Fusion categories and homotopy theory

By Pavel Etingof, D. Nikshych, V. Ostrik, with an appendix by Ehud Meir. 2009

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fec5de6603e6500d512c78faf99b0bd5f62194cf

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Citations in the corpus

On finite symmetries and their gauging in two dimensions (2017, background): “It is known [20] that this happens if and only if Z(C) ' Z(C ′), where Z(C) is the Drinfeld center of C.”
Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface (2015, methodology): “Given the objects ρ and [w] for a proposed symmetry fractionalization pattern, there is a well-defined procedure to obtain [O] from the properties of the topological phase [18, 26, 27], schematically represented by a grinder in Fig.”
Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model (2013, background): “[43] This structure is reminiscent of the notion of G action in a braided G-crossed category [38, 39] [44] It is actually known that the symmetric center Z(C) of any braided fusion category C comes in two types [40]: either it consists entirely of bosons and is isomorphic to the set of representatio”
State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter (2016, background): “It follows from the results of [36] that for any such long exact sequence there exists a fusion category with these fusion rules, provided a certain obstruction [O4] ∈ H4(G,R/Z) constructed from n2 and π1 vanishes.”
Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (2019, background): “There are certain anomaly-vanishing conditions for such a symmetry which were worked out in [19].”
On braided fusion categories I (2009, methodology): “, so we get a tensor functor F : Rep(G) → D. Thus we have deﬁned a 2-functor K2 → K1. Theorem 4.37. These 2-functors K1 → K2 and K2 → K1 are mutually inverse 2-equivalences. This theorem is proved in [ENO4]. In the following remarks we explain the idea of the proof. Theorem 4.37 will not be used in ”
The balanced tensor product of module categories (2014, background): “ategory M CN (for example, see [Mug03,ENO10,JL09,GS12b,GS12a,DNO13,FSV13]). For C= Vect, this agrees with Deligne’s [Del90] tensor product K Lof nite linear categories. Etingof, Nikshych, and Ostrik [ENO10] established the existence of a balanced tensor product M CNof nite semisimple module categori”
The Picard crossed module of a braided tensor category (2012, methodology): “Note that although the categories considered in [Etingof et al. 2010] were assumed to be semisimple the proof of this particular result does not use semisimplicity.”
Categorical Lagrangian Grassmannians and Brauer-Picard groups of pointed fusion categories (2013, methodology): “, 2018. 1 2 DMITRI NIKSHYCH AND BRIANNA RIEPEL 1. Introduction Let A be a fusion category. The Brauer-Picard group BrPic(A) of A consists of equivalence classesof invertible A-bimodule categories(see [ENO2]). Brauer-Picard groups play an important role in the theory of fusion categories. In particul”