Extensions of groups by braided 2-groups

📝 Abstract

We classify extensions of a group $G$ by a braided 2-group $\mathcal{B}$ as defined by Drinfeld, Gelaki, Nikshych, and Ostrik. We describe such extensions as homotopy classes of maps from the classifying space of $G$ to the classifying space of the 3-group of braided $\mathcal{B} $-bitorsors. The Postnikov system of the latter space contains a generalization of the classical Pontryagin square to the setting of local coefficients, which has been previously discussed by Baues and more recently, in a setting close to ours, by Etingof, Nikshych, and Ostrik. We give an explicit cochain-level description of this Pontryagin square for group cohomology.

💡 Analysis

We classify extensions of a group $G$ by a braided 2-group $\mathcal{B}$ as defined by Drinfeld, Gelaki, Nikshych, and Ostrik. We describe such extensions as homotopy classes of maps from the classifying space of $G$ to the classifying space of the 3-group of braided $\mathcal{B} $-bitorsors. The Postnikov system of the latter space contains a generalization of the classical Pontryagin square to the setting of local coefficients, which has been previously discussed by Baues and more recently, in a setting close to ours, by Etingof, Nikshych, and Ostrik. We give an explicit cochain-level description of this Pontryagin square for group cohomology.

📄 Content

arXiv:1106.0772v1 [math.CT] 3 Jun 2011 EXTENSIONS OF GROUPS BY BRAIDED 2-GROUPS EVAN JENKINS Abstract. We classify extensions of a group G by a braided 2-group B as defined by Drinfeld, Gelaki, Nikshych, and Ostrik. We describe such extensions as homotopy classes of maps from the classifying space of G to the classifying space of the 3-group of braided B-bitorsors. The Postnikov system of the latter space contains a generalization of the classical Pontryagin square to the setting of local coefficients, which has been previously discussed by Baues and more recently, in a setting close to ours, by Etingof, Nikshych, and Ostrik. We give an explicit cochain-level description of this Pontryagin square for group cohomology.

1. Introduction A classical problem in homological algebra is to classify extensions of a group G by another group K, i.e., groups E equipped with a surjection ∂: E ։ G and an identification K ∼= ∂−1(e). When K = A is an abelian group, the action of an element of E on an element of A by conjugation depends only on its image in G, so extensions of G by A include the data of an action of G as automorphisms of A. If we fix such an action, then extensions with that action are classified by H2(G, A) (see, for example, [Wei95, 6.6]). In [DGNO10, Appendix E], a categorification of the notion of an extension of a group by an abelian group is defined, in which the abelian group A is replaced by a braided 2-group B (see [BL04] for basic results about 2-groups, which have also been studied under the names gr-category and categorical group, and [JS93] for basic results about braided monoidal categories). Given such an extension one has an underlying action of G as braided autoequivalences of the braided 2-group B, and also an underlying extension of G by the abelian group A = π0(B) of isomorphism classes of objects of B. It is natural to ask under what circumstances these two pieces of data will determine an extension of G by B and how unique such an extension is. In this paper, we show that we can lift these data to an extension if and only if a certain cohomology class in H4(G, H) vanishes, where H = π1(B) is the abelian group of isomorphisms of the unit object in B, and the action of G on H is in- duced from the action of G on B. This cohomology class comes from a function H2(G, A) →H4(G, H) (determined by a fixed braided 2-group B and an action of G on B) that generalizes the classical Pontryagin square, which can be recovered in our language by taking the trivial action of G on B. If this obstruction vanishes, then extensions form a torsor for H3(G, H). These results are parallel to the results of [ENO10] on the related problem of classifying braided G-crossed fusion categories. In that paper, a braided G-crossed fusion category C is viewed as a G-indexed family of invertible bimodules over the neutral component. Analogously, we will view extensions of G by a braided 2-group 1 2 EVAN JENKINS B as G-indexed families of braided bitorsors over B. We then use obstruction theory to obtain our classification. Sections 2 through 4 set up the machinery of braided bitorsors. Section 5 relates extensions and bitorsors by a construction of Grothendieck. Section 6 contains our classification result. Section 7 contains an explicit cochain-level description of the Pontryagin square defined in Section 6. The author would like to thank his advisor, Vladimir Drinfeld, for suggesting this topic and providing inspiration, guidance, and careful reading of many drafts. The author would also like to thank Peter May, Daniel Sch¨appi, Mike Shulman, and Ross Street for helpful discussions.
2. Torsors for 2-groups We can define torsors for 2-groups much the same as we do for groups. We first review the notion of modules for monoidal categories. A monoidal category C can be viewed as a one-object bicategory, which we will denote by C[1]. This can be viewed as the “delooping” of C, a notion we will revisit in Section 6. For now, we note only that strong monoidal functors between monoidal categories correspond to pseudofunctors between their deloopings, and that C[1]op = (Crev)[1], where Crev denotes the category C with reversed tensor product. Definition 2.1. Let C be a monoidal category. A left (resp. right) module (or module category) over C is a pseudofunctor X : C[1] →Cat (resp. X : C[1]op → Cat). By abuse of notation, we denote the image of the unique object of C[1] by X. We denote by C Mod and ModC the 2-categories of left and right C-modules, respectively. We will write a left action of an element c ∈C on an element x ∈X by c⊲x, and a right action as x ⊳c. For the rest of this subsection, we will only deal with left C- modules. Completely analogous definitions and proofs work with “left” everywhere replaced by “right.” Lemma 2.2. Let F : X →Y be a morphism of left C-modules, and suppose further that F is an equivalence of categories. Then F is an equivalence of C-modules. Proof. The forgetful 2-functor C Mod →Cat is monadic, and hence

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