Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)

We study group-graded extensions of fusion 2-categories. As an application, we obtain a homotopy theoretic classification of fermionic strongly fusion 2-categories. We examine various examples in detail.

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2024
1. Verfasser:	Décoppet, Thibault Didier
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author	Décoppet, Thibault Didier
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citation_txt	Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd). Thibault Didier Décoppet. SIGMA 20 (2024), 092, 20 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We study group-graded extensions of fusion 2-categories. As an application, we obtain a homotopy theoretic classification of fermionic strongly fusion 2-categories. We examine various examples in detail.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 092, 20 pages Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd) Thibault Didier DÉCOPPET Mathematics Department, Harvard University, Cambridge, Massachusetts, USA E-mail: decoppet@math.harvard.edu URL: https://www.thibaultdecoppet.com Received March 11, 2024, in final form October 09, 2024; Published online October 17, 2024 https://doi.org/10.3842/SIGMA.2024.092 Abstract. We study group graded extensions of fusion 2-categories. As an application, we obtain a homotopy theoretic classification of fermionic strongly fusion 2-categories. We examine various examples in detail. Key words: extension theory; fusion 2-category; supercohomology 2020 Mathematics Subject Classification: 18M20; 18N25 1 Introduction Let us fix an algebraically closed field k of characteristic zero. In the theory of fusion 1-categories over k, one of the main tools for constructing new fusion 1-categories out of the ones that are already known is the concept of a group graded extension as introduced in [16]. Fusion 2-ca- tegories over k were introduced in [11], and categorify the notion of a fusion 1-category. It is then only natural to investigate the notion of group graded extension for fusion 2-categories. More specifically, setting up group graded extension theory for fusion 2-categories is motivated by the problem of classifying fermionic strongly fusion 2-categories. In more detail, recall that a bosonic (resp. fermionic) strongly fusion 2-category is a fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is Vect (resp. SVect). It was proven in [7] that every fusion 2-category over k is Morita equivalent to the 2-Deligne tensor product of a strongly fusion 2-category and an invertible fusion 2-category, the latter corresponding to the data of a non-degenerate braided fusion 1-category. This motivates the problem of classifying strongly fusion 2-categories. Now, it was shown in [24] that every simple object of such a fusion 2-category is invertible. In particular, strongly fusion 2-categories are group graded extensions of either 2Vect or 2SVect. On the one hand, the classification of bosonic strongly fusion 2- categories is then routine: It consists of a finite group together with a 4-cocycle [11, Section 2.1]. On the other hand, the classification of fermionic strongly fusion 2-categories is more subtle, and does require the full strength of extension theory for fusion 2-categories. We ought to mention that, at a Physical level of rigor and in the context of the classification of topological orders in (3 + 1) dimensions, related results were already discussed in [29] and [20]. 1.1 Extension theory for fusion 1-categories Let us fix a fusion 1-category C. We can then consider the associated space of invertible (fi- nite semisimple) C-C-bimodule 1-categories, invertible bimodule functors, and bimodule natural isomorphisms. We denote this space by BrPic(C), and refer to it as the Brauer–Picard space of C. This space admits a canonical product structure given by the relative Deligne tensor mailto:decoppet@math.harvard.edu https://www.thibaultdecoppet.com https://doi.org/10.3842/SIGMA.2024.092 2 T.D. Décoppet product over C. In fact, extension theory was the original motivation for the introduction of the relative Deligne tensor product of finite semisimple module 1-categories over a fusion 1- category [16]. The relative Deligne tensor product endows the Brauer–Picard space with the structure of a group-like topological monoid, so that we may consider its delooping BBrPic(C). The main result of [16] is that, given any finite group G, faithfully G-graded extensions of C are parameterised by homotopy classes of maps from BG to BBrPic(C). 1.2 Results Over an algebraically closed field k of characteristic zero, the existence of the relative 2-Deligne tensor product for finite semisimple module 2-categories over a fusion 2-category was established in [9]. In particular, for any fusion 2-category C, one can consider the associated space BrPic(C) of invertible C-C-bimodule 2-categories. We refer to this space as the Brauer–Picard space of C, and note that, by construction, it can be delooped. We obtain a categorification of the main theorem of [16] to fusion 2-categories, noting that a more general result was announced in [22]. Theorem (Theorem 3.11). For any finite group G, faithfully G-graded extensions of the fu- sion 2-category C are classified by homotopy classes of maps from BG to BBrPic(C), the de- looping of the Brauer–Picard space of C. We wish to remark that Tambara–Yamagami fusion 2-categories, which are Z/2-graded fu- sion 2-categories whose non-trivially graded piece is 2Vect, were studied in [10] via different methods. In fact, unlike in the decategorified setting, it seems difficult to use extension theory to classify Tambara–Yamagami fusion 2-categories, as the group H5(BZ/2;k×), in which a certain obstruction lies, is non-zero. Using results from [17] and [23], we perform a careful analysis of the structure of the Brauer– Picard space associated to the fusion 2-category 2SVect. When combined with the above theorem, it yields the following result. Theorem (Theorem 4.5). Fermionic strongly fusion 2-categories equipped with a faithful grad- ing are classified by a finite group G together with a class ϖ in H2(BG;Z/2) and a class π in SH4+ϖ(BG). In the physics literature, versions of this classification have already appeared in [29], and [20, Section V.D]. In particular, the notation SH4+ϖ(BG) refers to ϖ-twisted supercohomology, the twisted generalized cohomology theory associated to the space 2SVect× of invertible objects and morphisms in 2SVect. This generalized cohomology theory is well known in the litera- ture on symmetry protected topological phases [17, 26, 35], where it is referred to as extended supercohomology. It is instructive to briefly discuss how the above data can be recovered from a given fermionic strongly fusion 2-category C. The corresponding group G is the group of connected components of C. Further, the group of invertible objects of C is a central extension of G by Z/2. However, it is not necessarily the extension corresponding to ϖ, rather it depends on the class π. We end by examining many examples of the above classification. In particular, we find in Example 4.13 that there are exactly three fermionic strongly fusion 2-categories whose group of connected components is Z/2n for any positive integer n. We also study the fermionic strongly fusion 2-categories whose group of connected components is Z/2⊕Z/2 in Example 4.14. Finally, it was asked in [24] whether there exists a finite group G and a faithfully G-graded fermionic strongly fusion 2-category with trivial twist ϖ whose group of invertible objects is a non-trivial central extension of G. In Appendix A, joint with Theo Johnson-Freyd, we perform a computation in supercohomology in order to show that this behaviour does occur when G = Z/4⊕ Z/4. Extension Theory and Fermionic Strongly Fusion 2-Categories 3 2 Preliminaries 2.1 Fusion 2-categories Over an algebraically closed field k of characteristic zero, the notions of finite semisimple 2- category and fusion 2-category were introduced in [11]. We succinctly review these definitions here, and refer the reader to the aforementioned reference and also [5] for details. A (k-linear) 2-category C is locally finite semisimple if its Hom-1-categories are finite semisim- ple. An object C of C is called simple if IdC is a simple object of the finite semisimple 1- category EndC(C). Then, a finite semisimple 2-category is a locally finite semisimple 2-category that is Cauchy complete in the sense of [18] (see also [4]), has right and left adjoints for 1- morphisms, and has finitely many equivalence classes of simple objects. In a finite semisimple 2-category C, any two simple objects are called connected if there exists a non-zero 1-morphism between them. This defines an equivalence relation, and we write π0(C) for the corresponding set of equivalences classes; This is the set of connected components of C. Now, the definition of a monoidal 2-category is well known. In particular, using the notations of [34], a monoidal structure on the 2-category C involves a 2-functor 2 : C×C → C, the monoidal product, and a distinguished object I, the monoidal unit, together with various other coherence data. Given any object C of C, we can ask for the existence of its left dual ♯C and its right dual C♯. These two notions were studied in [32]. Categorifying the concept of a fusion 1-category is therefore straightforward, and we obtain the following definition. Definition 2.1. A multifusion 2-category is a finite semisimple monoidal 2-category C such that every object C admits a right dual C♯ and a left dual ♯C. A fusion 2-category is a multifusion 2-category whose monoidal unit I is a simple object. To every fusion 2-category C, there is an associated braided fusion 1-category ΩC := EndC(I), the 1-category of endomorphisms of the monoidal unit I of C. Conversely, given any braided fusion 1-category B, we can consider the fusion 2-category Mod(B) of finite semisimple left B- module 1-categories with monoidal structure given by ⊠B, the relative Deligne tensor product. Below, we give another class of examples that will be relevant for our purposes. Many more may be found in [11] and [10], and others will be discussed subsequently. Example 2.2. Given G a finite group, and π a 4-cocycle for G with coefficients in k×, we can then consider the fusion 2-category 2Vectπ(G) of G-graded finite semisimple 1-categories, also known as 2-vector spaces, with pentagonator twisted by π. Such fusion 2-category were completely characterized in [24] as those fusion 2-categories C for which ΩC ≃ Vect. We call these bosonic strongly fusion 2-categories. 2.2 Module 2-categories and the relative 2-Deligne tensor product Let C be a monoidal 2-category with monoidal product 2, and monoidal unit I. A right C- module 2-category is a 2-category M equipped with an action 2-functor 2 : M × C → M, and coherence data satisfying various axioms. Likewise, there is a notion of left C-module 2-category. Additionally, there are notions of C-module 2-functors, C-module 2-natural transformations, and C-module modifications. There are also notions of bimodule 2-categories, and maps between them. We refer the reader to [8, Section 2] for the precise definitions. Let us now assume that C is a fusion 2-category. If M is a left C-module 2-category, then we can consider EndC(M), the monoidal 2-category of left C-module 2-endofunctors on M. The next result combines [6, Theorem 5.3.2] with [7, Corollary 5.1.3]. Theorem 2.3. The monoidal 2-category EndC(M) is a multifusion 2-category. 4 T.D. Décoppet Moreover, the multifusion 2-category EndC(M) is fusion if and only if M is indecomposable as a finite semisimple left C-module 2-category. For our purposes, we need to consider another operation on finite semisimple module 2- categories. Let M be a finite semisimple right C-module 2-category and N be a finite semisimple left C-module 2-category. There are notions of C-balanced 2-functors, C-balanced 2-natural transformations, and C-balanced modifications out of M × N. We refer the reader to [9, Sec- tion 2.2] for the details. The next result is a combination of [9, Theorem 2.2.4] with [7, Corol- lary 5.1.3]. Theorem 2.4. There is a 3-universal C-balanced 2-functor ⊠C : M ×N → M ⊠C N to a finite semisimple 2-category. In fact, the proof of [9, Theorem 2.2.4] gives an explicit description of M ⊠C N. More precisely, if A is a (necessarily separable) algebra in C such that M is equivalent to LModC(A), the right C-module 2-category of left A-modules in C, and B is a (necessarily separable) algebra in C such that N is equivalent to ModC(B), the left C-module 2-category of right B-modules in C, then M⊠C N ≃ BimodC(A,B), the finite semisimple 2-category of A-B-bimodules in C. Finally, let us recall that, as was explained in [9, Section 3.2], the existence of the relative 2-Deligne tensor product allows us to consider the symmetric monoidal Morita 4-category F2C of (multi)fusion 2-categories, finite semisimple bimodule 2-categories, and bimodule morphisms. 3 Extension theory For fusion 1-categories over an algebraically closed field of characteristic zero, extension theory was developed in [16]. The key concept is that of an invertible bimodule 1-category, or, more precisely, the space formed by such objects together with their invertible morphisms. Proceeding in a similar fashion, we will discuss extension theory for fusion 2-categories, i.e., we will study group graded fusion 2-categories in the sense of the definition below. Definition 3.1. Let C be a fusion 2-category and G a finite group. A G-grading on C is a decomposition ⊞g∈GCg of C into a direct sum of finite semisimple 2-categories such that for every C in Cg and D in Ch, C2D lies Cgh. A G-grading on C is faithful if Cg is non-zero for every g in G. 3.1 Invertible bimodule 2-categories Let k be an algebraically closed field of characteristic zero. We begin by recalling a definition from [10]. Let C and D be two fusion 2-categories. We write Dmop for the fusion 2-category obtained by endowing the finite semisimple 2-category D with the opposite of the monoidal product of D. Definition 3.2. A finite semisimple C-D-bimodule 2-category M is invertible if the canonical monoidal 2-functor Dmop → EndC(M) is an equivalence. At the decategorified level, i.e., for fusion 1-categories, invertibility of a finite semisimple bimodule 1-category admits various equivalent characterizations as is explained in [16, Proposi- tion 4.2]. A categorified version of this result was given in [9], which we partially recall below. Proposition 3.3. Let M be a finite semisimple C-D-bimodule 2-category. The following are equivalent: (1) The C-D-bimodule 2-category M is invertible. (2) The C-D-bimodule 2-category M defines an invertible 1-morphism from C to D in F2C. Extension Theory and Fermionic Strongly Fusion 2-Categories 5 The next proposition gives preliminary insight into the relation between group graded fusion 2-categories and invertible bimodule 2-categories. We note that the first part of the result below has already appeared as [10, Proposition 3.1.7]. We also refer the reader to [16, Theorem 6.1] for the decategorified version of this proposition. Proposition 3.4. Let C be a faithfully G-graded fusion 2-category C. For any g ∈ G, the finite semisimple Ce-Ce-bimodule 2-category Cg is invertible. Furthermore, for any g, h ∈ G, the 2-functor 2 : C×C → C induces an equivalence Cg ⊠Ce Ch ≃−→ Cgh of Ce-Ce-bimodule 2-categories. Proof. The first part follows from the second, so we only prove the latter. For the second part, note that it is enough to show that the canonical 2-functor is an equivalence. To this end, pick any non-zero objects X in Cg−1 , and Y in Ch. It follows from the proof of [6, Theorem 5.4.3] that Y2♯Y is a separable algebra in C, and that the left C-module 2-functor C → ModC ( Y2♯Y ) given by C 7→ C2♯Y is an equivalence. Likewise, X2♯X is a separable algebra in C, and the right C-module 2-functor C → LModC ( X2♯X ) given by C 7→ X2C is an equivalence. In particular, we also find that the 2-functor C → BimodC ( X2♯X,Y2♯Y ) given by C 7→ X2C2♯Y is an equivalence. Under these identifications, as was recalled above, it follows from the proof of [9, Theorem 2.2.4] that the canonical 2-functor C×C → C⊠CC is identified with 2 : C×C → C, which is given by LModC ( X2♯X ) ×ModC ( Y2♯Y ) → BimodC ( X2♯X,Y2♯Y ) , (M,N) 7→ M2N. But, by considering the restriction Ce ↪→ C, the above equivalences restrict to equivalences Ch → ModCe ( Y2♯Y ) , Cg → LModCe ( X2♯X ) , and Cgh → BimodCe ( X2♯X,Y2♯Y ) . In particular, the canonical 2-functor Cg × Ch → Cg ⊠Ce Ch is identified with 2 : Cg × Ch → Cgh as claimed. ■ A similar argument as the one used in the proof of the above proposition yields the following result. Corollary 3.5. The canonical Ce-Ce-bimodule 2-functor Cg−1 → FunCe(Cg,Ce) given by D 7→ {C 7→ D2C} is an equivalence. 3.2 Brauer–Picard spaces and extensions Group-graded extensions of fusion 1-categories are parameterised by maps into the space of invertible bimodule 1-categories and their (invertible) higher morphisms [16]. Thanks to the existence of the 4-category F2C obtained in [9], the corresponding spaces for our 2-categorical purposes are easy to define. Definition 3.6. Let C be a fusion 2-category. The Brauer–Picard space of C consists of the invertible objects and the invertible morphisms in the monoidal 3-category of finite semisimple C-C-bimodule 2-categories, that is, BrPic(C) := ( EndF2C(C) )× . Remark 3.7. We will write BrPic(C) := π0(BrPic(C)), the Brauer–Picard group of C. It follows from [7, Lemma 2.2.1] that ΩEndF2C(C) ≃ Z (C), the Drinfeld center of C, as defined in [1]. Thus, the homotopy groups of the space BrPic(C) are given as follows: π0 π1 π2 π3 BrPic(C) Inv(Z (C)) Inv(ΩZ (C)) k× 6 T.D. Décoppet Example 3.8. Taking C = 2Vect, we have that BrPic(C) is the core of the symmetric monoidal 3-category of multifusion 1-categories and the homotopy groups of this core are well known π0 π1 π2 π3 1 0 0 k× Example 3.9. More generally, we may also take C = 2VectπG, in which case we have π0 π1 π2 π3 H3(G;k×)⋊Out(G) Z(G)⊕H2(G;k×) Ĝ k× We have used Ĝ to denote the group of multiplicative characters of G, [10, Proposition 2.4.1] for the description of π0, and the main theorem of [27] for π1 and π2 together with the identifi- cation Pic(Rep(G)) ∼= H2(G;k×), which is given, for instance, in [3, Proposition 6.1]. The following example will be relevant in the next section. Example 3.10. Let E be a symmetric fusion 1-category, and take C = Mod(E). In this case we have π0 π1 π2 π3 M̃ext(E)⋊Autbr(E) Z(Spec(E))⊕ Pic(E) Inv(E) k× We have used M̃ext(E) to denote the group of Witt-trivial minimal non-degenerate extensions of E : The description of π0 then follows from [7, Corollary 3.1.7]. The statement for π2 follows from of [23, Lemma 2.16]. Finally, the description of π1 follows from [3, Corollary 6.11]. Let us also point out that the group Pic(E) with E super-Tannakian was computed in [3, Theorem 6.5]. A more general version of the next result has been announced [22]. We are very grateful to them for outlining their proof to us, which has inspired the argument that we give below. Theorem 3.11. Let G be a finite group, and C be a fusion 2-category. Then, faithfully G-graded extensions of C are classified by homotopy classes of maps BG → BBrPic(C). Proof. For the purpose of this proof, it will be convenient to think of maps of spaces BG → BBrPic(C) as monoidal maps G → BrPic(C) of (∞, 1)-categories. We will also consider the (∞, 1)-category C obtained from EndF2C(C) by only considering the invertible n-morphisms when n ≥ 2. Then, by definition we have BrPic(C) = C× as monoidal (∞, 1)-categories. We begin the proof by some general nonsense. Recall that algebras in the monoidal (∞, 1)- category C correspond precisely to lax monoidal functors ∗ → C (see [30, Example 2.2.6.10]). We want a similar description for the notion of a (faithfully) G-graded algebra. In order to do so, consider the monoidal 1-category G⊔, which is the coproduct completion of G. Now, there is a canonical algebra A[G] in G⊔ given by A[G] := ∐ g∈G g, or equivalently a lax monoidal functor ∗ → G⊔. Then, giving a G-grading on an algebra A in C is equivalent to providing a factorization of lax monoidal functors ∗ C G⊔. A A[G] Extension Theory and Fermionic Strongly Fusion 2-Categories 7 But, given that C has direct sums, lax monoidal functors G⊔ → C correspond exactly to lax monoidal functors G → C . Now, given a map of spaces BG → BBrPic(C), or equivalently a strongly monoidal functor F : G → C , we obtain a G-graded algebra D := F (A[G]) in C. We also write D = ⊞g∈GDg = ⊞g∈GF (g). In fact, as F (e) = C by definition, the monoidal 2-category D is a faithfully G-graded extension of C. It is therefore enough to prove that D is a fusion 2-category. The only property that is not obvious is rigidity. We have that EndC(D) is a fusion 2-category thanks to Theorem 2.3 above. Let us fix g ∈ G. For any simple object X in Dg, we can consider the left C-module 2-functor RX : D → D given by D 7→ D2X on De ≃ C, and zero on Dh for any e ̸= h ∈ G. As the objects of the monoidal 2-category EndC(D) have duals, we can consider the right adjoint R∗ X of RX . But, we have that Dg−1 → FunDe(Dg,De) is an equivalence of De-De- bimodule 2-categories as F is strongly monoidal, and therefore preserves duals. Thus, we find that R∗ X ≃ Y2(−) : Dg → De = C for some Y in Dg−1 . This proves that X has a left dual Y in D. One shows analogously that X has a right dual. Conversely, given D = ⊞g∈GDg a G-graded extension of C = De, we can consider the cor- responding lax monoidal functor F : G → C . More precisely, we set F (g) = Dg with lax monoidal structure given by the monoidal structure of D. We want to show that F is strongly monoidal and factors through BrPic(C) ⊂ C . It follows from the definition of an extension that F is strongly unital. Proposition 3.4 above establishes that F has the remaining desired properties. ■ Remark 3.12. The classification of Tambara–Yamagami 1-categories may be recovered from extension theory for fusion 1-categories as explained in [16, Section 9.2]. It would similarly be interesting to understand the classification of Tambara–Yamagami 2-categories obtained in [10, Proposition 5.2.3] in terms of the extension theory of fusion 2-categories. Namely, Tambara– Yamagami 2-categories are by definition Z/2-graded extensions of 2Vect(A[1]×A[0]) by 2Vect for some finite abelian group A. However, there is a complication that arises with the case of Tambara–Yamagami 2-categories: The group H5(BZ/2;k×) ∼= Z/2 is non-zero. But, in order to construct a map of spaces BZ/2 → BBrPic(2Vect(A[1] × A[0])) one has to check that a certain obstruction class living in H5(BZ/2;k×) vanishes. We do not know how to do this directly. In fact, the vanishing of such obstructions is a well-known difficulty in the exten- sion theory of fusion 1-categories. However, for Tamabra–Yamagami 1-categories the relevant group H4(BZ/2;k×) = 0 is trivial. Relatedly, general results guaranteeing the vanishing of this obstruction for fusion 1-categories are known such as [16, Theorem 8.16]. We wonder whether such a criterion may be established for extensions of higher fusion categories. Remark 3.13. Over an arbitrary field, some results on extension theory for finite semisimple tensor 1-categories were obtained in [33]. Going up one categorical level, one can setup ex- tension theory for locally separable compact semisimple tensor 2-categories over an arbitrary field. Namely, the reference [9] does work at this level of generality, and the proofs of both Proposition 3.4 and Theorem 3.11 continue to hold up to the obvious modifications. 3.3 Extensions from crossed braided fusion 1-categories We now review [11, Construction 2.1.23], which will later allow us to give very concrete models for some fermionic strongly fusion 2-categories. A related construction appeared in [2, Section 6]. In a slightly different direction, we also refer the reader to [25] for a detailed discussion of the relation between G-crossed braided 1-categories and higher categories. 8 T.D. Décoppet Construction 3.14. Fix G a finite group, and let C be a (not necessarily faithfully graded) G-crossed braided fusion 1-category. Following the notations of [15, Section 8.24], C is a (not necessarily faithfully) G-graded fusion 1-category C = ⊞g∈GCg equipped with a G-action g 7→ Tg such that Tg(Ch) ⊆ Cghg−1 together with suitably coherent natural isomorphisms cW,X : W⊗X ∼= Tg(X)⊗W whenever W is in Cg. For simplicity, and without loss of generality, we will assume that the underlying monoidal 1-category of C is strict. We can then consider the monoidal 2-category D̂ whose set of objects is the finite group G, and with Hom-1-categories given by Hom D̂ (g, h) := Chg−1 . Composition of 1-morphisms is given by the tensor product ⊗ of C. Then, the monoidal 2-functor 2 is defined by 2 : Hom D̂ (g1, h1)×Hom D̂ (g2, h2) → Hom D̂ (g1g2, h1h2), (W,X) 7→ W ⊗ Tg1(X) and its naturality constraints are given by the G-crossed braided structure of C. More precisely, for any 1-morphisms W : g1 → h1, X : g2 → h2, Y : h1 → f1, Z : h2 → f2, the interchanger ϕ2 is given by ϕ2 (W,X),(Y,Z) : (Y Th1(Z))(WTg1(X)) c−1 W,Z−−−→ YWTg1(Z)Tg1(X) (µg1 )Z,X−−−−−→ (YW )Tg1(ZX). Then, the associator 2-natural equivalence α is given on objects by αg1,g2,g2 = Idg1g2g3 , and on 1-morphisms W : g1 → h1, X : g2 → h2, Y : g3 → h3 by αW,X,Y : WTg1(X)Tg1g2(Y ) (γ−1 g1,g2 )Y−−−−−−→ WTg1(X)Tg1(Tg2(Y )) (µg1 )X,Y−−−−−−→ WTg1(XTg2(Y )). Likewise, the 2-natural equivalences witnessing unitality are the obvious ones. The pentagonator as well as the other invertible modifications that have to be specified are all taken to be the identity ones. That these assignments yield a monoidal 2-category follow at once from the axioms of a G-crossed braided 1-category. Finally, we obtain a fusion 2-category D by taking the Cauchy completion of D̂ in the sense of [18] (see also [4, 11]). Remark 3.15. Let us write suppG(C) for the support of C in G, that is the subset of ele- ments g ∈ G such that Cg is non-zero. As C is G-crossed braided, suppG(C) is a normal subgroup of G. It follows from the above construction that π0(D) = G/suppG(C) as finite groups, and that D is faithfully G/suppG(C)-graded. 4 Fermionic strongly fusion 2-categories Recall that k is an algebraically closed field of characteristic zero. It is well known that bosonic strongly fusion 2-categories are classified by a finite group G and a 4-cocycle for G with coef- ficients in k×. In fact, once we know that every simple object of such a fusion 2-category is invertible, as follows from [24, Theorem A], the problem becomes a straightforward application of Theorem 3.11 with Example 3.8. At a Physical level of rigor, this was already observed in [28]. Below, we will establish a similar classification of fermionic strongly fusion 2-categories. Recall that a fermionic strongly fusion 2-category is a fusion 2-category C such that ΩC = SVect. Then, it follows from [24, Theorem B] that every simple object of C is invertible. This was first observed in the physics literature [29]. As a consequence of this last fact, we find that π0(C) is a finite group, and there is a central extension of finite groups 0 → Z/2 → Inv(C) → π0(C) → 1. Namely, we have Inv(2SVect) ∼= Z/2. Examples are known for which the associated short exact sequence is not split (see [11, Example 2.1.27] or Figure 1 below). Extension Theory and Fermionic Strongly Fusion 2-Categories 9 4.1 The Brauer–Picard space of 2SVect It follows from Example 3.10 above that the homotopy groups of the space BrPic(2SVect) are as follows π0 π1 π2 π3 1 Z/2⊕ Z/2 Z/2 k× In order to completely characterize the space BBrPic(2SVect), we have to understand its Post- nikov k-invariants. They are precisely those of the braided monoidal 2-category Z (2SVect)×, that is of the space B2Z (2SVect)×, and these k-invariants can be determined using [17, 21]. In particular, this space has non-zero homotopy groups exactly in degrees 2, 3, and 4. So as to do this, we review some notation; For the most part we follow those used in [23, Section 3.1]. � Given an abelian group A and a non-negative integer n, we use A[n] to denote the n-th Eilenberg–MacLane space associated to A. � The cohomology ring H•(Z/2[n];Z/2) is generated by an element in degree n under cup products and the action of the Steenrod operations Sqi. � We will also consider the cohomology groups H•(Z/2[n];k×). Via the map t 7→ (−1)t induced by the inclusion Z/2 ↪→ k×, we can give names to all of the cohomology classes that we will consider in these groups. Lemma 4.1. Let us write c2 for the generator of H2((Z/2⊕0)[2];Z/2) and m2 for the generator of H2((0 ⊕ Z/2)[2];Z/2). The first k-invariant of B2Z (2SVect)× is c22 + c2m2 in the group H4((Z/2⊕ Z/2)[2];Z/2). Let us write Y := Fib ( (Z/2⊕ Z/2)[2] c22+c2m2−−−−−→ Z/2[4] ) , for the (homotopy) fiber, and t3 for the generator of H3(Z/2[3];Z/2), the second k-invariant of B2Z (2SVect)×, which lives in H5(Y ;k×), is σ := (−1)Sq 2t3+t3m2. Proof. The underlying 2-category of Z (2SVect) is depicted below (see, for instance, [23]). I M C C2M Vect SVect Vect VectZ/2 Vect VectZ/2 Vect VectZ/2 We want to describe the space B2Z (2SVect)×. We begin by determining its first k-invariant. We have Z (2SVect)0 ≃ 2SVect as symmetric fusion 2-categories, so the restriction of the first k-invariant to the braided monoidal sub-2-category spanned by C has to be c22 by [17]. Furthermore, its restriction to both M and C2M has to be trivial by [23, Theorem 3.2] and its proof. This uniquely determines the first k-invariant as c22 + c2m2 in H4((Z/2⊕ Z/2)[2];Z/2). Now, let Y be the space in the statement of the lemma. We want to determine the group H5(Y ;k×) and its generators. In order to do so, we use the Serre spectral sequence for Y Ei,j 2 = H i((Z/2⊕ Z/2)[2], Hj(Z/2[3];k×)) ⇒ H i+j(Y ;k×). 10 T.D. Décoppet The first entries of the E2 page for this spectral sequence are given by j 5 Z/2 0 4 0 0 0 3 Z/2 0 Z/2⊕2 Z/2⊕2 2 0 0 0 0 0 1 0 0 0 0 0 0 0 k× 0 Z/2⊕2 0 Z/2⊕2 ⊕ Z/4 Z/2⊕3 Z/2⊕4 0 1 2 3 4 5 6 i. (4.1) We will abstain from giving generators for all the non-zero entries. Let us nonetheless point out that the (6, 0) entry is generated by (−1)c 3 2 , (−1)c 2 2m2 = (−1)c2m 2 2 , (−1)m 3 2 , (−1)Sq 1c2Sq 1m2 . The equality follows from the fact that (−1)Sq 1 is the trivial map. Now, the (5, 0) entry automat- ically survives to E∞ as no differential can hit it. Then, the d2 differential is trivial, and the d3 differential is given by d3 ( (−1)t3 ) = (−1)c 2 2+c2m2 due to the k-invariant of Y . In particular, we find that d3 ( (−1)t3c2 ) = (−1)c 3 2+c22m2 , d3 ( (−1)t3m2 ) = (−1)c 2 2m2+c2m2 2 = 0, so that the Z/2 summand of the (2, 3) entry that is generated by (−1)t3m2 does survive to the E∞ page. It remains to understand what happens to the entry in degree (0, 5). We will argue that this entry survives. Namely, we have that the space Y is a truncation of B2Z (2SVect)×. But, we know that ΩZ (2SVect)× = SVect×, and it is well known that the k-invariant of the space B3SVect× is the non-trivial class (−1)Sq 2t3 in H5(Z/2[3];k×). Thus, the edge map H5(Y ;k×) → H5(Z/2[3];k×) induced by the inclusion Z/2[3] ↪→ Y is non-zero, and the claim follows. Therefore, we find that H5(Y ;k×) is generated by (−1)Sq 2t3 , (−1)t3m2 , (−1)Sq 2Sq1c2 , (−1)c2Sq 1m2 = (−1)m2Sq 1c2 , and (−1)Sq 2Sq1m2 = (−1)m2Sq 1m2 . We now turn our attention towards describing the second k-invariant σ of B2Z (2SVect)×. To this end, let us write X := Fib ( Z/2[2] 0−→ Z/2[4] ) . Then, there is an inclusion f : X ↪→ Y induced by the inclusion of the second summand Z/2[2] ↪→ (Z/2 ⊕ Z/2)[2]. (This corresponds to picking out the object M of Z (2SVect)×.) Let us note that this map is compatible with the spectral sequence (4.1) above. It was computed in [23, Theorem 3.2] that the pullback f∗(σ) = (−1)Sq 2t3+t3m2 , so that σ must contain at least the factors (−1)Sq 2t3 and (−1)t3m2 , and cannot contain (−1)Sq 2Sq1m2 . In addition, there is another map g : X ↪→ Y induced by the diagonal inclusion Z/2[2] ↪→ (Z/2⊕Z/2)[2]. (This corresponds to picking out the object C2M of Z (2SVect)×.) Again, it follows from [23, Theorem 3.2] that the pullback g∗(σ) = (−1)Sq 2t3+t3m2 . But, by naturality of the Serre spectral sequence, we find that g∗ ( (−1)Sq 2Sq1c2 ) = (−1)Sq 2Sq1t2 = g∗ ( (−1)c2Sq 1m2 ) with t2 the generator of H2(Z/2[2];Z/2), so that σ = (−1)Sq 2t3+t3m2 or σ = (−1)Sq 2t3+t3m2+Sq2Sq1c2+c2Sq 1m2 . It turns out that these two possibilities are equivalent. More precisely, consider the homotopy autoequivalence ϕ : Y ≃ Y with ϕ∗(t3) = t3 + Sq1c2. Then, we have ϕ∗((−1)Sq 2t3+t3m2 ) = (−1)Sq 2t3+t3m2+Sq2Sq1c2+c2Sq 1m2 , which concludes the proof. ■ Extension Theory and Fermionic Strongly Fusion 2-Categories 11 4.2 Twisted supercohomology In order to be able to carry out computations, as well as to make contact with the existing literature, it is useful to unpack the data of the space BrPic(2SVect) in a different way and express the classification of fermionic strongly fusion 2-categories using supercohomology. Definition 4.2. Supercohomology is the generalized cohomology theory associated to the dou- ble loop space of the spectrum associated to 2SVect×. For any space X and integer n, we write SHn(X) for the group of homotopy classes of maps of spaces from X to Bn−22SVect×. This generalized cohomology theory first appeared in the literature on symmetry protected topological phases [17, 26, 35], where it is also referred to as extended supercohomology.1 More precisely, as we have already recalled in the proof of Lemma 4.1 above, the homotopy groups of 2SVect× are given by Z/2, Z/2, k×, and supercohomology is shifted so that SH0(pt) = k×. In particular, for any space X and integer n, it follows from the Atiyah–Hirzebruch spectral sequence Ei,j 2 = H i ( X;SHj(pt) ) ⇒ SH i+j(BG) that the group SHn(X) has a filtration with successive subquotients En,0 ∞ , En−1,1 ∞ , and En−2,2 ∞ . But, En−2,2 ∞ is a subgroup of En−2,2 2 = Hn−2(X;Z/2), En−1,1 ∞ is a subquotient of En−1,1 2 = Hn−1(X;Z/2), and En,0 ∞ is a quotient of En,0 2 = Hn(X;k×), so that a class π in SHn(X) may, by abusing notations, be written as a triple (α, β, γ) with α ∈ Hn−2(X;Z/2), β ∈ Hn−1(X;Z/2), and γ ∈ Hn(X;k×). Now, for the purposes of classifying fermionic strongly fusion 2-categories, we will also need to consider twisted supercohomology. More precisely, the spectrum 2SVect× has a non-trivial space of automorphisms. We will only be interested in the subspace given by A uttens(2SVect) ≃ A utbr(SVect) ≃ Z/2[1]. Concretely, this corresponds to the symmetric monoidal natural autoequivalence φ of the identity functor on SVect which takes the value −1 on purely odd vector spaces. Then, as is explained in [31, Section 4.1], the action of the higher group Z/2[1] on the spectrum 2SVect× is encoded by the fiber sequence of spaces Bn−22SVect× Bn−22SVect×//(Z/2[1]) BZ/2[1] = Z/2[2] (4.2) for large enough n. Definition 4.3. Let X be a space equipped with an action ϖ : X → Z/2[2], and write P → X for the bundle with fiber Bn−22SVect× obtained by pulling back (4.2) along ϖ. The ϖ-twisted n-th supercohomology of X is the group SHn+ϖ(X) := ΓX(P ) of homotopy classes of sections of the bundle P → X. Just as we have explained above in the untwisted case, by abusing notations, we will also write classes in twisted supercohomology groups as triples. We now identify the total space B22SVect×//(Z/2[1]). 1An earlier but distinct notion of supercohomology appeared in [19]. In our notations, what they are considering is the generalized cohomology theory associated to (the loop space of) the spectrum SVect×, whose homotopy groups are Z/2 and k×. This generalized cohomology theory is sometimes called restricted supercohomology. We will make no use of this notion. 12 T.D. Décoppet Lemma 4.4. The canonical fibration B22SVect×//(Z/2[1]) → Z/2[2] is isomorphic to the map B2Z (2SVect)×→Z/2[2] collapsing the connected component of the identity of Z (2SVect). Proof. As above, let M denote an invertible object of Z (2SVect) that is not in the con- nected component of the identity. In order to identify the spaces B22SVect×//(Z/2[1]) and B2Z (2SVect)×, it is enough to show that the induced action of the invertible object M in Z (2SVect) on Z (2SVect)0 ≃ 2SVect is the canonical one. The invertible object M induces a braided monoidal 2-natural equivalence of 2SVect. More precisely, let us use b to denote the braiding of Z (2SVect), which is an adjoint 2-natural equivalence equipped with a pseudo-inverse b•. Let us also write u : I ≃ M2M for an adjoint equivalence witnessing thatM is invertible. Then, the 2-natural autoequivalence t of the identity 2-functor on Z (2SVect) that is given on an object X in Z (2SVect) by tX : X X2u−−−→ X2M2M bX,M2M −−−−−−→ M2X2M M2b•X,M−−−−−−→ M2M2X u•2X−−−−→ X, can be canonically upgraded to a braided monoidal 2-natural equivalence. In particular, we can restrict t to a braided monoidal 2-natural equivalence of the identity 2-functor on 2SVect. Now, as A utbr(2SVect) ≃ A uttens(2SVect) ≃ Z/2[1], it is enough to check that this action is non-trivial. In order to see this, let e denote the non-identity invertible 1-morphism in Ω2SVect = SVect. Then, it was explained in [23, Section 3.1] that the double braiding of the object M and the 1-morphism e is given by bM,e · be,M = (−1) IdM2e. This shows that Ωt is the non-trivial braided monoidal autoequivalence of SVect, and therefore concludes the proof. ■ Combining the last lemma together with Theorem 3.11 yields the following result. We wish to point out that, relying on the as-of-yet incomplete theory of higher condensations [18], a version of the classification of fermionic strongly fusion 2-categories has already been given in [20, Section V.D]. In the physics literature, an even earlier, albeit slightly incorrect, version appeared in [29]. Theorem 4.5. Fermionic strongly fusion 2-categories equipped with a faithful grading are clas- sified by a finite group G together with a class ϖ in H2(BG;Z/2) and a class π in SH4+ϖ(BG). Proof. It follows from Theorem 3.11 that (faithfully) G-graded extensions of 2SVect are clas- sified by homotopy classes of maps BG → BBrPic(2SVect) ≃ B2Z (2SVect)×. The result then follows from the last lemma above. Namely, the class ϖ is given by the compos- ite BG → B2Z (2SVect)× → BZ/2[1], and endows BG with an action by Z/2[1]. In addition, the data of a map BG → B2Z (2SVect)× liftingϖ is precisely that of a class in SH4+ϖ(BG). ■ Remark 4.6. Without taking into account the faithful grading, fermionic strongly fusion 2- categories are classified by a finite group G together with a class in H2(BG;Z/2)/Out(G) and a class in SH4+ϖ(BG)/Out(G). Remark 4.7. As is clear from the proof, the finite group G corresponds to the group of con- nected components of the fermionic strongly fusion 2-category C. We emphasize that the class ϖ in H2(BG;Z/2) is not the extension class determining the group of invertible objects Inv(C) as a Z/2 extension of G! This was first observed in the physics literature [29] and then given a more mathematical treatment in [21], and transpires from Example 4.13 below, but also from the result of Appendix A. Rather, the extension class is given by the bottom layer α of the class π = (α, β, γ) in SH4+ϖ(BG). The class β supplies the 1-morphisms witnessing associativ- ity, and the class γ gives the pentagonator. As for the class ϖ, it follows from the proof of Theo- rem 4.5 that it corresponds to the action of G on C0 = 2SVect by conjugation. Said differently, Extension Theory and Fermionic Strongly Fusion 2-Categories 13 ϖ encodes the data of the interchanger, or, equivalently, the 2-naturality of the associator. That these two pieces of coherence data are intimately related can be seen from [12, Proposition 4.2, equation (Aâ2)]. We expect that the action of G on 2SVect can be detected at the level of the Drinfeld center Z (C) of C. More precisely, let us write ( G̃, z ) for the central extension of G by Z/2 parameterised by ϖ, then let Rep ( G̃, z ) be the subcategory of super-representations of G̃ on which z acts as the parity automorphism. We expect that ΩZ (C) ≃ Rep ( G̃, z ) as symmetric fusion 1-categories. We have checked this property for the fermionic strongly fusion 2-categories of Examples 4.10, 4.11 and 4.12 below. Remark 4.8. Let us momentarily work over the field of real numbers R. It is interesting to ask whether the classification of strongly fusion 2-categories remains valid at this level of generality. This turns out to be wrong, even in the bosonic case: The most general version of [24, Theorem A] does not hold over fields that are not algebraically closed. More precisely, it is not true that every fusion 2-category C over R with ΩC = VectR is a group graded extension of 2VectR. The following counterexample was pointed out to us by Theo Johnson-Freyd. Let us consider the fusion 2-category 2RepR(Z/3[2]) of real 2-representations of the 2-group Z/3[2], that is, finite semisimple R-linear 1-categories equipped with an action of Z/3[1]. Over the complex numbers C, the notion of a 2-representation was first considered in [13, Section 6] (see also [11, Section 1.4.5] for a recent account). The underlying 2-category and fusion rules of the fusion 2-category 2RepR(Z/3[2]) are as depicted below. C X I H VectσC(Z/2) VectC VectR VectH 2 I H C X I I H C X H H I C X C C C 2C 2X X X X 2X 2C ⊞X In particular, this shows that, unlike in the case of algebraically closed fields [11, Example 1.4.22], the fusion 2-categories 2RepR(Z/3[2]) and 2VectR(Z/3) are not monoidally equivalent. Never- theless, they become equivalent upon base extension to C. This suggests that it is possible to classify real strongly fusion 2-categories by combining the known classification of strongly fusion 2-categories over an algebraically closed field of characteristic zero together with a 2-categorical version of the descent techniques of [14]. 4.3 Examples We examine various special cases of the classification of fermionic strongly fusion 2-categories obtained above. Example 4.9. Let us take G a finite group. Then, for any 4-cocycle γ for G with coefficients in k×, we can consider the fermionic strongly fusion 2-category 2SVect ⊠ 2Vectγ(G). Their Drinfeld centers are completely understood thanks to [27] and [21], as taking Drinfeld centers commutes with 2-Deligne tensor products by [7]. Further, in the classification of Theorem 4.5 the corresponding data is ϖ = triv, and the class π in SH4(BG) is given by (triv, triv, γ). However, for a general group G, different 4-cocycles γ may yield equivalent fusion 2-categories (see Example 4.14 below). Finally, let us note that, when G has odd order, these are all of the fermionic strongly fusion 2-categories. Namely, in this case, we have H2(BG;Z/2) ∼= 0 and SH4(BG) ∼= H4(BG;k×). Example 4.10. Let ( G̃, z ) be a finite super-group, that is a finite group G̃ equipped with a central element z of order exactly 2. Let us write B for (any of) the braided Ising 1-categories. 14 T.D. Décoppet Such a braided fusion 1-category has a canonical Z/2-grading B = B0 ⊞ B1 with B0 = SVect and B1 = Vect. This allows us to consider B as a non-faithfully G̃-crossed braided 1-category by setting Be = B0, Bz = B1, Bg = 0 for any other g ∈ G̃, and taking the trivial G̃-action. Then, we can use [11, Construction 2.1.23], recalled above in Construction 3.14, so as to ob- tain a monoidal 2-category D̂ whose group of objects is G̃, and with Hom-1-categories given by Hom D̂ (g, h) := Bhg−1 . We then get a fusion 2-category D by taking the additive completion of D̂. (In the specific case under consideration, the additive completion is the Cauchy comple- tion.) Further, it is clear that Inv(C) ∼= G̃ and π0(C) = G̃/z. The fermionic strongly fusion 2-category corresponding to G̃ = Z/4 is depicted below in Figure 1. It is informative to understand how the fermionic strongly fusion 2-category D constructed above fits into the classification of Theorem 4.5. As was already pointed out, the correspond- ing group is G := G̃/z. Further, it follows from the construction of D that the class ϖ in H2(BG;Z/2) is the class corresponding to the central extension G̃ of G. More precisely, one computes that ΩZ (D) ≃ Rep ( G̃, z ) . Finally, the class π in SH4+ϖ(BG) is given by (ϖ, triv, triv). In particular, the construction above does not depend on the choice of braided Ising 1-category B. X2 X3 Vect X I VectZ/2 VectZ/2 VectZ/2 sVect Figure 1. An exotic fermionic strongly fusion 2-category. Example 4.11. Again, let us fix ( G̃, z ) a finite super-group. Then, we may consider the sym- metric fusion 1-category Rep ( G̃, z ) of finite dimensional super-representations of G on which z acts by the parity automorphism. We write 2Rep ( G̃, z ) := Mod ( Rep ( G̃, z )) for the corre- sponding (symmetric) fusion 2-category. Then, given that there is an essentially unique fiber functor Rep ( G̃, z ) → SVect, we can view 2SVect as a finite semisimple module 2-category for 2Rep ( G̃, z ) . We write C := End 2Rep(G̃,z) (2SVect) for its Morita dual fusion 2-category. By [7, Lemma 3.2.1], C is a fermionic strongly fusion 2-category. We claim that, in the formulation of Theorem 4.5 the data corresponding to the fusion 2- category C is the finite group G := G̃/z, the class ϖ in H2(BG;Z/2) is the class classifying the central extension G̃ of G, and the class π in SH4+ϖ(BG) is the trivial one. In partic- ular, we have Inv(C) ∼= G⊕ Z/2, so that C is distinct from the fermionic strongly fusion 2- category D constructed in the previous example. In order to see this, observe that there is a monoidal 2-functor C → 2SVect. Namely, by construction, C acts on the finite semisimple 2- category 2SVect, i.e., there is a monoidal 2-functor F : C → End(2SVect) ≃ Mod(Z(SVect)). But, this action commutes with the action of 2Rep ( G̃, z ) , so that the image of Fmust land in the monoidal sub-2-category 2SVect of Mod(Z(SVect)) as desired. This implies that the class π is trivial. On the other hand, it follows from [7, Theorem 2.3.2] that ΩZ (C) ≃ Rep ( G̃, z ) , so that the class ϖ is as claimed above. Example 4.12. Let G be a finite group, ϖ a 2-cocycle for G with coefficients in Z/2, and π = (α, β, γ) a class in SH4+ϖ(BG). Provided that α = triv, then the corresponding fermionic Extension Theory and Fermionic Strongly Fusion 2-Categories 15 strongly fusion 2-category is a fusion 2-category of twisted 2-group graded 2-vector spaces. More precisely, we can think of the 3-cocycle β for G with coefficients in Z/2 as the Postnikov class of a finite 2-group G = Z/2[1] · G[0]. Given a 4-cocycle ω in H4(BG,k×), we can consider the fusion 2-category 2Vectω(G) of ω-twisted G-graded 2-vector spaces. This is explained in detail in [11, Construction 2.1.16]. This is a fermionic strongly fusion 2-category exactly if the image of ω under the canonical map H4(BG,k×) → H4(B2Z/2,k×) is (−1)Sq 2 . Namely, in this case, we have Ω2Vectω(G) ≃ SVect. Furthermore, every fermionic strongly fusion 2-category with α = triv can be obtained via this construction. In particular, this includes all the fermionic strongly fusion 2-categories of Example 4.9. We now turn our attention to fermionic strongly fusion 2-categories whose group of connected components is a fixed 2-torsion group. Example 4.13. Let us examine the case G = Z/2n with n ≥ 1. Then, we have H2(BZ/2n;Z/2) ∼= Z/2. On one hand, we have SH4(BZ/2n) = 0 corresponding to the fermionic strongly fusion 2-category 2SVect(Z/2n). Namely, we can consider the Atiyah–Hirzebruch spectral sequence Ei,j 2 = H i ( BZ/2n;SHj(pt) ) ⇒ SH i+j(BZ/2n) with corresponding E2 page given by j 2 Z/2 Z/2 Z/2 Z/2 1 Z/2 Z/2 Z/2 Z/2 Z/2 0 k× Z/2n 0 Z/2n 0 Z/2n 0 1 2 3 4 5 i. The d2 differentials for this spectral sequence, for any finite group G and with ϖ = triv, are given by the k-invariants of 2SVect×, so that we have d2 = Sq2 : Ei,2 2 → Ei+2,1 2 and d2 = (−1)Sq 2 : Ei,1 2 → Ei+2,0 2 , (4.3) provided that i ≥ 1, and where, as above, t 7→ (−1)t is the homomorphism induced by Z/2 ↪→ k×. In particular, the terms in degrees (2, 2) and (3, 1) are both killed so that SH4(BZ/2n) = 0 as claimed. On the other hand, using ϖ to denote the non-trivial class in H2(BZ/2n,Z/2), we claim that SH4+ϖ(BZ/2n) = Z/2. Namely, in that case, the E2 page of the corresponding Atiyah– Hirzebruch spectral sequence is the same as above. However, the differentials d2 : E i,j 2 → Ei−1,j+2 2 may be different. We do not know how to describe these differentials, and will therefore use other techniques in order to compute this group. We have to understand whether or not the groups in degrees (2, 2) and (3, 1) survive to the E∞ page. We assert that the group in de- gree (2, 2) does. Namely, this follows from the fact that we have exhibited two non-equivalent fermionic strongly fusion 2-categories C as in Example 4.11 andD as in Example 4.10 with super- group ( Z/2(n+1), z ) classified by ϖ together with the classes (triv, triv, triv) and (ϖ, triv, triv). Now, let us assume that the group in degree (3, 1) survives. This would imply that there ex- ists a class in SH4+ϖ(BZ/2n) of the form (triv, β, triv) with non-trivial β. The corresponding fermionic strongly fusion 2-category then ought to be obtained via the construction of Exam- ple 4.12. However, it follows by inspection that there are exactly two fermionic strongly fusion 2-category that can be obtained that way, which must then be 2SVect(Z/2n) and C. We there- fore find that the group in degree (3, 1) must be killed by the d2 differential, thereby showing that SH4+ϖ(BZ/2n) = Z/2 as desired. We also wish to point out that it is expected that the two fusion 2-categories C and D have the same Drinfeld center. More specifically, if B is a braided Ising fusion 1-category, then it is predicted that D ⊠ Mod(B) is Morita equivalent to C. 16 T.D. Décoppet Example 4.14. We now take G = Z/2⊕ Z/2 with basis a = (1, 0) and b = (0, 1). In this case, we have H2(BG;Z/2) ∼= Z/2⊕ Z/2⊕ Z/2. More precisely, if c1, resp. d1, denotes the elements of H1(BG;Z/2) that restricts non-trivially to ⟨a⟩, resp. ⟨b⟩, then H2(BG;Z/2) has a basis given by c21, d 2 1, and c1d1. Let us consider the Atiyah–Hirzebruch spectral sequence Ei,j 2 = H i(BG;SHj(pt)) ⇒ SH i+j(BG) corresponding to the trivial class triv in H2(BG;Z/2). Its E2 page is given by j 2 Z/2 Z/2⊕2 Z/2⊕3 Z/2⊕4 1 Z/2 Z/2⊕2 Z/2⊕3 Z/2⊕4 Z/2⊕5 0 k× Z/2⊕2 Z/2 Z/2⊕3 Z/2⊕2 Z/2⊕4 0 1 2 3 4 5 i. The d2 differentials are as described in the formulas in equation (4.3). Beyond the potential differential d3, the other main difficulty lies in describing the differential d2 = (−1)Sq 2 , and, more precisely, the image of t 7→ (−1)t. In order to do so, consider the map of short exact sequences 0 Z/2 Z/4 Z/2 0 0 Z/2 k× k× 0. For i ≥ 2, this induces a commutative square H i−1(BG;Z/2) H i(BG;Z/2) H i−1(BG;k×) H i(BG;Z/2). Sq1 ∂ But, the bottom horizontal map is injective in the case G = Z/2 ⊕ Z/2 as every class in H i(BG;k×) is annihilated by the map on cohomology groups induced by x 7→ x2 on k×. This shows that the kernel of t 7→ (−1)t consists exactly of those classes that are in the image of the Bockstein homomorphism Sq1. In particular, we find that the group SH4(BG) has order 16. Namely, the d3 differentials on the entries (1, 2) and (2, 2) vanish. For the latter, this is immediate because the entry in degree (2, 2) is completely killed by d2. As for the former, this can be seen using the natu- rality of the Atiyah–Hirzebruch spectral sequence with respect to the various group homomor- phisms Z/2 ↪→ Z/2⊕ Z/2. A set of representatives for the classes in SH4(BG) is therefore given by (triv, β, γ) with γ in H4(BG;k×) ∼= Z/2 ⊕ Z/2 arbitrary, and β in the span of c21d1, c1d 2 1 in H3(BG;Z/2). The four fermionic strongly fusion 2-categories corresponding to (triv, triv, γ) are the ones of Example 4.9. The others are different, but all arise via the construction discussed in Example 4.12. It would be interesting to compute the twisted supercohomology groups SH4+ϖ(BG). Just as we have already seen in the preceding example, the main difficulty resides in describing the d2 differentials in the Atiyah–Hirzebruch spectral sequence in supercohomology. Extension Theory and Fermionic Strongly Fusion 2-Categories 17 A A computation in supercohomology. Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd∗ Let G = Z/4 ⊕ Z/4. We claim that the canonical map SH4(BG) → H2(BG;Z/2) is non-zero. Said differently, there exists a class (α, β, γ) in SH4(BG) with α ̸= triv. This proves that there exists a fermionic strongly fusion 2-category with trivial twistϖ whose group of invertible objects is a non-trivial central extension of the group of connected components, thereby answering a question of [24]. In order to prove the above claim, we will consider the Atiyah–Hirzebruch spectral sequence Ei,j 2 = H i(B(Z/4⊕ Z/4);SHj(pt)) ⇒ SH i+j(B(Z/4⊕ Z/4)), whose E2 page is depicted below j 2 Z/2 Z/2⊕2 Z/2⊕3 Z/2⊕4 1 Z/2 Z/2⊕2 Z/2⊕3 Z/2⊕4 Z/2⊕5 0 k× Z/4⊕2 Z/4 Z/4⊕3 Z/4⊕2 Z/4⊕4 0 1 2 3 4 5 i. The d2 differentials are given by (4.3). It will be necessary to give names to various of the classes in the groups above. To this end, recall that there is an isomorphism of graded rings H•(BZ/4;Z/2) ∼= Z/2[x1, x2]/x21, where x1 has degree 1 and x2 has degree 2. It then follows from the Künneth formula that H•(B(Z/4⊕ Z/4);Z/2) ∼= Z/2[x1, y1, x2, y2]/ ( x21, y 2 1 ) , where x1, y1 have degree 1, x2, y2 have degree 2, the classes x1, x2 are pulled back from Z/4⊕0, and the classes y1, y2 are pulled back from 0⊕Z/4. In particular, it follows that E2,2 3 , the kernel of the d2 differential d2 = Sq2 : E2,2 2 = H2(B(Z/4⊕ Z/4);Z/2) → E4,1 2 = H4(B(Z/4⊕ Z/4);Z/2), is spanned by the class x1y1. We also have to analyze E5,0 3 , the cokernel of the d2 differential d2 = (−1)Sq 2 : E3,1 2 = H3(B(Z/4⊕ Z/4);Z/2) → E5,0 2 = H5(B(Z/4⊕ Z/4);k×). We assert that the classes (−1)x1x2 2 , (−1)y1x 2 2 , (−1)x1y22 , (−1)y1y 2 2 (A.1) are linearly independent in H5(B(Z/4 ⊕ Z/4);k×) and span the image of d2. Namely, it fol- lows from the short exact sequence of coefficients 0 → Z/2 → k× → k× → 0, that the image of the map H5(B(Z/4 ⊕ Z/4);Z/2) → H5(B(Z/4 ⊕ Z/4);k×) is precisely the sub- group Z/2⊕4 ⊆ Z/4⊕4, so that E5,0 3 ∼= Z/2⊕4. Moreover, as (−1)Sq 1 is the trivial map, this image is indeed generated by the classes in (A.1): We can use this to give names to gener- ators of the group H5(B(Z/4⊕ Z/4);k×). We use √ x with x one of the classes in (A.1) to denote a class in H5(B(Z/4 ⊕ Z/4);k×) such that √ x · √ x = x. By definition, the classes √ x ∗Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada E-mail: theojf@pitp.ca Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, Canada E-mail: theojf@dal.ca mailto:theojf@pitp.ca mailto:theojf@dal.ca 18 T.D. Décoppet for x in (A.1) generate H5(B(Z/4 ⊕ Z/4);k×). Moreover, their images under the quotient map H5(B(Z/4⊕ Z/4);k×) ↠ E5,0 3 ∼= Z/2⊕2 are independent of any choices and generate the target. This means that the image of the class x1y1 under the d3 differential d3 : E 2,2 3 → E5,0 3 can be expressed as a linear combination of the classes √ x with x as in (A.1). It remains to argue that the class x1y1 in E2,2 3 ⊆ H2(B(Z/4⊕Z/4);Z/2) survives the d3 dif- ferential d3 : E 2,2 3 → E5,0 3 . In order to do so, we will use the naturality of the Atiyah–Hirzebruch spectral sequence. More precisely, we consider the epimorphism f : Z/4⊕Z/4 ↠ Z/4⊕Z/2, the Atiyah–Hirzebruch spectral sequence Ẽi,j 2 = H i(B(Z/4⊕ Z/2);SHj(pt)) ⇒ SH i+j(B(Z/4⊕ Z/2)), together with the induced map of spectral sequences f∗ : Ẽi,j 2 → Ei,j 2 . It follows from the Künneth formula that H•(B(Z/4⊕ Z/2);Z/2) ∼= Z/2[x1, z1, x2]/x21, where x1, z1 have degree 1, x2 has degree 2, the classes x1, x2 are pulled back from Z/4⊕0, and the class z1 is pulled back from 0⊕Z/2. Under the pullback map f∗, we have x1 7→ x1, x2 7→ x2, and z1 7→ y1. In particular, observe that the class x1y1 is in the image of the pullback f∗. Further, we have that Ẽ2,2 3 , the kernel of d2 = Sq2 : H2(B(Z/4 ⊕ Z/2);Z/2) → H4(B(Z/4 ⊕ Z/2);Z/2), is spanned by the class x1y1. Moreover, using an argument similar to the one given above, we find that the image of d2 = (−1)Sq 2 : H3(B(Z/4 ⊕ Z/2);Z/2) → H5(B(Z/4 ⊕ Z/2);k×) is Z/2⊕4 ⊆ Z/4⊕ Z/2⊕3 ∼= H5(B(Z/4⊕ Z/2);k×) and is generated by the classes (−1)x1x2 2 , (−1)z1x 2 2 , (−1)z 3 1x2 , (−1)z 5 1 . It follows from the naturality of the Künneth formula that there exists a class √ (−1)x1x2 2 in H5(B(Z/4⊕ Z/2);k×) such that√ (−1)x1x2 2 · √ (−1)x1x2 2 = (−1)x1x2 2 . In particular, the image of √ (−1)x1x2 2 under the quotient map H5(B(Z/4⊕Z/2);k×) ↠ Ẽ5,0 3 ∼= Z/2 is a generator. Moreover, as f∗((−1)x1x2 2 ) = (−1)x1x2 2 by naturality of t 7→ (−1)t, we must have f∗( √ (−1)x1x2 2) = √ (−1)x1x2 2 in E5,0 3 . But, if the d3 differential d3 : Ẽ 2,2 3 → Ẽ5,0 3 were non-zero, we would have d3(x1z1) = √ (−1)x1x2 2 , and, by naturality of the Atiyah–Hirzebruch spectral sequence, we would therefore have that d3(x1y1) = √ (−1)x1x2 2 in E5,0 3 . 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X 8 (2018), 011055, 29 pages, arXiv:1703.10937. https://doi.org/10.1090/jams/1023 https://arxiv.org/abs/2105.15167 https://doi.org/10.1017/S0004972721000095 https://arxiv.org/abs/2010.07950 https://doi.org/10.1112/jlms.12687 https://doi.org/10.1112/jlms.12687 https://arxiv.org/abs/2009.00405 https://doi.org/10.1007/jhep10(2017)080 https://doi.org/10.1007/jhep10(2017)080 https://arxiv.org/abs/1701.08264 https://doi.org/10.1016/j.aim.2019.106928 https://doi.org/10.1016/j.aim.2019.106928 https://arxiv.org/abs/1905.04644 https://doi.org/10.1103/PhysRevX.8.021074 https://arxiv.org/abs/1704.04221 https://doi.org/10.1103/PhysRevX.9.021005 https://arxiv.org/abs/1801.08530 https://www.math.ias.edu/~lurie/papers/HA.pdf https://doi.org/10.1007/s40062-014-0083-6 https://arxiv.org/abs/1207.0248 https://arxiv.org/abs/1411.6691 https://www.proquest.com/docview/2715420938/ACDC80FE6AEA477EPQ/1 https://arxiv.org/abs/1112.1000 https://doi.org/10.1103/PhysRevX.8.011055 https://arxiv.org/abs/1703.10937 1 Introduction 1.1 Extension theory for fusion 1-categories 1.2 Results 2 Preliminaries 2.1 Fusion 2-categories 2.2 Module 2-categories and the relative 2-Deligne tensor product 3 Extension theory 3.1 Invertible bimodule 2-categories 3.2 Brauer–Picard spaces and extensions 3.3 Extensions from crossed braided fusion 1-categories 4 Fermionic strongly fusion 2-categories 4.1 The Brauer–Picard space of 2SVect 4.2 Twisted supercohomology 4.3 Examples A A computation in supercohomology. Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd References
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spelling	Décoppet, Thibault Didier 2026-02-09T08:06:41Z 2024 Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd). Thibault Didier Décoppet. SIGMA 20 (2024), 092, 20 pages 1815-0659 2020 Mathematics Subject Classification: 18M20; 18N25 arXiv:2403.03211 https://nasplib.isofts.kiev.ua/handle/123456789/212614 https://doi.org/10.3842/SIGMA.2024.092 We study group-graded extensions of fusion 2-categories. As an application, we obtain a homotopy theoretic classification of fermionic strongly fusion 2-categories. We examine various examples in detail. I am particularly indebted to Theo Johnson-Freyd and David Reutter for sharing some of the ideas of their proof of the completely general version of group-graded extension theory, which have inspired our proof of Theorem 3.11, and to Matthew Yu for help with the cohomology computations of Section 4. I would also like to thank the referees for suggesting many invaluable improvements and clarifications. This work was supported in part by the Simons Collaboration on Global Categorical Symmetries. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd) Article published earlier
spellingShingle	Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd) Décoppet, Thibault Didier
title	Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
title_full	Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
title_fullStr	Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
title_full_unstemmed	Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
title_short	Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
title_sort	extension theory and fermionic strongly fusion 2-categories (with an appendix by thibault didier décoppet and theo johnson-freyd)
url	https://nasplib.isofts.kiev.ua/handle/123456789/212614
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Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)

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