Hochschild cohomology and extensions of triangulated categories
Alessandro Lehmann, Wendy Lowen
TL;DR
The paper develops a direct deformation theory for enhanced triangulated categories by introducing categorical first order deformations and proving a bijection with the second Hochschild cohomology $HH^2(\mathcal{T})$. It constructs these deformations via recollements and Yoneda $2$-extensions, and shows how curved Morita deformations of an algebra induce categorical deformations of its derived category, compatible with the associated Hochschild classes via the map $\chi_A$. A key application is that the $1$-derived category $D^\varepsilon(A_\varepsilon)$ arising from a curved deformation yields a categorical resolution of the uncurved derived category, connecting small and large models of noncommutative spaces. The framework also integrates extensions of $A_\infty$-functors, enhancements, and gluing techniques to relate algebras and categories, and discusses the need for higher categorical tools to capture morphisms and higher actions, laying groundwork for a full moduli-theoretic interpretation of the Hochschild complex. Overall, the work provides a cohesive, first-order, categorified viewpoint on deformations that links Hochschild theory, curved deformations, and derived-category techniques with potential for broader moduli-theoretic applications in noncommutative geometry.
Abstract
We define a notion of categorical first order deformations for (enhanced) triangulated categories. For a category $\mathcal{T}$, we show that there is a bijection between $\operatorname{HH}^2(\mathcal{T})$ and the set of categorical deformations of $\mathcal{T}$. We show that in the case of curved deformations of dg algebras considered in arXiv:2406.04945, the $1$-derived category of the deformation (introduced in arXiv:24020.8660) is a categorical deformation of the derived category of the base; the Hochschild class identified by this deformation is shown to restrict to the class defining the deformation of the algebra. As an application, we give a conceptual proof of the fact that (for a smooth base) the filtered derived category of a dg deformation yields a categorical resolution of the classical derived category.