This work describes tensor extriangulated categories and compatibility conditions, suggesting implications for thick tensor ideals.
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 A × B → 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.
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Bennett‐Tennenhaus et al. (2025) studied this question.
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