On 2-categories of extensions
D. Kaledin
TL;DR
The paper addresses refining the triangulated structure of derived categories by constructing a $2$-category of extensions supported in homological degrees $0$ and $1$. It provides a hands-on construction in the abelian setting using complexes of length $2$ and then shows compatibility with a general enhancement framework, proving a $2$-equivalence between the hand-built $C^{(2)}_{[0,1]}(\mathcal{A})$ and the enhanced, Segal-category-based construction $\Delta^h\mathcal{D}_{[0,1]}(\mathcal{A})^h$ when $\mathcal{A}$ has enough injectives. This demonstrates that higher homotopical data naturally captured by enhancements recovers the expected $2$-categorical structure of extensions, while noting that the full derived category $\mathcal{D}(\mathcal{A})^h$ may not be $1$-truncated, though its $1$-truncation yields a well-defined $2$-category. The results illustrate how homotopical enhancements encode extension data beyond triangulated operations, providing a robust framework for $2$-categorical refinement of derived categories.
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.