Fusion categories and homotopy theory (0909.3140v2)

Abstract: We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic(C) of invertible C-bimodule categories, called the Brauer-Picard groupoid of C, such that equivalence classes of G-extensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic(C). This gives rise to an explicit description of both the obstructions to existence of extensions and the data parametrizing them; we work these out both topologically and algebraically. One of the central results of the paper is that the 2-truncation of BrPic(C) is canonically the 2-groupoid of braided autoequivalences of the Drinfeld center Z(C) of C. In particular, this implies that the Brauer-Picard group BrPic(C) (i.e., the group of equivalence classes of invertible C-bimodule categories) is naturally isomorphic to the group of braided autoequivalences of Z(C). Thus, if C=Vec(A), where A is a finite abelian group, then BrPic(C) is the orthogonal group O(A+A^*). This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case G=Z/2, we rederive (without computations) the classical result of Tambara and Yamagami. Moreover, we explicitly describe the category of all (Vec(A1),Vec(A2))-bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences.

• The paper presents a novel homotopy-theoretic framework for classifying G-extensions of fusion categories using the Brauer-Picard groupoid.
• It establishes that the 2-truncation of BrPic(C) is canonically isomorphic to the group of braided autoequivalences of the Drinfeld center Z(C).
• The study offers a systematic topological and algebraic method for identifying obstructions and parameterizing data in fusion category extensions.

Analysis of "Fusion Categories and Homotopy Theory"

The paper "Fusion Categories and Homotopy Theory" by Pavel Etingof, Dmitri Nikshych, and Victor Ostrik presents a significant examination of the connections between fusion categories and classical homotopy theory. The authors explore the intricate classification problems of G-extensions of fusion and braided fusion categories by employing homotopy theoretic techniques. The paper explores applying higher groupoids and their classifying spaces, emphasizing the relationship between fusion categories and their homotopy classes of maps.

One pivotal construct introduced in the paper is the Brauer-Picard groupoid BrPic(C) associated with a fusion category C. This construct is a 3-groupoid that captures the group of invertible C-bimodule categories. The equivalence classes of G-extensions of C are correlated with the homotopy classes of maps from BG to the classifying space of the Brauer-Picard groupoid. This approach not only elucidates the structure of G-extensions but also provides a systematic method for identifying obstructions to their existence and for parameterizing their data both topologically and algebraically.

A key result established is that the 2-truncation of BrPic(C) corresponds canonically to the 2-groupoid of braided autoequivalences of the Drinfeld center Z(C) of a fusion category C. This implies that the group of invertible C-bimodule categories is naturally isomorphic to the group of braided autoequivalences of Z(C). In particular, for C = Vec ,Athe category of vector spaces graded by a finite abelian group A, the Brauer-Picard group equals the orthogonal group O(A ⊕ A ). This result allows for a detailed classification of extensions, exemplified by the Tambara-Yamagami categories.

The implications of these findings extend both theoretically and practically. Theoretical implications include a broader understanding of the categorical structures underlying fusion categories, while practical implications may involve applications in modular representation theory, topological quantum field theories (TQFTs), and other areas where braided fusion categories play a pivotal role.

The authors also provide a topological and algebraic approach to computing these extensions, employing the obstruction theory method within algebraic topology. By investigating the classifying space of categorical n-groups and its ramifications, the paper provides a more rigorous framework for dealing with extensions of fusion categories by finite groups. Notably, this framework shuns the traditional reliance on direct calculations, granting a systematic and theory-driven approach to extensions.

The paper concludes with an appendix written by Ehud Meir, which contextualizes the classification of these extensions within the familiar setting of G-extensions of group algebras. It elucidates the parallelism between the author's approach and classical group cohomological methods.

The power of this research lies in effectively combining the tools from higher category theory, representation theory, and homotopy theory to address and solve historically complex classification problems in the paper of fusion categories. Looking forward, future developments on this intersection of theoretical frameworks could yield new insights into the categorical structures related to quantum symmetries and topological quantum computation. This work sets the stage for further exploration into the homotopy-theoretic properties of more general monoidal categories.