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Triangulated structure for Bondarenko's Categories

Germán Benitez, Gustavo Costa, Lucas Q. Pinto

Abstract

V. Bondarenko and Y. Drozd gives a description of all indecomposable objects in a category of representations of posets, nowadays known as the Bondarenko's category. This category was essential for V. Bekkert and H. Merklen classify all indecomposable objects of the derived category of gentle algebras. In view of this connection with the derived category, which possess a triangulated structure, and of the fact that in this paper we show that the Bondarenko's category is not an abelian category, it is reasonable to contemplate the existence of a triangulated structure for the Bondarenko's category. In this paper we introduce a triangulated category structure over a quotient of Bondarenko's category, which will allow to use the techniques of triangulated category to study representations of posets.

Triangulated structure for Bondarenko's Categories

Abstract

V. Bondarenko and Y. Drozd gives a description of all indecomposable objects in a category of representations of posets, nowadays known as the Bondarenko's category. This category was essential for V. Bekkert and H. Merklen classify all indecomposable objects of the derived category of gentle algebras. In view of this connection with the derived category, which possess a triangulated structure, and of the fact that in this paper we show that the Bondarenko's category is not an abelian category, it is reasonable to contemplate the existence of a triangulated structure for the Bondarenko's category. In this paper we introduce a triangulated category structure over a quotient of Bondarenko's category, which will allow to use the techniques of triangulated category to study representations of posets.
Paper Structure (3 sections, 9 theorems, 42 equations)

This paper contains 3 sections, 9 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Bondarenko's Category
  3. Triangulated structure

Key Result

Lemma 2.3

$\mathbf{C}_T$ is an object in $s(\mathcal{Y},\Bbbk)$, for any morphism $T\in\textup{Hom}_{s(\mathcal{Y},\Bbbk)}(B,C)$.

Theorems & Definitions (20)

  • Definition 2.1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Remark 2.7
  • Example 2.8
  • ...and 10 more