Extension theory for braided-enriched fusion categories

For a braided fusion category $\mathcal{V}$, a $\mathcal{V}$-fusion category is a fusion category $\mathcal{C}$ equipped with a braided monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given a fixed $\mathcal{V}$-fusion category $(\mathcal{C}, \mathcal{F})$ and a fixed $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary fusion category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V} \to Z(\mathcal{D})$ of $\mathcal{D}$ which are compatible with the enrichment of $\mathcal{C}$. We show that G-crossed extensions of a braided fusion category $\mathcal{C}$ are G-extensions of the canonical enrichment of $\mathcal{C}$ over itself. As an application, we parameterize the set of $G$-crossed braidings on a fixed $G$-graded fusion category in terms of certain subcategories of its center, extending Nikshych's classification of the braidings on a fusion category.


I
In previous articles [MP19 ;MPP18] we de ned monoidal categories enriched in a braided monoidal category V, and showed this notion was equivalent to an oplax, strongly unital, braided monoidal functor from V into the Drinfeld center of an ordinary monoidal category. When the functor F Z : V → Z (C) is strong monoidal, this coincides with the notion of a 1-morphism V → Vec in a suitable Morita 4-category [BJS18] (see also §2.4 below), and with the module tensor categories of [HPT16b]. Recent work of Kong and Zheng uses monoidal categories enriched in a braided category to give a uni ed treatment of gapped and gapless edges for 2D topological orders [KZ18; CJKYZ19; KZ19]. Of particular importance is the case where V is a braided fusion category and F Z : V → Z (C) is a braided strong monoidal functor into the Drinfeld center of another fusion category C. We call such a pair (C, F Z ) a V-fusion category.
The extension theory for fusion categories of [ENO10] has proven to be an immensely important tool. Particular applications include the process of gauging a global symmetry on a modular tensor category [BBCW19;CGPW16], permutation symmetries on modular tensor categories [GJ19], rank niteness for (G-crossed) braided fusion categories [JMNR], and classi cation theorems for tensor categories generated by an object of small dimension [Edi19].
In this article, we de ne the notion of a G-graded extension of a V-fusion category. We begin by proving that G-gradings on a fusion category C are in bijective correspondence with liftings of a xed ber functor Rep(G) → Vec = 1 C ⊆ C to Z (C). Fixing such a G-grading C = ∈G C , we see that an object (c, σ ·,c ) ∈ Z (C) satis es c ∈ C e if and only if (c, σ ·,c ) ∈ Rep(G) ∩ Z (C). Given this, we de ne a G-graded V-fusion category to be a V-fusion category (C, F Z ) such that the underlying fusion category C = ∈G C is G-graded and F Z (V) ⊆ Rep(G) ∩ Z (C).
Theorem 1.1. Fix a G-graded extension C ⊆ D of ordinary fusion categories, and a V-fusion category structure (C, F Z ) on C. The following sets are in canonical bijection.
• For all ∈ V, extensions of the half-braiding for F Z ( ) with C to a half-braiding with all of D coherently with respect to morphisms in V. • Lifts π : G → BrPic V (C) of the 3-functor π : G → BrPic(C) a orded by the G-extension D, where BrPic V (C) := End 123 (C) is the endomorphism 3-category of C considered as a 1-morphism from V to Vec in a certain 4-category of braided fusion categories.
• Lifts F Z : V → Z (C) G such that Forget G • F Z = F Z where the categorical G-action ρ : G → Aut br ⊗ (Z (C)) comes from the G-extension C ⊆ D and Forget G : Z (C) G → Z (C) forgets the Gequivariant structure.
This theorem characterises the possible enrichments F Z : V → Z (D) of (C, F Z ) which are compatible with the xed G-graded extension C ⊆ D. The proof uses extension theory for fusion categories of [ENO10] together with the results of [GNN09].
The third description of compatible enrichments above bears many similarities to the classi cation from [BJLP19] of G-equivariant structures on a connected étale algebra in a nondegenerately braided fusion category. Adapting the arguments and techniques from [BJLP19], we see that there are two obstructions to lifting our V-enrichment. First, for every ∈ G, we must have that F Z • F Z as monoidal functors V → Z (C). We call the existence of such a monoidal natural isomorphism the rst obstruction to the equivariant functor lifting problem. When such monoidal natural isomorphisms exist, we say D passes the rst obstruction, or that the rst obstruction vanishes. In this case, similar to [BJLP19], we show that lifts ρ : G → Aut(Z (C)|F Z ) correspond to splittings of a certain exact sequence. Theorem 1.2. There is a short exact sequence where I : Z (C) → Z (C) G is the induction functor adjoint to the forgetful functor Forget G . Moreover, splittings of this exact sequence are in canonical bijection with lifts ρ : G → Aut(Z (C)|F Z ) as in the nal case of Theorem 1.1.
We call the exact sequence (1.1) the second obstruction to the equivariant functor lifting problem. We say the second obstruction vanishes when this short exact sequence splits, and a splitting is a witness of the vanishing of the second obstruction. In §5, we calculate the splittings of (1.1) for various examples.
In §6, we give an application of our two main theorems above to extend Nikshych's classi cation [Nik19] of braidings on a xed fusion category, classifying G-crossed braidings on a xed G-graded fusion category in Theorem 6.7. The main tool is the following theorem, which extends [Bis18,Prop. 2.4] in the unitary setting.
(c, γ •,c ) where c ∈ C and γ •,c = {γ a,c : a ⊗ c → c ⊗ a} a∈C is a family of half-braidings. In this convention, the braiding on Z (C) is given by β (c,γ •,c ),(d,δ •,d ) := δ c,d : c ⊗ d → d ⊗ c. When C is a monoidal subcategory of a monoidal category D, we use the notation Z C (D) for the relative Drinfeld center. This agrees with the notation of [GNN09], but is the reverse of the notation of [HP17].
2.1. Braided enriched monoidal categories. Recall from [Kel05] that given a monoidal category V, a V-category C has objects together with hom objects C(a → b) ∈ V for all a, b ∈ C. For every a, b, c ∈ C, we have a composition morphism − • C − ∈ V(C(a → b)C(b → c) → C(a → c)) which satis es an associativity axiom. For every a ∈ C, we have an identity element j a ∈ V(1 V → C(a → a)) which satis es a unitality axiom.
There are also notions of V-functors and (1 V -graded) V-natural transformations. We refer the reader to [Kel05] for more details. (See also the pedestrian exposition in [MP19,§2] or [MPP18,§2].) De nition 2.1. Given a V-category C, the underlying category C V has the same objects as C, and the hom-sets are given by C V (a → b) := V(1 V → C(a → b)). We leave the reader to work out the de nitions of composition and identity morphisms for C V .
De nition 2.3. A (strict) V-monoidal category is a V-category C equipped with an associative monoid structure on objects, denoted ab for a, b ∈ C, whose unit object is denoted by 1 C , together with a tensor product morphism − ⊗ C − ∈ V(C(a → c)C(b → d) → C(ab → cd)) for all a, b, c, d ∈ C satisfying strict associativity and unitality axioms. The tensor product and composition morphisms must further satisfy the braided interchange relation There are also notions of V-monoidal functors and (1 V -graded) V-monoidal natural transformations. We refer the reader to [MP19,§2]  In [MP19], we proved a classi cation theorem for (weakly) tensored rigid V-monoidal categories in terms of V-module tensor categories [HPT16a]. The tensored case was treated in [MPP18].
De nition 2.6. An (oplax) (strongly unital) V-module tensor category consists of a pair (T , F Z ) with T a monoidal category and F Z : V → Z (T ) a braided (oplax) monoidal (strongly unital) functor. We call an (oplax) (strongly unital) V-module tensor category: We have the following classi cation theorem, which has recently been extended to a 2-equivalence of 2-categories (pseudofunctor equivalence of bicategories) in [Del19].
In light of Theorem 2.7 together with the results of [ KZ19;KZ18] in the fusion setting, we make the following de nition.
De nition 2.8. A V-fusion category, for V a braided fusion category, consists of a fusion category C together with a braided strong monoidal functor F Z : V → Z (C). Observe as F Z is a functor between fusion categories, it automatically admits a left adjoint, and hence V-fusion categories are tensored.
We focus on the fusion setting in order to have access to the results of [ENO10] and [GNN09].
2.2. G-gradings on linear monoidal categories. Fix a nite group G. In this section, we prove a classi cation theorem for G-gradings on linear monoidal categories T , i.e., those monoidal categories which are tensored over Vec, the monoidal category of nite dimensional vector spaces over a eld k. For this section, we assume k is an algebraically closed eld of characteristic zero. (Although it is not necessary, we would even be happy to assume further that k = C.) De nition 2.9. We call a linear monoidal category T G-graded if T = ∈G T as linear categories, and if t ∈ T and t h ∈ T h , then t ⊗ t h ∈ T h . In this case, by [GNN09,p. 12] there is a canonical fully faithful strong monoidal functor I = I T : Rep(G) → Z (T ) de ned as follows. For a representation (H, π ) ∈ Rep(G), we consider the object I π := H ⊗ 1 T ∈ T . Notice that both I π ⊗ t and t ⊗ I π are canonically isomorphic to H ⊗ t. Thus we can endow I π with the half-braiding ζ t,I π := π ⊗ 1 t : For a morphism f : (H , π ) → (K, ρ), we get a morphism I f := f ⊗ 1 1 T : I π → I ρ . It is straightforward to verify that I is a fully faithful strong monoidal functor (using the obvious tensorator/strength) since T is tensored over Vec.
Observe that the forgetful functor Forget Z : Z (T ) → T restricted to this copy of Rep(G) ⊆ Z (T ) is monoidally naturally isomorphic to the canonical ber functor De nition 2.10. Suppose (V, F : V → Vec) is a linear monoidal category equipped with a xed faithful strong monoidal ber functor. A Vbered enrichment of a closed linear monoidal category T is a braided strong monoidal functor F Z : V → Z (T ) which admits a right adjoint such that Forget Z •F Z = F , where we identify Vec = 1 T ⊆ T .
The following important lemma is essentially in [GNN09].
Proof. It is clear that t ∈ T e implies (t, σ •,t ) ∈ Rep(G) . Suppose (t, σ •,t ) ∈ Rep(G) . If t = t , then for all (H, π ) ∈ Rep(G), Since T is tensored over Vec, the above holds if and only if t = 0 for all e.
Now starting with a Rep(G)bered enriched linear monoidal category (T , I Z ), we claim there is a canonical faithful G-grading on T . We expect this result is known to experts, but we are unaware of its existence in the literature.
Recall O(G) is the commutative algebra of k-valued functions on G. Moreover, O(G) is a Hopf algebra with comultiplication given by ∆(χ ) := h χ h −1 ⊗ χ h where χ denotes the indicator function at ∈ G, antipode given by S χ := χ −1 , and counit given by ϵ(χ ) = δ =e . Let Irr(Rep(G)) be a set of representatives for the simple objects of Rep(G). There is a unital isomorphism of Hopf algebras given on w * ⊗ ∈ (H, π ) * ⊗ (H, π ) by Φ(w * ⊗ )( ) := w * (π ( )). Multiplication on the left hand side is given on where {α } ⊆ Rep(G)(H 1 ⊗ H 2 → K) is a basis and {α * } ⊂ Rep(G)(K → H 1 ⊗ H 2 ) is the dual basis under the pairing α • α * = δ α =α 1 K . The unit on the left hand side is exactly 1 * C ⊗ 1 C ∈ C * ⊗ C where C ∈ Rep(G) is the trivial representation. Comultiplication on w * ⊗ ∈ H * ⊗ H is given by where {e i } is a basis for H and {e * i } is the dual basis. We will identify both sides of (2.1) under the isomorphism Φ below. Now given t ∈ T , we get a unital k-algebra homomorphism O(G) → T (t → t) (whose image lies in Z (T (t → t))) whose image on w * ⊗ ∈ H * ⊗ H is given by where we identify elements ∈ H as morphisms : k → H , which gives a map ∈ T (1 T → 1 T ⊗H = I π ), and similarly w * ∈ T (I π = 1 T ⊗H → 1 T ). Now O(G) k |G | is an abelian k-algebra, so for each t ∈ T and ∈ G, we have a canonical projector χ t ∈ T (t → t). The proof of the following lemma is straightforward using (2.2).
Lemma 2.12. For t ∈ T , the projectors χ t ∈ T (t → t) satisfy the relations • (direct sum) χ t • χ t h = δ =h χ t and ∈G χ t = 1 t , and • (compatibility with morphisms) for all s ∈ T with projectors χ s ∈ T (s → s) and all morphisms f ∈ T (s → t), we have χ s • f = f • χ t .
As T was assumed to be idempotent complete, for ∈ G, we may de ne t := im(χ t ). By the direct sum relation in Lemma 2.12 we have t = ∈G t . Moreover, for all f ∈ T (s → t), we see that T (s → t) = ∈G T (s → t ). Thus de ning T to be the subcategory whose objects are of the form t for t ∈ T , we have T = ∈G T , i.e., T is G-graded as a linear category. We now claim that this G-grading is compatible with the tensor product, i.e., if s ∈ T and t ∈ T h , then s ⊗ t ∈ T h . To show this, we observe that the map (2.2) endows each hom space T (s → t) with an O(G)-action Since for all s, t ∈ T , our O(G)-action satis es This immediately implies that the idempotent χ st ∈ T (st → st) decomposes as Thus the G-grading on T respects the tensor product of T . Putting together the constructions of this section, we get the following result.
Theorem 2.13. Fix a linear monoidal category T and a ber functor F : There is a bijective correspondence between (1) Rep(G)-bered enrichments I Z : Lemma 2.14. The G-grading on T induced from the Rep(G)-bered enrichment Proof. Expanding the de nition of ζ •,I π , we have The result follows.
With these results in hand, we make the following de nition.
De nition 2.15. A faithfully G-graded V-fusion category is a V-fusion category (D, F Z D ) such that D is faithfully G-graded as an ordinary fusion category, and We close this section with the following observation about Rep(G)bered enrichments. Given a fully faithful braided tensor functor Rep(G) → Z (C) where C is a fusion category, it is not necessarily the case that C is G-graded. For example, taking C = Rep(G), the universal grading group of C is Z (G). Note that this enrichment is as far as possible from a Rep(G)bered enrichment, since postcomposing the enrichment with the forgetful functor yields an equivalence. However, Rep(G) is Morita equivalent to Vec(G), the quintessential example of a G-graded fusion category. Our next result shows this behavior is generic. The proposition below shows that any fusion category with a Rep(G) enrichment is Morita equivalent to a G-graded fusion category whose associated Rep(G) enrichment (obtained from the canonical equivalence of centers) is bered. This can be interpreted as a partial converse to Theorem 2.13.
Proposition 2.16. Suppose C is a fusion category and F : Rep(G) → Z (C) is a fully faithful tensor functor. Then there exists a faithfully G-graded fusion category D which is Morita equivalent to C such that the associated enrichment is a multifusion category, and every center of a multifusion category is also the center of an ordinary fusion category [DMNO13, Rem. 5.2]. By [GNN09], there is a bijective correspondence between G-extensions of fusion categories F and G-crossed braided extensions of Z (F ) which is established by taking the relative center. Thus there is a G-graded fusion category D whose relative center with respect to its trivial component 2.3. Extension theory for fusion categories. We rapidly review the results of [ENO10] and [GNN09] on extension theory for fusion categories.
In [ENO10], Etingof-Nikshych-Ostrik give a recipe for constructing G-extensions of a xed fusion category C using cohomological obstruction theory.
De nition 2.17. Recall that a categorical n-group is an (n + 1)-category with one 0-morphism such that every k-morphism is invertible up to a (k+1)-isomorphism for k ≤ n, and all (n + 1)-morphisms are invertible. Typically, we indicate the categorical group number by adding that number of underlines below. We denote the k < n truncation obtained by inductively identifying higher isomorphism classes by simply removing underlines.
Example 2.18. Given a xed fusion category C, its Brauer-Picard groupoid BrPic(C) is the categorical 2-group whose unique 0-morphism is C, whose 1-morphisms are invertible C − C bimodule categories, whose 2-morphisms are C − C bimodule equivalences, and whose 3-morphisms are bimodule functor natural isomorphisms.
Theorem 2.19 ( [ENO10]). Equivalence classes of G-extensions D of the fusion category C are in bijective correspondence with equivalence classes of 3-functors π : BG → BrPic(C).
The main tool of [ENO10] gives a cohomological prescription for constructing G-graded extensions by lifting a group homomorphism, or symmetry action, ρ : G → BrPic(C) to G → BrPic(C). We can lift ρ to a categorical action ρ : G → BrPic(C) if and only if the obstruction o 3 (ρ) ∈ H 3 (G, Inv(Z (C))) vanishes. In this case, the set of equivalence classes of liftings form a torsor over H 2 (G, Inv(Z (C))). Given ρ : G → BrPic(C), there is a lift ρ : G → BrPic(C) if and only if the obstruction o 4 (ρ) ∈ H 4 (G, C × ) vanishes. In this case, the equivalence classes of liftings form a torsor over H 3 (G, C × ).
We now recall the main results of [GNN09]. Suppose we have a G-extension D = ∈G D of C. (Note that the convention C ⊆ D is opposite to the convention of [GNN09] which uses D ⊆ C.) The 4-category of braided tensor categories. Recall from [BJS18] there is a 4-category of braided tensor categories BrTens, and the sub-4-category BrFus of braided fusion categories is 4-dualizable.
De nition 2.20. The 4-category BrFus is de ned as follows.
• 1-morphisms BrFus 1 (A → B) are multifusion categories C together with a braided monoidal functor F C : A B rev → Z (C). Sometimes we denote C ∈ BrFus 1 (A → B) by A C B . The composite of A 1 C A 2 and A 2 D A 3 is de ned as follows. First, we look at the Deligne tensor product C D, which comes equipped with a braided monoidal functor F : A 2 is the commutative algebra obtained by taking I (1 A 2 ), where I is the left adjoint to the canonical tensor product functor ⊗ : A rev 2 A 2 → A 2 , given by ⊗(a b) := a ⊗ b and using the braiding for the tensorator. This algebra is commutative since ⊗ is a central functor [DMNO13, Lemma 3.5]. If A 2 is nondegenerate, this algebra is identi ed with the canonical Lagrangian algebra under the standard equivalence A rev 2 A 2 Z (A 2 ). To see that C A 2 D has the structure of a 1-morphism in An explicit example calculation of the composite Ad E 8 Fib Ad E 8 appears in [Row19]. (2.6) The de nitions of horizontal and vertical composition of 2-morphisms are given in [BJS18,. For our purposes, we need to know that vertical composition is given by relative Deligne tensor product C M D N E , where we use the model Here, we use the leg numbering convention as in Sweedler notation for the underlying objects in M D N , i.e., x (1) x (2) = k i=1 m i n i . We endow C M D N E with the isomorphisms η which are given by the following composite: • Let M and N be two 2-morphisms with source C and target D. Then a 3-morphism is a bimodule functor G : M → N such that the following diagram commutes: (2.10) • 4 morphisms are bimodule natural transormations with no extra compatibility required! Remark 2.21. Observe that we may consider a fusion category C ∈ BrFus 1 (Vec → Vec) where we suppress the obvious braided central functor F Z : Vec → Z (C). Then BrPic(C) is exactly the core (consisting of only the invertible morphisms) of the endomorphism 3-category End 123 (C) which has • a single 0-morphism C • 1-morphisms BrFus 2 (C → C) • 2-morphisms the 3-morphisms in BrFus, and • 3-morphisms the 4-morphisms in BrFus.
Recall that non-degenerate braided fusion categories A, B are said to be Witt equivalent [DMNO13, Def. 5.1 and Rem. 5.2] if there exist multifusion categories C, D such that A Z (C) B Z (D). We conclude this section with the following observation.
Theorem 2.23. Suppose A, B are non-degenerate braided fusion categories and C ∈ BrFus 1 (A → B). The following statements are equivalent.
Before proving the theorem, we observe that the existence of C as in (2)  Proof. Suppose C is an invertible 1-morphism in BrFus(A → B). First, since A and B rev are nondegenerate, every braided tensor functor out of A B rev is fully faithful. Hence Z (C) A D 1 B rev for some non-degenerate braided fusion category D 1 . Let C −1 ∈ BrFus(B → A) be an inverse for C such that A (C C −1 ) L as 1-morphisms in BrFus 1 (A → A), where L ∈ B rev B is the canonical Lagrangian algebra. By a similar argument as before, Z (C −1 ) B D 2 A rev for some non-degenerate braided fusion category D 2 . Observe now that This means that by But since Z (C C −1 ) loc L Z ((C C −1 ) L ) Z (A) A A rev as A is non-degenerate, we must have D 1 and D 2 are trivial, and thus Z (C) A B rev .
Conversely, if Z (C) A B rev , then observe that Z (C mp ) B A rev , where C mp is the monoidal opposite of C. For the canonical Lagrangian algebra L ∈ B rev B, and so (C C mp ) L A as 1-morphisms in BrFus 1 (A → A). Similarly, we have that (C mp C) L B as 1-morphisms in BrFus 1 (B → B), where L is the canonical Lagrangian algebra in A rev A.

L V G
For this section, we x a braided fusion category V, a V-fusion category (C, F Z ) ∈ BrFus 1 (V → Vec), and a G-graded extension D = ∈G D of C as an ordinary fusion category. We now give several equivalent characterizations of the set of V-module category structures F Z on D which are compatible with the V-module category structure F Z of C. That is, for every ∈ V, F Z ( ) ∈ Z (C) → Z C (D), so we need an extension of the half-braiding for F Z ( ) with C to a half-braiding with all of D, and these extensions must be coherent with respect to morphisms in V.
One can state this compatibility condition in terms of equality of certain tensor functors. Observe that we have tensor functors The set of coherent lifts of the half-braidings for F Z ( ) with C to all of D can be canonically identi ed with the set of lifts denotes the forgetful functor. Note that for such a lift, we automatically have De nition 3.1. Given a xed V-fusion category (C, F Z ) and a xed G-graded extension D = ∈G D of C as an ordinary fusion category, we say that a lifting of the V-enrichment F Z : We now give several equivalent characterizations of this compatibility using extension theory for fusion categories [ENO10], together with the results from [GNN09].   Proof. Suppose we can lift the V-enrichment of C to D. We de ne morphisms η ,m : m F C ( ) → F C ( ) m for each m ∈ D , where F C : V → Z (C) → C as follows. A lift F : V → Z (D) applied to a ∈ V can be viewed as F ( ) = (F C ( ), σ •,F C ( ) ), where σ •,F C ( ) is a half-braiding for F C ( ) with d ∈ D. We de ne η ,d := σ d,F C ( ) : d ⊗ F C ( ) → F C ( ) ⊗ d. The fact that F : V → Z (D) is a braided monoidal functor ensures that η ,d makes the diagrams (2.6), (2.7), and (2.8) commute. This means we can lift the image of the 3-functor G → BrPic(C) to BrPic V (C) at the level of 1-morphisms. To lift at the level of 2-morphisms, recall that ⊗ induces a bimodule equivalence D C D h → D h . We need to show that this bimodule equivalence is a morphism in BrPic V (C). Given objects d ∈ D , d h ∈ D h , we need to check the following diagram commutes: where the top isomorphism is that from (2.9). This now follows immediately from the associativity of a half-braiding. Conversely, given a π V : BG → BrPic V (C, F Z ) which forgets to π : G → BrPic(C), we need to extend the half-braiding of F Z ( ) with C to all of D. We simply use η on D as our half-braiding: Now one uses the commutativity of (2.6), (2.7), (2.8) and (3.2) to verify that this is a well-de ned halfbraiding with all of D. Finally, one veri es these two constructions are mutually inverse.

Classi cation in terms of
We insert the commutative diagram (2.5) based on [GNN09] into (3.1) to get the following diagram: Since i is faithful on both objects and morphisms, we can cancel it from the left on both sides of the equation, and the result follows.
Thus to classify enriched extensions, we must solve the equivariant lifting problem for the data given by the initial enrichment and the extension. In other words, given an (oplax) braided (strongly unital) monoidal functor F Z : V → Z (C) and a categorical action ρ : G → Aut br ⊗ (Z (C)), we need to nd all the G-equivariant structures on F Z . We will formalize this notion in De nition 4.3 in the next section.

T
In this section, we study the equivariant functor lifting problem, showing lifts are in bijection with splittings of a certain exact sequence. Our approach is similar to [BJLP19, §3]. We do so in greater generality than needed for (3.3) above, since our results are signi cantly more general.
We further suppose (ρ, µ) : G → Aut ⊗ (W) is a categorical action of the nite group G. We write = ρ for notational simplicity, and we write ψ for its tensorator. Our convention for the tensorator µ for ρ is µ ,h : • h ⇒ h.

The rst obstruction.
De nition 4.2. We consider the following categorical groups.
De nition 4.3. Let ρ : G → Aut ⊗ (W), → ρ be a categorical action, and F : V → W an oplax monoidal functor. A G-equivariant structure on F is a lifting (4.1) Forget F ρ ρ which satis es Forget F • ρ = ρ on the nose.
Hence in order to nd a lifting ρ : G → Aut(W|F ), it is necessary that for each ∈ G, there exists a monoidal natural isomorphism λ : F ⇒ • F . We call the existence of such a λ for each ∈ G the rst obstruction to the equivariant functor lifting problem. We say the rst obstruction vanishes if such a λ exists for each ∈ G.
4.2. The second obstruction. We now assume that the rst obstruction to the equivariant lifting problem vanishes, i.e., for every ∈ G, there exists a monoidal natural isomorphism λ : F ⇒ • F . We now give a necessary and su cient condition for the isomorphisms (λ ) ∈G to assemble to a lift ρ : G → Aut ⊗ (W|F ). We call this condition the second obstruction to the equivariant functor lifting problem.
Recall that the adjoint to the forgetful functor Forget G : W G → W is I : W → W G by w → (w) and f ∈ W(w 1 → w 2 ) maps to I (f ) ,h := δ ,h · (f ). Observe that given w ∈ W, f : I (w) → I (w) is G-equivariant if and only if the following diagram commutes for all , h, k ∈ G: The functor I is endowed with an oplax monoidal structure ν I w 1 ,w 2 ∈ W G (I (w 1 ⊗ w 2 ) → I (w 1 ) ⊗ I (w 2 )) given componentwise by ∈G ψ w 1 ,w 2 : Remark 4.4. In addition to F (1 V ) being a coalgebra with comultiplication ∆ (see Assumption 4.1), notice that (I • F )(1 V ) ∈ W G is also a coalgebra object with comultiplication given on components by and counit given on components by ε := (ε F ) : (F (1 V )) → 1 W .
We de ne ι : Aut ⊗ (F ) → Aut ⊗ (I • F ) by ι(f ) := I (f ) ∈ W G (I (F ( )) → I (F ( ))). To verify that ι(f ) is oplax monoidal, we see the outside square of the following diagram commutes, as the inner squares both commute: The following lemma is similar to [BJLP19, Lem. 3.2]. We provide a proof for completeness and convenience of the reader.
(3) For every h ∈ G, there are unique , k ∈ G such that η ,h 0 η h,k for all ∈ V. These , k are independent of ∈ V.
Proof. To prove (1), since η : Now setting = k gives the desired formula.
Lemma 4.6. The function π : Aut ⊗ (I • F ) → G given by setting π (η) to be the unique such that η 1 V −1 ,e 0 gives a well-de ned group homomorphism.
Theorem 4.9. The set of G-equivariant structures on F as in (4.1) is in bijective correspondence with splittings of the exact sequence (4.3).
4.3. The braided case. We now assume V, W are braided monoidal categories and F : V → W is an oplax braided monoidal functor. We again use Assumption 4.1 that F (1 V ) is a connected coalgebra in W.
Indeed, the left face commutes since η is natural, the right face commutes since β W is natural, the top and bottom faces commute since η is monoidal, and the front face commutes since α is braided. We conclude the back face must also commute. • Aut br ⊗ (W|F ) is a the full categorical subgroup of Aut ⊗ (W|F ) whose objects are triples (α,ψ α , λ α ), where (α,ψ α ) ∈ Aut br ⊗ (W). • Stab br ⊗ (F ) is the full categorical subgroup of Aut br ⊗ (W) generated by the image of Aut br ⊗ (W|F ) under the forgetful functor (α,ψ α , λ α ) → (α,ψ α ).
In this setting, we make the following de nition.

E
In this section, we work out examples of our main Theorems 3.4 and 4.9 above in the V-fusion setting. 5.1. Fully faithful enrichment. Suppose (C, F Z ) is a V-fusion category such that F Z is fully faithful. This type of example is particularly important, since every enrichment can be "pushed forward" to a fully faithful enrichment by considering the enrichment over the full subcategory generated by the image of V in Z (C). We will see that in the fully faithful setting, the G-action on the normal subgroup Aut ⊗ (F Z ) is trivial, and thus splitting of the short exact sequence (4.3) becomes a 2-cocycle obstruction. Now suppose D is any G-graded extension of C as an ordinary fusion category, so we get a categorical action ρ : G → Aut br ⊗ (Z (C)). Assume that ρ passes the rst obstruction, so that for each ∈ G, there exists a monoidal natural isomorphism λ : F ⇒ • F . By a direct computation, we see that is an element of Aut ⊗ (F ) Aut ⊗ (1 V ), which is in turn isomorphic to the group U(V) of characters on the universal grading group of V. In fact ω ∈ Z 2 (G, U(V)). Any other choice of λ for ∈ G will give a cohomologous 2-cocycle. We see directly that the second obstruction vanishes if and only if [ω] = 0 in H 2 (G, U(V)). Hence the exact sequence (4.3) is exactly which splits if and only if [ω] = 0. Observe that when ρ passes the rst obstruction, the 2-cocycle ω in (5.1) automatically vanishes if U(V) is trivial, in which case there is a unique splitting.
Corollary 5.1. Suppose (C, F Z ) is a V-fusion category with F Z fully faithful. Let D be an arbitrary G-graded extension of C for which the rst obstruction vanishes. If U(V) is trivial, then the V-enrichment has a unique lifting to D.
Example 5.2. If (C, F Z ) is a Fib-fusion category and D is a G-graded extension of C for which the rst obstruction vanishes, then there is unique lift of the Fib enrichment to D. For an explicit example, one may consider C = Ad(E 8 ) and D = E 8 . 5.2. Zesting a trivial extension. For convenience, we assume that H 4 (G, C × ) = (1). Recall a braided categorical action of G on Z (C) is called G-stable if each ∈ G acts by the identity functor. Such actions are given by twisting the trivial action by a 2-cocycle ω ∈ H 2 (G, Aut ⊗ (1 Z (C) ) = H 2 (G, Inv(Z (C))) [ENO10]. Since H 4 (G, C × ) = (1), we get a G-graded extension D of C called C ω Vec(G), which is C Vec(G) as a linear category with the tensor product functor twisted by ω. Twisting the monoidal product by a 2-cocycle in this manner is sometimes called zesting c.f. [Bru+17].
For such extensions, for any enrichment (C, F Z ), the rst obstruction always vanishes, namely . If in addition F Z is fully faithful (or more generally sends simple objects to simple objects), then we get a restriction map R : Aut ⊗ (1 Z (C) ) Inv(Z (C)) → Aut ⊗ (1 V ) U(V), and the 2-cocycle (5.1) corresponds to the push forward of R * ω ∈ H 2 (G, U(V)). Thus we can extend the enrichment (C, F Z ) if and only if R * ω is trivial.
For a slightly more explicit example, when V = Rep(N ) and C = Vec(N ), we have Inv(Z (C)) N × Z (N ) and U(V) Z (N ). Then the push-forward map R : N × Z (N ) → Z (N ) is the canonical projection to the factor Z (N ). In particular, for any group with Z (N ) = (1) and for any ω ∈ H 2 (G, N ) (with the trivial action of G on N ), we can lift the Rep(N ) enrichment on Vec(N ) to the zested extension Vec(N ) ω Vec(G). In this case, the latter category is actually equivalent to Vec(N × G, ω ), where ω ∈ Z 3 (N ×G, C × ) is possibly a non-trivial 3-cocyle obtained from ω via a connecting map in the Lyndon-Hochschild-Serre spectral sequence associated to the short exact sequence 1 → N → N × G → G → 1 (see [ENO10,Appendix]). 5.3. Fibered enrichments and group theoretical extensions. In this example, we focus on Rep(N )bered enrichments (recall De nition 2.10) with C = Vec(N ) and D = Vec(E) for some normal subgroup N ≤ E corresponding to a xed exact sequence We now analyze when we can extend the Rep(N )bered enrichment to Vec(E). The rst step will be to analyze the categorical action of G on the center, and in particular how it restricts to the bered enrichment. First, from the extension above we directly de ne a braided categorical action on Rep(N ). Pick a set theortical section λ : G → E of the quotient map E → G which we will denote → λ ∈ E. Then we have λ λ h = λ h n ,h for some n ,h ∈ N . For each ∈ G, we de ne α ∈ Aut br ⊗ (Rep(N )) by α (π , V ) := (π (λ −1 · λ ), V ) on objects, and we set α to be the identity on morphisms. This has the obvious structure of a (braided) monoidal functor. We now de ne monoidal natural isomorphisms µ ,h : α • α h → α h . For each (π , V ) ∈ Rep(N ), consider the linear map π (n ,h ) on the vector space V . Then we have Setting µ ,h := {µ (π ,V ) ,h := π (n ,h )} (π ,V )∈Rep(N ) , we see µ ,h : α • α h → α h gives a monoidal natural isomorphism of functors.
Lemma 5.3. The assignment → α ∈ Aut br ⊗ (Rep(G)) together with the monoidal natural isomorphisms µ ,h : α • α h → α h described above assembles into a categorical action α : G → Aut br ⊗ (Rep(N )). Proof. A quick computation shows that the equation we need to verify for all , h, k ∈ G and all representations (π , V ) is the cocycle-type equation π (n h,k λ −1 k n ,h λ k ) = π (n ,hk n h,k ). From the de nition of n ,h , we have λ λ h λ k = λ h n ,h λ k = λ h λ k λ −1 k n ,h λ k = λ hk n h,k λ −1 k n ,h λ k . On the other hand, we also have λ λ h λ k = λ λ hk n h,k = λ hk n ,hk n h,k .
Comparing these two expressions, we see n h,k λ −1 k n ,h λ k = n ,hk n h,k in N , so (5.3) holds for any representation of N . Now, we consider Vec(E) as a G-extensions of Vec(N ). This yields a braided categorical action which we denoteα : G → Aut br ⊗ (Z (C)). Lemma 5.4. The categorical actionα restricts on the canonical copy of Rep(N ) ⊆ Z (Vec(N )) to α de ned in Lemma 5.3.

Proof. Recall that as a Vec(N ) bimodule, Vec(E)
∈G Vec(N ), where here Vec(N ) can be viewed as the linear category of vector spaces graded by elements of the coset indexed by ∈ G. Let us consider the section G → λ ∈ E chosen above. We can identify the simple objects of Vec(N ) as elements λ n for n ∈ N . Furthermore Vec(N ) Vec(N ) as a right N module, where λ n n := λ n n, but the left action of N on Vec(N ) is given by n λ n = λ (λ −1 nλ )n . In other words, the left action is twisted by the auto-equivalence λ −1 · λ ∈ Aut(N ). From the de nition of the categorical actionα [ENO10,Eq. 24] and the canonical copy of Rep(N ) ⊆ Z (Vec(N )), the result follows.
Corollary 5.5. The canonical Rep(N )bered enrichment of Vec(N ) extends to Vec(E) if and only if E N × G. In this case, these extensions form a torsor over H 1 (G, Z (N )).
Proof. Since the canonical bered enrichment is fully faithful, by the previous lemma we can lift the enrichment if and only if the categorical action α : G → Aut br ⊗ (Rep(G)) is isomorphic to the trivial categorical action. This would imply, in particular, that each α is trivial, namely that π (λ −1 nλ ) = π (n) for all n ∈ N , ∈ G, and (π , V ) ∈ Rep(N ). Applying this to the regular representation implies λ −1 nλ = n, and thus we have a decomposition E N × ω G for some 2-cocycle ω ∈ Z 2 (G, Z (N )), where the action on the latter coe cient module is trivial. Furthermore, we see this 2-cocycle ω ,h is precisely the n ,h associated to our choice of λ. But since the tensorator for the action α is given by µ (π ,V ) ,h = π (n ,h ) by de nition, we see that the action α is precisely the trivial action twisted by ω. Therefore, α is isomorphic to the trivial action precisely when [ω] is trivial in H 2 (G, Z (N )), which happens precisely when E splits as N × G. The nal claim follows easily.

A : G
An interesting point of view we wish to advocate is that various sorts of structures on a G-graded extension can be equivalent to extensions of an enrichment on the base category. In particular, a braided fusion category can be canonically enriched over itself. In this section, our goal is to show that (equivalence classes of) G-crossed braidings on a G-graded fusion category D which restrict on the trivial graded component C to some xed braiding are exactly classi ed by extensions of the corresponding self enrichment of C to D.
While this proof essentially boils down to results in [ENO10; DN13], we believe our point of view sheds new light on G-crossed braidings while simultaneously providing intuition for enriched extensions as being 'something like a G-crossed braiding'. We then apply our earlier results to give a classi cation of G-crossed braidings generalizing the results of Nikshych [Nik19]. This allows us to classify G-crossed braidings on a G-graded fusion category D in terms of full subcategories of its Drinfeld center, satisfying some conditions. Lemma 6.1. Suppose F is a full and replete fusion subcategory of a fusion category T . If t ∈ T and f ∈ F such that t ⊗ f ∈ F , then t ∈ F .
and thus t ∈ F . Lemma 6.2. Suppose E is a a G-crossed braided fusion category and F ⊆ E is a full and replete faithfully G-graded subcategory. Then the G-action preserves F , and thus F ⊆ E is a G-crossed braided subcategory.
Proof. Suppose f ∈ F and pick f ∈ F . Then (f ) ⊗ f f ⊗ f ∈ F (via the G-crossed braiding), and so (f ) ∈ F by Lemma 6.1.
De nition 6.3. Suppose E = ∈G E is a faithfully G-graded fusion category and (ρ 1 , β 1 ), (ρ 2 , β 2 ) are two G-crossed braidings on E. We say (ρ 1 , β 1 ), (ρ 2 , β 2 ) are equivalent if there is an equivalence η : ρ 1 ⇒ ρ 2 of monoidal functors G → Aut ⊗ (E) such that for all x ∈ E and ∈ E, Observe that there is at most one equivalence between any two G-crossed braidings as the monoidal natural isomorphism η is completely determined by β 1 , β 2 if it exists. (Indeed, β 1 x , is invertible, and − ⊗ 1 c is injective on hom spaces for every fusion category using [HPT16a, Lem. A.5].) Proposition 6.4. Let C ⊆ D be a G-graded extension of fusion categories. The set of equivalence classes of G-crossed braidings on D are in bijection with the set of lifts of D to Z C (D).
Proof. Suppose we have a G-crossed braiding on D. Then for all d ∈ D and c ∈ C, our G-crossed braiding gives us a half-braiding c ⊗ d e(d) ⊗ c = d ⊗ c, so we have a lift of D to Z C (D). Conversely, given a lift D to Z C (D), observe that this lift of D is full, replete, and faithfully G-graded. By Lemma 6.2, the G-crossed braiding on Z C (D) from [GNN09] restricts to a G-crossed braiding on D.
It is easy to see that starting with a lift of D to Z C (D), the G-crossed braiding of Z C (D) restricted to D gives the same lift of D to Z C (D). Conversely, given a G-crossed braiding on D, the induced lift of D to Z C (D) gives an equivalent G-crossed braiding on D by [ENO10, Proof of Thm. 7.12].
We now x a braided fusion category C together with a G-extension C ⊆ D as ordinary fusion categories corresponding to a 2-functor ρ : G → BrPic(C) from [ENO10]. We are now ready to prove Theorem 1.3, which is equivalent to the following corollary. Proof. Equivalence classes of G-crossed braidings on D compatible with the braiding of C are in bijection with lifts of D to Z C (D) by Proposition 6.4. But observe that lifts from D to Z C (D) are in bijection with lifts of C to Z (D) by taking the inverse half-braiding. Remark 6.6. Observe that by Theorem 3.3, equivalence classes of G-crossed braidings are also in bijection with lifts BrPic C (C) G BrPic(C).
Forget ρ ρ A more conceptual way to prove Proposition 6.4 and Theorem 1.3 using [ENO10, Thm. 7.12] would be to construct a monoidal 2-equivalence Pic(C) BrPic C (C). We know how to write down a 2-equivalence, and equip it with a tensorator, but we have not attempted to construct the associator isomorphisms for the tensorator, or check the coherences. This might be an interesting project for someone interested in monoidal 2-categories.
We can use Theorem 1.3 to obtain a classi cation of G-crossed braidings on a G-graded fusion category generalizing a similar style of classi cation by Nikshych of braidings on a fusion category [Nik19]. Recall that if A ∈ C is an algebra object, a subcategory D ⊆ C is called transverse to A if for all objects d ∈ D, C(d → A) = C(d → 1). Theorem 6.7. Let D = D be a faithfully G-graded fusion category and Rep(G) ⊆ Z (D) the canonical subcategory of the center. Then G-crossed braidings on D are classi ed by full and replete fusion subcategories A ⊆ Z (C) satisfying the following properties: (1) A ⊆ Rep(G) .
(3) A is transverse to I (1), i.e. for any a ∈ A, Z (D)(a → I (1)) = Z (D)(a → 1), where I is the right adjoint of the forgetful functor and thus Forget Z | A is an equivalence. Thus we can transport the half-braidings induced from A onto D e , to obtain a braiding which lifts to the center. It is clear these two constructions are mutually inverse.
• a pair L, M of commuting normal subgroups of G, and • a G-invariant ω-bicharacter B : L × M → C × .
Given such an abstract fusion subcategory A, the subgroup L is determined by the normal subgroup of G generated by the image of the forgetful functor, while M is determined by Rep(G/M) = A ∩ Rep(G), where Rep(G) denotes the canonical copy of Rep(G) ⊂ Z (Vec(G, ω)). See [NNW09] for an explanation of the role of the bicharacter B.
We denote the subcategory associated to the above data as S(L, M, B). In this notation, the canonical subcategory Rep(G) is S (1, 1, 1), and the trivial subcategory Vec is S (1, G, 1). We further recall the following facts from [NNW09]. But since L ≤ ker(π ), we must have equality, which concludes the proof.
As a special case, we recover the following well known corollary.
Remark 6.12. Recall that a braiding on a G-graded fusion category D can be viewed as a G-crossed braiding together with an extra piece of data, namely a trivialization of the categorical action G → Aut ⊗ (D). For example, when G is abelian, we have a unique G-crossed braiding on Vec(G), where the G action is by conjugation, and the G-braiding is the identity. However, we have several di erent braidings on Vec(G) which correspond to distinct trivializations of the conjugation action, which it is easy to show correspond to bicharacters on G.
6.2. Example: Rep(G). Now we consider the case where D = Rep(G), and we consider its center in terms of Z (Vec(G)), where we can use the convenient description as above. In this case, the universal grading group is the dual group Z (G). The copy of Rep( Z (G)) Vec(Z (G)) sitting inside Z (Vec(G)) is identi ed with the objects which are direct sums of objects (z, 1) where z ∈ Z (G) represents a conjugacy class, and 1 is the trivial representation of the centralizer subgroup of z (which is G). Note that all (normal) subgroups of Z (G) are of the form for some H ≤ Z (G). Thus faithful grading groups are given by quotients Z (G)/H ⊥ , and Rep( Z (G)/H ⊥ ) Vec(H ) ⊆ Vec(Z (G)).
We have the following result: