## 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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### References:

 [1] B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, Univ. Lecture Ser. 21, Amer. Math. Soc., Providence, 2001. · Zbl 0965.18002 [2] M. Barr and C. Wells, Toposes, Triples and Theories, Grundlehren Math. Wiss. 278, Springer, New York, 1985. [3] D. Ben-Zvi, A. Brochier, and D. Jordan, Integrating quantum groups over surfaces: quantum character varieties and topological field theory, to appear in J. Topol., preprint, arXiv:1501.04652v5 [math.QA]. · Zbl 1409.14028 [4] P. Deligne, “Catégories tannakiennes” in The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Birkhäuser Boston, Boston, 1990, 111–195. [5] A. Davydov and D. Nikshych, The Picard crossed module of a braided tensor category, Algebra Number Theory 7 (2013), 1365–1403. · Zbl 1284.18015 [6] A. Davydov, D. Nikshych, and V. Ostrik, On the structure of the Witt group of braided fusion categories, Selecta Math. (N.S.) 19 (2013), 237–269. · Zbl 1345.18005 [7] C. L. Douglas, C. Schommer-Pries, and N. Snyder, Dualizable tensor categories, preprint, arXiv:1312.7188v2 [math.QA]. [8] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor categories, Lecture notes for MIT course 18.769, http://www-math.mit.edu/ etingof/tenscat1.pdf. · Zbl 1365.18001 [9] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Math. Surveys Monogr. 205, Amer. Math. Soc., Providence, 2015. · Zbl 1365.18001 [10] P. Etingof, D. Nikshych, and V. Ostrik, An analogue of Radford’s $$S^{4}$$ formula for finite tensor categories, Int. Math. Res. Not. 2004, no. 54, 2915–2933. · Zbl 1079.16024 [11] P. Etingof, D. Nikshych, and V. Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), 581–642. · Zbl 1125.16025 [12] P. Etingof, D. Nikshych, and V. Ostrik, Fusion categories and homotopy theory, with an appendix by E. Meir, Quantum Topol. 1 (2010), 209–273. · Zbl 1214.18007 [13] P. Etingof and V. Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004), 627–654, 782–783. · Zbl 1077.18005 [14] J. Fuchs, C. Schweigert, and A. Valentino, Bicategories for boundary conditions and for surface defects in 3-d TFT, Comm. Math. Phys. 321 (2013), 543–575. · Zbl 1269.81169 [15] J. Greenough, Relative centers and tensor products of tensor and braided fusion categories, J. Algebra 388 (2013), 374–396. · Zbl 1297.18003 [16] P. Grossman and N. Snyder, Quantum subgroups of the Haagerup fusion categories, Comm. Math. Phys. 311 (2012), 617–643. · Zbl 1250.46042 [17] P. Grossman and N. Snyder, The Brauer-Picard group of the Asaeda-Haagerup fusion categories, Trans. Amer. Math. Soc. 368, no. 4 (2016), 2289–2331. · Zbl 1354.46057 [18] T. Johnson-Freyd and C. Scheimbauer, (Op)lax natural transformations, twisted quantum field theories, and the “even higher” Morita categories, preprint, arXiv:1502.06526v3 [math.CT]. · Zbl 1375.18043 [19] D. Jordan and E. Larson, On the classification of certain fusion categories, J. Noncommut. Geom. 3 (2009), 481–499. · Zbl 1208.18004 [20] G. M. Kelly, Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Note Ser. 64, Cambridge Univ. Press, New York, 1982. · Zbl 0478.18005 [21] G. M. Kelly, Structures defined by finite limits in the enriched context, I, Cahiers Topologie Géom. Différentielle 23 (1982), 3–42. · Zbl 0538.18006 [22] I. López Franco, Tensor products of finitely cocomplete and abelian categories, J. Algebra 396 (2013), 207–219. · Zbl 1305.18042 [23] M. Müger, From subfactors to categories and topology, I: Frobenius algebras in and Morita equivalence of tensor categories, J. Pure Appl. Algebra 180 (2003), 81–157. · Zbl 1033.18002 [24] V. Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003), 177–206. · Zbl 1044.18004 [25] D. Tambara, A duality for modules over monoidal categories of representations of semisimple Hopf algebras, J. Algebra 241 (2001), 515–547. · Zbl 0987.16034
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