The naive approach for constructing the derived category of a $d$-abelian category fails

TL;DR

This paper provides a counterexample demonstrating that the straightforward method for constructing the derived category of a 2-abelian category does not always succeed, highlighting limitations in current approaches.

Contribution

It introduces a specific 2-abelian category where the naive derived category construction method fails, revealing a gap in existing theory.

Findings

Counterexample of a 2-abelian category where naive construction fails

Shows limitations of current derived category construction methods

Highlights need for alternative approaches in higher abelian categories

Abstract

Let $k$ be a field. In this short note we give an example of a $2$-abelian $k$-category, realized as a $2$-cluster-tilting subcategory of the category $\operatorname{mod}\,A$ of finite dimensional (right) $A$-modules over a finite dimensional $k$-algebra $A$, for which the naive idea for constructing its "bounded derived category" as $2$-cluster-tilting subcategory of the bounded derived category of $\operatorname{mod}\,A$ cannot work.