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The balanced tensor product of module categories. (English) Zbl 1478.18006

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.

MSC:

18C40 Structured objects in a category (group objects, etc.)
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
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