The balanced tensor product of module categories
Let \(\mathcal{C}\) be a finite, rigid, monoidal linear category, \(\mathcal{M}\) a finite, right \(\mathcal{C}\)-module category and \(\mathcal{N}\)-a finite, left \(\mathcal{C}\)-module category. It is a well-known result by Etingof, Gelaki, Nikshych and Ostrik that any such \(\mathcal{M}\) is equivalent to the category of right modules over some algebra object \(A\) in \(\mathcal{C}\) (and analogously for \(\mathcal{N}\)). This elegantly written, clear article provides an alternative construction of the balanced tensor product of these module categories \(\mathcal{M} \boxtimes_\mathcal{C} \mathcal{N}\) in terms of bimodules over the algebra objects associated to each module category by EGNO's theorem. This result allows for an easier, explicit construction of the balanced product -- important for working in higher categorical constructions involving tensor categories.
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