Tensor extriangulated categories

TL;DR

This paper introduces tensor extriangulated categories, combining extriangulated and monoidal structures, and extends Balmer's classification of thick tensor ideals to this new setting.

Contribution

It defines tensor extriangulated categories, introduces biextriangulated functors with compatibility conditions, and generalizes Balmer's classification to these categories.

Findings

Defined tensor extriangulated categories.

Introduced biextriangulated functors with compatibility conditions.

Generalized Balmer's classification of thick tensor ideals.

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.