Tensor extriangulated categories
Raphael Bennett-Tennenhaus, Isambard Goodbody, Janina C. Letz, Amit Shah
TL;DR
The paper develops a comprehensive framework for tensor extriangulated categories, unifying monoidal and extriangulated structures via biextriangulated and strong biextriangulated functors, and extends higher-extension theory through coends and (co)syzygies. It defines a ring structure on higher extensions, examines closed and stabilised variants, and provides a wealth of examples including matrix factorisations and Hopf actions. A central contribution is the Balmer-style classification of radical thick tensor ideals in this setting, yielding Balmer spectra and a robust functorial picture. The work thus generalises tensor triangulated approaches to a broader extriangulated context, enabling new avenues in representation theory, homological algebra, and categorical geometry with concrete computational tools for spectra and ideals. It also connects purity, dualisability, and substructures to structural properties of tensor extriangulated categories, highlighting potential applications to both algebraic and geometric contexts.
Abstract
A tensor extriangulated category is an extriangulated category with a symmetric monoidal structure that is compatible with the extriangulated structure. To this end we define a notion of a biextriangulated functor $\mathcal{A} \times \mathcal{B} \to \mathcal{C}$, with compatibility conditions between the components. We have two versions of compatibility conditions, the stronger depending on the higher extensions of the extriangulated categories. We give many examples of tensor extriangulated categories. Finally, we generalise Balmer's classification of thick tensor ideals to tensor extriangulated categories.