The paper is essentially an illustration to the general technique of homotopical enhancements developed recently in 6. Taking a derived category of an abelian category, we consider its full subcategory generated by complexes of length 2. It has a natural refinement to a 2-category, which we call the ‘2-category of extensions’. However, it is not possible to construct this refinement only by using a triangulated structure on the derived category. In this short note, first, we construct a 2-category of enhancements by hand, using the techniques of abelian categories, and then we show how it can quite easily and naturally be recovered in the framework of the enhanced formalism of 6. Bibliography: 8 titles.
D. Kaledin (Wed,) studied this question.