Braided Picard groups and graded extensions of braided tensor categories
aa r X i v : . [ m a t h . QA ] J un Braided Picard groupsand graded extensions of braided tensor categories
Alexei Davydov a ) , Dmitri Nikshych b ) June 16, 2020 a ) Department of Mathematics, Ohio University, Athens, OH 45701, USA b ) Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA
Abstract
We classify various types of graded extensions of a finite braided tensor category B in terms of its2-categorical Picard groups. In particular, we prove that braided extensions of B by a finite group A correspond to braided monoidal 2-functors from A to the braided 2-categorical Picard group of B (consisting of invertible central B -module categories). Such functors can be expressed in termsof the Eilnberg-Mac Lane cohomology. We describe in detail braided 2-categorical Picard groups ofsymmetric fusion categories and of pointed braided fusion categories. Contents
Mod br ( B ) as a braided monoidal 2-category . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Examples and basic properties of braided module categories . . . . . . . . . . . . . . . . . 494.4 The symmetric monoidal 2-category of symmetric module categories . . . . . . . . . . . . 51 -categorical Picard groups 52 -categorical Picard group of a symmetric fusion category 60 In this paper we work over an algebraically closed field k . All tensor categories are assumed to be k -linearand finite [18].Let B be a tensor category. An extension of B is an embedding of B into a tensor category C i.e., afully faithful tensor functor ι : B → C . We will identify B with its image in C and use notation B ⊂ C to denote an extension. An isomorphism between extensions ι , ι : B → C is a tensor autoequivalence F : C → C such that F ◦ ι = ι .When B is braided (or symmetric) there are several types of extensions reflecting different “amountsof commutativity” of C . Namely, we say that an extension ι : B → C is • central if there is a lifting tensor functor L : B → Z ( C ) such that ι coincides with the composition B L −→ Z ( C ) Forget −−−−→ C , where Z ( C ) is the center of C and Forget : Z ( C ) → C is the forgetful functor; • braided if C is braided; • symmetric if C is symmetric.The extending tensor category C can be viewed as a B -module category. Furthermore, the tensorproduct of C equips it with a structure of a pseudo-monoid [12] in a monoidal 2-category M consisting ofcertain B -modules. Commutativity properties of B , C and the type of an extension B ⊂ C are reflectedin the choice of B -modules in M and in the properties of the pseudo-monoid C . These properties aresummarized in Table 1.In this paper, for a tensor category D we denote Bimod ( D ) the monoidal 2-category of D -bimodulecategories [22]. For a braided tensor category B we denote Mod ( B ) the monoidal 2-category of B -module categories [10, 22] (it can be viewed as a monoidal 2-subcategory of Bimod ( B )) and Mod br ( B )the braided monoidal 2-category of braided B -module categories. For a symmetric tensor category E we3xtension B ⊂ C M of B -modules C tensor monoidal 2-category Bimod ( B ) pseudo-monoidcentral monoidal 2-category Mod ( B ) pseudo-monoidbraided braided monoidal 2-category Mod br ( B ) braided pseudo-monoidsymmetric symmetric monoidal 2-category Mod sym ( B ) symmetric pseudo-monoidTable 1: Extensions B ⊂ C as pseudo-monoids in a monoidal 2-category M .denote Mod sym ( E ) the symmetric monoidal 2-category of symmetric E -module categories . By definition[5, 2], a braided B -module category is equipped with a natural collection of isomorphisms coherentlyextending the braiding on B , see Definition 4.1. Braided module categories turn out to play an importantrole in 4d topological field theory and factorization homology. In Theorem 4.10 we show that the 2-category Mod br ( B ) of braided module categories is braided 2-equivalent to the center (in the sense of[1]) of Mod ( B ): Mod br ( B ) ∼ = Z ( Mod ( B )) . (1.1) -functors to -categorical groups We focus on extensions of finite braided tensor categories graded by finite groups.Let G be a group. A G -extension of B is an extension B ⊂ C along with a faithful G -grading of C such that B is the trivial component. In other words, C admits a decomposition C = M g ∈ G C g (1.2)such that C = B and the tensor multiplication of C maps C x × C y to C xy for all x, y ∈ G . An isomorphismof G -extensions is an isomorphism of extensions respecting the grading.In [22] G -extensions of a tensor category D were classified by means of the Brauer-Picard 2-categoricalgroup BrPic ( D ) of invertible D -bimodule categories. Namely, it was shown that extensions (1.2) corre-spond to monoidal 2-functors G → BrPic ( D ). Equivalently, isomorphism classes of such extensions canbe described in terms of certain cohomology groups associated to a homomorphism G → BrP ic ( D ).This paper provides a classification of various types of G -extensions (where G is an Abelian group)of a braided tensor category B .By a 2 -categorical group (respectively, braided or symmetric -categorical group ) we understand amonoidal (respectively, braided or symmetric monoidal 2 category) in which every 0-cell is invertiblewith respect to the tensor product, every 1-cell is an equivalence, and every 2-cell is an isomorphism. For For fusion categories, these 2-categories of module categories are fusion 2-categories [16]. M the set of its invertible objects is a 2-categorical group which we will denote Inv ( M ).For the monoidal 2-categories Bimod ( B ), Mod ( B ), Mod br ( B ) and Mod sym ( B ) (for symmetric B )introduced above the 2-categorical groups of invertible objects BrPic ( B ) = Inv ( Bimod ( B )) (1.3) Pic ( B ) = Inv ( Mod ( B )) , (1.4) Pic br ( B ) = Inv ( Mod br ( B )) , (1.5) Pic sym ( B ) = Inv ( Mod sym ( B )) , (1.6)are called the Brauer-Picard , Picard , braided Picard , and symmetric Picard the groupoid of G -extensions of B of a given type the groupoid ofcorresponding monoidal 2-functorsbetween 2-categorical groups G → G (1.7)for an appropriate 2-categorical group G . These categorical 2-groups and the correspondence betweendifferent types of G -extensions and monoidal 2-functors G → G are given in Table 2.Extensions B ⊂ C G G → G tensor 2-categorical group BrPic ( B ) monoidalcentral 2-categorical group Pic ( B ) monoidalbraided braided 2-categorical group Pic br ( B ) braidedsymmetric symmetric 2-categorical group Pic ( B ) symmetricTable 2: G -extensions B ⊂ C and corresponding monoidal 2-functors. -categorical groups Let G be a 2-categorical group with the identity object I . We introduce its homotopy groups as follows: π ( G ) = the group of isomorphism classes of objects (0-cells) of G , (1.8) π ( G ) = Aut G ( I ) , the group of isomorphism classes of 1-automorphisms of I , (1.9) π ( G ) = Aut (id I ) , the group of 2-automorpshisms of the identity 1-automorphism of I . (1.10)5he multiplication of π ( G ) is given by the tensor product of G and the multiplication of π ( G ) , π ( G )is the composition of automorphisms.These homotopy groups come equipped with additional structure, which we refer to as the standardinvariants , namely a π ( G )-action on π m ( G ), π ( G ) × π m ( G ) → π m ( G ) m = 0 , , X for X ∈ π ( G ) (this action is used while making sense of the cohomologygroups below) and the first and the second canonical classes α G ∈ H ( π ( G ) , π ( G )) and q G ∈ H br ( π ( G ) , π ( G )) . (1.12)Part of the properties of the standard invariants is that the second canonical class is invariant under the π ( G )-action.Here and in what follows we denote by H nbr ( A, M ) := H n +1 ( A, M ) and H nsym ( A, M ) := H n +3 ( A, M ) (1.13)the Eilenberg-Mac Lane cohomology [19] of level 2 and 4, respectively. Note that H br ( A, M ) is isomorphicto the group of quadratic functions from A to M .For a braided 2-categorical group G the π ( G )-action (1.11) is trivial. The canonical classes getpromoted to α G ∈ H br ( π ( G ) , π ( G )) and q G ∈ H sym ( π ( G ) , π ( G )) . (1.14)An additional structure is the Whitehead products π n ( G ) × π m ( G ) → π n + m +1 ( G ) , n, m = 0 , , . (1.15)Note that the product π ( G ) × π ( G ) → π ( G ) is determined by the first canonical class (as the polar-ization of the quadratic function α ).For a symmetric 2-categorical group G all Whitehead products are zero and the canonical classes are α G ∈ H sym ( π ( G ) , π ( G )) and q G ∈ H sym ( π ( G ) , π ( G )) . (1.16)For a tensor category D the homotopy groups and standard invariants of the 2-categorical group BrPic ( D ) were examined in [22]. One has π ( BrPic ( D )) = BrP ic ( D ) , π ( BrPic ( D )) = Inv ( Z ( D )) , π ( BrPic ( D )) = k × . It was shown there that the
BrP ic ( D )-action on Inv ( Z ( D )) (i.e., the π -action on π ) comes from theisomorphism BrP ic ( D ) ≃ Aut br ( Z ( D )) and that the second canonical class is given by the quadratic6unction π = Inv ( Z ( D )) → π = k × : Z c Z,Z , where c denotes the braiding of Z ( D ).The homotopy groups of 2-categorical groups introduced in (1.4) - (1.6) are π ( Pic ( B )) = P ic ( B ) , π ( Pic ( B )) = Inv ( B ) , π ( Pic ( B )) = k × ,π ( Pic br ( B )) = P ic br ( B ) , π ( Pic br ( B )) = Inv ( Z sym ( B )) , π ( Pic br ( B )) = k × ,π ( Pic sym ( E )) = P ic sym ( E ) , π ( Pic br ( E )) = Inv ( E ) , π ( Pic br ( E )) = k × , where B is a braided tensor category, and E is a symmetric tensor category.We investigate the standard invariants of the braided 2-categorical group Pic br ( B ) and of the symmet-ric 2-categorical group Pic ( E ). For a braided tensor B we describe the Whitehead product (Proposition5.3) π × π = P ic br ( B ) × Inv ( Z sym ( B )) → π = k × (1.17)and the first canonical class (viewed as a quadratic function) Q B : π = P ic br ( B ) → π = Inv ( Z sym ( B )) . (1.18)For a symmetric tensor category E the first canonical class becomes a homomorphism Q E : P ic sym ( E ) → Inv ( E ) (1.19)into the 2-torsion of the group of invertible objects of E . -functors In view of the identification (1.7) it is desirable to have a good description of various types of monoidal2-functors G → G . We present one in Section 2 in terms of the Eilenberg-Mac Lane cohomology.Let G be a 2-categorical group (respectively, braided, symmetric 2-categorical group). Denoteby - Fun ( G, G ) (respectively, - Fun br ( G, G ), - Fun sym ( G, G )) the 2-groupoid of monoidal (respec-tively, braided, symmetric) 2-functors G → G . Such a functor restricts on objects to a map from π ( - Fun ( G, G )) (respectively, from π ( - Fun br ( G, G )), π ( - Fun sym ( G, G ))) to Hom ( G, π ( G ))), i.e.,from the set of isomorphism classes of 2-functors to the set of group homomorphisms. A homomorphism φ : G → π ( G ) is in the image of this map (i.e., φ can be lifted to a monoidal (respectively, braided,symmetric) 2-functor if and only if the following two obstructions vanish.7he first obstruction is the image of φ under the homomorphism o : Hom ( G, π ( G )) → H ( G, π ( G )) (1.20)(respectively, Hom ( G, π ( G )) → H br ( G, π ( G )), Hom ( G, π ( G )) → H sym ( G, π ( G ))), given by the pull-back along φ of the first canonical class α G defined in (1.12) (respectively, in (1.14), (1.16)). Theobstruction o ( φ ) vanishes if and only if φ can be lifted to a monoidal (respectively, braided, symmetric)functor from G to the 1-categorical truncation Π ≤ ( G ) of G .Suppose that a lifting F : G → Π ≤ ( G ) of φ is chosen. Then the second obstruction is the image of F under the map o : Fun ( G, Π ≤ ( G ) → H ( G, π ( G )) (1.21)(respectively, Fun br ( G, Π ≤ ( G )) → H br ( G, π ( G )), Fun sym ( G, Π ≤ ( G )) → H sym ( G, π ( G ))). Theobstruction o ( F ) measures the failure of extending F to a monoidal (respectively, braided, symmetric)2-functor G → G . When o ( F ) vanishes, the equivalence classes of such 2-functors extending F form atorsor over the cokernel of a certain group homomorphism H ( G, π ( G )) → H ( G, π ( G )) (respectively, H ( G, π ( G )) → H br ( G, π ( G )), H ( G, π ( G )) → H sym ( G, π ( G ))) depending on F . For a non-degenerate braided fusion category B there is a monoidal 2-equivalence Mod ( Vect ) =
Mod br ( B ),see Proposition 4.16. In particular, the braided 2-categorical Picard group Pic br ( B ) is “trivial” in thiscase and so (as is well known) is the extension theory: any braided graded extension of B splits into thetensor product of B and a pointed braided fusion category.Thus, the most interesting braided Picard 2-categorical groups come from degenerate tensor categories.In Section 6 we compute the homotopy groups and standard invariants of symmetric fusion categories.For example, the homotopy groups of the braided 2-categorical group Pic br ( Rep ( G )), where G is a finitegroup, are π = H ( G, k × ) × Z ( G ) , π = H ( G, k × ) , π = H ( G, k × ) = k × , where Z ( G ) denotes the center of G . The first canonical class (1.18) is the quadratic function H ( G, k × ) × Z ( G ) → H ( G, k × ) , ( γ, z ) χ ( − ) = γ ( z, − ) γ ( − , z )and the second canonical class is trivial.The Whitehead product (1.17) is( H ( G, k × ) × Z ( G )) × H ( G, k × ) → k × , ( γ, z ) × χ χ ( z ) .
8e determine the corresponding homotopy groups and maps for a general (not necessarily Tannakian)symmetric fusion category in Section 6.3.We show that the groupoid of symmetric A -extensions of a symmetric tensor category E has a structureof a symmetric 2-categorical group Ex sym ( A, E ). We describe an exact sequence that can be used tocompute π ( Ex sym ( A, E )) in Section 2.8. We also determine the group of symmetric extensions of asymmetric fusion category in Theorem 8.26. Section 2 contains the technical tools we need. We include a detailed description of the Eilneberg-MacLane cohomology [19] in low degrees and the notions of braided and symmetric monoidal 2-categories and2-functors between them [12, 27, 32]. An important observation is that the axioms of such categories andfunctors can be viewed as “non-commutative versions” of the higher Eilneberg-Mac Lane cocycle equations(e.g., compare equations (2.7) -(2.10) with commuting polytopes (2.26) - (2.29)). This is parallel to thepentagon axiom of a monoidal category being a non-commutative version of a 3-cocycle equation. Of aspecial use to us are (braided, symmetric) 2-categorical groups, characterized by invertibility of their cellswith respect to the tensor product. Monoidal (braided, symmetric) 2-functors from a finite group (viewedas discrete 2-categorical group) to (braided, symmetric) 2-categorical groups can be obtained as liftings ofusual (braided, symmetric) monoidal functors, provided that certain cohomological obstructions vanish.These obstructions for monoidal (respectively, braided, symmetric) 2-functors and parameterization ofliftings are described in Section 2.5 (respectively, Section 2.6, Section 2.7). Symmetric monoidal 2-functorsas above form a symmetric 2-categorical group. Its group of isomorphism classes of objects fits into acertain exact sequence (Theorem 2.36).In Section 3 we recall the 2-category
Mod ( B ) of module categories over a finite tensor category B .When B is braided, Mod ( B ) is a monoidal 2-category. Its tensor product can be defined either by auniversal property or by an explicit construction, see Remark 3.6.Section 4 deals with braided module categories over a braided tensor category B introduced andstudied by Brochier [5] and Ben-Zvi, Brochier, and Jordan [2]. In such categories the action of B has anadditional symmetry compatible with the braiding of B (Definition 4.1). Equivalently, a module braidingon a B -module category M is the same thing as a natural tensor isomorphism between the α -inductions[4] α ± M : B op → End B ( M ) (Proposition 4.9). The 2-category Mod br ( B ) of braided B -module categoriesis 2-equivalent to the 2-center of Mod ( B ) (Theorem 4.10). In particular, Mod br ( B ) is braided. Theeasiest examples of braided B -module categories come from tensor automorphisms of id B , we describethese in Section 4.3. We also prove in Proposition 4.16 that Mod br ( B ) ∼ = Mod br ( Vect ) when B is a9on-degenerate braided fusion category. Finally, module categories over a symmetric tensor category E can be equipped with the identity E -module braiding and so they form a symmetric monoidal 2-category Mod sym ( E ). We prove in Proposition 4.20 that the induction Mod sym ( Z sym ( B )) → Mod br ( B ) ofbraided module categories from the symmetric center of B is a braided monoidal 2-functor.In Section 5 we describe various 2-categorical Picard groups associated to tensor categories. Theseare parts of the corresponding monoidal 2-categories consisting of invertible module categories. Thenew ones are the braided Picard 2-categorical group Pic br ( B ) = Inv ( Mod br ( B )) of a braided tensorcategory B and the symmetric Picard 2-categorical group Pic sym ( E ) = Inv ( Mod sym ( E )) of a symmetrictensor category E . We describe their homotopy groups, canonical classes, and Whitehead brackets.Proposition 5.4 provides an exact sequence featuring the group π ( Pic br ( B )) that can be seen as asequence of homotopy groups of a certain fibration. Here we also describe Azumaya algebras in braidedtensor categories, as they give a convenient description of invertible module categories.Section 6 is dedicated to the braided 2-categorical Picard group of a symmetric fusion category E .We recall the computation of P ic ( E ) due to Carnovale [7] and use it to describe the braided categoricalPicard group of E and its canonical classes.In Section 7 we compute the braided categorical Picard group of a pointed braided fusion category B . We show that in this case there is a braided monoidal equivalence of braided categorical groups Pic br ( B ) ∼ = Pic br ( Z sym ( B )), where Z sym ( B ) is the symmetric center of B , see Proposition 7.1.Finally, Section 8 contains a classification of graded extensions. Tensor (respectively, central, braided,and symmetric) graded extensions are classified in Theorem 8.5 (respectively, Theorem 8.13, Theo-rem 8.18, and Theorem 8.24). We compute the group of symmetric extensions of a symmetric fusioncategory in Theorem 8.26. Here we also explain that the zesting procedure studied in [14] can be under-stood as a deformation of a braided monoidal functor A → Pic br ( B ) and compute Pontryagin-Whiteheadobstructions to existence of extensions in this case. We thank Pavel Etingof, C´esar Galindo, Corey Jones, David Jordan, Liang Kong, Victor Ostrik, andMilen Yakimov for many useful discussions. The first author thanks the Simons foundation for partialsupport. The work of the second author was supported by the National Science Foundation under GrantNo. DMS-1801198. This material is based upon work supported by the National Science Foundation underGrant No. DMS-1440140, while the authors were in residence at the Mathematical Sciences ResearchInstitute in Berkeley, California, during the Spring 2020 semester.10
Higher categorical groups and group cohomology
We denote by C ∗ ( A, M ) the normalised standard complex of the abelian group A with coefficients in thetrivial A -module M .We denote by C ∗ br ( A, M ) = C ∗ +1 ( K ( A, , M ) the normalised standard complex computing the secondEilenberg-Mac Lane cohomology [19]. The first few terms of the cochain complex C ∗ br ( A, M ) are as follows: C br ( A, M ) = C ( A, M ) =
M, C br ( A, M ) = C ( A, M ) , C br ( A, M ) = C ( A, M ) ,C br ( A, M ) = C ( A, M ) ⊕ C ( A, M ) = { ( a ( − , − , − ) , a ( −|− )) } ,C br ( A, M ) = C ( A, M ) ⊕ C ( A, M ) ⊕ C ( A, M ) = { ( a ( − , − , − , − ) , a ( − , −|− ) , a ( −|− , − )) } ,C br ( A, M ) = C ( A, M ) ⊕ C ( A, M ) ⊕ C ( A, M ) ⊕ C ( A, M ) ⊕ C ( A, M )= { ( a ( − , − , − , − , − ) , a ( − , − , −|− ) , a ( − , −|− , − ) , a ( −|− , − , − ) , a ( −| − |− )) } with the differentials d : C br ( A, M ) → C br ( A, M ) d ( a )( x, y, z ) = a ( y, z ) − a ( xy, z ) + a ( x, yz ) − a ( x, y ) , (2.1) d ( a )( x | y ) = a ( y, x ) − a ( x, y ); (2.2) d : C br ( A, M ) → C br ( A, M ) d ( a )( x, y, z, w ) = a ( y, z, w ) − a ( xy, z, w ) + a ( x, yz, w ) − a ( x, y, zw ) + a ( x, y, z ) , (2.3) d ( a )( x, y | z ) = a ( y | z ) − a ( xy | z ) + a ( x | z ) + a ( x, y, z ) − a ( x, z, y ) + a ( z, x, y ) , (2.4) d ( a )( x | y, z ) = a ( x | y ) − a ( x | yz ) + a ( x | z ) − a ( x, y, z ) + a ( y, x, z ) − a ( y, z, x ) , (2.5)and d : C br ( A, M ) → C br ( A, M )11 ( a )( x, y, z, w, u ) = a ( y, z, w, u ) − a ( xy, z, w, u ) + a ( x, yz, w, u ) − a ( x, y, zw, u ) + a ( x, y, z, wu ) − a ( x, y, z, w ) , (2.6) d ( a )( x | y, z, w ) = a ( x | z, w ) − a ( x | yz, w ) + a ( x | y, zw ) − a ( x | y, z ) − a ( x, y, z, w ) + a ( y, x, z, w ) − a ( y, z, x, w ) + a ( y, z, w, x ) , (2.7) d ( a )( x, y, z | w ) = a ( y, z | w ) − a ( xy, z | w ) + a ( x, yz | w ) − a ( x, y | w ) − a ( x, y, z, w ) + a ( x, y, w, z ) − a ( x, w, y, z ) + a ( w, x, y, z ) , (2.8) d ( a )( x, y | z, w ) = a ( y | z, w ) − a ( xy | z, w ) + a ( x | z, w ) − a ( x, y | w ) + a ( x, y | zw ) − a ( x, y | z )+ a ( x, y, z, w ) − a ( x, z, y, w ) + a ( z, x, y, w )+ a ( x, z, w, y ) − a ( z, x, w, y ) + a ( z, w, x, y ) , (2.9) d ( a )( x | y | z ) = − a ( x, y | z ) + a ( y, x | z ) − a ( x | y, z ) + a ( x | z, y ) , x, y, z, w ∈ A. (2.10) Example 2.1.
The first few terms of the cochain complex C ∗ br ( Z / Z , M ) are M / / M d / / M / / M d / / M d / / M / / ... where M n is the direct sum of n copies of M and d ( m ) = 2 md ( m, l ) = (2 m, l + m, l − m ) d ( m, l, k ) = (0 , , m − l + k ) , , H br ( Z / Z , M ) = M,H br ( Z / Z , M ) = M ,H br ( Z / Z , M ) = M/ M,H br ( Z / Z , M ) = M ,H br ( Z / Z , M ) = M ⊕ M/ M. Here M s = { m ∈ M | sm = 0 } .We denote by C ∗ syl ( A, M ) = C ∗ +2 ( K ( A, , M ) the normalised standard complex computing the thirdEilenberg-Mac Lane cohomology [19]. The first few terms of the cochain complex C ∗ syl ( A, M ) are as12ollows: C syl ( A, M ) =
M, C syl ( A, M ) = C ( A, M ) ,C syl ( A, M ) = C ( A, M ) , C syl ( A, M ) = C br ( A, M ) ,C syl ( A, M ) = C br ( A, M ) ⊕ C ( A, M ) = { ( a ( − , − , − , − ) , a ( − , −|− ) , a ( −|− , − ) , a ( −||− )) } ,C syl ( A, M ) = C br ( A, M ) ⊕ C ( A, M ) ⊕ C ( A, M ) == { ( a ( − , − , − , − , − ) , a ( − , − , −|− ) , a ( − , −|− , − ) , a ( −|− , − , − ) ,a ( −| − |− ) , a ( − , −||− ) , a ( −||− , − )) } with the additional differentials d : C syl ( A, M ) → C syl ( A, M ) d ( a )( x || y ) = a ( x | y ) + a ( y | x ) , (2.11)and d : C syl ( A, M ) → C syl ( A, M ) d ( a )( x || y, z ) = a ( x | y, z ) + a ( y, z | x ) + a ( x || y ) + a ( x || z ) − a ( x || yz ) , (2.12) d ( a )( x, y || z ) = a ( x, y | z ) + a ( z | x, y ) + a ( x || z ) + a ( y || z ) − a ( xy || z ) . (2.13) Example 2.2.
The first few sylleptic cohomology groups of Z / Z are H syl ( Z / Z , M ) = M,H syl ( Z / Z , M ) = M ,H syl ( Z / Z , M ) = M/ M,H syl ( Z / Z , M ) = M ,H syl ( Z / Z , M ) = M ⊕ M/ M. We denote by C ∗ sym ( A, M ) = C ∗ +3 ( K ( A, , M ) the normalised standard complex computing thefourth Eilenberg-Mac Lane cohomology [19]. The first few terms of the cochain complex C ∗ sym ( A, M ) areas follows: C sym ( A, M ) =
M, C sym ( A, M ) = C ( A, M ) , C sym ( A, M ) = C ( A, M ) ,C sym ( A, M ) = C br ( A, M ) , C sym ( A, M ) = C syl ( A, M ) ,C sym ( A, M ) = C syl ( A, M ) ⊕ C ( A, M ) == { ( a ( − , − , − , − , − ) , a ( − , − , −|− ) , a ( − , −|− , − ) , a ( −|− , − , − ) ,a ( −| − |− ) , a ( − , −||− ) , a ( −||− , − ) , a ( −|||− )) } d : C sym ( A, M ) → C sym ( A, M ) d ( a )( x ||| y ) = a ( x || y ) − a ( y || x ) , x, y ∈ A. (2.14) Example 2.3.
The first few level 4 cohomology groups of Z / Z are the same as the symmetric coho-mology H nsym ( Z / Z , M ) = H nsyl ( Z / Z , M ) , n ≤ . Example 2.4.
It is immediate from the definitions that H br ( A, M ) = H syl ( A, M ) = H sym ( A, M ) ∼ = M,H br ( A, M ) = H syl ( A, M ) = H sym ( A, M ) ∼ = Hom ( A, M ) ,H br ( A, M ) = H syl ( A, M ) = H sym ( A, M ) ∼ = Ext ( A, M ) . It was shown in [20] that there are isomorphisms H br ( A, M ) ∼ = Quad ( A, M ) , H syl ( A, M ) = H sym ( A, M ) ∼ = Hom ( A, M ) , given by ( a ( − , − , − ) , a ( −|− )) q, where q ( x ) = a ( x | x ) , x ∈ A. Here
Quad ( A, M ) is the group of quadratic maps and M = { m ∈ M | m = 0 } is the 2-torsion subgroupof M .The 4th cohomology groups are especially important for our purposes. Let us now assume that M isdivisible. The following results are from [20]. Example 2.5.
There is an isomorphism θ sym : H sym ( A, M ) ∼ −→ Hom ( A , M ) (2.15)assigning to a symmetric 4-cocycle ( a ( − , − , − , − ) , a ( − , −|− ) , a ( −|− , − ) , a ( −||− )) the homomorphism A → M : x a ( x, x | x ) − a ( x | x, x ) − a ( x, x, x, x ) . (2.16)There is an isomorphism θ syl : H syl ( A, M ) ∼ −→ Hom ( A , M ) ⊕ Hom ( Λ A, M ) , (2.17)whose first component is given by (2.16) and the second component assigns to a sylleptic 4-cocycle( a ( − , − , − , − ) , a ( − , −|− ) , a ( −|− , − ) , a ( −||− )) the homomorphism Λ A → M : x ∧ y a ( x || y ) − a ( y || x ) , (2.18)14hich is the obstruction for a sylleptic 4-cocycle to be symmetric.Finally, there is a homomorphism θ br : H br ( A, M ) −→ Ext ( A, Hom ( A, M )) , (2.19)which is the obstruction for a braided 4-cocycle to have a sylleptic structure. It is defined as follows.Let ( a ( − , − , − , − ) , a ( − , −|− ) , a ( −|− , − )) be a braided 4-cocycle. For any x ∈ A define a function b x ∈ C ( A, M ) by b x ( y, z ) = a ( x | y, z ) + a ( y, z | x ) , x, y, z ∈ A. It follows from formulas (2.6) -(2.10) and divisibility of M that b x is a 2-coboundary. That is there existsa function a ( −||− ) ∈ C ( A, M ) such that (2.12) vanishes. The function A → M : ( x, y, z ) a ( x || y ) − a ( y || x ) + a ( x || z ) − a ( z || x ) − a ( x || yz ) + a ( yz || x )is multiplicative in x and, hence, defines a symmetric 2-cocycle g on A with values in Hom ( A, M ). This2-cocycle is cohomologically trivial if and only if the given braided 4-cocycle admits a sylleptic structure.We set θ br ( a ( − , − , − , − ) , a ( − , −|− ) , a ( −|− , − )) to be the class of g in Ext ( A, Hom ( A, M )).The kernel of (2.19) is isomorphic to
Hom ( A , M ) via (2.16). Semistrict monoidal 2-categories were defined by Kapranov and Voevodsky [27] and also by Day andStreet [12] under the name of Gray monoids. It was shown that every monoidal 2-category is equivalentto a semistrict one. We refer the reader to these papers and to the work of Schommer-Pries [32] for basicdefinitions. All monoidal 2-categories considered in this paper will be assumed semistrict.Let F , H : M → N be 2-functors between 2-categories. Recall that a pseudo-natural transformation P : F → H is a collection of 1-morphisms P M : F ( M ) → H ( M ) and invertible 2-cells F ( M ) F ( F ) (cid:15) (cid:15) P M / / H ( M ) H ( F ) (cid:15) (cid:15) F ( N ) P N / / H ( N ) , P F + (2.20)for all objects M and 1-morphisms F : M → N in M such that P id M = id P M and P F ◦ G = P F ◦ P G (2.21)for all composable 1-morphisms F and G . 15et P, Q : F → H be pseudo-natural transformations between 2-functors. A modification η : P → Q is a collection of 2-cells F ( M ) P M + + Q M H ( M ) , η M (cid:11) (cid:19) (2.22)for all objects M in M , natural in 1-morphisms in M . Definition 2.6. A (semistrict) braided monoidal -category [27, 8, 1] consists of a (semistrict) monoidal2-category ( M , ⊠ , I ), where ⊠ is the tensor product, equipped with invertible 2-cells M ⊠ N M ⊠ W (cid:15) (cid:15) Z ⊠ N / / M ′ ⊠ N M ′ ⊠ W (cid:15) (cid:15) M ⊠ N ′ Z ⊠ N ′ / / M ′ ⊠ N ′ , ⊠ Z,W + (2.23)for any Z ∈ M ( M , M ′ ) , W ∈ M ( N , N ′ ), and I is the unit object,together with a pseudo-natural equivalence ( braiding ) B M , N : M ⊠ N → N ⊠ M , M , N ∈ M , invertible 2-cells M ⊠ N B M , N (cid:15) (cid:15) Z ⊠ N / / M ′ ⊠ N B M ′ , N (cid:15) (cid:15) M ⊠ N B M , N (cid:15) (cid:15) M ⊠ W / / M ⊠ N ′ B M ′ , N (cid:15) (cid:15) N ⊠ M N ⊠ Z / / N ⊠ M ′ , N ⊠ M W ⊠ M / / N ′ ⊠ M , B Z, N + B M ,W + (2.24)satisfying B Z , N ◦ B Z , N = B Z ⊗ Z , N and B M ,W ◦ B M ,W = B M ,W ⊗ W , and two invertible modifications LNM B L , N ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ MLN B L , N ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ LMN B M , N ♠♠♠♠♠♠♠♠♠♠♠♠♠ B LM , N / / NLM LMN B L , M ♠♠♠♠♠♠♠♠♠♠♠♠♠ B L , MN / / MNL , β L , M | N (cid:11) (cid:19) ✤✤✤✤✤✤ β L | M , N (cid:11) (cid:19) ✤✤✤✤✤✤ (2.25) Below we omit the identity functors and the tensor product symbol ⊠ , so we write MN for M ⊠ N . : LKMN B K , M (cid:15) (cid:15) KLMN B K , L o o B K , LMN (cid:15) (cid:15) B K , LM x x qqqqqqqqqqqqqqqqqqqqqqqq LKMN B K , M (cid:15) (cid:15) B K , MN & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ KLMN B K , L o o B K , LMN (cid:15) (cid:15) = LMKN B K , N / / LMNK LMKN B K , N / / LMNK , β K | L , M " * ▼▼▼▼▼▼ ▼▼▼▼▼▼ β K | LM , N + β K | M , N < qqqqqq qqqqqq β K | L , MN + (2.26) KLNM B L , N (cid:15) (cid:15) KLMN B M , N o o B KLM , N (cid:15) (cid:15) B LM , N x x qqqqqqqqqqqqqqqqqqqqqqqq KLNM B L , N (cid:15) (cid:15) B KL , N & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ KLMN B M , N o o B KLM , N (cid:15) (cid:15) = KNLM B K , N / / NKLM LMKN B K , N / / NKLM , β L , M | N " * ▼▼▼▼▼▼ ▼▼▼▼▼▼ β K , LM | N + β K , L | N < qqqqqq qqqqqq β KL , M | N + (2.27) KLMN B L , M x x qqqqqqqqqqq (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ (cid:25) (cid:25) ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ B KL , M & & ▼▼▼▼▼▼▼▼▼▼▼ KLMN B L , M x x ♣♣♣♣♣♣♣♣♣♣ B KL , M & & ◆◆◆◆◆◆◆◆◆◆ KMLN (cid:15) (cid:15)
MKLN (cid:15) (cid:15)
KMLN (cid:15) (cid:15) / / MKLN (cid:15) (cid:15) (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ = KMNL / / & & ▼▼▼▼▼▼▼▼▼▼▼ MNKL KMNL & & ▼▼▼▼▼▼▼▼▼▼▼ MNKL
MKNL qqqqqqqqqqq MKNL , qqqqqqqqqqq β L | M , N + ❖❖❖❖ ❖❖❖❖ β KL | M , N s { ♦♦♦♦♦♦♦♦♦♦ β K , L | MN < ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ β K | M , N K S β K , L | M . ❡❡❡❡❡❡❡❡❡❡❡❡❡ ❡❡❡❡❡❡❡❡❡❡❡❡❡ ⊠ B K , M ,B L , N ( ❨❨❨❨❨❨❨❨❨❨❨ ❨❨❨❨❨❨❨❨❨❨❨ β K , L | N < qqqq qqqq (2.28) Equalities of 2-cell compositions in this paper can be used to represent commuting polytopes [26]. These polytopes arerecovered by gluing the diagrams on both sides of equality along the perimeter LM y y ttttttttt (cid:6) (cid:6) ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍ % % ❏❏❏❏❏❏❏❏❏ KLM y y sssssssss (cid:24) (cid:24) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ % % ❑❑❑❑❑❑❑❑❑ LKM (cid:15) (cid:15)
KML (cid:15) (cid:15) (cid:7) (cid:7) ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍
LKM (cid:15) (cid:15) (cid:24) (cid:24) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵
KML (cid:15) (cid:15) = LMK $ $ ❏❏❏❏❏❏❏❏❏❏ MKL z z tttttttttt LMK % % ❏❏❏❏❏❏❏❏❏❏ MKL y y tttttttttt MLK MLK , β K | L , M " * ▲▲▲▲▲▲ β K | M , L b j ▼▼▼▼▼▼▼▼ β L , K | M < qqq qqq β K , L | M t | qqqqqq B K ,B L , N (cid:3) (cid:11) ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍ B B K , L , M (cid:20) (cid:28) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ (2.29)for all K , L , M , N ∈ M . Definition 2.7. A symmetric monoidal 2-category is a braided monoidal 2-category with an additionalstructure (in contrast with ordinary monoidal categories, for which being symmetric is a property).Namely, a symmetric structure on a braided monoidal 2-category M as above is an invertible modification MN B M , N + + NM , B N , M k k τ M , N (cid:11) (cid:19) (2.30)i.e., τ M , N is an invertible modification between B N , M B M , N and id M ⊠ N such that LNM B L , N (cid:15) (cid:15) LNM B L , N (cid:16) (cid:16) B N , M | | ②②②②②②②②②②②②②②②②②②② = LMN B M , N - - B LM , N / / NLM B N , LM c c LMN B M , N - - NLM , B N , LM c c B N , L b b ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ β L , M | N (cid:11) (cid:19) ✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤ τ LM , N (cid:11) (cid:19) β N | L , M (cid:11) (cid:19) ✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤ τ N , M X ` ✾✾✾✾✾✾✾✾✾✾✾✾ τ N , L > F ✆✆✆✆✆✆ ✆✆✆✆✆✆ (2.31) MLN B L , N (cid:16) (cid:16) MLN B L , N (cid:16) (cid:16) B M , L | | ②②②②②②②②②②②②②②②②②②② = LMN B L , M - - B L , MN / / MNL B MN , L c c LMN B L , M - - MNL , B MN , L c c B N , L b b ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ β L | M , N (cid:11) (cid:19) ✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤ τ L , MN (cid:11) (cid:19) β M , N | L (cid:11) (cid:19) ✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤✤ τ M , L X ` ✾✾✾✾✾✾✾✾✾✾✾✾ τ N , L ? G ✝✝✝✝✝✝ ✝✝✝✝✝✝ (2.32)18nd MN B M , N ! ! B M , N NM B N , M o o = MN B M , N ! ! B M , N NM τ M , N (cid:11) (cid:19) τ N , M (cid:11) (cid:19) id M ⊠ N (cid:11) (cid:19) (2.33)commute for all objects L , M , N in M . Definition 2.8. A monoidal -functor F : M → M ′ between monoidal 2-categories is a 2-functor alongwith a pseudo-natural equivalence F M , N : F ( M ) F ( N ) → F ( MN ) , (2.34)an equivalence U : F ( I ) → I , and invertible modifications F ( L ) F ( M ) F ( N ) F L , M (cid:15) (cid:15) F M , N / / F ( L ) F ( MN ) F L , MN (cid:15) (cid:15) F ( LM ) F ( N ) F LM , N / / F ( LMN ) , α L , M , N + (2.35) F ( I ) F ( M ) F I , M (cid:15) (cid:15) U / / IF ( M ) L F ( M ) (cid:15) (cid:15) F ( M ) F ( I ) F M , I (cid:15) (cid:15) U / / F ( M ) I , R F ( M ) (cid:15) (cid:15) and F ( IM ) F ( L M ) / / F ( M ) , F ( MI ) F ( R M ) / / F ( M ) , λ M + ρ M + (2.36)where L and R denote the unit constraints of M , such that F ( K ) F ( L ) F ( M ) F ( N ) F K , L t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ F M , N * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F ( K ) F ( L ) F ( M ) F ( N ) F K , L t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ F M , N * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F L , M (cid:15) (cid:15) F ( KL ) F ( M ) F ( N ) F KL , M (cid:15) (cid:15) F M , N * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F ( K ) F ( L ) F ( MN ) F L , MN (cid:15) (cid:15) F K , L t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ F ( KL ) F ( M ) F ( N ) F KL , M (cid:15) (cid:15) F ( K ) F ( L ) F ( MN ) F L , MN (cid:15) (cid:15) F ( KL ) F ( MN ) F KL , MN (cid:15) (cid:15) = F ( K ) F ( LM ) F ( N ) F K , LM t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ F LM , N * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F ( KLM ) F ( N ) F KLM , N * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F ( K ) F ( LMN ) F K , LMN t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ F ( KLM ) F ( N ) F KLM , N * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F ( K ) F ( LMN ) F K , LMN t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ F ( KLMN ) F ( KLMN ) α KL , M , N & . ❯❯❯❯❯❯❯❯❯❯❯❯ ❯❯❯❯❯❯❯❯❯❯❯❯ α K , L , MN (cid:11) (cid:19) ⊠ F K , L ,F M , N + α K , L , M (cid:11) (cid:19) α L , M , N & . ❯❯❯❯❯❯❯❯❯❯❯❯ ❯❯❯❯❯❯❯❯❯❯❯❯ α K , LM , N + (2.37)19nd F ( M ) F ( I ) F ( N ) F M , I u u ❧❧❧❧❧❧❧❧❧❧❧❧❧ F I , N ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( M ) F ( I ) F ( N ) F M , I u u ❧❧❧❧❧❧❧❧❧❧❧❧❧ F I , N ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ U (cid:15) (cid:15) F ( MI ) F ( N ) F ( R M ) (cid:15) (cid:15) F MI , N ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( M ) F ( IN ) F ( L N ) (cid:15) (cid:15) F M , IN u u ❧❧❧❧❧❧❧❧❧❧❧❧❧ F ( MI ) F ( N ) F ( R M ) (cid:15) (cid:15) F ( M ) F ( IN ) F ( L N ) (cid:15) (cid:15) F ( MIN ) F ( R M ) F ( L N ) (cid:15) (cid:15) = F ( M ) IF ( N ) R F ( M ) u u ❧❧❧❧❧❧❧❧❧❧❧❧❧ L F ( N ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ F ( M ) F ( N ) F M , N ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( M ) F ( N ) F M , N u u ❧❧❧❧❧❧❧❧❧❧❧❧❧ F ( M ) F ( N ) F M , N ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( K ) F ( M ) F ( N ) F M , N u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ F ( MN ) F ( MN ) F R M , id N % - ❘❘❘❘❘❘❘❘❘ ❘❘❘❘❘❘❘❘❘ F id M ,L N q y ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ α M , I , N + ρ M (cid:11) (cid:19) λ N (cid:11) (cid:19) (2.38)for all K , L , M , N ∈ M . Definition 2.9. A braided monoidal -functor F : M → M ′ between braided monoidal 2-categories is amonoidal functor along with an invertible modification F ( M ) F ( N ) F M , N (cid:15) (cid:15) B F ( M ) ,F ( N ) / / F ( N ) F ( M ) F N , M (cid:15) (cid:15) F ( MN ) F ( B M , N ) / / F ( NM ) δ M , N + (2.39)such that F ( L ) F ( MN ) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( L ) F ( MN ) ' ' PPPPPPPPPPPP (cid:15) (cid:15) F ( L ) F ( M ) F ( N ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ (cid:15) (cid:15) (cid:29) (cid:29) ❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁ F ( LMN ) (cid:26) (cid:26) ✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺ F ( L ) F ( M ) F ( N ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ (cid:15) (cid:15) F ( LMN ) (cid:25) (cid:25) ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ (cid:15) (cid:15) F ( LM ) F ( N ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( ML ) F ( N ) ' ' PPPPPPPPPPPP F ( M ) F ( L ) F ( N ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ (cid:1) (cid:1) ✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂ F ( MLN ) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ = F ( MLN ) (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ F ( M ) F ( LN ) ♥♥♥♥♥♥♥♥♥♥♥♥ (cid:2) (cid:2) ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ F ( MN ) F ( L ) ' ' PPPPPPPPPPPP F ( M ) F ( N ) F ( L ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ F ( MNL ) F ( M ) F ( N ) F ( L ) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ ❧❧❧❧❧❧❧❧❧❧❧❧❧ F ( MNL ) F ( M ) F ( NL ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ F ( M ) F ( NL ) ♥♥♥♥♥♥♥♥♥♥♥♥ α − L , M , N (cid:11) (cid:19) α M , L , N (cid:11) (cid:19) δ L , M (cid:11) (cid:19) F B L , M , N (cid:11) (cid:19) F M ,B L , N (cid:11) (cid:19) δ L , N q y ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ β ′ F ( L ) | F ( M ) ,F ( N ) k s α M , N , L (cid:11) (cid:19) F ( β L | M , N ) k s δ L , MN + PPPPPPPP PPPPPPPP B ′ F ( L ) ,F M , N q y ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ (2.40)20nd F ( LM ) F ( N ) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( LM ) F ( N ) ' ' PPPPPPPPPPP (cid:15) (cid:15) F ( L ) F ( M ) F ( N ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ (cid:15) (cid:15) (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ F ( LMN ) (cid:26) (cid:26) ✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺ F ( L ) F ( M ) F ( N ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ (cid:15) (cid:15) F ( LMN ) (cid:25) (cid:25) ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ (cid:15) (cid:15) F ( L ) F ( MN ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( L ) F ( NM ) ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ F ( L ) F ( N ) F ( M ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ F ( LNM ) (cid:4) (cid:4) ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ = F ( LNM ) (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ F ( LN ) F ( M ) ♦♦♦♦♦♦♦♦♦♦♦♦ (cid:2) (cid:2) ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ F ( N ) F ( LM ) ' ' PPPPPPPPPPP F ( N ) F ( L ) F ( M ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ F ( NLM ) F ( N ) F ( L ) F ( M ) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ ❧❧❧❧❧❧❧❧❧❧❧❧❧ F ( NLM ) F ( NL ) F ( M ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ F ( NL ) F ( M ) ♥♥♥♥♥♥♥♥♥♥♥ α L , M , N (cid:11) (cid:19) α − L , N , M (cid:11) (cid:19) δ M , N (cid:11) (cid:19) F L ,B N , M (cid:11) (cid:19) F B L , N , M (cid:11) (cid:19) δ L , N q y ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ β ′ F ( L ) ,F ( M ) | F ( N ) k s α − N , L , M (cid:11) (cid:19) F ( β L , M | N ) k s δ LM , N + PPPPPPPP PPPPPPPP B ′ F L , M ,F ( N ) r z ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ (2.41)for all objects L , M , N in M . Here B, B ′ denote the braidings on M , M ′ . We omit 1-cell labels to keepthe diagrams readable. Definition 2.10.
A braided monoidal 2-functor F : M → M ′ between symmetric monoidal 2-categories M and M ′ is symmetric if F ( M ) F ( N ) B ′ F ( M ) , F ( N ) , , F M , N (cid:15) (cid:15) F ( N ) F ( M ) F N , M (cid:15) (cid:15) F ( M ) F ( N ) B ′ F ( M ) ,F ( N ) , , F M , N (cid:15) (cid:15) F ( N ) F ( M ) F N , M (cid:15) (cid:15) B F ( N ) , F ( M ) l l = F ( MN ) F ( B M , N ) + + F ( NM ) F ( B N , M ) k k F ( MN ) F ( NM ) , F ( B N , M ) k k δ M , N + δ N , M k s F ( τ M , N ) (cid:11) (cid:19) τ ′ F ( M ) ,F ( N ) (cid:11) (cid:19) (2.42)for all M , N in M . Here τ, τ ′ are the modification defined in (2.30). Definition 2.11.
Let F , F ′ : M → M ′ be two monoidal functors. A monoidal pseudo-natural transfor- ation P : F → F ′ is a pseudo-natural transformation along with an invertible modification F ( L ) F ( M ) F L , M (cid:15) (cid:15) P L P M / / F ′ ( L ) F ′ ( M ) F ′ L , M (cid:15) (cid:15) F ( LM ) P LM / / F ′ ( LM ) µ L , M + (2.43)such that F ( LM ) F ( N ) F LM , N ( ( PPPPPPPPPPPP F ( LM ) F ( N ) F LM , N ( ( PPPPPPPPPPPP P LM P N (cid:15) (cid:15) F ( L ) F ( M ) F ( N ) F L , M ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ F M , N ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ P L P M P N (cid:15) (cid:15) F ( LMN ) P LMN (cid:15) (cid:15) F ( L ) F ( M ) F ( N ) F L , M ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ P L P M P N (cid:15) (cid:15) F ( LMN ) P LMN (cid:15) (cid:15) F ( L ) F ( MN ) F L , MN ♥♥♥♥♥♥♥♥♥♥♥♥ P L P MN (cid:15) (cid:15) = F ′ ( LM ) F ′ ( N ) F ′ LM , N ( ( PPPPPPPPPPPP F ′ ( L ) F ′ ( M ) F ′ ( N ) F ′ M , N ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ F ′ ( LMN ) F ′ ( L ) F ′ ( M ) F ′ ( N ) F ′ M , N ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ F ′ L , M ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ F ′ ( LMN ) F ′ ( L ) F ′ ( MN ) F ′ L , MN ♥♥♥♥♥♥♥♥♥♥♥♥ F ′ ( L ) F ′ ( MN ) F ′ L , MN ♥♥♥♥♥♥♥♥♥♥♥♥ α L , M , N $ , ❘❘❘❘❘❘❘❘❘ ❘❘❘❘❘❘❘❘❘ α ′ L , M , N $ , ❘❘❘❘❘❘❘❘❘ ❘❘❘❘❘❘❘❘❘ µ M , N (cid:11) (cid:19) µ L , MN (cid:11) (cid:19) µ L , M (cid:11) (cid:19) µ LM , N (cid:11) (cid:19) (2.44)for all L , M , N in M . Here α and α ′ denote the monoidal structures of F and F ′ . Definition 2.12.
A monoidal pseudo-natural transformation P : F → F ′ between braided monoidal2-functors is braided if F ( LM ) P LM ' ' ❖❖❖❖❖❖❖❖❖❖❖ F ( LM ) P LM ' ' ❖❖❖❖❖❖❖❖❖❖❖ F ( B L , M ) (cid:15) (cid:15) F ( L ) F ( M ) F L , M ♥♥♥♥♥♥♥♥♥♥♥♥ P L P M ' ' PPPPPPPPPPPP B ′ F ( L ) , F ( M ) (cid:15) (cid:15) F ′ ( LM ) F ′ ( B L , M ) (cid:15) (cid:15) F ( L ) F ( M ) F L , M ♥♥♥♥♥♥♥♥♥♥♥♥ B ′ F ( L ) , F ( M ) (cid:15) (cid:15) F ′ ( LM ) F ′ ( B L , M ) (cid:15) (cid:15) F ′ ( L ) F ′ ( M ) F ′ L , M ♦♦♦♦♦♦♦♦♦♦♦ B F ′ ( L ) , F ′ ( M ) (cid:15) (cid:15) = F ( ML ) P ML ' ' ❖❖❖❖❖❖❖❖❖❖❖ F ( M ) F ( L ) P M P L ' ' PPPPPPPPPPPP F ′ ( ML ) F ( M ) F ( L ) P M P L ' ' PPPPPPPPPPPP F M , L ♥♥♥♥♥♥♥♥♥♥♥♥ F ′ ( ML ) F ′ ( M ) F ′ ( L ) F ′ M , L ♦♦♦♦♦♦♦♦♦♦♦ F ′ ( M ) F ′ ( L ) F ′ M , L ♦♦♦♦♦♦♦♦♦♦♦ µ L , M + ❖❖❖❖❖❖❖❖ ❖❖❖❖❖❖❖❖ µ M , L + ❖❖❖❖❖❖❖❖ ❖❖❖❖❖❖❖❖ B ′ P L ,P M + PPPPPPPP PPPPPPPP δ ′ L , M (cid:11) (cid:19) δ M , L (cid:11) (cid:19) P B L , M + ❖❖❖❖❖❖❖ ❖❖❖❖❖❖❖ (2.45)for all L , M in M . Here δ and δ ′ denote the braided structures on F and F ′ . Definition 2.13.
A modification η : P → Q between two monoidal pseudo-natural transformations22 , Q : F → F ′ is monoidal if F ( M ) F ( N ) F M , N / / P M P N (cid:31) (cid:31) F ( MN ) Q MN (cid:127) (cid:127) P MN (cid:31) (cid:31) F ( M ) F ( N ) F M , N / / P M P N (cid:31) (cid:31) Q M Q N (cid:127) (cid:127) F ( MN ) Q MN (cid:127) (cid:127) = F ′ ( M ) F ′ ( N ) F ′ M , N / / F ′ ( MN ) F ′ ( M ) F ′ ( N ) F ′ M , N / / F ′ ( MN ) , η MN + η M η N + µ M , N " * ν M , N t | (2.46)for all M , N ∈ M , where µ and ν are modifications (2.43) defining the monoidal structures on P and Q , respectively. -category Let M be a braided monoidal 2-category. Its center Z ( M ) is a braided monoidal 2-category definedas follows [1, Section 3]. Objects of Z ( M ) are triples ( N , S, γ ) where N is an object of M , S is apseudo-natural collection of equivalences (called a half-braiding ) S M : MN → NM , M ∈ A , and γ is an invertible modification LNM S L ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ LMN S M ❧❧❧❧❧❧❧❧❧❧❧❧❧ S LM / / NLM γ L , M (cid:11) (cid:19) ✤✤✤✤✤✤ (2.47)such that KLNM S L (cid:15) (cid:15) KLMN S M o o S KLM (cid:15) (cid:15) S LM x x qqqqqqqqqqqqqqqqqqqqqqqq KLNM S L (cid:15) (cid:15) S KL & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ KLMN S M o o S KLM (cid:15) (cid:15) = KNLM S K / / NKLM LMKN S K / / NKLM , γ L , M " * ▼▼▼▼▼▼ ▼▼▼▼▼▼ γ K , LM + γ K , L < qqqqqq qqqqqq γ KL , M + (2.48)for all K , L , M ∈ M .A morphism between ( N , S, γ ) and ( N ′ , S ′ , γ ′ ) in Z ( M ) is pair ( F, σ ), where F : N → N ′ is amorphism in M and σ is an invertible modification MN S M / / F (cid:15) (cid:15) NM F (cid:15) (cid:15) MN ′ S ′ M / / N ′ M σ M + (2.49)23uch that LNM S L ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ LNM S L ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ F (cid:15) (cid:15) LMN S LM / / S M ❧❧❧❧❧❧❧❧❧❧❧❧❧ F (cid:15) (cid:15) NLM F (cid:15) (cid:15) = LMN S M ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ F (cid:15) (cid:15) LN ′ M S ′ L ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ NLM F (cid:15) (cid:15) LMN ′ S ′ LM / / N ′ LM LMN ′ S ′ LM / / S ′ M ❧❧❧❧❧❧❧❧❧❧❧❧❧ N ′ LM . σ LM + σ M ❧❧❧❧❧❧❧❧❧ ❧❧❧❧❧❧❧❧❧ σ L % - ❘❘❘❘❘❘❘❘❘ ❘❘❘❘❘❘❘❘❘ γ L , M (cid:11) (cid:19) γ ′ L , M (cid:11) (cid:19) ✥✥✥✥✥✥ for all L , M in M .A 2-morphism in Z ( M ) between ( F, σ ) and ( F ′ , σ ′ ) is a 2-cell N F * * F ′ N ′ , α (cid:11) (cid:19) (2.50)such that MN S M / / F NM F ′ ~ ~ F MN S M / / F F ′ ~ ~ NM F ′ ~ ~ = MN ′ S ′ M / / N ′ M MN ′ S ′ M / / N ′ M , α id M + id M α + σ M " * σ ′ M t | (2.51)for all M in M .The tensor product in Z ( M ) is given by( N , S, γ ) ⊠ ( N ′ , S ′ , γ ′ ) = ( NN ′ , SS ′ , γγ ′ ) , (2.52)where NN ′ is the tensor product in M , ( SS ) ′ M is defined as the composition( SS ) ′ M : MNN ′ S M N ′ −−−−→ NMN ′ N S ′ M −−−→ NN ′ M , and ( γγ ′ ) L , M is given by the following composition of 2-cells LNN ′ M S L ' ' ◆◆◆◆◆◆◆◆◆◆◆ LNMN ′ S ′ M ♣♣♣♣♣♣♣♣♣♣♣ S L ' ' ◆◆◆◆◆◆◆◆◆◆◆ NLN ′ M S ′ L ' ' ◆◆◆◆◆◆◆◆◆◆◆ LMNN ′ S M ♣♣♣♣♣♣♣♣♣♣♣ S LM / / NLMN ′ S ′ M ♣♣♣♣♣♣♣♣♣♣♣ S ′ LM / / NN ′ LM γ L , M (cid:11) (cid:19) γ ′ L , M (cid:11) (cid:19) ⊠ S ′ M ,S L (cid:11) (cid:19) (2.53)for all L , M ∈ M .The braiding between ( N , S, γ ) and ( N ′ , S ′ , γ ′ ) is given by( S ′ N , Σ) : ( N , S, γ ) ⊠ ( N ′ , S ′ , γ ′ ) → ( N ′ , S ′ , γ ′ ) ⊠ ( N , S, γ ) , (2.54)24here Σ M is the following composition 2-cell: MNN ′ S M / / S ′ N (cid:15) (cid:15) S ′ MN ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ NMN ′ S ′ M / / S ′ NM ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ NN ′ M S ′ N (cid:15) (cid:15) MNN ′ S ′ M / / N ′ MN S M / / N ′ NM . γ ′ M , N ❧❧❧❧ ❧❧❧❧ γ ′ N , M q y ❧❧❧❧❧❧❧❧ S ′ S M (cid:11) (cid:19) Let M be a braided monoidal 2 category with braiding B M , N and structure modifications β and γ (2.25). There is a braided monoidal 2-functor F : M → Z ( M ) : N ( N , B − , N , β − , −| N )with F N,N ′ : F ( N ) F ( N ′ ) → F ( NN ′ ) given by (id NN ′ , β −| N , N ′ ) and identity 2-cells α (2.35) and δ (2.39).See [1, 8] for details. -categorical groups Recall that a categorical group is a monoidal category in which every object is invertible with respect tothe tensor product and each morphism is an isomorphism.We call an object P of a monoidal 2-category M invertible if there is another object Q together withan equivalence P ⊠ Q → I , where I is the unit object of M . Note that in this case the object Q is uniqueup to an equivalence.Note that the tensor products − ⊠ P and P ⊠ − with an invertible object P ∈ M are 2-autoequivalencesof M . In particular, each of them defines an equivalence of monoidal categories M ( I , I ) → M ( P , P ),where I is the unit object of M . Definition 2.14.
A 2 -categorical group is a monoidal 2-category whose objects are invertible with respectto the tensor product, whose 1-morphisms are equivalences, and whose 2-cells are isomorphisms.
Example 2.15.
Let A be a 2-category. Then the monoidal 2-category Aut ( A ) of autoequivalences of A with pseudo-natural equivalences as 1-morphisms and isomorphisms as 2-morphisms is a 2-categoricalgroup. Example 2.16.
Let M be a monoidal 2-category. Then the monoidal 2-category Inv ( M ) of invertibleobjects in M with equivalences as 1-morphisms and isomorphisms as 2-morphisms is a 2-categoricalgroup.Let G be a braided 2-categorical group with the tensor product ⊠ and unit object I . Below we discusssome invariants of G . 25et Π ≤ ( G ) denote the 1-categorical truncation of G , i.e., the categorical group whose objects areobjects of G and morphisms are isomorphism classes of 1-cells in G . Let Π ≤ ( G ) = G ( I , I ) be thebraided categorical group of autoequivalences of I . Its braiding is given by the naturality 2-cells I ⊠ I id ⊠ g / / f ⊠ id (cid:15) (cid:15) I ⊠ I f ⊠ id (cid:15) (cid:15) I ⊠ I id ⊠ g / / I ⊠ I ⊠ f,g + (2.55)for all f, g ∈ G ( I , I ). Definition 2.17.
The homotopy groups of G are defined as follows. • the 0th homotopy group π ( G ) is the group of equivalence classes of objects of G , • the 1st homotopy group π ( G ) is the group of isomorphism classes of autoequivalences in G ( I , I ), • the 2nd homotopy group π ( G ) is the group of automorphisms of the identity 1-morphism id I .Since Π ≤ ( G ) is braided, the homotopy groups π ( G ) , π ( G ) are abelian. Definition 2.18.
The first and second canonical classes of G , α G ∈ H ( π ( G ) , π ( G )) and q G ∈ H br ( π ( G ) , π ( G )) (2.56)are, respectively, the associator of the categorical group Π ≤ ( G ) and the braided associator of the braidedcategorical group Π ≤ ( G ). Proposition 2.19.
There is a monoidal functor a : Π ≤ ( G ) → A ut br ( Π ≤ ( G )) (2.57) canonically defined up to a natural isomorphism.Proof. For any object P in G the corresponding autoequivalence a ( P ) of G ( I , I ) is given by composingthe monoidal equivalence G ( I , I ) → G ( P , P ) : f f ⊠ P , α α ⊠ P with the quasi-inverse of G ( I , I ) → G ( P , P ) : f P ⊠ f, α P ⊠ α. That a ( P ) is a braided autoequivalence and that a is a monoidal functor follow the naturality propertiesof ⊠ . 26 emark 2.20. The action (2.57) of Π ≤ ( G ) on Π ≤ ( G ) can also be recovered from the adjoint action of the 2-categorical group G on itself, i.e., a monoidal 2-functor Ad : G → Aut ( G ) characterized (upto an equivalence) by a coherent collection of equivalences P ⊠ X → Ad P ( X ) ⊠ P , pseudo-natural in X ∈ G .The action (2.57) yields canonical group homomorphisms π ( G ) → Aut ( π ( G )) , π ( G ) → Aut ( π ( G )) , and π ( G ) → Hom ( π ( G ) , π ( G ))corresponding, respectively, to the actions of objects of Π ≤ ( G ) on objects and morphisms of Π ≤ ( G )and to the action of the group of automorphisms of I by automorphisms of id Π ≤ ( G ) . We will refer tothe corresponding maps π ( G ) × π ( G ) → π ( G ) , π ( G ) × π ( G ) → π ( G ) , and π ( G ) × π ( G ) → π ( G ) (2.58)as the Whitehead brackets .Note that the first canonical class α G is invariant with respect to the action of π ( G ) and that thebimultiplicative pairing π ( G ) × π ( G ) → π ( G ) is given by the polarization of the second canonicalclass q G .Suppose that G is a braided 2-categorical group. In this case Π ≤ ( G ) is a braided categorical groupand Π ≤ ( G ) is a symmetric categorical group. Hence, the canonical classes (2.56) get promoted to α G ∈ H br ( π ( G ) , π ( G )) and q G ∈ H sym ( π ( G ) , π ( G )) . (2.59)The braiding of G gives a trivialization of the functor (2.57) which implies that Whitehead brackets(2.58) are trivial and yields a new bilinear pairing[ , ] : π ( G ) × π ( G ) → π ( G ) (2.60)constructed as follows. For each object P in G , or an element of π ( G ), we have a canonical monoidalautomorphism of a ( P ) = id I , i.e., a homomorphism π ( G ) → π ( G ). This gives a homomorphism π ( G ) → Hom ( π ( G ) , π ( G )) identified with (2.60).For a symmetric 2-categorical group G the canonical classes are α G ∈ H sym ( π ( G ) , π ( G )) and q G ∈ H sym ( π ( G ) , π ( G )) . (2.61)All Whitehead brackets are trivial in this case. -functors between 2-categorical groups Let G be a group. We consider it as a 2-categorical group with identity 1- and 2-morphisms.27et G be a 2-categorical group (viewed as a semistrict monoidal 2-category) with the correspondingcanonical classes α G ∈ H ( π ( G ) , π ( G )) and ( ω G , c G ) ∈ H br ( π ( G ) , π ( G )) . Let C : G → Π ≤ ( G ) : x → C x be a monoidal functor. This means that there are 0-cells C x in G ,1-isomorphisms M x,y : C x C y → C xy , and invertible 2-cells C x C y C zM x,y (cid:15) (cid:15) M y,z / / C x C yzM x,yz (cid:15) (cid:15) C xy C z M xy,z / / C xyz , α x,y,z + (2.62)for all x, y, z ∈ G .Note that C gives rise to an action of G on π ( G ) (obtained by composing the underlying grouphomomorphism G → π ( G ) with the action of π ( G ) on π ( G )). We denote this action by ( g, Z ) Z g .Following [22] define a 4-cochain p C : G → π ( G ) by setting p C ( x, y, z, w ) to be the composition offaces of the following cube: C x C y C z C wM x,y z z ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ M y,z (cid:15) (cid:15) ✤✤✤✤✤✤ M z,w $ $ ■■■■■■■■■■■■■■■■■■■ C xy C z C wM xy,z (cid:15) (cid:15) M z,w " " ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ C x C yz C wM x,yz | | ② ② ② ② ② ② ② ② ② ② ② M yz,w " " ❊❊❊❊❊❊❊❊❊❊❊ C x C y C zwM x,y | | ②②②②②②②②②②②②②②②②②②②②② M y,zw (cid:15) (cid:15) p C ( x, y, z, w ) = C xyz C w M xyz,w $ $ ■■■■■■■■■■■■■■■■■■■ C xy C zwM xy,zw (cid:15) (cid:15) C x C yzwM x,yzw z z ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ C xyzw , α x,yz,w (cid:31) ' ❋❋❋❋ ❋❋❋❋ α x,y,z w (cid:127) ✇ ✇ ✇ ✇ ✇✇ ✇ ✇ ✇ ✇ α y,z,w ( ■■■■■■ ■■■■■■ α xy,z,w " * ▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲ α x,y,zw z (cid:2) ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ (2.63)where the top face is given by ⊠ M x,y ,M z,w . That is, we view p C ( x, y, z, w ) as a 2-automorphism ofthe composition of morphisms between opposite corners, e.g., of M xyz,w M xy,z M x,y , x, y, z, w ∈ G (the2-automorphisms of other compositions are conjugate to this one). We use this convention for otherpolytopes in this paper. 28 roposition 2.21. p C is a -cocycle whose cohomology class in H ( G, π ( G )) depends only on theisomorphism class of C . A monoidal functor C : G → Π ≤ ( G ) extends to a monoidal -functor G → G if and only if p C = 0 in H ( G, π ( G )) .Proof. Consider the following polytope (its planar projection is pictured): C x C y C z C w C u v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ } } ④④④④④④④④④④④④④④④④④ ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ C xy C z C w C u } } ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ (cid:15) (cid:15) ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ C x C yz C w C u v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ C x C y C zw C u t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (cid:15) (cid:15) ( ( PPPPPPPPPPPPPPPPPPPPPPPPPP C x C y C z C wu t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (cid:15) (cid:15) ❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆ C xyz C w C u ! ! ❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ C x C y C z C w C u (cid:15) (cid:15) * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ C xy C z C wu (cid:15) (cid:15) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ C x C yzw C u t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ! ! ❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇ C x C yz C wu (cid:15) (cid:15) t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ C x C y C zwu ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ C xyzw C u ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ C xyz C wu ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ C xy C zwu } } ④④④④④④④④④④④④④④④④④ C x C yzwu v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ C xyzwu . (2.64)The edges of this polytope are isomorphisms M x,y . The faces are cells α x,y,z , x, y, z ∈ G, (2.62) and ⊠ f,g .The polytope (2.64) consists of 8 cubes (four containing the top vertex and four containing the bottomone) glued together in such a way that each of their 48 faces belongs to exactly two cubes (so that theboundary is empty). Six of these cubes are of the form (2.63); their composition is the differential of p C . The remaining two cubes commute due to the naturality of the tensor product in G . Namely, M x,y commutes with the 2-cell α z,w,u and M w,u commutes with the 2-cell α x,y,z . Thus, d ( p C ) = 1.A different choice of 2-cells (2.62) results in multiplying p C by a 4-coboundary, so its class in H ( G, π ( G )) is well-defined.Finally, C extends to a monoidal 2-functor if the 2-cells (2.62) can be chosen in such a way that (2.37)is satisfied. This is equivalent to commutativity of the cube (2.63) (the latter is obtained by gluing thetwo sides of (2.37)), i.e., to p C being cohomologically trivial.For L ∈ H ( G, π ( G )) the monoidal functor L · C : G → Π ≤ ( G ) is obtained by multiplying M x,y by L x,y for all x, y ∈ G .Let C : G → Π ≤ ( G ) be a monoidal functor with the monoidal structure M x,y : C x C y → C xy , x, y ∈ G .The group Aut ( C ) of automorphisms of C is isomorphic to H ( G, π ( G )). Explicitly, P ∈ Aut ( C )29orresponds to a collection of equivalences P x : C x → C x such that there are invertible 2-cells C x C yM x,y (cid:15) (cid:15) P x P y / / C x C yM x,y (cid:15) (cid:15) C xy P xy / / C xyµ x,y + (2.65)for all x, y ∈ G .Suppose that a monoidal functor C : G → Π ≤ ( G ) extends to a monoidal 2-functor C : G → G . Thatis, there is a choice of invertible 2-cells (2.62) such that the cubes (2.63) commute, i.e., p C = 1. Let P be a monoidal automorphism of C .Define a function p C ( P ) : G → π ( G ) by C x C y C zM x,y | | ②②②②②②②②②②②②②②②②②② P x P y P z (cid:15) (cid:15) ✤✤✤✤✤✤ M y,z " " ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ C xy C zP xy P z (cid:15) (cid:15) M xy,z ❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇ C x C y C zM x,y ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ M y,z ❇❇❇❇❇❇❇❇❇❇ C x C yzM x,yz ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ P x P yz (cid:15) (cid:15) p C ( P )( x, y, z ) = C xy C z M xy,z " " ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ C xyzP xyz (cid:15) (cid:15) C x C yzM x,yz | | ②②②②②②②②②②②②②②②②②② C xyz , α x,y,z (cid:29) % ❉❉❉❉ ❉❉❉❉ µ x,y y (cid:1) ④ ④ ④ ④④ ④ ④ ④ µ yz,w ( ❍❍❍❍❍❍ ❍❍❍❍❍❍ µ xy,z ( ❏❏❏❏❏❏❏❏❏❏❏❏ ❏❏❏❏❏❏❏❏❏❏❏❏ µ x,yz { (cid:3) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (2.66)for all x, y, z ∈ G . Here the top and bottom faces are α x,y,z . Proposition 2.22. p C ( P ) is a -cocycle and the map p C : Aut ( C ) = H ( G, π ( G )) → H ( G, π ( G )) : P p C ( P ) (2.67) is a well defined homomorphism. The automorphism P extends to a monoidal pseudo-natural automor-phism of C if and only if p C ( P ) = 0 in H ( G, π ( G )) . roof. Consider the following polytope (its planar projection is pictured): C x C y C z C w w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ' ' C xy C z C w (cid:0) (cid:0) ✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂ (cid:15) (cid:15) * * C x C yz C w x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ (cid:15) (cid:15) * * C x C y C zw t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ' ' C x C y C z C w (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:15) (cid:15) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ C xyz C w (cid:30) (cid:30) ❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁ & & C xy C zw (cid:15) (cid:15) * * C x C yzw (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) * * C xy C z C w w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ (cid:15) (cid:15) C x C yz C w t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (cid:15) (cid:15) C x C y C zw w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ C xyzw ' ' C xyz C w (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ C xy C zw (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ C x C yzw w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ C xyzw . (2.68)The solid arrows are isomorphisms M x,y and the dotted ones are products of isomorphisms P x . The facesare cells α x,y,z (2.62), µ x,y (2.65), and ⊠ x,y , x, y, z ∈ G .The polytope (2.68) consists of 8 cubes (four containing the top vertex and four containing the bottomone) glued together in such a way that each of their 48 faces belongs to exactly two cubes (so that theboundary is empty). Five of these cubes are of the form (2.66); their composition is the differential of p C ( P ). Two cubes consisting of solid arrows are the cubes (2.63) and so they commute by assumption.The remaining cube commutes due to the naturality of the tensor product of G . Thus, d ( p C ( P )) = 1.A different choice of 2-cells (2.65) results in multiplying p C ( P ) by a coboundary, so its class in H ( G, π ( G )) is well-defined.The equality p C ( P Q ) = p C ( P ) p C ( Q ) , P, Q ∈ Aut ( C )is proved directly by gluing two cubes (2.66) for P and Q along the face α x,y,z .Finally, P extends to a monoidal pseudo-natural automorphism of C if 2-cells (2.65) can be chosen insuch a way that (2.44) is satisfied. This is equivalent to commutativity of the cube (2.66), i.e., to p C ( P )being cohomologically trivial.Let C : G → G be a monoidal 2-functor. For any ω ∈ Z ( G, π ( G )) let C ω be a monoidal 2-functorobtained from C by multiplying each 2-cell α x,y,z by ω ( x, y, z ) , x, y, z ∈ G . The monoidal 2-equivalenceclass of C ω depends only on the cohomology class of ω in H ( G, π ( G )). If C , C ′ are extensions of thesame monoidal functor C : G → Π ( G ) if and only if C ′ ∼ = C ω for some ω .31 orollary 2.23. Monoidal -functors C ω , C ω : G → G are isomorphic if and only if ω = p C ( P ) ω for some P ∈ Aut ( C ) = H ( G, π ( G )) .Proof. Let α x,y,z , x, y, z ∈ G, be the cells (2.62) for C . A monoidal pseudo-natural isomorphism between C ω and C ω consists of 1-automorphisms P x : C x → C x such that the cube (2.66) (with the top andbottom faces being, respectively, ω ( x, y, z ) α x,y,z and ω ( x, y, z ) α x,y,z ) commutes. This is equivalent to ω /ω = p C ( P ), where C : G → Π ≤ ( G ) is the underlying monoidal functor of C . Example 2.24.
Let I : G → Π ≤ ( G ) : x I denote the trivial monoidal functor. Then p I ( P )( x, y, z ) = ω G ( P x , P y , P z ) , x, y, z ∈ G. (2.69)The next Corollary summarizes our description of monoidal 2-functors G → G . Corollary 2.25.
Let C : G → Π ( G ) be a monoidal functor. An extension of C to a monoidal -functor C : G → G exists if and only if p C = 0 in H ( G, π ( G )) . Equivalence classes of such extensions of C form a torsor over Coker (cid:0) p C : H ( G, π ( G )) → H ( G, π ( G )) (cid:1) . -functors between 2-categorical groups Let A be an Abelian group. We consider it as a 2-categorical group with identity 1- and 2-morphisms.Let G be a braided 2-categorical group with the corresponding canonical classes α G ∈ H br ( π ( G ) , π ( G )) and q G = ( ω G , c G ) ∈ H sym ( π ( G ) , π ( G )) . Let C : A → Π ≤ ( G ) : x → C x be a braided monoidal functor. This means that there is a 0-cell C x in G for each x ∈ G , 1-isomorphisms M x,y : C x C y → C xy , invertible 2-cells α x,y,z (2.62), and invertible2-cells C x ⊠ B C y B x,y / / M x,y ( ( PPPPPPPPPPPPP C y ⊠ B C xM y,x v v ♥♥♥♥♥♥♥♥♥♥♥♥♥ C xyδ x,y K S (2.70)for all x, y, z ∈ G . Let β x | y,z and β x,y | z denote the invertible modifications (2.25) with L = C x , M = C y , N = C z . 32efine a braided 4-cochain p C ∈ C br ( A, π ( G )) by taking p br ( C )( x, y, z, w ) from (2.63), C x ⊠ B C y ⊠ B C z B x ⊠ y,z M y,z / / M x,y " " ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ B y,z ' ' C x ⊠ B C yzM x,yz (cid:15) (cid:15) C x ⊠ B C z ⊠ B C yM z,y o o M x,z (cid:15) (cid:15) B x,z | | p C ( x, y | z ) = C xy ⊠ B C z M xy,z / / B xy,z (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ C xyz C xz ⊠ B C yM xz,y o o C z ⊠ B C xyM z,xy O O C z ⊠ B C x ⊠ B C y , M x,y o o M z,x O O δ y,z T \ ✶✶✶✶✶✶ β x,y | z | (cid:4) ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ δ x,z + δ xy,z { (cid:3) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ α x,y,z , ❛❛❛❛❛❛❛❛❛❛❛ ❛❛❛❛❛❛❛❛❛❛❛ α x,z,y k s α z,x,y k s B Mx,y, C z E M ✓✓✓✓✓ ✓✓✓✓✓ (2.71)and C x ⊠ B C y ⊠ B C z B x,y ⊠ z M x,y / / M y,z " " ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ B x,y ( ( C xy ⊠ B C zM xy,z (cid:15) (cid:15) C y ⊠ B C x ⊠ B C zM y,x o o M x,z (cid:15) (cid:15) B x,z | | p C ( x | y, z ) = C x ⊠ B C yz M x,yz / / B x,yz ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ C xyz C y ⊠ B C xzM y,xz o o C yz ⊠ B C x , M yz,x O O C y ⊠ B C z ⊠ B C x . M y,z o o M z,x O O δ x,y T \ ✶✶✶✶✶✶ β x | y,z | (cid:4) ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ δ x,z + δ x,yz z (cid:2) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ α x,y,z (cid:12) (cid:20) ✥✥✥✥✥✥✥✥✥✥ α y,x,z (cid:11) (cid:19) α y,z,x K S B C x,My,z F N ✕✕✕✕✕ ✕✕✕✕✕ (2.72)where the plane projections of the octahedra are pictured. Remark 2.26.
The octahedra (2.71) and (2.72) are special cases of those from the definition of a braidedpseudomonoid in a braided Gray monoid [12, Definition 13].
Proposition 2.27. p C is a braided -cocycle whose cohomology class in H br ( A, π ( G )) depends only onthe isomorphism class of C . A braided monoidal functor C : A → Π ≤ ( G ) extends to a braided monoidal -functor A → G if and only if p C = 0 in H br ( A, π ( G )) . roof. We need to verify vanishing of the shuffle differentials (2.6) – (2.10). That the differential (2.6)is zero follows from the construction of polytope (2.64). Vanishing of the differentials (2.7) (respectively,(2.8), (2.9), and (2.10)) is proved in a similar way. Namely, we form polytopes by gluing the octahedra(2.71), (2.72) to cubes (2.63) and to the commuting polytopes (2.26) (respectively, (2.27), (2.28), and(2.29)) so that the faces of the octahedra labelled by β ’s in octahedra and polytopes are glued to eachother. In the resulting large polytopes each face labelled by α or δ is glued to its inverse. This impliescommutativity of the polytopes, i.e., vanishing of the differentials.A different choice of 2-cells (2.62) and (2.70) results in multiplying p C by a braided 4-coboundary, soits class in H ( G, π ( G )) is well-defined.Note that C extends to a braided monoidal 2-functor A → G if the cells α x,y,z and δ x,y can be chosenin such a way that (2.37), (2.40), and (2.41) are satisfied. This is equivalent to commutativity of cubes(2.63) and octahedra (2.71) and (2.72), i.e., to p C = 1. Indeed, these polytopes are obtained by gluingthe two sides of (2.37), (2.40), and (2.41).Suppose that a braided monoidal functor C : A → Π ≤ ( G ) extends to a braided monoidal 2-functor C : A → G . That is, there is a choice of invertible 2-cells (2.62) and (2.70) such that the cubes (2.63)and octahedra (2.71) and (2.72) commute, i.e., p C = 1. Let P be a monoidal automorphism of C .Let p C ( P )( x, y, z ) be defined by (2.66) and let p C ( P )( x | y ) be the composition of the faces of theprism C x C yP x P y } } ④④④④④④④④④④④④④④④④④ B x,y (cid:15) (cid:15) ✤✤✤✤✤✤ M x,y ( ( PPPPPPPPPPPPPPPPPPPPPPPPPPPPPP p C ( P )( x | y ) = C x C yB x,y (cid:15) (cid:15) M x,y ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ C y C xP y P x } } ④ ④ ④ ④ ④ ④ ④ ④ ④ M y,x / / ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ C xyP xy ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ C y C x M y,x / / C xy , B Px,Py y (cid:1) ④ ④ ④ ④ ④④ ④ ④ ④ ④ µ x,y y (cid:1) ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ δ x,y (cid:9) (cid:17) ✛✛✛✛ δ x,y (cid:9) (cid:17) ✛✛✛✛✛✛✛✛ µ y,x y (cid:1) ⑤ ⑤ ⑤ ⑤ ⑤⑤ ⑤ ⑤ ⑤ ⑤ (2.73)for all x, y ∈ A . Proposition 2.28. p C ( P ) is a braided -cocycle and the map p C : Aut ⊗ ( C ) = H ( A, π ( G )) → H br ( A, π ( G )) : P p C ( P ) (2.74) is a well defined group homomorphism.The natural automorphism P extends to a braided monoidal pseudo-natural automorphism of C if andonly if p C ( P ) = 0 in H br ( A, π ( G )) . roof. The differential (2.3) vanishes by Proposition 2.22. The vanishing of the differential (2.4) (respec-tively,(2.5)) is established by gluing the faces of three cubes (2.66), three prisms (2.73), and two copiesof the octahedron (2.71) (respectively, (2.72)) in such a way that the result has the empty boundary.A different choice of 2-cells (2.65) results in multiplying p C ( P ) by a braided coboundary, so its classin H br ( G, π ( G )) is well-defined.The multiplicative property of p C is a direct consequence of the definition of prisms (2.73).Finally, a monoidal pseudo-natural automorphism of C obtained by extending P is braided if 2-cells(2.65) can be chosen in such a way that the cube (2.45) is satisfied. But this cube becomes a prism (2.73)in our situation since the structural 2-cells of A , viewed as a braided 2-categorical group, are trivial. Sothe braided property is equivalent to the braided cohomology class of p C ( P ) being trivial.Let C : A → G be a monoidal 2-functor. For any ( ω, c ) ∈ Z br ( A, π ( G )) let C ( ω,c ) be a monoidal2-functor obtained from C by multiplying each 2-cell α x,y,z by ω ( x, y, z ) and each 2-cell δ x,y by c ( x, y )for all x, y, z ∈ A . The isomorphism class of C ( ω,c ) depends only on the class of ( ω, c ) in H br ( A, π ( G )).If C , C ′ are extensions of the same braided monoidal functor C : A → Π ≤ ( G ) if and only if C ′ ∼ = C ( ω,c ) for some ( ω, c ). Corollary 2.29.
Braided monoidal -functors C ( ω ,c ) , C ( ω ,c ) : A → G are isomorphic if and only if ( ω , c ) = ( ω , c ) p C for some P ∈ Aut ( C ) = H ( A, π ( G )) .Proof. This is similar to the proof of Corollary 2.23, where a criterion for isomorphism of monoidal2-functors C ω and C ω was established. The braided property of such an isomorphism translates tocommutativity the prism (2.73) with the front and back faces being, respectively, c ( x, y, z ) δ x,y and c ( x, y ) δ x,y . This is equivalent to c /c = p C ( P ), where C : A → Π ≤ ( G ) is the underlying monoidalfunctor of C .The next Corollary summarizes our description of braided monoidal 2-functors A → G . Corollary 2.30.
Let C : G → Π ≤ ( G ) be a braided monoidal functor. An extension of C to a monoidal -functor A → G exists if and only if p C = 0 in H br ( A, π ( G )) . Equivalence classes of such extensionsform a tosrsor over Coker (cid:0) p C : H ( A, π ( G )) → H br ( A, π ( G )) (cid:1) . -functors between 2-categorical groups Let A be an Abelian group. Here we will consider it as a symmetric 2-categorical group with identity 1-and 2-morphisms. 35et G be a symmetric 2-categorical group with the corresponding canonical classes α G ∈ H sym ( π ( G ) , π ( G )) and q G ∈ H sym ( π ( G ) , π ( G )) . One can extend the obstruction theory from Section 2.6 to symmetric monoidal 2-functors A → G .Namely, let C : A → Π ≤ ( G ) be a symmetric monoidal functor. We define p C ∈ C sym ( A, π ( G ))by extending the braided 4-cocycle from Proposition 2.27 as follows. The components p C ( − , − , − , − ), p C ( − | − , − ), and p C ( − , − , | − ) are given by (2.63) , (2.71), and (2.72), respectively, and C x ⊠ B C y B x,y M x,y (cid:28) (cid:28) ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ C y ⊠ B C xB y,x v v M y,x (cid:2) (cid:2) ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ p C ( x || y ) = C xy , τ x,y (cid:11) (cid:19) δ x,y K S δ y,x K S ✤✤✤✤✤✤ ✤✤✤✤✤✤ (2.75)for all x, y ∈ A . Proposition 2.31.
The above p C is a symmetric -cocycle whose cohomology class in H sym ( A, π ( G )) depends only on the isomorphism class of C . A symmetric monoidal functor C : A → Π ≤ ( G ) extendsto a symmetric monoidal -functor A → G if and only if p C = 0 in H sym ( A, π ( G )) .Proof. We need to check vanishing of the differentials (2.12), (2.13), and (2.14). Vanishing of the differ-entials (2.12) and (2.13) is checked by gluing polytopes (2.71) and (2.72) along their associativity faces α and gluing their braiding faces δ to two sides of cones (2.75). The differential (2.14) vanishes thanksto axiom (2.33) of a symmetric monoidal 2-category.Proposition 2.27 gives a criterion for C to have an extension to a braided monoidal 2-functor. Thisextension admits a symmetric monoidal 2-functor structure if and only if the cells (2.39) are chosen insuch a way that the cone (2.75) commutes. This is equivalent to p C being trivial in H sym ( A, π ( G )).Suppose that a symmetric monoidal functor C : A → Π ≤ ( G ) extends to a symmetric monoidal2-functor C : A → G . For any P ∈ Aut ⊗ ( C ) the braided 3-cocycle from Proposition 2.28 is symmetric,i.e., p C ( P )( x | y ) p C ( P )( y | x ) = 1for all x, y ∈ A . This can be seen gluing boundaries of two prisms (2.73) and two cones (2.75).36 orollary 2.32. Let ( ω , c ) , ( ω , c ) ∈ Z sym ( A, π ( G )) be symmetric -cocycles. Symmetric monoidal -functors C ( ω ,c ) , C ( ω ,c ) : A → G are isomorphic if and only if ( ω , c ) = ( ω , c ) p C ( P ) for some P ∈ Aut ( C ) = H ( A, π ( G )) .Proof. This is the same as Corollary 2.29, since there is no difference between isomorphisms of braidedand symmetric monoidal 2-functors.The next Corollary summarizes our description of braided monoidal 2-functors A → G . Corollary 2.33.
Let C : G → Π ≤ ( G ) be a symmetric monoidal functor. An extension of C to amonoidal -functor A → G exists if and only if p C = 0 in H sym ( A, π ( G )) . Equivalence classes of suchextensions form a torsor over Coker (cid:0) p C : H ( A, π ( G )) → H sym ( A, π ( G )) (cid:1) . -categorical group of symmetric monoidal -functors Let A be a finite Abelian group and let G be a symmetric 2-categorical group.Let C, C ′ : A → Π ≤ ( G ) be symmetric monoidal 2-functors, where C is given by x C x with themonoidal structure M x,y : C x ⊠ C y ∼ −→ C xy and C ′ is given by x C ′ x with the monoidal structure M ′ x,y : C ′ x ⊠ C ′ y ∼ −→ C ′ xy , x, y ∈ A .Define a symmetric monoidal functor˜ C := C ⊠ C ′ : A → Π ≤ ( G ) : x C x ⊠ B C ′ x . (2.76)with the monoidal structure˜ M x,y : C x ⊠ C ′ x ⊠ C y ⊠ C ′ y B x ′ ,y −−−→ C x ⊠ C y ⊠ C ′ x ⊠ C ′ y M x,y ⊠ M ′ x,y −−−−−−−−→ C xy ⊠ C ′ xy , x, y ∈ A. (2.77)Here and below we denote B x ′ ,y the braiding between C ′ x and C y .Suppose that C and C ′ extend to symmetric monoidal 2-functors C , C ′ : A → G . The associativityand braiding 2-cells (2.35) and (2.39) for C and C ′ will be denoted α, α ′ and δ, δ ′ , respectively.Our goal is to construct a canonical braided monoidal 2-functor ˜ C extending ˜ C .37efine the associativity 2-cells ˜ α x,y,z ( x, y, z ∈ A ) by C x C ′ x C y C ′ y C z C ′ z B y ′ ,z / / B x ′ ,y (cid:15) (cid:15) C x C ′ x C y C z C ′ y C ′ z M y,z M ′ y,z / / B x ′ ,y ⊠ z (cid:15) (cid:15) B x ′ ,y v v ♠♠♠♠♠♠♠♠♠♠♠♠ C x C ′ x C yz C ′ yzB x ′ ,yz (cid:15) (cid:15) C x C y C ′ x C z C ′ y C ′ z B x ′ ,yz ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ C x C y C ′ x C ′ y C z C ′ z B ( x ⊠ y ) ′ ,z / / M x,y M ′ x,y (cid:15) (cid:15) B y ′ ,z ♠♠♠♠♠♠♠♠♠♠♠♠ C x C y C z C ′ x C ′ y C ′ z M y,z M ′ y,z / / M x,y M ′ x,y (cid:15) (cid:15) C x C yz C ′ x C ′ yzM x,yz M ′ x,yz (cid:15) (cid:15) C xy C ′ xy C z C ′ z B ( xy ) ′ ,z / / C xy C z C ′ xy C ′ z M xy,z M ′ xy,z / / C xyz C ′ xyzα x,y,z α ′ x,y,z + ⊠ Mx,y,BM ′ x,y, C z + ⊠ B C ′ x,My,z ,M ′ y,z (cid:11) (cid:19) ⊠ Bx ′ ,y,By ′ ,z $ , ◗◗◗◗◗◗◗◗◗ ◗◗◗◗◗◗◗◗◗ β x ′ ,y ′| z (cid:11) (cid:19) β x ′| y,z + (2.78)and the braiding 2-cells ˜ δ x,y ( x, y ∈ A ) by C x C ′ x C y C ′ y B x ⊠ x ′ ,y ⊠ y ′ / / B x ′ ,y (cid:15) (cid:15) B x ⊠ x ′ ,y $ $ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ C y C ′ y C x C ′ x B y ′ ,x y y C x C y C ′ x C ′ y B x,y / / M x,y M ′ x,y $ $ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ C y C x C ′ x C ′ y B x ′ ,y ′ / / B x ⊠ x ′ ,y ′ : : ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ C y C x C ′ y C ′ xB x,y ′ O O M y,x M ′ y,x z z ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ C xy C ′ xy . δ x,y δ ′ x,y K S β x,x ′| y > ✈✈✈✈✈ ✈✈✈✈✈ β x,x ′| y ′ ` h ❍❍❍❍❍❍❍❍❍❍ β x ⊠ x ′| y,y ′ K S τ x,y ′ + (2.79)Here we write β x ⊠ x ′ | y,y ′ as a shorthand for β C x ⊠ C ′ x | C y , C ′ y etc. Proposition 2.34.
The -cells (2.78) and (2.79) make ˜ C = C ⊠ C ′ a symmetric monoidal -functor.Proof. The proof is tedious but straightforward. It extends the corresponding argument for symmetricmonoidal functors and consists of decomposing the commuting cube (2.63) and octahedra (2.71), (2.72)formed by 2-cells (2.78) and (2.79) into unions of commuting polytopes glued together.For the cube (2.63) for ˜ C one gets commuting polytopes obtained by gluing both sides of (2.26) -(2.29), the polytopes commuting due to the naturality of braiding and the naturality of cells β and τ ,and cubes (2.63) for C and C ′ . For the octahedra (2.71), (2.72) one gets commuting polytopes as above,38he corresponding polytopes for C and C ′ , and the symmetry polytopes (2.31), (2.32), and (2.33)) of G .It follows that ˜ C is a braided monoidal 2-functor.The cone (2.75) corresponding to the property of ˜ C being symmetric is comprised from δ x,y , δ ′ x,y and τ x,y for x, y ∈ A . This cone decomposes into the union of several commuting polytopes, namely the pairof corresponding cones for C and C ′ and the symmetry polytopes (2.31), (2.32), and (2.33). Hence, itcommutes. Proposition 2.35.
The above product of functors turns - Fun sym ( A, G ) into a symmetric -categoricalgroup.Proof. For C , C ′ , C ′′ ∈ - Fun sym ( A, G ) there is a pseudo-natural equivalence between ( C ⊠ C ′ ) ⊠ C ′′ and C ⊠ ( C ′ ⊠ C ′′ ). This can be seen to be monoidal by comparing the associativities (2.78) for both 2-functors.The unit object of - Fun sym ( A, G ) is the trivial symmetric 2-functor (with C x = I for all x ∈ A and allstructure morphisms and cells being identities).The braiding of C and C ′ is a pseudo-natural isomorphism given by B C , C ′ ( x ) : C x ⊠ C ′ x B x,x ′ −−−→ C ′ x ⊠ C x , x ∈ A, (2.80)with 2-cells (2.11) being the following compositions: C x C ′ x C y C ′ y B x ′ ,y " " B yy ′ (cid:15) (cid:15) C x C y C ′ x C ′ y M x,y ⊠ M x ′ ,y ′ / / B x ⊠ y,x ′ ⊠ y ′ (cid:15) (cid:15) B y,x ′ ⊠ y ′ z z ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ B y,x ′ o o C xy C ′ xyB xy,x ′ y ′ (cid:15) (cid:15) C x C ′ x C ′ y C yB x,x ′ (cid:15) (cid:15) B x,x ′ ⊠ y ′ $ $ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ C ′ x C x C ′ y C y B x,y ′ / / C ′ x C ′ y C x C y M x ′ ,y ′ ⊠ M x,y / / C ′ xy C xy . τ x ′ ,y (cid:11) (cid:19) β y | x ′ ,y ′ ( ❍❍❍❍❍❍ ❍❍❍❍❍❍ β x,y | x ′ ⊠ y ′ + β x | x ′ ,y ′ > ✈✈✈✈✈✈ ✈✈✈✈✈✈ δ x,y ⊠ δ ′ x,y (cid:11) (cid:19) (2.81)The 2-cells (2.25) are β x,x ′ | x ′′ and β x | x ′ x ′′ , x ∈ A . One can directly verify commutativity of the cubes(2.44) and (2.45).Finally, the symmetry 2-cell τ of G provides an invertible modification between B C ′ , C ◦ B C , C ′ andid C ⊠ C ′ satisfying (2.31), (2.32), and (2.33). 39 heorem 2.36. There is an exact sequence of group homomorphisms: H ( A, π ( G )) → H sym ( A, π ( G )) → π ( - Fun sym ( A, G )) → π ( Fun sym ( A, Π ≤ ( G ))) → H sym ( A, π ( G )) . (2.82) Proof.
The first three arrows are described in Section 2.7. That they are homomorphisms follows fromthe definition of the tensor product in - Fun sym ( A, G ).We need to check that π ( Fun sym ( A, Π ≤ ( G ))) → H sym ( A, π ( G )) : C p C , where the components of the symmetric 4-cocycle p C are given by the values of polytopes (2.63), (2.71),(2.72), and (2.75), is a group homomorphism. This is achieved by decomposing each of these polytopesfor C ⊠ C ′ , where C , C ′ ∈ - Fun sym ( A, G ), into the union of the corresponding polytopes for C and C ′ and commuting polytopes satisfied by the structure 2-cells of G as well as those of C , C ′ , glued togetherin such a way that the resulting boundary is empty.The exactness of this sequence follows from Corollary 2.33. Le C be a tensor category with the associativity constraint a X,Y,Z : ( X ⊗ Y ) ⊗ Z ∼ −→ X ⊗ ( Y ⊗ Z ). Let C op denote the tensor category with the opposite multiplication X ⊗ op Y = Y ⊗ X and the associativityconstraint a op X,Y,Z = a − Z,Y,X : Z ⊗ ( Y ⊗ X ) ∼ −→ ( Z ⊗ Y ) ⊗ X for X, Y, Z ∈ C . Below we recall definitionsfrom [30, 21]. Definition 3.1.
A (left) C -module category is a finite Abelian k -linear category M together with abifunctor C × M → M , ( X, M ) X ∗ M, exact in each variable, and a collection of isomorphisms ( C -module associativity constraint ) m X,Y,M : ( X ⊗ Y ) ∗ M ∼ −→ X ∗ ( Y ∗ M ) , X, Y ∈ C , M ∈ M and such that the diagram(( X ⊗ Y ) ⊗ Z ) ∗ M m X ⊗ Y,Z,M ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ a X,Y,Z ∗ id M u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ( X ⊗ ( Y ⊗ Z )) ∗ M m X,Y ⊗ Z,M (cid:15) (cid:15) ( X ⊗ Y ) ∗ ( Z ∗ M ) m X,Y,Z ∗ M (cid:15) (cid:15) X ∗ (( Y ⊗ Z ) ∗ M ) id X ∗ m Y,Z,M / / X ∗ ( Y ∗ ( Z ∗ M )) (3.1)commutes for all X, Y, Z ∈ C , M ∈ M .A right C -module category is a C op -module category. A C -bimodule category is a ( C ⊠ C op )-modulecategory. Remark 3.2. A C -bimodule category M can be equivalently described as a category with both left andright C -module structures and a collection of isomorphisms (a middle associativity constraint ) m X,M,Y : ( X ∗ M ) ∗ Y ∼ −→ X ∗ ( M ∗ Y ) (3.2)natural in X, Y ∈ C , M ∈ M compatible in a certain way [21, Definition 7.1.7]. Definition 3.3. A C -module functor F : M → N between C -module categories is a functor along witha collection of isomorphisms F X,M : X ∗ F ( M ) ∼ −→ F ( X ∗ M ) natural in X ∈ C , M ∈ M such that thefollowing diagram ( X ⊗ Y ) ∗ F ( M ) F X ⊗ Y,M ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ m X,Y,F ( M ) u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ X ∗ ( Y ∗ F ( M )) id X ∗ F Y,M (cid:15) (cid:15) F (( X ⊗ Y ) ∗ M ) F ( m X,Y,M ) (cid:15) (cid:15) X ∗ F ( Y ∗ M ) F X,Y ∗ M / / F ( X ∗ ( Y ∗ M )) (3.3)commutes for X, Y ∈ C , , M ∈ M . Definition 3.4.
A natural C -module transformation between C -module functors F, F ′ : M → N is anatural transformation η : F → F ′ such that X ∗ F ( M ) F X,M / / id X ∗ η M (cid:15) (cid:15) F ( X ∗ M ) η X ∗ M (cid:15) (cid:15) X ∗ F ′ ( M ) F ′ X,M / / F ′ ( X ∗ M ) (3.4)commutes for all X ∈ C , M ∈ M . 41et F : L → M and F ” : M → N be C -module functors then F ′ ◦ F has a canonical structure of C -module functor( F ′ ◦ F ) X,M : X ∗ F ′ ( F ( M )) F ′ X,F ( M ) −−−−−→ F ′ ( X ∗ F ( M )) F ′ ( F X,M ) −−−−−−→ F ′ ( F ( X ∗ M )) , X ∈ C , M ∈ M . (3.5)Thus, C -module categories, C -module functors, and C -module natural transformations form a strict 2-category Mod ( C ). Let C be a tensor category, let M be a right C -module category, and let N be a left C -module category.The (relative) tensor product M ⊠ C N [22] is an abelian category M ⊠ C N along with a functor M × N → M ⊠ C N universal among C -balanced and right exact in each variable functors from M × N to Abeliancategories.An explicit description is given as follows. Objects of M ⊠ C N are pairs ( V, γ ), where V ∈ M ⊠ N and γ X : V ∗ ( X ⊠ ) ∼ −→ ( ⊠ X ) ∗ V, (3.6)is a balancing isomorphism natural in V ∈ M ⊠ N , X ∈ C and such that the following diagram V ∗ (( X ⊗ Y ) ⊠ ) γ X ⊗ Y / / m V,X,Y (cid:15) (cid:15) ( ⊠ ( X ⊗ Y )) ∗ V n X,Y,V (cid:15) (cid:15) ( V ∗ ( X ⊠ )) ∗ ( Y ⊠ ) γ X / / ( ⊠ X ) ∗ ( V ∗ ( Y ⊠ )) γ Y / / ( ⊠ X ) ∗ (( ⊠ Y ) ∗ V ) , (3.7)commutes. Here m and n are the module associativity constraints in M and N .A morphism between ( V, { γ X } X ∈ C ) and ( V ′ , { γ ′ X } X ∈ C ) in M ⊠ C N is a morphism f : V → V ′ in M ⊠ N such that the diagram V ∗ ( X ⊠ ) f ∗ ( X ⊠ ) / / γ X (cid:15) (cid:15) V ′ ∗ ( X ⊠ ) γ ′ X (cid:15) (cid:15) ( ⊠ X ) ∗ V ( ⊠ X ) ∗ f / / ( ⊠ X ) ∗ V ′ (3.8)commutes for all X ∈ C .If M is a C -bimodule category then M ⊠ C N inherits the left C -module category structure from M : Y ∗ ( V, { γ X } ) = (( Y ⊠ ) ∗ V, { γ X } ) , (3.9)where (( Y ⊠ ) ∗ V ) ∗ ( X ⊠ ) γ X / / m − Y,V,X (cid:15) (cid:15) ( ⊠ X ) ∗ (( Y ⊠ ) ∗ V )( Y ⊠ ) ∗ ( V ∗ ( X ⊠ )) γ X / / ( Y ⊠ ) ∗ (( ⊠ X ) ∗ V ) (3.10)42or all X, Y ∈ C . Similarly, if N is a C -bimodule category then M ⊠ C N inherits the right C -modulecategory structure from N .Thus, there is a monoidal 2-category Bimod ( C ) of C -bimodule categories. Its 1-cells are C -bimodulefunctors and 2-cells are natural transformations of C -bimodule functors. The regular C -bimodule category C is the identity for ⊠ C . Let B be a braided tensor category with the braiding c X,Y : X ⊗ Y ∼ −→ Y ⊗ X, X, Y ∈ B . The braiding of B allows to turn a left B -module category M into a B -bimodule category as follows.Let m X,Y,M : ( X ⊗ Y ) ⊗ M ∼ −→ X ⊗ ( Y ⊗ M ) denote the left B -module associativity constraint of M .Define the right action of B on M by M ∗ X := X ∗ M for all X ∈ B and M ∈ M . The right B -moduleassociativity constraint is given by the composition M ∗ ( X ⊗ Y ) m M,X,Y / / ( M ∗ X ) ∗ Y ( X ⊗ Y ) ∗ M c X,Y / / ( Y ⊗ X ) ∗ M m Y,X,M / / Y ∗ ( X ∗ M ) (3.11)and the middle associativity constraint is given by( X ∗ M ) ∗ Y m X,Y,M / / X ∗ ( M ∗ Y ) Y ∗ ( X ∗ M ) m − Y,X,M / / ( Y ⊗ X ) ∗ M c Y,X / / ( X ⊗ Y ) ∗ M m X,Y,M / / X ∗ ( Y ∗ M ) , (3.12)for all X, Y ∈ B and M ∈ M . Remark 3.5.
Since B -module functors and their B -module natural transformations extend to B -bimodulefunctors and B -bimodule transformations in an obvious way, there is a 2-embedding Mod ( B ) → Bimod ( B ).Using the B -bimodule structure of M we define the tensor product M ⊠ B N of B -module categories M , N , as in Section 3.2. It has a canonical structure of a left B -module category. This makes Mod ( B )a monoidal 2-category. Remark 3.6.
From (3.7) we see that objects of M ⊠ B N are pairs ( V, { γ X } X ∈ B ), where V ∈ M ⊠ N and γ X : ( X ⊠ ) ∗ V → ( ⊠ X ) ∗ V, V ∈ M ⊠ N , X ∈ B ,
43s a natural balancing isomorphism satisfying(( X ⊗ Y ) ⊠ ) ∗ V γ X ⊗ Y / / c X,Y (cid:15) (cid:15) ( ⊠ ( X ⊗ Y )) ∗ V n X,Y,V (cid:15) (cid:15) (( Y ⊗ X ) ⊠ ) ∗ V m Y,X,V (cid:15) (cid:15) ( Y ⊠ ) ⊗ (( X ⊠ ) ∗ V ) γ X / / ( Y ⊠ X ) ∗ V γ Y / / ( ⊠ Y ) ⊗ (( ⊠ X ) ∗ V ) . (3.13)The vertical composition on the left side is the right B -module associativity constraint of M . Proposition 3.7.
Let C ⊂ B be a tensor subcategory. The induction Mod ( C ) → Mod ( B ) : N B ⊠ C N (3.14) is a monoidal -functor.Proof. The monoidal structure on the 2-functor (3.14) is given by the canonical equivalence( B ⊠ C M ) ⊠ B ( B ⊠ C N ) ∼ = B ⊠ C ( M ⊠ B B ) ⊠ C N ∼ = B ⊠ C ( M ⊠ C N ) , M , N ∈ Mod ( C ) . The verification of axioms is straightforward and is left to the reader.
Let B be a braided tensor category. The following definition appeared in [5, 2]. Definition 4.1. A braided B -module category is a pair ( M , σ ), where M is a B -module category and σ = { σ X,M : X ∗ M → X ∗ M } X ∈ B , M ∈ M is a natural isomorphism (called a B -module braiding ) with σ ,M = 1 M such that the diagrams X ∗ ( Y ∗ M ) σ X,Y ∗ M / / m − X,Y,M (cid:15) (cid:15) X ∗ ( Y ∗ M ) m − X,Y,M (cid:15) (cid:15) ( X ⊗ Y ) ∗ M c X,Y (cid:15) (cid:15) ( X ⊗ Y ) ∗ M c − Y,X (cid:15) (cid:15) ( Y ⊗ X ) ∗ M m Y,X,M (cid:15) (cid:15) ( Y ⊗ X ) ∗ M m Y,X,M (cid:15) (cid:15) Y ∗ ( X ∗ M ) σ X,M / / Y ∗ ( X ∗ M ) ( X ⊗ Y ) ∗ M σ X ⊗ Y,M / / c − Y,X (cid:15) (cid:15) ( X ⊗ Y ) ∗ M c X,Y (cid:15) (cid:15) ( Y ⊗ X ) ∗ M m Y,X,M (cid:15) (cid:15) ( Y ⊗ X ) ∗ M m Y,X,M (cid:15) (cid:15) Y ∗ ( X ∗ M ) σ X,M ( ( PPPPPPPPPPPP Y ∗ ( X ∗ M ) Y ∗ ( X ∗ M ) σ Y,X ∗ M ♥♥♥♥♥♥♥♥♥♥♥♥ (4.1)commute for all X, Y ∈ B and M ∈ M . 44 efinition 4.2. A B -module functor F : M → N between braided B -module categories is braided if thediagram X ∗ F ( M ) σ X,F ( M ) / / F X,M (cid:15) (cid:15) X ∗ F ( M ) F X,M (cid:15) (cid:15) F ( X ∗ M ) F ( σ X,M ) / / F ( X ∗ M ) (4.2)commutes for all X ∈ B and M ∈ M .A morphism between braided B -module functors is a B -module natural transformation.Let Mod br ( B ) denote the 2-category of braided B -module categories. Example 4.3.
Let C be a braided tensor category containing B . Then C is a braided B -module categorywith the B -module braiding σ X,Y = c Y,X c X,Y , X, Y ∈ B , (4.3)where c denotes the braiding of C . The commutativity of diagrams in Definition 4.1 follows directly fromthe hexagon identities and naturality of braiding. Example 4.4.
A special case of the previous example is C = B , the regular B -module category, withthe module braiding (4.3) The category of braided module endofunctors of B is the symmetric tensorcategory Z sym ( B ). Remark 4.5.
The name braided in Definition 4.1 is justified as follows. Recall that the pure Artin braidgroup of type B is the group P n generated by elements ς , . . . , ς n and relations ς n − ς n ς n − ς n = ς n ς n − ς n ς n − ,ς i ς j = ς i ς j , | i − j | ≥ ,ς i ς i +1 ς i = ς i +1 ς i ς i +1 , i = 1 , . . . , n − . Given objects X , . . . , X n − in a braided tensor category B and an object M in a braided B -modulecategory M there is a group homomorphism ρ : P n → Aut M ( X ⊗ · · · ⊗ X n − ∗ M ) given by ς i = id X ⊗ · · · ⊗ id X i − ⊗ c X i ,X i +1 ⊗ id X i +2 ⊗ · · · ⊗ id X n − ∗ id M ,ς n = id X ⊗ · · · ⊗ id X n − ⊗ σ X n − ,M , where we omit the associativity constraints. Remark 4.6.
A braided B -module category is indecomposable if and only if it is indecomposable as a B -module category. 45he α -inductions [4] for a a left B -module category M are tensor functors α ± M : B op → End B ( M ) , α ± ( X )( M ) = X ∗ M, X ∈ B , M ∈ M . (4.4)Here End B ( M ) is the category of right exact B -module endofunctors of M , The B -module structures on α ± ( X )( are given by the compositions Y ⊗ α + M ( X )( M ) α + M ( X ) Y,M / / α + M ( X )( Y ∗ M ) Y ∗ ( X ∗ M ) m − Y,X,M (cid:15) (cid:15) X ∗ ( Y ∗ M )( Y ⊗ X ) ∗ M c − X,Y ∗ id M / / ( X ⊗ Y ) ∗ M, m X,Y,M O O Y ⊗ α − M ( X )( M ) α − M ( X ) Y,M / / α − M ( X )( Y ∗ M ) Y ∗ ( X ∗ M ) m − Y,X,M (cid:15) (cid:15) X ∗ ( Y ∗ M )( Y ⊗ X ) ∗ M c Y,X ∗ id M / / ( X ⊗ Y ) ∗ M, m X,Y,M O O for all X, Y ∈ C , M ∈ M respectively.The monoidal structures of α ± M are α + M ( Y )( α + M ( X )( M )) = Y ∗ ( X ∗ M ) m − Y,X,M −−−−−→ ( Y ⊗ X ) ∗ M c Y,X −−−→ ( X ⊗ Y ) ∗ M = α + M ( X ⊗ Y )( M ) ,α − M ( Y )( α − M ( X )( M )) = Y ∗ ( X ∗ M ) m − Y,X,M −−−−−→ ( Y ⊗ X ) ∗ M c − X,Y −−−→ ( X ⊗ Y ) ∗ M = α − M ( X ⊗ Y )( M ) . Remark 4.7.
For every B -module functor F : M → N there are natural transformations of B -modulefunctors M α ± M ( X ) (cid:15) (cid:15) F / / N α ± N ( X ) (cid:15) (cid:15) M F / / N , F ± X, − + (4.5)for all X ∈ B . Remark 4.8.
Let M be a B -module category. The B -bimodule category M constructed in Section 3.3can be conveniently described by means of the functor α + M : B op → End B ( M ). Indeed, this functor turnsa canonical ( B ⊠ End B ( M ))-module category M into a B -bimodule category. Note that the functor α − M gives rise to a different B -bimodule category obtained from M using the reverse braiding of B .Let A ( B ) denote the 2-category whose objects are pairs ( M , η ), where M is a B -module categoryand η : α + M ∼ −→ α − M is an isomorphism of tensor functors, 1-cells are B -module functors F : M → N such46hat M F / / α + M ( X ) N α − N ( X ) ~ ~ α + N ( X ) M F / / α + M ( X ) α − M ( X ) ~ ~ N α − N ( X ) ~ ~ = M F / / N M F / / N , η X + η X + F + X, − & . F − X, − p x (4.6)for all X ∈ B , where F ± X, − are natural isomorphisms from (4.5), and 2-cells are B -module naturaltransformations. Proposition 4.9.
There is canonical -equivalence Mod br ( B ) ∼ = A ( B ) .Proof. A module braiding σ X,M on M is the same thing as a natural isomorphism η : α + M ∼ −→ α − M via η X ( M ) = σ X,M : α + M ( X )( M ) = X ∗ M → X ∗ M = α − M ( X )( M ) , X ∈ B , M ∈ M . The first diagram in (4.1) is equivalent to η X : α + M ( X ) ∼ −→ α − M ( X ) being an isomorphism of left B -modulefunctors and the second diagram expresses the tensor property of the natural isomorphism η M . On thelevel of 1-cells, the commuting square (4.2) is equivalent to the identity (4.6). br ( B ) as a braided monoidal -category Theorem 4.10.
There is a canonical -equivalence Mod br ( B ) ∼ = Z ( Mod ( B )) . In particular, Mod br ( B ) has a canonical structure of a braided monoidal -category.Proof. In view of Proposition 4.9 it suffices to construct a 2-equivalence A ( B ) ∼ = Z ( Mod ( B )).We construct a 2-functor A ( B ) → Z ( Mod ( B )) as follows. Let ( N , η : α + N ∼ −→ α − N ) be an object of A ( B ). Let A be an algebra in B and let M = Mod B ( A ) be the category of A -modules in B (any B -modulecategory is of this form). Then M ⊠ B N ∼ = Mod N ( α + N ( A )), where α + N ( A ) is an algebra in End B ( N ) andits module in N is an object N ∈ N along with an action α + N ( A ) ∗ N → N satisfying usual axioms.Similarly, N ⊠ B M ∼ = Mod N ( α − N ( A )). Hence, the isomorphism η A : α + N ( A ) ∼ −→ α − N ( A ) of algebras in End B ( N ) yields a pseudo-natural B -module equivalence S M : M ⊠ B N ∼ −→ N ⊠ B M .Let L = Mod B ( A ) and M = Mod B ( A ). The invertible modification γ L , M (2.47) comes from thecommutative diagram of algebra isomorphisms α + N ( A ) ⊗ α + N ( A ) η A ⊗ η A / / ( α + N ) A ⊗ A (cid:15) (cid:15) α − N ( A ) ⊗ α − N ( A ) ( α − N ) A ⊗ A (cid:15) (cid:15) α + N ( A ⊗ A ) η A ⊗ A / / α − N ( A ⊗ A ) . (4.7)47ote that since α ± N is a central functor, α ± N ( A ) ⊗ α ± N ( A ) are algebras in End B ( N ) and η A ⊗ η A isan algebra isomorphism. The coherence condition (2.48) follows from the tensor property of η . Thus,we have an object ( N , S = { S M } , γ = { γ L , M } ) of Z ( Mod ( B )), see Section 2.3. This gives rise to a2-functor A ( B ) → Z ( Mod ( B )) : ( N , η ) ( N , S, γ ) . (4.8)To construct a 2-functor in the opposite direction, note that for any X ∈ End B ( I Mod ( B ) ) ∼ = B op and N ∈ Mod ( B ) the tensor functors B op → End B ( N ) : X L N ◦ ( X ⊠ B id N ) ◦ L − N , (4.9) B op → End B ( N ) : X R N ◦ (id N ⊠ B X ) ◦ R − N , (4.10)where L N , R N are the unit constraint 1-cells in Mod ( B ), are isomorphic to α + N and α − N , respectively.For an object ( N , S, γ ) in Z ( Mod ( B )) consider the following composition of invertible 2-cells: N L − N v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ R − N ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ B ⊠ B N X ⊠ B id N (cid:15) (cid:15) S B / / N ⊠ B B id N ⊠ B X (cid:15) (cid:15) B ⊠ B N L N ( ( PPPPPPPPPPPPPP S B / / N ⊠ B B R N v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ N , + + S X + (4.11)where the top and bottom triangles are canonical 2-cells coming from the unit constraints of Mod ( B )and S X is the half-braiding 2-cell. The outside compositions are (4.9) and (4.10). Thus, 2-cells (4.11)give an isomorphism of B -module functors η X : α + N ( X ) ∼ −→ α − N ( X ) , X ∈ B . The multiplicative property(2.21) of the pseudo-natural transformation S implies that η is an isomorphism of tensor functors.This gives a 2-functor Z ( Mod ( B )) → A ( B ) : ( N , S, γ ) ( N , η N ) (4.12)quasi-inverse to (4.8).This resulting 2-equivalence Mod br ( B ) ∼ = Z ( Mod ( B )) obtained using Proposition 4.9 induces on Mod br ( B ) a structure of a braided monoidal 2-category.For braided B -module categories M := ( M , σ M ) and N := ( N , σ N ) the braiding B M , N : M ⊠ B N ∼ −→ N ⊠ B M in Mod br ( B ) is given by the half braiding of N . 48 roposition 4.11. A braided tensor functor F : B → C induces a braided monoidal -functor Mod br ( C ) → Mod br ( B ) : ( M , σ ) ( ˜ M , ˜ σ ) , where ˜ M = M with the action X ∗ M = F ( X ) ∗ M and ˜ σ X,M = σ X,M , X ∈ B , M ∈ M .Proof. This is verified directly using the C -module braiding axioms of σ and the braided property of F . Remark 4.12.
The braided monoidal 2-category structure on
Mod br ( B ) constructed in Theorem 4.10can also be described explicitly as follows. Let ( M , σ M ) and ( N , σ N ) be braided B -module categories.Recall that objects of M ⊠ B N are pairs ( V, γ ), where V ∈ M ⊠ N and γ X : ( X ⊠ ) ∗ V ∼ −→ ( ⊠ X ) ∗ V, X ∈ B , is a balancing isomorphism satisfying (3.7).The tensor product of Mod br ( B ) is( M , σ M ) ⊠ B ( N , σ N ) = ( M ⊠ B N , σ M ⊠ B N ) , where σ M ⊠ B N X, ( V,γ ) : ( X ⊠ ) ∗ ( V, γ ) → ( ⊠ X ) ∗ ( V, γ ) , X ∈ B , ( V, γ ) ∈ M ⊠ B N , is given by the composition( X ⊠ ⊗ V σ M X,V −−−→ ( X ⊠ ⊗ V γ X −−→ (1 ⊠ X ) ⊗ V σ N X,V −−−→ (1 ⊠ X ) ⊗ V γ − X −−→ ( X ⊠ ⊗ V. The unit object for this tensor product is the regular braided B -module category from Example 4.4.The braiding is B ( M , σ M ) , ( N , σ N ) : ( M , σ M ) ⊠ B ( N , σ N ) ∼ −→ ( N , σ N ) ⊠ B ( M , σ M ) , ( V, γ ) ( V t , ˜ γ ) , where M ⊠ N → N ⊠ M : V V t is the transposition functor, i.e., V t = N ⊠ M for V = M ⊠ N (thisextends to M ⊠ N thanks to the universal property of ⊠ ) and˜ γ X : ( X ⊠ ) ∗ V t σ N X,V −−−→ ( X ⊠ ) ∗ V t ( γ tX ) − −−−−−→ ( ⊠ X ) ∗ V t , X ∈ B . Let B be a braided tensor category. Example 4.13.
The regular braided B -module category B from Example 4.4 is the unit object of Mod br ( B ). It generates a braided monoidal 2-category Mod ( B ) whose objects are direct sums of49opies of B (identified with natural numbers), 1-cells are matrices of objects in Z sym ( B ), and 2-cells arematrices of morphisms in B . The tensor product is given by the Kronecker product of such matriceswhile the braiding 2-cells are given by the braiding of Z sym ( B ). Example 4.14.
Note that B can have other B -module braidings, in addition to one from Example 4.4.Namely, it follows from the first diagram in (4.1) that a B -module braiding σ on B satisfies σ X,Y = σ X, c Y,X c X,Y , X, Y ∈ B . The second diagram in (4.1) is equivalent to σ X, being a tensor automorphism of id B . Conversely, any ν ∈ Aut ⊗ (id B ) yields a module braiding σ νX,Y = ( ν X ⊗ ) c Y,X c X,Y , X, Y ∈ B . (4.13)Let B ν := ( B , σ ν ) denote the corresponding braided B -module category.There is an exact sequence [18, 3.3.4] of groups1 → Inv ( Z sym ( B )) → Inv ( B ) α −→ Aut (id B ) , (4.14)where α ( Z ) X = c Z,X c X,Z ∈ Aut ( X ⊗ Z ) = k × (4.15)for every simple object X ∈ B . Two braided B -module categories B ν and B ν , ν , ν ∈ Aut (id B ) , areequivalent if and only if ν = ν α ( Z ) for some Z ∈ Inv ( B ). Thus, the group of equivalence classes ofbraided B -module categories of the form B ν is isomorphic to Coker ( Inv ( B ) α −→ Aut (id B )).Let Mod ( B ) denote the full braided monoidal 2-subcategory of Mod br ( B ) generated by braided B -module categories B ν . Example 4.15.
Let M be an exact B -module category. The 2-categorical half-braiding M ∼ = M ⊠ B B ν S ν M −−→ B ν ⊠ B M ∼ = M (4.16)is identified with the image of ν under the composition Aut ⊗ (id B ) ι −→ Inv ( Z ( B )) ∼ = Inv ( Z ( End B ( M )) → Inv ( End B ( M )) = Aut B ( M ) , (4.17)where ι ( ν ) = as an object of B with the half-braiding ν X id X : X ∼ = X ⊗ ι ( ν ) ∼ −→ ι ( ν ) ⊗ X ∼ = X, X ∈ B . For the trivial tensor category B = Vect we have
Mod br ( Vect ) =
Mod ( Vect ) =
Mod ( Vect ), the2-category of 2-vector spaces. The objects of this category are natural numbers, 1-cells are matrices ofvector spaces, and 2-cells are matrices of linear transformations.The following result was established in [29] using different methods and terminology.50 roposition 4.16.
Let B be a non-degenerate braided fusion category. There is an equivalence of braidedmonoidal -categories Mod br ( B ) ∼ = Mod br ( Vect ) . (4.18) Proof.
Let M be an indecomposable B -module category. The tensor functors α ± M : B op → End B ( M ) aregiven by the compositions B op ∼ −→ B → Z ( B ) ∼ = Z ( End B ( M )) → End B ( M ) , (4.19)where the first functor is the equivalence provided by the braiding of B , the second functor is the em-bedding of B (respectively, B rev ) into Z ( B ), and the last one is the forgetful functor. The images of B and B rev generate Z ( End B ( M )). If M has a B -module braiding, it follows from Proposition 4.9 that thefull images of α + M ( B ) and α − M ( B ) in End B ( M ) coincide. Since the forgetful functor is surjective, we have α ± M ( B ) = End B ( M ). Thus, M is an invertible B -module category such that the braided autoequivalence ∂ M := ( α + M ) − ◦ α − M ∈ Aut br ( B ) is trivial. It follows from [10, 22] that M ∼ = B as a B -module category,i.e., Mod br ( B ) ∼ = Mod ( B ). Since the homomorphism α from Example 4.14 is an isomorphism fora non-degenerate category B , we have Mod ( B ) ∼ = Mod ( B ). Since Z sym ( B ) = Vect , Mod ( B ) is2-equivalent to Mod br ( Vect ), and the statement follows. -category of symmetric module categories Let E be a symmetric tensor category. For any E -module category M we have α + M = α − M . In particularany E -module category M has the identity module braiding id X ⊗ M . Definition 4.17.
A braided E -module category ( M , σ ) is called symmetric if σ X,M = id X ⊗ M for all X ∈ E and M ∈ M . Example 4.18.
Let C be a symmetric braided tensor category containing E . Then C is a symmetric E -module category.Clearly, the tensor product of symmetric module categories is symmetric. We will denote Mod sym ( E )the symmetric monoidal 2-category of symmetric E -module categories (its double braiding 2-cells (2.30)are identities). Note that Mod sym ( E ) = Mod ( E ) as a monoidal 2-category and can also be viewed as abraided monoidal 2-subcategory of Mod br ( E ). Example 4.19.
Let B be a braided tensor category. It follows from Example 4.15 that the braidedmonoidal 2-category Mod ( B ) has a symmetric structure.51 roposition 4.20. Let E be a tensor subcategory of Z sym ( B ) . The induction Mod ( E ) → Mod br ( B ) : N B ⊠ E N (4.20) is a braided monoidal -functor.Proof. Let N = Mod E ( A ) ∈ Mod ( E ) for some algebra A ∈ E . Then B ⊠ E N = Mod B ( A ). For any M ∈ Mod ( B ) the composition of B -module equivalences M ⊠ B ( B ⊠ E N ) ∼ = Mod M ( α − M ( A )) = Mod M ( α + M ( A )) ∼ = ( B ⊠ E N ) ⊠ B M , where the equality in the middle is due to the fact that A ∈ Z sym ( B ), defines a half braiding on B ⊠ E N .It follows that B ⊠ E N is a braided B -module category and the monoidal induction 2-functor fromProposition 3.7 lifts to a braided monoidal 2-functor (4.20). -categorical Picard groups In this Section we describe categorical 2-groups of module categories over tensor categories in termsintroduced in Section 2.4. -categorical Brauer-Picard group of a tensor category Let D be a braided tensor category. Recall from [22, Section 4.1] that a D -bimodule category M isinvertible with respect to ⊠ D if and only if M o ⊠ D M ∼ = D and M ⊠ D M o ∼ = D , where M o is theopposite Abelian category of M with the left (respectively, right) D -module actions of X ∈ D given bythe right (respectively, left) actions of ∗ X (see also [17]).The 2 -categorical Brauer-Picard group [22] of a tensor category D is BrPic ( D ) = Inv ( Bimod ( D )) . (5.1)Its objects are invertible D -bimodule categories, 1-cells are D -bimodule equivalences, and 2-cells areisomorphisms of D -bimodule equivalences. The tensor product is ⊠ D and the unit object is the regular D -bimodule category. Let BrPic ( D ) denote the categorical group obtained by truncating BrPic ( D ) andlet BrP ic ( D ) denote the group of isomorphism classes of objects.The homotopy groups of BrPic ( D ) are π ( BrPic ( D )) = BrP ic ( D ) ∼ = Aut br ( Z ( D )) , (5.2) π ( BrPic ( D )) = Inv ( Z ( D )) , (5.3) π ( BrPic ( D )) = k × . (5.4)52he 1-categorical truncations of BrPic ( D ) are Π ≤ ( BrPic ( D )) = BrPic ( D ) ∼ = Aut br ( Z ( D )) , (5.5) Π ≤ ( BrPic ( D )) = Inv ( Z ( D )) . (5.6)The first canonical class is the associator α BrPic ( D ) ∈ H ( Aut br ( Z ( D )) , Inv ( Z ( D ))) of the categoricalgroup Aut br ( Z ( D )). The second canonical class is the braided associator q BrPic ( D ) ∈ H br ( Inv ( Z ( D )) , k × )of the braided categorical group Inv ( Z ( D )), corresponding to the quadratic form q BrPic ( D ) : Inv ( Z ( D )) → k × : Z c Z,Z . The monoidal functor Π ≤ ( BrPic ( D )) → A ut br ( Π ≤ ( BrPic ( D ))) coincides with the composition BrPic ( D ) ∼ = A ut br ( Z ( D )) → A ut br ( Inv ( Z ( D ))) [10, 22].The non-trivial Whitehead brackets are the maps π × π → π and π × π → π given by Aut br ( Z ( D )) × Inv ( Z ( D )) → Inv ( Z ( D )) : ( F, Z ) F ( Z ) , (5.7) Inv ( Z ( D )) × Inv ( Z ( D )) → k × : ( Z, W ) c W,Z c Z,W . (5.8)Here c denotes the braiding of Z ( D ). -categorical Picard group of a braided tensor category Let B be a braided tensor category. Recall [10, 22] that a B -module category M is invertible if and onlyif the α -induction tensor functors α ± M : B op → End B ( M ), see (4.4), are equivalences. Here End B ( M )denotes the category of right exact B -module endofunctors of M .Recall that a B -module category M is exact [23] if for any projective object P ∈ B and any object M ∈ M the object P ⊗ M ∈ M is projective. For an exact M , the dual category End B ( M ) is amultitensor category. The tensor product of functors is their composition and the left and right duals ofa C -module functor F : M → M are its left and right adjoints.The following result was explained to us by Victor Ostrik. Proposition 5.1.
An invertible B -module category is exact.Proof. Let M be an invertible B -module category. It is equivalent to Mod B ( A ) for some algebra A in B and Bimod B ( A ) ∼ = End B ( M ) ∼ = B op . So the tensor product over A is exact on the category of A -bimodules. This implies that it is exact forright modules tensored with left modules (as any right A -module M can be made into a bimodule A ⊗ M Hom functor M → B : M Hom(
N, M ) for all N ∈ Mod B ( A )and, hence, to exactness of M . Corollary 5.2.
Let C be a finite tensor category. An invertible C -bimodule category is exact.Proof. A canonical monoidal 2-equivalence between
Pic ( Z ( C )) and BrPic ( C ) preserves exactness by [23,Theorem 3.31].The 2 -categorical Picard group [10, 22] of B is Pic ( B ) = Inv ( Mod ( B )) . (5.9)Its objects are invertible B -module categories, 1-cells are B -module equivalences, and 2-cells are isomor-phisms of B -module equivalences. The tensor product is ⊠ B and the unit object is the regular B -modulecategory. Let Pic ( B ) denote the categorical group obtained by truncating Pic ( B ) and let P ic ( B ) denotethe group of isomorphism classes of objects.The homotopy groups of Pic ( B ) are π ( Pic ( B )) = P ic ( B ) , (5.10) π ( Pic ( B )) = Inv ( B ) , (5.11) π ( Pic ( B )) = k × . (5.12)The 1-categorical truncations of Pic ( B ) are Π ≤ ( Pic ( B )) = Pic ( B ) , (5.13) Π ≤ ( Pic ( B )) = Inv ( B ) . (5.14)The first canonical class is the associator α Pic ( B ) ∈ H ( P ic ( B ) , Inv ( B )) of the categorical group Pic ( B ). The second canonical class is the braided associator q Pic ( B ) ∈ H br ( Inv ( B ) , k × ) of the braidedcategorical group Inv ( B ), corresponding to the quadratic form q Pic ( B ) : Inv ( B ) → k × : Z c Z,Z . The monoidal functor Π ≤ ( Pic ( B )) → Aut br ( Π ≤ ( Pic ( B ))) coincides with the composition Pic ( B ) → A ut br ( B ) → A ut br ( Inv ( B )) [10, 22]. 54he non-trivial Whitehead brackets are the maps π × π → π and π × π → π given by Aut br ( B ) × Inv ( B ) → Inv ( B ) : ( F, Z ) F ( Z ) , (5.15) Inv ( B ) × Inv ( B ) → k × : ( Z, W ) c W,Z c Z,W . (5.16)Here c denotes the braiding of B .For any tensor category D there is a monoidal 2-equivalence BrPic ( D ) ∼ = Pic ( Z ( D )) [22, Theorem5.2]. Thus, 2-categorical Picard groups generalize Brauer-Picard groups. -categorical Picard group of a braided tensor category The braided -categorical Picard group of a braided tensor category B is Pic br ( B ) = Inv ( Mod br ( B )) ∼ = Inv ( Z ( Mod ( B ))) , (5.17)where the last 2-equivalence is by Theorem 4.10. Its objects are invertible braided B -module categories,1-cells are braided B -module equivalences, and 2-cells are natural isomorphisms of B -module equivalences.The tensor product is ⊠ B and the unit object is the regular braided B -module category (see Example 4.1).Let Pic br ( B ) denote the braided categorical group obtained by truncating Pic br ( B ) and let P ic br ( B )denote the group of isomorphism classes of objects.The homotopy groups of Pic br ( B ) are π ( Pic ( B )) = P ic br ( B ) , (5.18) π ( Pic ( B )) = Inv ( Z sym ( B )) , (5.19) π ( Pic ( B )) = k × . (5.20)The 1-categorical truncations of Pic br ( B ) are Π ≤ ( Pic ( B )) = Pic br ( B ) , (5.21) Π ≤ ( Pic ( B )) = Inv ( Z sym ( B )) . (5.22)The first canonical class is the braided associator α Pic br ( B ) ∈ H br ( P ic br ( B ) , Inv ( Z sym ( B ))) of thebraided categorical group Pic br ( B ) corresponding to the quadratic function Q Pic br ( B ) : P ic br ( B ) → Inv ( Z sym ( B )) : M B M , M , where B denotes the braiding of Pic br ( B ). The second canonical class is the symmetric associator q Pic br ( B ) ∈ H sym ( Inv ( Z sym ( B )) , k × ) of the symmetric categorical group Inv ( Z sym ( B )) corresponding tothe homomorphism q Pic br ( B ) : Inv ( Z sym ( B )) → {± } ⊂ k × : Z c Z,Z . roposition 5.3. Let ( M , σ M ) be an indecomposable braided B -module category and let Z be an invert-ible object in Z sym ( B ) . The Whitehead bracket [ , ] : π × π → π (2.60) of Pic br ( B ) satisfies σ M Z, M = [ M , Z ] id Z ∗ M (5.23) for all objects M ∈ M .Proof. For any simple M in M we identify σ M Z,M ∈ Aut ( Z ⊗ M ) with a non-zero scalar. It suffices tocheck that this scalar does not in fact depend on M . Note that σ M Z, X ⊗ M = c Z,X c X,Z σ M Z,M = σ M Z,M for any simple object X ∈ B . Since every simple object N of M is contained in some X ⊗ M we concludethat σ M Z, M = σ M Z, N .Recall from [10, 22] a monoidal functor ∂ : Pic ( B ) → Aut br ( B ) : M ( α + M ) − ◦ α − M , (5.24)where α ± M : B op → End B ( M ) are equivalences (4.4). Proposition 5.4.
There is an exact sequence → Inv ( Z sym ( B )) −→ Inv ( B ) α −→ Aut ⊗ ( id B ) ε −→ P ic br ( B ) φ −→ P ic ( B ) ∂ −→ Aut br ( B ) , (5.25) where α is defined in (4.15) , ε ( ν ) = B ν (see Example 4.15), and φ ( M , σ ) = M .Proof. This is an immediate consequence of the definitions.
Remark 5.5.
By the fiber of the monoidal functor F : G → H between groupoids we mean the categoryof pairs ( X, x ), where X ∈ G and x : F ( G ) → I for the unit object I ∈ H . It follows from Proposition 4.9that the fiber of the monoidal functor Pic ( B ) → Aut ( B ) coincides with Pic br ( B ). The exact sequence(5.25) can be seen as the Serre exact sequence of homotopy groups of the fibration of categorical groups Pic br ( B ) → Pic ( B ) → Aut ( B ) . Example 5.6.
Let
Pic ( B ) be the braided 2-categorical subgroup of Pic br ( B ) consisting of braided B -module categories whose underlying B -module category is the regular B -module category B . That is, Pic ( B ) = Inv ( Mod ( B )), see Example 4.14.The objects of Pic ( B ) are braided B -module categories B ν , ν ∈ Aut ⊗ (id B ). The module braidingof B ν is σ νX,M = ( ν X ⊗ id M ) ◦ c M,X c X,M , X, M ∈ B . (5.26)56he homotopy groups of Pic ( B ) are π ( Pic ( B )) = Coker( Inv ( B ) ∂ −→ Aut ⊗ (id B )) , (5.27) π ( Pic ( B )) = Inv ( Z sym ( B )) , (5.28) π ( Pic ( B )) = k × . (5.29)The following is a convenient “non-skeletal” description of Pic ( B ). The objects B ν correspond toelements of Aut ⊗ (id B ), 1-morphisms are given by Pic ( B )( B µ , B ν ) = { Z ∈ Inv ( B )) | c Z,X ◦ c X,Z ◦ ( µ X ⊗ id Z ) = ν X ⊗ id Z , X ∈ B } , (5.30)and 2-cells are isomorphisms between invertible objects of B . The tensor product is B µ ⊠ B ν := B µµ forall µ, ν ∈ Aut ⊗ (id B ). The associativity and braiding 2-cells are identities and the pseudo-naturality 2-cellfor the tensor product is ⊠ Z,W = c Z,W , Z, W ∈ Inv ( B ) , where c denotes the braiding of B . We also have B Z, B ν = ν Z , Z ∈ Inv ( B ) , ν ∈ Aut ⊗ ( B ) . All other structural 2-cells are identities. -categorical Picard group of a symmetric tensor category Let E be a symmetric tensor category. The symmetric -categorical Picard group of E is Pic sym ( E ) = Inv ( Mod sym ( E )) = Inv ( Mod ( E )) = Pic ( E ) . (5.31)Its objects are invertible symmetric E -module categories, 1-cells are E -module equivalences, and 2-cellsare natural isomorphisms of E -module equivalences. Let Pic sym ( E ) denote the categorical group obtainedby truncating Pic sym ( E ) and let P ic sym ( E ) denote the group of isomorphism classes of objects.The homotopy groups of Pic sym ( E ) are π ( Pic ( E )) = P ic sym ( E ) = P ic ( E ) , (5.32) π ( Pic ( E )) = Inv ( E ) , (5.33) π ( Pic ( E )) = k × . (5.34)The 1-categorical truncations of Pic sym ( E ) are Π ≤ ( Pic ( E )) = Pic sym ( E ) = Pic ( E ) , (5.35) Π ≤ ( Pic ( E )) = Inv ( E ) . (5.36)57he first canonical class is the symmetric associator α Pic sym ( B ) ∈ H sym ( P ic ( E ) , Inv ( E )) of the sym-metric categorical group Pic ( B ) corresponding to the homomorphism Q Pic sym ( E ) : P ic ( E ) → Inv ( E ) : M B M , M , where B denotes the braiding of Pic ( E ). The second canonical class is the symmetric associator q Pic ( E ) ∈ H sym ( Inv ( E ) , k × ) of the braided categorical group Inv ( E ) corresponding to the homomorphism q Pic sym ( E ) : Inv ( E ) → {± } ⊂ k × : Z c Z,Z . Let R be an algebra in a braided tensor category B , i.e., an object together with morphisms µ : R ⊗ R → R (the product ) and ι : I → R (the unit map) satisfying the associativity and unit conditions.Denote by Aut alg ( R ) the group of algebra automorphisms of R . Remark 5.7.
The assignment
Aut ⊗ (id B ) → Aut alg ( R ) : a a R (5.37)is a group homomorphism.Let M be a right R -module in B with the structural map ρ : M ⊗ R → M . For any X ∈ B there isan R -module structure on X ⊗ M defined byid X ⊗ ρ : X ⊗ M ⊗ R → X ⊗ M. Thus, the category Mod B ( R ) of right R -modules in B is a left B -module category via B × Mod B ( R ) → Mod B ( R ) , ( X, M ) X ⊗ M. The α -induction functors (4.4) for Mod B ( R ) are α ± Mod B ( R ) : B → Bimod B ( R ) = End B (Mod B ( R )) op : X X ⊗ R, (5.38)with the obvious right R -module structures and the left R -module structures given by R ⊗ X ⊗ R c R,X ⊗ id R −−−−−−→ X ⊗ R ⊗ R id X ⊗ µ −−−−→ X ⊗ R,R ⊗ X ⊗ R c − X,R ⊗ id A −−−−−−→ X ⊗ R ⊗ R id X ⊗ µ −−−−→ X ⊗ R, X ∈ B for α +Mod B ( R ) and α − Mod B ( R ) , respectively. 58he tensor product R ⊗ S of two algebras R, S ∈ B has an algebra structure, with the multiplicationmap µ R ⊗ S defined as R ⊗ S ⊗ R ⊗ S id R ⊗ c S,R ⊗ id S −−−−−−−−−→ R ⊗ R ⊗ S ⊗ S µ R ⊗ µ S −−−−−→ R ⊗ S, where µ R and µ S are multiplications of algebras R and S , respectively (here we suppress the associativityconstraints in B ). We have Mod B ( R ) ⊠ B Mod B ( S ) ∼ = Mod B ( R ⊗ S ) . Let R op = R denote the algebra with the multiplication opposite to that of R : R ⊗ R c R,R −−−→ R ⊗ R µ −→ R. Following [33], we say that an algebra R in a braided monoidal category B is Azumaya if the morphism R ⊗ R op ⊗ R id R ⊗ c R,R −−−−−−→ R ⊗ R ⊗ R µ ⊗ id R −−−−→ R ⊗ R µ −→ R induces an isomorphism R ⊗ R op → R ⊗ R ∗ . The B -module category Mod B ( R ) is invertible in Pic ( B ) ifand only if R is an Azumaya algebra (in which case α ± Mod B ( R ) are equivalences).Thus, the 2-categorical Picard group of Pic ( B ) is monoidally 2-equivalent to the group of Moritaequivalence classes of exact Azumaya algebras in B (the latter group was called in [33] the Brauer groupof B ).It was shown in [33, Theorem 3.1] that for an Azumaya algebra R the functor B → Bimod B ( R ) , X X ⊗ R (5.39)is a monoidal equivalence.For an Azmaya algebra R ∈ B and an automorphism φ ∈ Aut alg ( R ) let φ R be an invertible R -bimoduleobtained from R by twisting the right R -action by φ . Under the equivalence α ± Mod B ( R ) it corresponds toan invertible object P φ ∈ B and we have a group homomorphism Aut alg ( R ) → Inv ( B ) : φ P φ . (5.40) Remark 5.8.
An isomorphism of R -bimodules f : P ⊗ R → φ R is completely determined by themorphism g : P → R defined by g = f (1 ⊗ ι ). Indeed, f = µ ( g ⊗ R -module propertyof such f is automatic, the left R -module property amounts to the condition µ ( φ ⊗ g ) = µ ( g ⊗ c R,P . Remark 5.9.
An Azumaya algebra R ∈ B gives rise to a homomorphism ς R : Aut ⊗ ( Id B ) → Inv ( B ),which is the composition of the homomorphisms (5.37) and (5.40).59ote that ς R ⊗ S ( a ) = ς R ( a ) ⊗ ς S ( a ), so that we have a homomorphism ς : P ic ( B ) → Hom ( Aut ⊗ (id B ) , Inv ( B )) , (5.41)or, equivalently, a pairing h− , −i : P ic ( B ) × Aut ⊗ (id B ) → Inv ( B ) . (5.42)This can be interpreted in terms of module categories as follows. For any M ∈ Pic ( B ) there is anisomorphism Aut B ( M ) ∼ = Inv ( B ) given by α + M . For ν ∈ Aut ⊗ (id B ) the value of h M , ν i is the image of ν under the composition Aut ⊗ (id B ) ι −→ Inv ( Z ( B )) ∼ = Inv ( Z ( End B ( M ))) → Inv ( End B ( M )) ∼ = Inv ( B ) . (5.43)Note that the object h M , ν i coincides with the central structure of the braided module B -category B ν (see Example 5.6), i.e., with the value of the half-braiding M ⊠ B B ν → B ν ⊠ B M viewed as an object of Aut B ( M ) ∼ = Inv ( B ). -categorical Picard group of a symmetric fusioncategory Let G be a finite group and let Rep ( G ) denote the category of representations of G . It was proved byDeligne [15] that a symmetric fusion category is equivalent to the following “super” generalization of Rep ( G ). Namely, let G be a finite group and let t ∈ G be a central element such that t = 1. Then Rep ( G ) has a braiding defined by c V,W : V ⊗ W ∼ −→ W ⊗ V : v ⊗ w − w ⊗ v if tv = − v , tw = − w,w ⊗ v otherwise. (6.1)The fusion category Rep ( G ) equipped with the above braiding will be denoted Rep ( G, t ). Any symmetricfusion category is equivalent to
Rep ( G, t ) for a unique up to an isomorphism pair (
G, t ). Under thisnotation
Rep ( G,
1) is nothing but
Rep ( G ) with its usual transposition braiding. We call Rep ( G, t ) Tannakian if t = 1 and super-Tannakian if t = 1.The Picard group of P ic ( Rep ( G, t )) was computed by Carnovale in [7]. We recall this descriptionin Sections 6.1 and 6.2 and describe the symmetric categorical group
Pic ( Rep ( G, t )), i.e., the quadraticcharacter Q Rep ( G, t ) : P ic ( Rep ( G, t )) → b G. In Sections 6.3 and 6.4 we describe the braided categorical Picard group
Pic br ( Rep ( G, t )).60 .1 The -categorical Picard group of a Tannakian category Let B = Rep ( G ) be the category of finite dimensional representations of a finite group G with its standardsymmetric braiding. For a 2-cocycle γ ∈ Z ( G, k × ) denote by Rep γ ( G ) the category of γ -projectiverepresentations of G .The first statement of the following Proposition is well known [7]. Proposition 6.1.
The assignment H ( G, k × ) → P ic ( Rep ( G )) : γ Rep γ ( G ) is an isomorphism. The quadratic character Q Rep ( G ) : H ( G, k × ) → ( b G ) is trivial.Proof. Since an Azumaya algebra in
Rep ( G ) is also an Azumaya algebra in Vect it should have the form
End k ( V ) for some vector space V . The G -action on End k ( V ) corresponds to the structure of a projective G -representation on V . Its Schur multiplier (as a class in H ( G, k × )) is the only Morita invariant of the G -algebra End k ( V ) .We describe Q Rep ( G ) as follows. The transposition automorphism c End k ( V ) ,End k ( V ) of the tensorsquare End k ( V ) ⊗ of the Azumaya algebra End k ( V ) is inner, i.e., is given by conjugation with aninvertible element ζ ∈ End k ( V ) ⊗ . The value Q Rep ( G ) ( End k ( V )) is the character χ ∈ b G defined by g ( ζ ) = χ ( g ) ζ for all g ∈ G . Under the isomorphism End k ( V ) ⊗ ∼ = End k ( V ⊗ ) the element ζ correspondsto c V,V ∈ End k ( V ⊗ ). Clearly, c V,V is G -invariant, which makes the character χ trivial. -categorical Picard group of a super-Tannakian category Example 6.2.
Let G = Z / Z and let t be the nontrivial element of G . Then Rep ( G, t ) = sVect ,the category of super vector spaces. It goes back to [34] that
P ic ( sVect ) = Z / Z . Let Π denote thenon-identity simple object of sVect .The homomorphism Q sVect : P ic ( sVect ) = Z / Z → Inv ( sVect ) = Z / Z is the identity map. In-deed, the Azymaya algebra R = k h x | x = 1 i = I ⊕ Π (with x odd) represents the non-trivial classin P ic ( sVect ) ≃ Z / Z . Its tensor square in sVect is R ⊗ = k h x, y | x = y = 1 xy + yx = 0 i . Thebraiding c R,R is the algebra automorphism τ sending x to y and back. Note that τ is inner, namely τ ( r ) = ( x + y ) r ( x + y ) − . Since the conjugating element x + y ∈ R ⊗ is odd, the invertible object in sVect corresponding to τ (determined as in the remark 5.8) is the non-trivial element Π of Inv ( sVect ) ≃ Z / Z .We say that Rep ( G, t ) is split super-Tannakian if h t i is a direct summand of G and non-split super-Tannakian otherwise. 61he following definition was given and Theorem 6.4 below was proved by Carnovale [7]. We includethe argument for the sake of completeness and to set up notation for subsequent computations.Define the group H ( G, t, k × ) to be the second cohomology H ( G, k × ) as a set, with the groupoperation (on the level of cocycles) given by( γ ∗ ν )( f, g ) = ( − ξ γ ( f ) ξ ν ( g ) γ ( f, g ) ν ( f, g ) f, g ∈ G, γ, ν ∈ H ( G, k × ) . (6.2)where ξ γ : G → Z / Z is the homomorphism defined by( − ξ γ ( g ) = γ ( t, g ) γ ( g, t ) . (6.3) Remark 6.3.
It was explained in [7] that H ( G, t, k × ) is (non-canonically) isomorphic to H ( G, k × ). Theorem 6.4.
The Picard group of a split super-Tannakian symmetric fusion category is
P ic ( Rep ( G, t )) ≃ H ( G, t, k × ) × Z / Z . (6.4) The Picard group of a non-split super-Tannakian symmetric fusion category is
P ic ( Rep ( G, t )) ≃ H ( G, t, k × ) . (6.5) The homomorphism Q Rep ( G,t ) : P ic ( Rep ( G, t )) → Inv ( Rep ( G, t )) = ( b G ) restricted to H ( G, t, k × ) hasthe form γ χ γ , χ γ ( g ) = γ ( t, g ) γ ( g, t ) . (6.6) In the split case, the homomorphism Q Rep ( G,t ) restricted to Z / Z is the isomorphism Z / Z → c h t i .Proof. Consider the homomorphism
P ic ( Rep ( G, t )) → P ic ( sVect ) induced by the restriction functor Rep ( G, t ) → sVect . We start by showing that this homomorphism is surjective if and only if Rep ( G, t )is split. Indeed, a splitting of the restriction functor
Rep ( G, t ) → sVect induces a splitting of the homo-morphism P ic ( Rep ( G, t )) → P ic ( sVect ). Conversely, an Azumaya algebra R ∈ Rep ( G, t ), which classis mapped to the class of I ⊕ Π ∈ sVect , has the form ( I ⊕ Π) ⊗ End k ( U ) for some vector space U . Inparticular its classical center (computed in Vect ) coincides with I ⊕ Π. The G -action descends from R toits center and gives a splitting G → Aut alg ( I ⊕ Π) = Z / Z .The restriction of the homomorphism Q Rep ( G,t ) to P ic ( sVect ) is described in Example 6.2.In the following we argue that the kernel of the homomorphism P ic ( Rep ( G, t )) → P ic ( sVect ) isisomorphic to H ( G, t, k × ). This kernel consists of classes of Azumaya algebras of the form End k ( V )for a projective G -representation V . It is straightforward to see (e.g., by computing the left center [9]in Rep ( G, t )) that
End k ( V ) is an Azumaya algebra in Rep ( G, t ) for any projective G -representation62 . Thus we have a set-theoretic bijection H ( G, t, k × ) → Ker ( P ic ( Rep ( G, t )) → P ic ( sVect )) sending γ ∈ Z ( G, k × ) to the (class of) End k ( V ), where V is a projective G -representation with the Schurmultiplier γ . To show that this is a group isomorphism we need a few facts.Let U and V be super vector spaces. Define a map φ : End k ( U ) ⊗ End k ( V ) → End k ( U ⊗ V ) : a ⊗ b φ a,b , (6.7)by φ a,b ( u ⊗ v ) = ( − | b || u | a ( u ) ⊗ b ( v ) for all homogeneous maps a ∈ End k ( U ) , b ∈ End k ( V ) and homoge-neous vectors u ∈ U, v ∈ V , where | a | (respectively, | u | ) denotes the degree of a (respectively, the degreeof u ).The map (6.7) is an isomorphism of algebras in sVect , i.e. φ a,b ◦ φ c,d = ( − | b || c | φ a ◦ c,b ◦ d . Indeed, φ a,b ( φ c,d ( u ⊗ v )) = ( − | d || u | φ a,b ( c ( u ) ⊗ d ( v ))= ( − | d || u | + | b || c ( u ) | a ( c ( u )) ⊗ b ( d ( v ))= ( − | d || u | + | b || c | + || b | u | ( a ◦ c )( u ) ⊗ ( b ◦ d )( v )= ( − | b || c | φ a ◦ c,b ◦ d ( u ⊗ v ) . Let now U be a projective G -representation with the Schur multiplier γ ∈ Z ( G, k × ). Let t ∈ G be a central involution. Denote by U = U ⊕ U the Z / Z -grading corresponding to t , i.e., u ∈ U is homogeneous of degree | u | iff t ( u ) = ( − | u | u . Then the G -action is related to the grading in thefollowing way | g.u | = ξ γ ( g ) + | u | , where ξ γ is defined in (6.3). Indeed, t. ( g.u ) = γ ( t, g )( tg ) .u = γ ( t, g )( gt ) .u = γ ( t, g ) γ ( g, t ) g. ( t.u ) = ( − | u | γ ( t, g ) γ ( g, t ) g.u. Now let U and V be projective G -representations with the Schur multipliers γ U , γ V ∈ Z ( G, k × ) corre-spondingly. Note that End k ( U ) is a G -algebra with g ( a ) defined by g ( a )( u ) = g. ( a ( g − .u )) (and similarlyfor End k ( V )). Then the homomorphism φ : End k ( U ) ⊗ End k ( V ) → End k ( U ⊗ V ) has the following G -equivariance property: φ g ( a ) ,g ( b ) = ρ ( g ) ◦ φ a,b ◦ ρ ( g ) − , where ρ ( g ) : U ⊗ V → U ⊗ V is given by ρ ( g )( u ⊗ v ) = ( − ξ γ ( g ) | v | g.u ⊗ g.v . Indeed, the relation63 g ( a ) ,g ( b ) ◦ ρ ( g ) = ρ ( g ) ◦ φ a,b can be checked directly: φ g ( a ) ,g ( b ) ( ρ ( g )( u ⊗ v )) = ( − ξ γ ( g ) | v | φ g ( a ) ,g ( b ) ( g.u ⊗ g.v )= ( − ξ γ ( g ) | v | + | g ( b ) || g.u | g ( a )( g.u ) ⊗ g ( b )( g.v )= ( − | b || u | + ξ γ ( g ) | b ( v ) | g. ( a ( u )) ⊗ g. ( b ( v ))= ( − | b || u | + ξ γ ( g ) | b ( v ) | g. ( a ( u )) ⊗ g. ( b ( v ))= ( − | b || u | ρ ( g )( a ( u ) ⊗ b ( v ))= ρ ( g )( φ a,b ( u ⊗ v )) . The map ρ : G → GL ( U ⊗ V ) is a projective representation with the Schur multiplier γ ( f, g ) = ( − ξ γ ( f ) ξ V ( g ) γ U ( f, g ) γ V ( f, g ) , f, g ∈ G. (6.8)To see this, we compute ρ ( f g )( u ⊗ v ) = ( − ξ γ ( fg ) | v | ( f g ) .u ⊗ ( f g ) .v = ( − ξ γ ( fg ) | v | γ U ( f, g ) γ V ( f, g ) f. ( g.u ) ⊗ f. ( g.v )= ( − ξ γ ( g ) | v | + ξ γ ( f ) | g.v | γ ( f, g ) f. ( g.u ) ⊗ f. ( g.v )= ( − ξ γ ( g ) | v | γ ( f, g ) ρ ( f )( g.u ⊗ g.v )= γ ( f, g ) ρ ( f )( ρ ( g )( u ⊗ v )) . In the rest of the proof we describe the restriction of the homomorphism Q Rep ( G,t ) to H ( G, t, k × ). Letagain V be a projective G -representation with the Schur multiplier γ ∈ Z ( G, k × ). The automorphism c End k ( V ) ,End k ( V ) of the algebra End k ( V ) ⊗ in Rep ( G, t ) c End k ( V ) ,End k ( V ) ( a ⊗ b ) = ( − | a || b | b ⊗ a, transported (along φ ) to an automorphism of the algebra End k ( V ⊗ ), is inner. More precisely, we have c V,V ◦ φ a,b ◦ c − V,V = ( − | a || b | φ b,a , since c V,V ( φ a,b ( u ⊗ v )) = ( − | b || u | c V,V ( a ( u ) ⊗ b ( v ))= ( − | b || u | + | a ( u ) || b ( v ) | b ( u ) ⊗ a ( v )= ( − | a || b | + | u || v | + | a || v | b ( u ) ⊗ a ( v )= ( − | a || b | + | u || v | φ b,a ( v ⊗ u )= ( − | a || b | φ b,a ( c V,V ( u ⊗ v )) . c V,V ∈ End k ( V ) ⊗ has the following G -equivariance property: g ( c V,V ) = χ γ ( g ) c V,V ,where χ γ is defined in (6.6). That is, ρ ( g ) ◦ c V,V ◦ ρ ( g ) − = χ ( g ) c V,V since we have ρ ( g )( c V,V ( u ⊗ v )) = ( − | u || v | ρ ( g )( v ⊗ u )= ( − | u || v | + ξ ( g ) | u | g.v ⊗ g.u = ( − ξ ( g ) + ξ ( g ) | v | + | g.u || g.v | g.v ⊗ g.u = ( − ξ ( g )+ ξ ( g ) | v | c V,V ( g.u ⊗ g.v )= χ γ ( g ) c V,V ( ρ ( g )( u ⊗ v )) . The formula for Q Rep ( G,t ) on H ( G, t, k × ) now follows from Remark 5.8. Proposition 6.5.
Let E be a symmetric tensor category. There is a group isomorphism π ( Pic br ( E )) ∼ = P ic ( E ) × Aut ⊗ ( id E ) , (6.9) such that the braiding between categories corresponding to ( M , ν ) and ( M , ν ) is given by the pairing h M , ν i defined in (5.42) . In particular, the first canonical class of Pic br ( E ) is Q E ( M , ν ) = h M , ν i . (6.10) Proof.
Since every E -module category admits an identity E -module braiding it follows that Pic br ( E ) isgenerated by symmetric categorical subgroups Pic ( E ) and Aut ⊗ (id E ) (the latter consists of invertiblecategories in Mod ( E )). These subgroups intersect trivially, so P ic br ( E ) is their direct product. Theresult follows from Example 4.15.The Whitehead bracket π × π → π (2.60) of Pic br ( E ) is given by( P ic ( E ) × Aut ⊗ (id E )) × Inv ( E ) → k × : [( M , x ) , Z ] = x | Z . (6.11)We can now describe the braided categorical Picard group of a Tannakian category. Corollary 6.6.
We have π ( Pic br ( Rep ( G ))) = H ( G, k × ) × Z ( G ) , (6.12) π ( Pic br ( Rep ( G ))) = b G, (6.13) where b G denotes the group of characters of G . he pairing (5.42) in this case is given by the alternator H ( G, k × ) × Z ( G ) → b G : ( γ, z ) Alt γ ( z, − ) := γ ( z, − ) γ ( − , z ) . (6.14) The first canonical class is the quadratic form Q Pic br ( Rep ( G )) ( µ, z ) = Alt γ ( z, − ) , z ∈ Z ( G ) , γ ∈ H ( G, k × ) . (6.15) The second canonical class is trivial.The Whitehead bracket (6.11) is given by [( γ, z ) , χ ] = χ ( z ) , χ ∈ b G, z ∈ Z ( G ) . (6.16) Proof.
Follows from Proposition 6.1 and Theorem 6.5.
Now we deal with the super-Tannakian case. We start with the basic example of the symmetric fusioncategory sVect of super vector spaces. As before, Π denotes the non-identity simple object of sVect . Example 6.7.
The group π ( Pic br ( sVect ) ≃ P ic br ( sVect ) × Aut ⊗ (id sVect ) ∼ = Z / Z × Z / Z consists ofpairs ( I , id ) , ( I , π ) , ( R , id ) , ( R , π ), where I is the regular sVect -module category, R = Vect viewed asan sVect -module category (i.e., R = Mod sVect ( R ), where R is the algebra from Example 6.2), π is thenatural automorphism of the identity functor of sVect such that π = id and π Π = − id Π . The quadraticfunction Q sVect : P ic br ( sVect ) → Inv ( sVect ) is given by Q sVect ( I , id ) = Q sVect ( I , π ) = Q sVect ( R , π ) = I , Q sVect ( R , id ) = Π . Corollary 6.8. π ( Pic br ( Rep ( G, t ))) ∼ = H ( G, t, k × ) × Z ( G ) in the non-split case ,H ( G, t, k × ) × Z / Z × Z ( G ) in the split case . (6.17) π ( Pic br ( Rep ( G, t ))) ∼ = b G. (6.18) The first canonical class of
Pic br ( Rep ( G, t )) is given by the quadratic form Q Rep ( G,t )) ( γ, z )( x ) = γ ( z, x ) γ ( x, z ) in the non-split case , (6.19) Q Rep ( G,t )) ( γ, ε, z )( x ) = γ ( z, x ) γ ( x, z ) ρ t ( x ) ε in the split case , (6.20) for all γ ∈ H ( G, k × ) , z ∈ Z ( G ) , and x ∈ G , where ρ t is the composition G → h t i → k × of the non-trivialcharacter on h t i with a (chosen) splitting G → h t i . he second canonical class is the quadratic homomorphism b G → {± } : Z Z ( t ) . (6.21) The Whitehead bracket is given by (6.16) (it does not depend on t ).Proof. Follows from Theorems 6.4 and 6.5.
Recall [25, 18] that a pointed braided fusion category B is determined by a quadratic form q : A → k × ,where A is the finite Abelian group of isomorphisms classes of simple objects of B and q ( x ) = c x,x , x ∈ A ,where c denotes the braiding of B . Proposition 7.1.
Let B be a pointed braided fusion category. There is an equivalence Pic br ( B ) ∼ = Pic br ( Z sym ( B )) (7.1) of braided categorical groups.Proof. The group
P ic br ( B ) can be computed using the exact sequence (5.25). Namely, we have a shortexact sequence0 → Coker(
Inv ( B ) α −→ Aut ⊗ (id B )) → P ic br ( B ) → Ker(
P ic ( B ) ∂ −→ Aut br ( B )) → . (7.2)By [10, Proposition 5.17], the inductionInd : P ic ( Z sym ( B )) → P ic ( B ) : M B ⊠ Z sym ( B ) M establishes a group isomorphism P ic ( Z sym ( B )) ∼ = Ker( ∂ ). Note that (4.20) provides a splitting for (7.2).The homomorphism α is the map A b A coming from the bilinear form on A associated to q . Its cokernelis c A ⊥ ∼ = Aut ⊗ (id Z sym ( B ) ).Thus, the split exact sequence (7.2) yields a group isomorphism (7.1) by Proposition 6.5: P ic br ( Z sym ( B )) ∼ = P ic ( Z sym ( B )) × Aut ⊗ (id Z sym ( B ) ) → P ic br ( B ) : ( M , ν ) Ind( M ) ν , cf. Example 5.6. The value of the quadratic form q Pic br ( B ) on Ind( M ) x is given by the half braidingInd( M ) ⊠ B B ν → B ν ⊠ B Ind( M ) . The latter coincides with the value of the pairing h M , x | Z sym ( B ) i (5.42), see Remark 5.8. So the resultfollows from Proposition 6.5. 67ote that τ := q | A ⊥ is an element of c A ⊥ of order at most 2. Corollary 7.2.
P ic br ( C ( A, q )) ∼ = Hom ( Λ b A, k × ) × c A ⊥ if τ = 1 , Hom ( Λ b A, k × ) × c A ⊥ if τ = 1 and A = Ker ( τ ) × Z / Z , Hom ( Λ b A, k × ) × Z / Z × c A ⊥ if τ = 1 and A = Ker ( τ ) × Z / Z . (7.3) Proof.
This follows from Proposition 7.1 and the description of the Picard group of a symmetric fusioncategory, see Sections 6.1 and 6.2.
Let D be a tensor category and let G be a finite group. Definition 8.1. A tensor G -graded extension (or, simply, a G -extension ) of a tensor category D is atensor category C = M x ∈ G C x , C e = D , (8.1)such that C x = 0 and the tensor product of C maps C x × C y to C xy for all x, y ∈ G . Definition 8.2. An equivalence between two G -graded extensions, C = L x ∈ G C x and ˜ C = L x ∈ G ˜ C x of D is a tensor equivalence F : C ∼ −→ ˜ C such that F | D = id D and F ( C x ) = ˜ C x for all x ∈ G . An isomorphism between equivalences of G -extensions F, F ′ : C ∼ −→ ˜ C is a tensor isomorphism η : F ∼ −→ F ′ whose restrictionon F | D = id D is the identity isomorphism.Thus, G -extensions of D form a 2-groupoid Ex ( G, D ) whose objects are extensions, 1-cells are equiv-alences of extensions, and 2-cells are isomorphisms of equivalences. Example 8.3. G -extensions of Vect are precisely pointed fusion categories
Vect ωG , where ω ∈ Z ( G, k × )is a 3-cocyle. Equivalences between extensions Vect ωG and Vect ˜ ωG correspond to 2-cochains µ ∈ C ( G, k × )such that d ( µ ) = ˜ ω/ω . Thus, π ( Ex ( G, Vect )) = H ( G, k × ). Remark 8.4.
Example 8.3 shows that there exist equivalent tensor categories that are not equivalent asextensions. Indeed, if the cohomology classes of ω and ˜ ω are in the same Aut ( G )-orbit then Vect ωG ∼ = Vect ˜ ωG as tensor categories. 68he following theorem is essentially proved in [22]. We include its proof for the reader’s convenience.Our arguments for central, braided, and symmetric extensions in subsequent sections will follow the samepattern. Theorem 8.5.
There is an equivalence of -groupoids Ex ( G, D ) ∼ = - Fun ( G, BrPic ( D )) .Proof. We construct a 2-functor M : Ex ( G, D ) → - Fun ( G, BrPic ( D )) (8.2)as follows. Given a G -extension C = L x ∈ G C x of D , each homogeneous component C g is an invertible D -bimodule category. The restrictions ⊗ x,y : C x × C y → C xy , x, y ∈ G, of the tensor product of C are D -balanced functors and so give rise to D -bimodule equivalences M x,y : C x ⊠ D C y ∼ −→ C xy . (8.3)The associativity constraints of C restricted to C x × C y × C z can be viewed as natural isomorphisms of D -balanced functors and so give rise to natural isomorphisms of D -bimodule functors C x ⊠ D C y ⊠ D C z M y,z / / M x,y (cid:15) (cid:15) C x ⊠ D C yzM x,yz (cid:15) (cid:15) C xy ⊠ D C z M xy,z / / C xyz , α x,y,z + (8.4)for all x, y, z ∈ G , cf. (2.35). The pentagon identity for the associativity constraints of C implies that (2.37)is satisfied (equivalently, the cubes (2.63) commute for all x, y, z, w ∈ G ). This means that the above dataconsisting of D -bimodule categories C x , equivalences M x,y , and natural isomorphisms α x,y,z , x, y, z ∈ G, determine a monoidal 2-functor M ( C ) : G → BrPic ( D ).Suppose that there is another G -extension ˜ C = L g ∈ G ˜ C g of D and an equivalence of extensions F : C → ˜ C . It restricts to D -bimodule equivalences F x : C x ∼ −→ ˜ C x . (8.5)The tensor structure of F restricted to C x × C y gives rise to an invertible 2-cell C x ⊠ D C y F x ⊠ D F y / / M x,y (cid:15) (cid:15) ˜ C x ⊠ D ˜ C y ˜ M x,y (cid:15) (cid:15) C xy F xy / / ˜ C xy , µ x,y + (8.6) x, y ∈ G , and the coherence axiom for the tensor structure of F implies that (2.44) is satisfied (equivalently,that the cubes (2.66) commute for all x, y, z ∈ G ), so that we have a pseudo-natural isomorphism M ( F ) : M ( C ) → M (˜ C ). 69iven an isomorphism η between a pair of equivalences F, F ′ of extensions C and C ′ its componentsare natural isomorphisms of D -bimodule functors: C x F x * * F ′ x ˜ C x . η x (cid:11) (cid:19) (8.7)The tensor property of η implies that (2.46) is satisfied, i.e., the cylinder C x ⊠ D C y F x F y , , F ′ x F ′ y M x,y (cid:15) (cid:15) ˜ C x ⊠ D ˜ C y ˜ M x,y (cid:15) (cid:15) C xy F gh + + ❢ ❡ ❞ ❝ ❛ ❵ ❴ ❫ ❪ ❬ ❩ ❨ ❳ F ′ xy ˜ C gh . η g η h (cid:11) (cid:19) η gh (cid:11) (cid:19) µ g,h (cid:11) (cid:19) ✤✤✤✤✤✤✤✤✤✤ µ ′ g,h (cid:11) (cid:19) ✣✣✣✣✣✣✣✣✣✣✣✣✣✣✣✣✣✣ (8.8)commutes for all x, y ∈ G . So we get an invertible modification M ( η ) between pseudo-natural isomor-phisms M ( F ) and M ( F ′ ). This completes the construction of a monoidal 2-functor (8.2).A 2-functor L : - Fun ( G, BrPic ( D )) → Ex ( G, D ) (8.9)quasi-inverse to (8.2) can be constructed by reversing the above constructions. Namely, let C : G → BrPic ( D ) : x C x be a monoidal 2-functor. Form a D -bimodule category L ( C ) := L x ∈ G C x withthe tensor product given by composing C x × C y → C x ⊠ D C y with 1-cells (8.3) and the associativityconstraints coming from 2-cells (8.4). The commuting polytopes (2.63) give the pentagon identity for theassociativity constraint.To check that L ( C ) is rigid, note that by Corollary 5.2 it is exact as a D -module category. Hence,the dual category End D ( L ( C )) is a tensor category (i.e., is rigid). Given a homogeneous object X in C x ⊂ L ( C ) , x ∈ G, define a D -module endofuctor L ( X ) ∈ End D ( L ( C )) by setting L ( X ) = X ⊗ − on D and L ( X ) = 0 on C g , g = 0. Its adjoints are given by functors X ∗ ⊗ − , ∗ X ⊗ − : C x → D for someobjects X ∗ , ∗ X ∈ C x − . These objects are the duals of X . Thus, L ( C ) is a tensor category and so it is a G -extension of D .From the universal property of ⊠ D , a pseudo-natural isomorphism of functors C, C ′ : G → BrPic ( D )gives an equivalence of extensions with the tensor structure coming from (8.6) and that a modificationof pseudo-natural isomorphisms gives a natural isomorphism of equivalences of extensions. Remark 8.6.
The proof of Theorem 8.5 is based on the correspondences (coming from the universalproperty of ⊠ D ) between the structure functors and morphisms of graded tensor categories and the70xioms they satisfy and the structure 1- and 2-cells of monoidal 2-functors and the commutative polytopessatisfied by them. We summarize these correspondences in Table 3 (cf. the table from [3, Section 2.3]).Tensor G -extensions C of D Monoidal 2-functors M : G → BrPic ( D )homogeneous components C g M ( g ) := C g tensor products C g × C h → C gh monoidal 1-cells M g,h : C g ⊠ D C h → C gh associativity constraints a X,Y,Z associativity 2-cells α f,g,h (8.4)commuting pentagon diagram for a commuting cubes (2.63) for α equivalence F : C → ˜ C of extensions 1-cells F g : C g → ˜ C g tensor structure of F monoidal 2-cells µ g,h (8.6)commuting tensor property diagram for F commuting cubes (2.66) for µ isomorphism η : F → F ′ of equivalences modification 2-cells η g : F g → F ′ g (8.7)commuting tensor property diagram for η commuting cylinders (8.8) for η Table 3: A correspondence between tensor G -extensions and monoidal 2-functors.We can describe G -graded extensions of D in terms of group cohomology. It follows from constructionsof Section 2.5 that given a monoidal functor M : G → BrPic ( D ) there exist a canonical cohomology class p M ∈ H ( G, k × ) and a canonical group homomorphism p M : H ( G, Inv ( Z ( D ))) → H ( G, k × )defined in (2.63) and (2.66), respectively. Corollary 8.7.
A monoidal functor M : G → BrPic ( D ) gives rise to a G -graded extension of D if andonly if p M = 0 in H ( G, k × ) . Equivalence classes of such extensions form a torsor over the cokernelof p M .Proof. This follows from Theorem 8.5 and Corollary 2.25.
Remark 8.8.
In [22] the notion of an equivalence of graded extensions was not explicitly defined andextensions were parameterized by a torsor over H ( G, k × ). We would like to point that the map p M isnon-trivial in general. Here is a simple example. Let D = Vect ω Z / Z , where ω is the non-trivial element of H ( Z / Z , k × ). Then Inv ( Z ( D )) ∼ = Z / Z × Z / Z with the asscoaitor ω × ω − . Take the trivial monoidalfunctor M : Z / Z → BrPic ( D ). The homomorphism p M is given by (2.69), i.e., p M : Hom ( Z / Z , Z / Z × Z / Z ) → H ( Z / Z , k × ) : P ( ω × ω − ) ◦ ( P × P × P ) , Let B be a braided tensor category. Definition 8.9. A central G -extension of B is a pair ( C , ι ), where C is a G -extension and ι : B ֒ → Z ( C )is a braided tensor functor whose composition with the forgetful functor Z ( C ) → C coincides with theinclusion B ֒ → C . Definition 8.10.
Let ( C , ι : B → Z ( C )) and (˜ C , ˜ ι : B → Z (˜ C )) be two central G -extensions of B . An equivalence between these extensions is an equivalence F : C ∼ −→ ˜ C of G -extensions such that ˜ ι = ind( F ) ◦ ι ,where ind( F ) : Z ( C ) ∼ −→ Z (˜ C ) is the braided equivalence induced by F .Central G -extensions of B form a 2-groupoid Ex ctr ( G, B ).Recall that a G -crossed braided tensor category is a G -graded tensor category C = L x ∈ G C x equippedwith the action of G on C , i.e., a monoidal functor G → Aut ⊗ ( C ), such that x ( C y ) = C xyx − and with a G -crossed braiding c X,Y : X ⊗ Y → g ( Y ) ⊗ X, X ∈ C x , Y ∈ C , (8.10)satisfying certain natural axioms. Note that the trivial component of the grading C e is a braided tensorcategory. Let Ex cr − br ( G, B ) denote the 2-groupoid of G -crossed braided fusion categories whose trivialcomponent is B .The next Proposition was essentially proved in [24]. It shows that the notions of a central extensionand a G -crossed extension coincide. Proposition 8.11.
There is a -equivalence Ex cr − br ( G, B ) ∼ = Ex ctr ( G, B ) .Proof. We need to explain how a G -crossed braided structure translates into a central functor and viceversa.Let C be a G -crossed braided tensor category with C e = B . The restriction of the crossed braiding(8.10) provides X ∈ C e with the structure of a central object of C and ι : B = C e → Z ( C ) : X ( X, c − X, − )is a braided tensor functor whose composition with teh forgetful functor Z ( C ) → C is identity.In the opposite direction, a central G -extension ( C , ι : B → Z ( C )) yields a natural isomorphism c X,Z : X ⊗ Z ∼ −→ Z ⊗ X, X ∈ B , Z ∈ C , (8.11)72atisfying the hexagon axioms. This turns each C x into an invertible B -module category. Furthermore,there are B -module equivalences C y → Fun B ( C x , C xy ) : Y
7→ − ⊗ Y, C xyx − → Fun B ( C x , C xy ) : Y Y ⊗ − , for all x, y ∈ G . Here the functor categories consist of right exact B -module functors. Combining theseequivalences for a fixed x ∈ G we obtain a tensor autoequivalence x ∈ Aut ⊗ ( C ) such that x ( C y ) = C xyx − and there is a natural isomorphism x ( Y ) ⊗ X ∼ = X ⊗ Y for all X ∈ C x , Y ∈ C . The latter is a crossed braiding on C .These constructions are inverses of each other and are compatible with equivalences of G -crossedbraided and central extensions, i.e., define a 2-equivalence between the corresponding 2-groupoids. Remark 8.12.
Let C be a central G -extension of B . The braided tensor category C G obtained from C as the equivariantization with respect to the canonical action of G constructed in the proof of Proposi-tion 8.11 coincides with the centralizer of the image of B in Z ( C ).Recall that the Picard group of B was introduced in Section 5.2. The following result is essentially aconsequence of Proposition 8.11 and [22, Theorem 7.12]. We include the proof for the sake of completeness. Theorem 8.13.
There is an equivalence of -groupods Ex ctr ( G, B ) ∼ = - Fun ( G, Pic ( B )) .Proof. We adjust the proof of Theorem 8.5 to the present setting (with D -bimodule categories, functors,and isomorphisms replaced by B -module ones).A central structure on a G -extension C = L x ∈ G C x of B consists of isomorphisms (8.11) that turnevery component C x into an invertible B -module category, i.e., C x belongs to Pic ( B ). Equivalences(8.3) coming from tensor products C x × C y → C xy are B -module equivalences in this case and naturalisomorphisms (8.4) are isomorphisms of B -module functors.An equivalence of central G -extensions of B yields B -module equivalences (8.5) between homogeneouscomponents and isomorphisms (8.6) of B -module functors. An isomorphism between equivalences ofcentral G -extensions yields an isomorphism (8.7) of B -module functors.Diagrams (2.63), (2.66), and (8.8) commute for the same reason as in the proof of Theorem 8.5.Thus, (8.2) becomes a 2-functor Ex ctr ( G, B ) → - Fun ( G, Pic ( B )) . (8.12)73onversely, given a monoidal 2-functor G → Pic ( B ), consider its composition with the inclusion Pic ( B ) → BrPic ( B ). By Theorem 8.5, this yields a G -extension C of B . The B -bimodule structure of C comes from its left B -module structure, so there is a natural isomorphism between the functors of leftand right tensor multiplication by X ∈ B : c X,Z : X ⊗ Z ∼ −→ Z ⊗ X, Z ∈ C . (8.13)The hexagon for (8.13) follow from the above definition of a B -bimodule category structure of C and fromthe monoidal property of the 2-functor Mod ( B ) → Bimod ( B ). Thus, (8.13) is a central structure onthe G -extension C of B and there is a 2-functor - Fun ( G, Pic ( B )) → Ex ctr ( G, B ) (8.14)quasi-inverse to (8.12). Remark 8.14.
It follows from Theorem 8.13 that central G -extensions of B can be described in termsof monoidal functors G → Pic ( B ) and group cohomology analogously to Corollary 8.20. Let B be a braided tensor category and let A be an Abelian group. Definition 8.15. A braided A -extension of B is a braided tensor category C that is an A -extension of B . Definition 8.16.
An equivalence between braided A -extensions C , ˜ C of B is an equivalence of A -extensions that is a braided functor. Example 8.17.
Braided A -extensions of Vect are precisely pointed braided fusion categories
Vect ωA , where ω ∈ Z br ( G, k × ) is an abelian 3-cocycle. Equivalences between extensions Vect ωA and Vect ˜ ωA correspondto abelian 2-cochains µ ∈ C br ( A, k × ) such that d ( µ ) = ω/ ˜ ω . Thus, the set of isomorphism classes Ex br ( A, Vect ) of braided A -extensions of Vect is in bijection with H br ( A, k × ) ∼ = Quad ( A, k × ).Let - Fun br ( A, Pic br ( B )) denote the 2-groupoid of braided monoidal 2-functors from A to Pic br ( B ). Theorem 8.18.
There is a -equivalence Ex br ( A, B ) ∼ −→ - Fun br ( A, Pic br ( B )) .Proof. Let C = L x ∈ A C x be a braided A -extension of B . Each homogeneous component C y , y ∈ A, is aninvertible B -module category. The squared braiding σ X,Y = c Y X c XY , X ∈ B , Y ∈ C y , B -module category, i.e., C y ∈ Pic br ( B ). The equivalences M x,y : C x ⊠ B C y ∼ −→ C xy from (8.3) are braided module equivalences. Indeed, commutativity of thediagram (4.2) is a consequence of the identity c Y ⊗ Y ,X c X,Y ⊗ Y = ( c Y ,X ⊗ id Y )(id Y ⊗ c Y ,X c X,Y )( c X,Y ⊗ id Y ) , X, Y , Y ∈ B , where we omit the associativity constraints in C .As in the proof of Theorem 8.5, equivalences M x,y along with the associativity 2-cells α x,y,z from (8.4)define a monoidal structure on the 2-functor M ( C ) : A → Pic br ( B ) : x C x . Furthermore, the commutativity constraint of C gives rise to invertible 2-cells C x ⊠ B C y B x,y / / M x,y ( ( PPPPPPPPPPPPP C y ⊠ B C xM y,x v v ♥♥♥♥♥♥♥♥♥♥♥♥♥ C xyδ x,y K S (8.15)for all x, y ∈ A . The conditions (2.40) and (2.41) in the definition of a braided monoidal 2-functor(i.e., commutativity of the octahedra (2.71) and (2.72)) follow from the hexagon axioms satisfied by thebraiding of C .Thus, M ( C ) : A → Pic br ( B ) is a braided monoidal 2-functor.Suppose there is another braided A -extension ˜ C = L x ∈ A ˜ C x of B and an equivalence of braided A -extensions F : C → ˜ C . The B -module equivalences F x : C x ∼ −→ ˜ C x between the homogeneous componentsare braided B -module equivalences. Indeed, commutativity of diagram (4.2) is a consequence of thebraided property of F . We have invertible 2-cells (8.6) satisfying (2.63) as in the proof of Theorem 8.5.The condition (2.45) (i.e., commutativity of the prism (2.73)) follows from the braided property of F .Thus, we have a pseudo-natural isomorphism M ( F ) : M ( C ) → M (˜ C ) of braided monoidal 2-functors.Given an isomorphism η between a pair of equivalences F, F ′ of braided extensions C and C ′ oneconstructs an invertible modification M ( η ) between M ( F ) and M ( F ′ ) as in (8.7).Thus, we have a 2-functor M : Ex br ( A, B ) ∼ −→ - Fun br ( A, Pic br ( B )). In the opposite direction,the 2-fumctor (8.9) constructed in the proof of Theorem 8.5 carries a braided structure on a 2-functor C : A → Pic br ( B )) to a braiding on L ( C ) = L x ∈ A C x . Namely, 2-cells (8.15) give rise to the braidingconstraints for L ( C ) while commuting octahedra (2.71), (2.72) ensure that they satisfy give the hexagonidentities. Remark 8.19.
The proof of Theorem 8.18 extends that of Theorem 8.5. So it extends the correspon-dences in Table 3 as follows: 75raided tensor A -extensions C of B Monoidal 2-functors M : A → Pic br ( B )braiding constraints c X,Y braiding 2-cells δ x,y (8.15)commuting hexagon diagrams for c commuting octahedra (2.71), (2.72)braided property diagram for F commuting prism (2.73)Table 4: A correspondence between braided extensions and braided monoidal 2-functors.We can describe A -graded extensions of B in terms of braided group cohomology. It follows fromconstructions of Section 2.6 that given a braided monoidal functor M : A → BrPic ( D ) there exista canonical braided cohomology class p M ∈ H br ( A, k × ) and a canonical group homomorphism p M : H ( A, Inv ( B )) → H br ( A, k × ). Corollary 8.20.
A braided monoidal functor M : A → Pic br ( B ) gives rise to an A -graded extension of B if and only if p M = 0 in H br ( A, k × ) . Equivalence classes of such extensions of form a torsor over thecokernel of p M .Proof. This follows from Theorem 8.18 and Corollary 2.30.
Example 8.21.
Let B be a non-degenerate braided fusion category. By Proposition 4.16, Pic br ( B ) = Pic br ( Vect ) and so Ex br ( A, B ) ∼ −→ Ex br ( A, Vect ). Thus, any braided A -extension of B is equivalent toone of the form B ⊠ C ( A, q ) for some q ∈ Quad ( A, k × ) = H br ( A, k × ).Thus, the only braided fusion categories that admit interesting extensions are degenerate ones. Let E be a symmetric tensor category and let A be an Abelian group. Definition 8.22. A symmetric A -extension of E is a symmetric tensor category C that is an A -extensionof E .Equivalences of symmetric A -extensions are the same as for braided A -extensions. The 2-groupoid Ex sym ( A, E ) of symmetric A -extensions of E is a 2-subgroupoid of Ex br ( A, E ). Example 8.23.
Symmetric A -extensions of Vect are precisely pointed braided fusion categories
Vect ωA ,where ω ∈ Z sym ( G, k × ) is a symmetric 3-cocycle. Equivalences between extensions Vect ωA and Vect ˜ ωA correspond to symmetric 2-cochains µ ∈ C sym ( A, k × ) such that d ( µ ) = ω/ ˜ ω . Thus, the set of iso-morphism classes Ex sym ( A, Vect ) of braided A -extensions of Vect is in bijection with H sym ( A, k × ) ∼ = Hom ( A, k × ) = Hom ( A, Z ). 76et C = L x ∈ A C x and C ′ = L x ∈ A C ′ x be symmetric A -extensions of E . Then C ⊠ E C ′ is an ( A × A )-extension of E . Define the tensor product of these extensions to be the diagonal subcategory of C ⊠ E C ′ : C ⊙ E C ′ = M x ∈ A C x ⊠ E C ′ x . (8.16)This equips the 2-groupoid Ex sym ( A, E ) of symmetric A -extensions of E with a structure of a symmetric2-categorical group.Recall that the symmetric 2-categorical group Pic sym ( E ) of symmetric E -module categories is equi-valent to Pic ( E ), the Picard group of E . Let - Fun sym ( A, Pic ( E )) denote the 2-groupoid of symmetricmonoidal 2-functors from A to Pic ( E ). Theorem 8.24.
There is a symmetric monoidal -equivalence Ex sym ( A, E ) ∼ −→ - Fun sym ( A, Pic ( E )) .Proof. We extend Theorem 8.18 to the symmetric setting. Observe that the homogeneous components ofa symmetric extension C = ⊕ x ∈ A C x of E are necessarily symmetric E -module categories. Commutativityof the cones (2.75) is equivalent to the squared braiding of C being identity, i.e., to the braided monoidal2-functor x C x being symmetric.The monoidal structure of this 2-equivalence is established by comparing the tensor products, asso-ciativities, and braidings of Ex sym ( A, E ) and - Fun sym ( A, Pic ( E )). Corollary 8.25.
There is an exact sequence of group homomorphisms: H ( A, Inv ( E )) → H sym ( A, k × ) → π ( Ex sym ( A, E )) → π ( Fun sym ( A, Pic ( E ))) → H sym ( A, k × ) . (8.17) Proof.
This follows from Theorem 2.36.
Let G be a finite abelian group and let t ∈ G be a central element such that t = 1. Let A be a finiteAbelian group. In this Section we compute the group Ex sym ( A, E ) := π ( Ex sym ( A, E ))of symmetric A -extensions of E = Rep ( G, t ). Theorem 8.26.
There are group isomorphisms Ex sym ( A, Rep ( G )) ∼ = H ( G, b A ) ⊕ H ( A, Z ) , (8.18) Ex sym ( A, Rep ( G, t )) ∼ = H ( G/ h t i , b A ) if t = 1 . (8.19)77 roof. Let us first consider symmetric A -extensions of Rep ( G ). They are of the form Rep ( ˜ G, ˜ t ), where˜ G is a central extension 1 → b A → ˜ G π −→ G → t is a central element of ˜ G such that ˜ t = 1 and π (˜ t ) = 1. Thus, every symmetric A -extension of Rep ( G ) is completely determined by the pair consisting of a cohomology class in H ( G, b A ) correspondingto the isomorphism class of the group extension (8.27) and ˜ t ∈ ( b A ) = H ( A, Z ). The correspondingmap Ex sym ( A, Rep ( G )) → H ( G, b A ) ⊕ H ( A, Z ) is a group isomorphism. It is clearly injective. To seethat it is surjective note that the elements of H ( G, b A ) form a subgroup Ex T an ( A, Rep ( G )) of Tannakian A -extensions of Rep ( G ) while the elements of H ( A, Z ) form the subgroup of split extensions.Now consider a symmetric A -extension C of Rep ( G, t ) with t = 1. It contains a unique maximalTannakian subcategory C of index 2 which is a Tannakian A -extension of Rep ( G/ h t i ). We have a grouphomomorphism f : Ex sym ( A, Rep ( G, t )) → Ex T an ( A, Rep ( G/ h t i )) = H ( G/ h t i , b A ) : C C . (8.21)We claim that f has an inverse given by the induction g : Ex T an ( A, Rep ( G/ h t i )) → Ex sym ( A, Rep ( G, t )) : T Rep ( G ) ⊠ Rep ( G/ h t i ) T , (8.22)where the tensor product of fusion categories over a symmetric fusion category is defined in [11, Section2.5].Indeed, we have f ◦ g = id since the maximal Tannakian subcategory of Rep ( G ) ⊠ Rep ( G/ h t i ) T is Rep ( G/ h t i ) ⊠ Rep ( G/ h t i ) T ∼ = T . To check that g ◦ f = id we observe that there is a surjective symmetrictensor functor F : C ⊠ Rep ( G, t ) → C given by embedding of factors. Since the intersection of C and Rep ( G, t ) in C is Rep ( G/ h t i ) we see that F factors through C ⊠ Rep ( G/ h t i ) Rep ( G, t ). The latter fusioncategory has the same Frobenius-Perron dimension as C so that C ∼ = C ⊠ Rep ( G/ h t i ) Rep ( G, t ). Corollary 8.27. Ex sym ( A, sVect ) = 0 . Below we describe the exact sequence (8.17) computing the group of symmetric extensions. This ismeant to illustrate our obstruction theory and give an alternative proof of Theorem 8.26.
Proposition 8.28.
There are group isomorphisms π ( Fun sym ( A, Pic sym ( Rep ( G )))) ∼ = H ( G, b A ) , (8.23) π ( Fun sym ( A, Pic sym ( Rep ( G, t )))) ∼ = Ker (cid:16) H ( G, b A ) Ξ t −→ Hom ( G/ h t i , ( b A ) ) (cid:17) , (8.24) where Ξ t : H ( G, b A ) → Hom ( G/ h t i , ( b A ) ) : m m ( t, − ) m ( − , t ) − . (8.25)78 roof. Let us consider the Tannakian case first. Let G ′ = [ G, G ] and b G = Hom ( G, k × ). We have ahomomorphism of short exact sequences0 / / Ext ( G/G ′ , b A ) / / α (cid:15) (cid:15) H ( G, b A ) / / β (cid:15) (cid:15) Hom ( A, H ( G, k × )) / / / / H sym ( A, b G ) / / π ( Fun sym ( A, Pic sym ( Rep ( G )))) / / Hom ( A, H ( G, k × )) / / . (8.26)Here α is the duality isomorphism. The homomorphism β is defined as follows. An element m ∈ H ( G, b A )gives rise to a central group extension 1 → b A → ˜ G π −→ G → . (8.27)The category Rep ( ˜ G ) is a symmetric A -extension of Rep ( G ) and, therefore, yields a symmetric monoidalfunctor α ( m ) : A → Pic sym ( Rep ( G )). The first row of (8.26) is split exact [28, Theorem 2.1.19] andthe second row comes from assigning to a symmetric functor a group homomorphism. Hence, β is anisomorphism. This proves (8.23).In the super-Tannakian case we have an exact sequence0 → H sym ( A, b G ) → F un sym ( A, Pic sym ( Rep ( G, t ))) → Hom ( A, H ( G, k × )) q ∗ −→ Hom ( A, ( b G ) ) , (8.28)where q ∗ is induced by the second canonical class of Pic sym ( Rep ( G, t )), q : P ic sym ( Rep ( G, t )) = H ( G, k × ) → Inv ( Rep ( G, t )) = ( b G ) : µ µ ( t, − ) µ ( − , t )see Theorem 6.4. Combining (8.28) with the commuting square H ( G, b A ) Ξ t / / (cid:15) (cid:15) Hom ( A, ( b G ) ) (cid:15) (cid:15) Hom ( A, H ( G, k × )) q ∗ / / Hom ( G, ( b A ) ) (8.29)we obtain (8.24).Recall that isomorphism (2.15) identifies H sym ( A, k × ) with Hom ( A , k × ) = d ( A ). Combining thiswith the isomorphisms H ( Z , b A ) ∼ = b A/ ( b A ) ∼ = d ( A ) we obtain H sym ( A, k × ) ∼ = H ( Z , b A ) . (8.30) Proposition 8.29.
The obstruction homomorphism π ( Fun sym ( A, Pic sym ( E ))) → H sym ( A, k × ) in (8.17) is given by F un sym ( A, Pic sym ( E )) ∼ = Ker (Ξ t ) ֒ → H ( G, b A ) res −−→ H ( h t i , b A ) ∼ = H sym ( A, k × ) , (8.31)79 here the first isomorphism is (8.24) , the last one is (8.30) , Ξ t is defined in (8.25) , and r es is therestriction map in cohomology.Proof. For a symmetric monoidal functor F : A → Pic sym ( E )) let a = a ( F ) ∈ H sym ( A, k × ) be theobstruction to lifting it to a symmetric monoidal 2-functor. The isomorphism (2.15) expresses a as anelement of d ( A ) in terms of its components a ( x, x, x, x ), a ( x, x | x ), and a ( x | x, x ), x ∈ A . We have a ( x, x | x ) = a ( x | x, x ) = 1 while the value of a ( x, x, x, x ) is found as follows. Let us view the E -moduleequivalence M x,x : F ( x ) ⊠ E F ( x ) ∼ −→ E coming from the monoidal functor structure of F as an elementof Inv ( E ) = b G . Then a ( x, x, x, x ) is equal to the value of the self-braiding of M x,x , i.e., to the evaluation M x,x ( t ). Note that the map A → b G : x M x,x | h t i is a homomorphism, since M x,x M y,y = M xy,xy M x,y M y,x , x, y ∈ A , and M x,y M y,x | h t i = M xy | h t i = 1.By (2.15), a is identified with the homomorphism A → k × : x M x,x ( t ) . (8.32)That this map coincides with the restriction map Ker(Ξ t ) ֒ → H ( G, b A ) Res −−→ H ( h t i , b A ) follows fromcommutativity of the following diagram H ( G, b A ) s / / Res (cid:15) (cid:15) H sym ( G/ [ G, G ] , b A ) ∼ / / H sym ( A, b G ) (cid:15) (cid:15) H ( h t i , b A ) ∼ / / H sym ( A, c h t i ) , (8.33)where s denotes a splitting of the first row of (8.26).Thus, for t = 1 the exact sequence (8.17) gives rise to a split short exact sequence0 → Hom ( A, Z ) → Ex sym ( A, Rep ( G )) → H ( G, b A ) → , (8.34)while for t = 1 it becomes H ( G, b A ) Res −−→ H ( h t i , b A ) → Ex sym ( A, Rep ( G, t )) → Ker (cid:16) H ( G, b A ) Ξ t −→ H ( G/ h t i , H ( h t i , b A )) (cid:17) Res −−→ H ( h t i , b A ) . (8.35)The isomorphism Ex sym ( A, Rep ( G, t )) ∼ = H ( G/ h t i , b A ) from (8.19) can be recovered by comparing thesequence (8.35) with the exact sequence coming from the Lyndon-Hochschild-Serre spectral sequence8013, 31]: H ( G, b A ) Res −−→ H ( h t i , b A ) → H ( G/ h t i , b A ) Inf −−→
Ker (cid:16) H ( G, b A ) Res −−→ H ( h t i , b A ) (cid:17) Ξ t −→ H ( G/ h t i , H ( h t i , b A )) . (8.36) Let B be a braided tensor category and let A be an Abelian group. Fix a homomorphism f : A → P ic br ( B ) : x C x (8.37)that extends to a braided monoidal 2-functor A → Pic br ( B ). That is, there is a braided extension C = M x ∈ A C x . Let c denote the braiding of C .Let Ex fbr ( A, B ) ⊂ Ex br ( A, B ) be the 2-subgroupoid of extensions corresponding to f . Our goal hereis to describe π ( Ex fbr ( A, B )).An extension of (8.37) to a braided monoidal functor A → Pic br ( B ) amounts to choosing B -equivalences C x ⊠ B C y ∼ −→ C xy , x, y ∈ A, satisfying coherence conditions. Any two such equivalences differ by a tensormultiplication by an invertible object L x,y ∈ Z sym ( B ). Hence, any extension ˜ C ∈ Ex fbr ( A, B ) is equal to C as an abelian category and has the tensor product X ˜ ⊗ Y = L x,y ⊗ X ⊗ Y, X ∈ C x Y ∈ C y , x, y, ∈ A. (8.38)To get associativity and braiding constraints of ˜ C it is necessary to have isomorphisms ξ x,y,z : L xy,z ⊗ L x,y ∼ −→ L x,yz ⊗ L y,z , κ x,y : L x,y ∼ −→ L y,x , x, y, z ∈ A, (8.39)i.e., L = { L x,y } x,y ∈ A must be a 2-cocycle in Z br ( A, Inv ( Z sym ( B ))). These constraints are given by( X ˜ ⊗ Y ) ˜ ⊗ Z = L xy,z ⊗ L x,y ⊗ X ⊗ Y ⊗ Z ξ x,y,z −−−→ L x,yz ⊗ L y,z ⊗ X ⊗ Y ⊗ Z c Ly,z,X −−−−−→ L x,yz ⊗ X ⊗ L y,z ⊗ Y ⊗ Z = X ˜ ⊗ ( Y ˜ ⊗ Z ) (8.40)and X ˜ ⊗ Y = L x,y ⊗ X ⊗ Y κ x,y −−−→ L y,x ⊗ X ⊗ Y c X,Y −−−→ L y,x ⊗ Y ⊗ X = Y ˜ ⊗ X (8.41)for all objects X ∈ C x , Y ∈ C y , Z ∈ C z , x, y, z ∈ A , where we omit the associativity constraints of C .This is a braided version of the construction introduced in [22, Section 8.7]. Such extensions wereconsidered in [6] where they were called zestings of C . Recently, a more general construction was studied81n great detail in [14]. In Propositions 8.32 and 8.33 below we compute obstructions and give a parame-terization of such extensions. Our treatment of equivalence classes of zesting extensions and obstructionsseems to be different from that of [14, Section 4].By Proposition 5.3 the Whitehead bracket[ − , − ] : P ic br ( B ) × Inv ( Z sym ( B )) → k × satisfies c M,Z c Z,M = [ M , Z ]id Z ⊗ M for all Z ∈ Inv ( Z sym ( B )) and M ∈ M , where M ∈ P ic br ( B ).Define a group homomorphism P W C : H br ( A, Inv ( Z sym ( B ))) → H br ( A, k × ) : Z Q Z , (8.42)where Q Z is identified with the quadratic from Q Z ( x ) = [ C x , Z ( x )] , c Z ( x ) ,Z ( x ) , x ∈ A. (8.43)Define a quadratic function P W C : H br ( A, Inv ( Z sym ( B ))) → H br ( A, k × ) , (8.44)by setting the components of P W C ( L ) for a braided 2-cocycle L to be P W C ( L )( x, y, z, w ) = c L x,y ,L z,w , (8.45) P W C ( L )( x, y | z ) = 1 , (8.46) P W C ( L )( x | y, z ) = [ C x , L y,z ] , x, y, z, w ∈ A. (8.47) Definition 8.30.
We will call (8.42) and (8.44) the first and second Pontryagin-Whitehead maps , cf. [22,Section 8.7].
Remark 8.31.
The maps
P W C and P W C depend on the homomorphism f : A → P ic br ( B ) : x C x . Proposition 8.32.
Let L be a -cocycle in Z br ( A, Inv ( Z sym ( B ))) . One can choose isomorphisms (8.39) so that the associativity and braiding isomorphisms (8.40) , (8.41) satisfy the pentagon and hexagon axioms(i.e., give rise to a tensor category) if and only if P W C ( L ) is trivial in H br ( A, k × ) .Proof. Define a cochain a ∈ C br ( A, k × ) by a ( x, y, z ) = ξ x,y,z , a ( x | y ) = κ x,y for all x, y, z ∈ A , where ξ and κ are isomorphisms (8.39).In the diagrams below we will omit the the tensor product sign and the associativity constraints of C .82he pentagon for the associativity constraint (8.40) becomes the diagram L xyz,w L xy,z L x,y XY ZW ξ x,y,z t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ξ xy,z,w / / L xy,zw L z,w L x,y XY ZW c Lz,w,Lx,y * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ L xyz,w L x,yz L y,z XY ZW c Ly,z,X (cid:15) (cid:15) ξ x,yz,w * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ L xy,zw L x,y L z,w XY ZW ξ x,y,zw t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ c Lz,w,XY (cid:15) (cid:15) L xyz,w L x,yz XL y,z Y ZW ξ x,yz,w (cid:15) (cid:15) L x,yzw L yz,w L y,z XY ZW c Ly,z,X t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ξ y,z,w / / L x,yzw L y,zw L z,w XY ZW c Lz,w,X (cid:15) (cid:15) c Lz,w,XY * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ L xy,zw L x,y XY L z,w ZW ξ x,y,zw (cid:15) (cid:15) L x,yzw L yz,w XL y,z Y ZW c Lyz,w,X (cid:15) (cid:15) L x,yzw L y,zw XL z,w Y ZW c Ly,zw,X (cid:15) (cid:15) L x,yzw L y,zw XY L z,w ZW c Ly,zw,X (cid:15) (cid:15) L x,yzw XL yz,w L y,z Y ZW ξ y,z,w / / L x,yzw XL y,zw L z,w Y ZW c Lz,w,Y / / L x,yzw XL y,zw Y L z,w
ZW, (8.48)while the hexagons are the diagrams L xy,z L x,y XY Z c Lx,yXY,Z / / L xy,z ZL x,y XY κ xy,z / / c − Lx,y,Z (cid:15) (cid:15) L z,xy ZL x,y XY c − Lx,y,Z (cid:15) (cid:15) L x,yz L y,z XY Z ξ − x,y,z O O c XY,Z * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ L xy,z L x,y ZXY κ xy,z / / L z,xy L x,y ZXY ξ − z,x,y (cid:15) (cid:15) L x,yz XL y,z Y Z c − Ly,z,X O O c Y,Z (cid:15) (cid:15) L x,yz L y,z ZXY ξ − x,y,z O O κ y,z (cid:15) (cid:15) L zx,y L z,x ZXYL x,yz XL y,z ZY κ y,z (cid:15) (cid:15) L x,yz L y,z ZXY ξ − x,z,y / / L xz,y L x,z ZXY κ x,z O O L x,zy XL z,y ZY c − Lz,y,X / / L x,yz L y,z XZY ξ − x,z,y / / c X,Z O O L xz,y L x,z XZY, c X,Z O O (8.49)and L x,yz XL y,z Y Z κ x,yz / / L yz,x XL y,z Y Z c X,Ly,zY Z / / c X,Ly,z (cid:6) (cid:6) L yz,x L y,z Y ZX ξ y,z,x (cid:15) (cid:15) L x,yz L y,z XY Z c Ly,z,X O O κ x,yz / / L yz,x L y,z XY Z ξ y,z,x (cid:15) (cid:15) c Ly,z,X G G L y,zx L z,x Y ZX c Lz,x,Y (cid:15) (cid:15) L xy,z L x,y XY Z ξ x,y,z O O κ x,y (cid:15) (cid:15) L y,zx L z,x XY Z c X,Y Z ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ L y,zx Y L z,x
ZXL xy,z L y,x XY Z ξ y,x,z / / c X,Y (cid:15) (cid:15) L y,zx L x,z XY Z κ x,z O O c X,Y (cid:15) (cid:15) L y,zx Y L z,x XZ c X,Z O O L xy,z L y,x Y XZ ξ y,x,z / / L y,xz L x,z Y XZ c Lx,z,Y / / L y,xz Y L x,z
XZ, κ x,z O O (8.50)for all x, y, z, w ∈ A and X ∈ C x , Y ∈ C y , Z ∈ C z , W ∈ C w .83fter cancelling internal polygones commuting by the functoriality of the tensor product of C , natu-rality of c , and the Yang-Baxter equation, we see that the clockwise compositions given by the perimetersof (8.48), (8.49), and (8.50) are c L x,y ,L z,w d ( a )( x, y, z, w ) , d ( a )( x, y | z ) , and c X,L y,z c L y,z ,X d ( a )( x | y, z ), respectively.Comparing this with the definition of P W C ( L ) we get the result. Proposition 8.33.
There is a fibration F → π ( Ex fbr ( A, B )) → B , where the base B is the set of zeroesof P W C and the fiber F is the cokernel of P W C .Proof. The assertion about the base follows from Proposition 8.32.Let C , ˜ C be A -extensions of B corresponding to the same braided monoidal functor A → Pic br ( B ).Then ˜ C = C ( ω,ς ) for some ( ω, ς ) ∈ H ( A, k × ). An equivalence of extensions C ( ω,ς ) ∼ −→ C is given onhomogeneous component C x , x ∈ A , by X Z ( x ) ⊗ X, X ∈ C x for Z ( x ) ∈ Inv ( Z sym ( B )). The tensorproperty of this equivalence means that Z : A → Inv ( Z sym ( B )) is a homomorphism, while its braidedproperty translates to commutativity of the diagram Z ( x ) ⊗ X ⊗ Z ( y ) ⊗ Y c Z ( x ) ⊗ X,Z ( y ) ⊗ Y / / c X,Z ( y ) (cid:15) (cid:15) Z ( y ) ⊗ Y ⊗ Z ( x ) ⊗ X c Y,Z ( x ) (cid:15) (cid:15) Z ( xy ) ⊗ X ⊗ Y ς ( x,y ) c X,Y / / Z ( xy ) ⊗ Y ⊗ X, (8.51)for all x, y ∈ A, X ∈ C x Y ∈ C y . Here c denotes the braiding of C .Comparing the compositions in (8.51) we see that ς ( x, y ) = c Z ( x ) ,Z ( y ) c Y,Z ( x ) c Z ( x ) ,Y , for all Y ∈ C y , and so the corresponding quadratic form is ς ( x, x ) = [ C x , Z ( x )] c Z ( x ) ,Z ( x ) = Q Z ( x ) , x ∈ A . Let B be a braided tensor category. We saw in Example 5.6 that the braided 2-categorical group Pic br ( B )contains a full 2-categorical subgroup Pic ( B ) consisting of braided B -module categories M such that M ∼ = B as a B -module category. Definition 8.34.
Let A be a finite group. We say that a braided A -graded extension of B is quasi-trivial if it contains an invertible object in every homogeneous component.Equivalently, an A -extension of B is quasi-trivial if the corresponding homomorphism A → P ic br ( B )factors through P ic br ( B ). 84 emark 8.35. A quasi-trivial extension is a special type of a braided zesting considered in [14]. Namely,it is a zesting of C ( A, ⊠ B .Let Ex br − qt ( A, B ) denote the 2-groupoid of quasi-trivial braided A -extensions of B . We have anequivalence of 2-groupoids Ex br − qt ( A, B ) ∼ = - Fun br ( A, Pic ( B )) . Since objects of
Pic ( B ) are of the form B ν , ν ∈ Aut ⊗ (id B ) (see Example 5.6), any braided monoidal2-functor A → Pic ( B ) (and any extension in Ex br − qt ( A, B ) comes from a group homomorphism f : A → Aut ⊗ (id B ) . Example 8.36.
Given f as above, there is a canonical quasi-trivial A -graded braided extension B ( f )of B such that B ( f ) = B ⊠ Vect A as a tensor category and its braiding is given by c X ⊠ x,Y ⊠ y = f ( x ) Y c X,Y , X, Y ∈ B , x, y ∈ A, where x ∈ A denote the simple objects of Vect A .Hence, Ex br − qt ( A, B ) = _ f ∈ Hom ( A, Aut ⊗ (id B )) Ex f br − qt ( A, B ) , where Ex f br − qt ( A, B ) is the 2-subgroupoid of quasi-trivial extensions corresponding to f . Furthermore, Ex f br − qt ( A, B ) = Ex f br − qt ( A, B ) if and only if f = f ∂ ( Z ) for some Z ∈ Inv ( B ).The Pontryagin-Whitehead maps (8.42) and (8.44) in this situation are given by P W B ( f ) ( Z )( x ) = f ( x ) Z ( x ) c Z ( x ) ,Z ( x ) , Z ∈ Hom ( A, Inv ( Z sym ( B ))) , (8.52)and P W B ( f ) ( L )( x, y, z, w ) = c L x,y ,L z,w , (8.53) P W B ( f ) ( L )( x, y | z ) = 1 , (8.54) P W B ( f ) ( L )( x | y, z ) = f ( x ) L y,z , L ∈ H br ( A, Inv ( Z sym ( B ))) , (8.55)for all x, y, z, w ∈ A . Corollary 8.37.
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