Behavior of tensor product and module action under equivalences with finite dimensional algebra module categories

Determine how the monoidal tensor product of a finite tensor category C and the associated module product (action) on a finite exact module category M transform under the equivalences between M and module categories of finite dimensional algebras arising from projective generators, and characterize whether these structures are preserved or how they are realized under such equivalences.

Background

A key ingredient in the paper’s arguments is the tensor product on the finite tensor category and the module action on its module category. While abelian equivalences to module categories of finite dimensional algebras exist, the categorical monoidal and action structures may not have direct analogues under these equivalences.

The authors explicitly note that they do not know what becomes of these structures under the equivalences, marking a technical gap that affects the transfer of representation type results from the algebraic to the categorical setting.