8. Two descriptions of Derived category

Key Result

Lemma \oldthetheorem

Let $\mathcal{A}$ be an abelian model category and $f,g:X\rightarrow Y$ be two morphisms. If $X$ is cofibrant and $Y$ is fibrant, then $f$ and $g$ are homotopic (we denote by $f\sim g)$ if and only if $f-g$ factor through a trivially fibrant and cofibrant object.

Theorems & Definitions (25)

  • Lemma \oldthetheorem
  • Proof 1
  • Example \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • Proof 2
  • Lemma \oldthetheorem
  • Proof 3
  • Corollary \oldthetheorem
  • Proof 4
  • ...and 15 more