On 2-categories of extensions

TL;DR

This paper constructs a 2-category of extensions within derived categories of abelian categories, illustrating how homotopical enhancements can refine classical categorical structures.

Contribution

It introduces a concrete construction of the 2-category of extensions using abelian techniques and connects it to recent homotopical enhancement methods.

Findings

Constructs the 2-category of extensions explicitly.

Shows the connection between classical and homotopical approaches.

Demonstrates the natural recovery of the 2-category in the enhanced formalism.

Abstract

This is essentially an illustration for the general technology of homotopical enhancements developed recently in arxiv:2409.17489. We take the derived category of an abelian category, and we look at the full subcategory spanned by complexes of length 2. This has a natural refinement to a 2-category that we call "the 2-category of extensions". However, just using the triangulated structure on the derived category is not enough to obtain this refinement. In this short note, we first construct the 2-category of extensions by hand -- that is, using abelian category techniques -- and then show how it can be recovered very easily and naturally in the enhanced formalism of arxiv:2409.17489.