Equivalences of graded fusion categories
Fusion categories can be seen as a categorical generalization of the representation theory of finite groups. Many fusion categories possess an additive decomposition into components indexed by elements of a finite group \(G\) such that the fusion product respects this decomposition. A fusion category with this decomposition is called \(G\)-graded, and can be thought of as a \(G\)-graded extension of the trivial component, which is itself a fusion category by definition. Equivalence classes of \(G\)-graded extensions of fusion categories were described in terms of module theory for fusion categories and cohomological data of finite groups in [\textit{P. Etingof} et al., Quantum Topol. 1, No. 3, 209--273 (2010; Zbl 1214.18007)]. The study of tensor autoequivalences of \(G\)-graded fusion categories was initiated in [\textit{I. Marshall}, PhD Thesis, University of New Hampshire (2016)]; the overlap of the reviewed manuscript with the content of this thesis is discussed at the end of Section 1. Here the author studies a weak notion of \(G\)-graded equivalence between \(G\)-graded fusion categories which are only required to preserve the trivial component of the respective \(G\)-gradings. The benefit to this weakening of the notion of \(G\)-graded equivalence is a more computable set of data describing monoidal equivalence classes of \(G\)-graded extension in this sense (Theorem 1.1) than is described in [\textit{P. Etingof} et al., Quantum Topol. 1, No. 3, 209--273 (2010; Zbl 1214.18007)]. But applications of Theorem 1.1 (Section 9) are still limited by the need to compute intractable higher categorical obstructions. The author relies solely on graphical computations to justify main statements which differs from the arguments in the previously mentioned literature on the subject.
scientific article; zbMATH DE number 7357195
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