On 2-categories of extensions

Аннотация

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 by only using a triangulated structure. 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].