Fully Exact Module Categories Advance Stability Under Deligne Product for Finite Braided Tensor Categories

Key Takeaways

• Fully exact categories stay stable under relative Deligne product.
• They strictly contain invertible and separable module categories.
• Every internal algebra becomes projectively separable.
• In semisimple case, fully exact ⇔ separable.
• Perfect categories form rigid monoidal 2‑subcategory.

Pulse Analysis

Module categories are central objects in modern representation theory, encoding actions of tensor categories on linear categories. The relative Deligne product, a categorical analogue of tensoring over a base, has long been a tool for building new module categories, yet its behavior across the broad class of exact modules was poorly understood. By isolating the fully exact subclass, Gainutdinov and Laugwitz provide a clear criterion for when this product preserves exactness, thereby offering mathematicians a reliable method to compose and analyze complex categorical structures without losing essential properties.

The authors demonstrate that fully exact categories form a dense, hierarchical layer within all exact modules, strictly extending the well‑studied invertible and separable cases. A key technical advance is the proof that any internal algebra in a fully exact module is projectively separable, generalizing classical separability to incorporate projective objects. In semisimple environments the equivalence between fully exact and separable categories supplies a practical test for researchers, while the identification of dualizable fully exact modules as “perfect” establishes a rigid monoidal 2‑subcategory poised to serve as a model for finite tensor 2‑categories.

Concrete classifications reinforce the theory’s relevance. The team fully describes fully exact (and thus perfect) modules over the symmetric tensor category derived from Sweedler’s four‑dimensional Hopf algebra, and they analyze quasi‑triangular Hopf algebras, showing that the category of finite‑dimensional vector spaces fails to be fully exact for both Sweedler’s algebra and Lusztig’s small quantum group at odd roots of unity. These findings delineate the boundaries of the new framework and suggest pathways for future work, including extensions to small quantum groups of arbitrary Lie type and potential applications in topological quantum field theory and quantum computing architectures.