Categorical framework and coherence of tensor products of Fell-bundle actions

Develop a categorical framework for the tensor product constructions of actions of Fell bundles, and establish whether suitable forms of associativity and, where applicable, commutativity hold.

Background

The paper constructs tensor products of actions of Fell bundles on Hilbert bundles, including both the same-bundle construction and the more general construction associated with homomorphisms between the indexing groups. These constructions are shown to produce new actions and to interact with correspondences and induction procedures.

The authors explicitly leave unresolved whether these constructions admit a categorical formulation and whether they satisfy coherence properties analogous to associativity or commutativity in settings where such properties are meaningful.

References

Can the tensor product construction(s) in Section \ref{tensoraction} be put in a categorical framework? Of course, here one should at least check the general validity of some (probably weak) form of associativity and, in more specific situations, also commutativity whenever it makes sense.

— Actions of Fell bundles, II: tensor products and Fourier-Stieltjes algebras  (2610.13646 - Bédos et al., 9 Oct 2026) in Section 7, “Some questions” (Section \ref{final})