Monoidal 2-structure of bimodule categories

By J. Greenough. 2009

106 citations recorded by the source corpus.

Abstract We define a notion of tensor product of bimodule categories and prove that with this product the 2-category of C -bimodule categories for fixed tensor C is a monoidal 2-category in the sense of Kapranov and Voevodsky (1991) [1] . We then provide a monoidal-structure preserving 2-equivalence between the 2-category of C -bimodule categories and Z ( C ) -module categories (module categories over the center of C ). For a finite group G we show that de-equivariantization is equivalent to the tensor product over Rep ( G ) . We derive Rep ( G ) -module fusion rules and show that the group of invertible Rep ( G ) -module categories is isomorphic to H 2 ( G , k × ) , extending results in Etingof et al. [2] .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cb9e6ccea0a15b4f3558e07c1c1a221f0ccbbde9

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Citations in the corpus

Universal Non‐Invertible Symmetries (2022, background): “2 Relation to the Twisted Burnside Ring The fusion structure of the objects in 2Rep(Γ) that we encountered has an alternative representation-theoretic description in terms of the twisted Burnside ring [30].”
Categorical Lagrangian Grassmannians and Brauer-Picard groups of pointed fusion categories (2013, methodology): “irect product of N and H. We can apply Proposition 6.8 to centers of Tannakian categories. Let G be a ﬁnite group. Corollary 6.9. We have (21) Stab(Rep(G)) ∼= H2(G, k×) ⋊Out(G) Proof. It was shown in [Gr] that Pic(Rep(G)) ∼= H2(G, k×). Combining this with Proposition 6.1 we get the result. Let us de”
Fusion categories and homotopy theory (2009, unknown): “The next proposition is proved in [19].”