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Relationship between weak-braided monoidal categories and pseudo-braided tensor categories

Clarify the relationship between weak-braided monoidal categories (algebras over the operad wPaB) and Soibelman’s pseudo-braided tensor categories, particularly in view of results establishing pseudo-braiding structures for categories of vertex operator algebra modules that are C1-cofinite with finitely generated duals.

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Background

The paper’s weak-braiding formalism is presented as a generalization apt for capturing non-invertible symmetries. Pseudo-braided tensor categories, as introduced by Soibelman, offer another relaxed notion of braiding.

Recent progress (e.g., Moriwaki 2024) provides pseudo-braiding for certain VOA module categories, prompting a direct comparison: whether weak braiding and pseudo-braiding coincide, differ, or embed into one another under precise conditions.

References

The present work leaves open the following questions: How is the present construction related to the pseudo-braided tensor categories of , especially in light of the work of , which gives pseudo-braiding structure for categories of VOA modules which are $C_1$ cofinite with finitely generated duals?

— Weak braiding for algebras in braided monoidal categories (2410.23027 - Stockall, 30 Oct 2024) in Subsection “Future questions”