2.1 Finite semisimple 2-categories

We begin by reviewing the definition of a 2-condensation monad introduced in [Reference Gaiotto and Johnson-FreydGJ19a] as a categorification of the notions of an idempotent. More precisely, we recall the unpacked version of this definition given in § 1.1 of [Reference DécoppetDéc22a].

Categorifying the notion of split surjection, [Reference Gaiotto and Johnson-FreydGJ19a] gave the definition of a 2-condensation, which we recall below.

We say that a 2-condensation monad splits if it can be extended to a 2-condensation (see definition 1.1.4 of [Reference DécoppetDéc22a] for details).

Let us now fix a field $\mathbb{k}$ . Following [Reference Gaiotto and Johnson-FreydGJ19a], we will call a $\mathbb{k}$ -linear 2-category locally Cauchy complete if its Hom-categories are Cauchy complete, that is, they have direct sums and idempotents split. Furthermore, a Cauchy complete linear 2-category is a locally Cauchy complete 2-category that has direct sums for objects and such that 2-condensation monads split. Given an arbitrary locally Cauchy complete 2-category $\mathfrak{C}$ , one can form its Cauchy completion $Cau(\mathfrak{C})$ , which is Cauchy complete (see [Reference Gaiotto and Johnson-FreydGJ19a], and also [Reference Douglas and ReutterDR18] for a slightly different approach). Furthermore, it was shown in Proposition 1.2.5 of [Reference DécoppetDéc22a] that $Cau(\mathfrak{C})$ is 3-universal with respect to linear 2-functors from $\mathfrak{C}$ to a Cauchy complete linear 2-category. Let us also mention that, given two Cauchy complete linear 2-categories $\mathfrak{C}$ and $\mathfrak{D}$ , one can form their Cauchy completed tensor product $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ , which is 3-universal with respect to bilinear 2-functor from $\mathfrak{C}\times\mathfrak{D}$ to a Cauchy complete linear 2-category (see Proposition 3.4 of [Reference DécoppetDéc24]).

From now on, we will assume that $\mathbb{k}$ is an algebraically closed field of characteristic zero. We are ready to review the definitions of a semisimple 2-category and of a finite semisimple 2-category introduced in [Reference Douglas and ReutterDR18] (more precisely, we review the variants given in [Reference DécoppetDéc22a]).

An object C of a semisimple 2-category $\mathfrak{C}$ is called simple if the identity 1-morphism $Id_C$ is a simple object of the semisimple 1-category $End_{\mathfrak{C}}(C)$ .

Given a multifusion 1-category $\mathcal{C}$ , the 2-category $\mathbf{Mod}(\mathcal{C})$ of finite semisimple right $\mathcal{C}$ -module 1-categories is a finite semisimple 2-category (see Theorem 1.4.8 of [Reference Douglas and ReutterDR18]). Moreover, by Theorem 1.4.9 of [Reference Douglas and ReutterDR18], every finite semisimple 2-category is of this form. If $\mathfrak{C}$ and $\mathfrak{D}$ are finite semisimple, then it follows from Theorem 3.7 of [Reference DécoppetDéc24] that $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ is a finite semisimple 2-category, which we denote by $\mathfrak{C}\boxtimes\mathfrak{D}$ and call the 2-Deligne tensor product of $\mathfrak{C}$ and $\mathfrak{D}$ .

Given two simple objects C and D of a finite semisimple 2-category $\mathfrak{C}$ , the finite semisimple 1-category $Hom_{\mathfrak{C}}(C,D)$ may be non-zero. If this is the case, we say that C and D are connected. We emphasize that two non-equivalent simple objects may be connected. This is in sharp contrast to the decategorified setting, in which the Schur lemma prevents such behavior. Furthermore, it was shown in § 1.2.3 of [Reference Douglas and ReutterDR18] that being connected defines an equivalence relation on the set of simple objects.

We say that a finite semisimple 2-category $\mathfrak{C}$ is connected if $\pi_0(\mathfrak{C})$ is a singleton. For instance, if $\mathcal{C}$ is a fusion 1-category, then $\mathbf{Mod}(\mathcal{C})$ is connected. Furthermore, in an abuse of terminology, we will identify a connected component (i.e., an equivalence class in $\pi_0(\mathfrak{C})$ ) with the finite semisimple sub-2-category of $\mathfrak{C}$ that is spanned by the simple objects in this equivalence class and full on 1-morphisms.

2.2 Fusion 2-categories

We recall the definition of a fusion 2-category first introduced in [Reference Douglas and ReutterDR18] (more precisely, we review the variant considered in [Reference DécoppetDéc22b]) and give a number of key examples. Throughout, we work over $\mathbb{k}$ , an algebraically closed field of characteristic zero.

The following classes of fusion 2-categories were first singled out in [Reference Johnson-Freyd and YuJY21], and will play a key role in the proof of our main theorem.

2.3 Rigid and separable algebras

Let us fix a fusion 2-category $\mathfrak{C}$ (over an algebraically closed field of characteristic zero), with monoidal product $\Box$ and monoidal unit I. An algebra, also called a pseudo-monoid in [Reference Day and StreetDS97], in $\mathfrak{C}$ is an object A of $\mathfrak{C}$ equipped with a multiplication 1-morphism $m:A\Box A\rightarrow A$ , a unit 1-morphism $i:I\rightarrow A$ , and coherence 2-isomorphisms ensuring that m is associative and unital. We focus our attention on the following refinement of the notion of an algebra, which was introduced in [Reference GaitsgoryGai12], and was first considered in the context of fusion 2-categories in [Reference Johnson-Freyd and ReutterJR24].

For our purposes, it is necessary to single out the following class of rigid algebras introduced in [Reference Johnson-Freyd and ReutterJR24], whose origin can be traced back to [Reference Gaiotto and Johnson-FreydGJ19a].

As we will briefly recall below, separable algebras can be used to define the notion of Morita equivalence between fusion 2-categories. This is thanks to the following result, which is a combination of Theorem 3.1.6 of [Reference DécoppetDéc23d] and Corollary 2.2.3 of [Reference DécoppetDéc23a].

It is in general difficult to prove that a rigid algebra is separable. Nonetheless, if A is a connected rigid algebra, meaning that $i:I\rightarrow A$ is a simple 1-morphism, then one can associate to A a scalar $\mathrm{Dim}_{\mathfrak{C}}(A)$ (see § 3.2 of [Reference DécoppetDéc23d]). It was established in Theorem 3.2.4 of [Reference DécoppetDéc23d] that A is separable if and only if $\mathrm{Dim}_{\mathfrak{C}}(A)$ is non-zero. This criterion can be effectively used to give examples of separable algebras.

Namely, let $\mathcal{C}$ be a $\mathcal{B}$ -central multifusion 1-category. We use $\mathcal{C}^{mop}$ to denote the multifusion 1-category obtained from $\mathcal{C}$ by reversing the order of the monoidal product. Note that there is a canonical $\mathcal{B}$ -central structure on $\mathcal{C}^{mop}$ . Then inspection shows that there is an equivalence

\[\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})\simeq\mathbf{Mod}(\mathcal{C}^{mop}\boxtimes_{\mathcal{B}}\mathcal{C})\]

of 2-categories. Moreover, it follows from Theorem 6.2 of [Reference GreenoughGre10] that $\mathcal{C}^{mop}\boxtimes_{\mathcal{B}}\mathcal{C}$ is a rigid monoidal 1-category. Furthermore, it is finite semisimple by Proposition 3.5 of [Reference Etingof, Nikshych and OstrikENO09]. This shows that $\mathcal{C}$ is indeed separable.

2.4 The Morita theory of fusion 2-categories

We end this section by reviewing the notion of Morita equivalence between fusion 2-categories introduced in [Reference DécoppetDéc23b]. Given a fusion 2-category $\mathfrak{C}$ , a left $\mathfrak{C}$ -module 2-category is a 2-category $\mathfrak{M}$ equipped with a coherent left action by $\mathfrak{C}$ (see Definition 2.1.3 of [Reference DécoppetDéc21] for details). If $\mathfrak{M}$ is a finite semisimple 2-category, then it follows from Theorem 5.3.4 of [Reference DécoppetDéc21] that there exists a rigid algebra A in $\mathfrak{C}$ , together with an equivalence $\mathfrak{M}\simeq \mathbf{Mod}_{\mathfrak{C}}(A)$ of left $\mathfrak{C}$ -module 2-categories. We say that $\mathfrak{M}$ is separable if A is separable, and note that this property does not depend on the algebra A (see Proposition 5.1.3 of [Reference DécoppetDéc23b]).

Given a separable left $\mathfrak{C}$ -module 2-category $\mathfrak{M}$ , we use $\mathfrak{C}^*_{\mathfrak{M}}$ to denote $\mathbf{End}_{\mathfrak{C}}(\mathfrak{M})$ , the monoidal 2-category of left $\mathfrak{C}$ -module 2-endofunctors on $\mathfrak{M}$ (see Theorem 4.1.7 of [Reference DécoppetDéc23b]), and call it the dual tensor 2-category to $\mathfrak{C}$ with respect to $\mathfrak{M}$ . On the other hand, given a separable algebra A in $\mathfrak{C}$ , it follows from Theorem 3.2.8 of [Reference DécoppetDéc23b] that the finite semisimple 2-category $\mathbf{Bimod}_{\mathfrak{C}}(A)$ has a canonical monoidal structure given by the relative tensor product over A. The next result is given by combining Theorem 5.3.2 of [Reference DécoppetDéc23b] with Corollary 2.2.3 of [Reference DécoppetDéc23a].

Theorem 2.23. Let A be a separable algebra in a fusion 2-category $\mathfrak{C}$ . Then

\[\mathbf{End}_{\mathfrak{C}}(\mathbf{Mod}_{\mathfrak{C}}(A))\simeq\mathbf{Bimod}_{\mathfrak{C}}(A)^{mop}\]

is a multifusion 2-category.

In order to ensure that the multifusion 2-category of Theorem 2.23 is fusion, we need to impose a condition on the algebra A. More precisely, we say that an algebra is indecomposable if it cannot be written as the direct sum of two non-trivial algebras. For instance, any connected algebra is indecomposable. We use an analogous terminology for module 2-categories. The next result is obtained by combing Theorem 5.4.3, which characterizes Morita equivalence between multifusion 2-categories, and § 5.2 of [Reference DécoppetDéc23b].

1. 1. There exist an indecomposable separable left $\mathfrak{C}$ -module 2-category $\mathfrak{M}$ , and an equivalence of monoidal 2-categories $\mathfrak{D}^{mop}\simeq\mathfrak{C}_{\mathfrak{M}}^*$ .

2. 2. There exist an indecomposable separable algebra A in $\mathfrak{C}$ , and an equivalence of monoidal 2-categories $\mathfrak{D}\simeq \mathbf{Bimod}_{\mathfrak{C}}(A)$ .

If either of the above conditions is satisfied, we say that $\mathfrak{C}$ and $\mathfrak{D}$ are Morita equivalent. Furthermore, Morita equivalence defines an equivalence relation between fusion 2-categories.

For later use, we also record the following technical lemmas.

Lemma 2.25. Let $\mathfrak{C}$ and $\mathfrak{D}$ be two multifusion 2-categories. For any separable algebras A in $\mathfrak{C}$ and B in $\mathfrak{D}$ , there is an equivalence

\[\mathbf{Mod}_{\mathfrak{C}}(A)\boxtimes\mathbf{Mod}_{\mathfrak{D}}(B)\simeq\mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{D}}(A\boxtimes B)\]

of left $\mathfrak{C}\boxtimes\mathfrak{D}$ -module 2-categories.

Proof. Thanks to the 3-universal property of the 2-Deligne tensor product obtained in Theorem 3.7 of [Reference DécoppetDéc24], the bilinear 2-functor

\[\begin{array}{ccc}\mathbf{Mod}_{\mathfrak{C}}(A)\times\mathbf{Mod}_{\mathfrak{D}}(B)&\rightarrow & \mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{D}}(A\boxtimes B)\\(M,N) & \mapsto & M\boxtimes N\end{array}\]

induces a linear 2-functor $\mathbf{K}:\mathbf{Mod}_{\mathfrak{C}}(A)\boxtimes\mathbf{Mod}_{\mathfrak{D}}(B)\rightarrow\mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{D}}(A\boxtimes B).$ Furthermore, this 2-functor is compatible with the left $\mathfrak{C}\boxtimes\mathfrak{D}$ -module structures. Now, it follows from the proof of Proposition 3.1.2 of [Reference DécoppetDéc23d] that $\mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{D}}(A\boxtimes B)$ is generated under Cauchy completion by the full sub-2-category on the objects of the form $(C\Box D)\boxtimes (A\Box B)$ with C in $\mathfrak{C}$ and D in $\mathfrak{D}$ . Likewise, it follows from its construction that $\mathbf{Mod}_{\mathfrak{C}}(A)\boxtimes\mathbf{Mod}_{\mathfrak{D}}(B)$ is generated under Cauchy completion by the full sub-2-category on the objects of the form $(C\Box A)\boxtimes (D\Box B)$ for every C in $\mathfrak{C}$ and D in $\mathfrak{D}$ . But we have that $\mathbf{K}((C\Box A)\boxtimes (D\Box B))=(C\Box D)\boxtimes (A\Box B)$ . Furthermore, it follows by combining Proposition 4.1 of [Reference DécoppetDéc24] and Lemma 3.2.13 of [Reference DécoppetDéc21] that $\mathbf{K}$ induces an equivalence between these two full sub-2-categories. This establishes that $\mathbf{K}$ is an equivalence as desired.

Lemma 2.26. Let $\mathfrak{C}$ and $\mathfrak{D}$ be two multifusion 2-categories. For any separable algebras A in $\mathfrak{C}$ and B in $\mathfrak{D}$ , there is an equivalence

\[\mathbf{Bimod}_{\mathfrak{C}}(A)\boxtimes\mathbf{Bimod}_{\mathfrak{D}}(B)\simeq\mathbf{Bimod}_{\mathfrak{C}\boxtimes\mathfrak{D}}(A\boxtimes B)\]

of multifusion 2-categories.

3.1 Definition

Let $\mathfrak{C}$ be a monoidal 2-category in the sense of [Reference Schommer-PriesSch11] with monoidal product $\Box$ and monoidal unit I. Thanks to [Reference GurskiGur13], we may assume without loss of generality that $\mathfrak{C}$ is strict cubical in that it satisfies Definition 2.26 of [Reference Schommer-PriesSch11]. This ensures that almost all of the coherence data for $\mathfrak{C}$ is given by identities, so that we will often omit the symbols $\Box$ and I. The only exception being the interchanger 2-isomorphism $\phi^{\Box}$ witnessing that the 2-functor $\Box$ respects the composition of 1-morphisms. We now recall from [Reference CransCra98] the definition of the Drinfeld center $\mathscr{Z}(\mathfrak{C})$ of $\mathfrak{C}$ , which is a braided monoidal 2-category (see also [Reference Baez and NeuchlBN95]). We do so using the variant of the graphical calculus of [Reference Garner and SchulmanGS16] introduced in [Reference DécoppetDéc21]. For the definition of the Drinfeld center of a general monoidal 2-category, we refer the reader to [Reference Kong, Tian and ZhouKTZ20].

An object of $\mathscr{Z}(\mathfrak{C})$ consists of an object Z of $\mathfrak{C}$ equipped with a half-braiding, that is, an adjoint 2-natural equivalence $b_Z$ given on C in $\mathfrak{C}$ by

\[(b_Z)_C:Z\Box C\rightarrow{\simeq} C\Box Z,\]

together with an invertible modification $R_Z$ given on C, D in $\mathfrak{C}$ by

\[(b_Z)_C:Z\Box C\rightarrow{\simeq} C\Box Z,\]

such that, for every objects C,D,E in $\mathfrak{C}$ , we have

(1)

in $Hom_{\mathfrak{C}}(ZCDE, CDEZ)$ .

A 1-morphism in $\mathscr{Z}(\mathfrak{C})$ from $(Y,b_Y,R_Y)$ to $(Z,b_Z,R_Z)$ consists of a 1-morphism $f:Y\rightarrow Z$ in $\mathfrak{C}$ together with an invertible modification $R_{f}$ given on C in $\mathfrak{C}$ by

such that, for every objects C,D in $\mathfrak{C}$ , we have

in $Hom_{\mathfrak{C}}(YCD, CDZ)$ .

A 2-morphism in $\mathscr{Z}(\mathfrak{C})$ from $(f,R_f)$ to $(g,R_g)$ , two 1-morphisms from $(Y,b_y,R_Y)$ to $(Z,b_Z,R_Z)$ , is a 2-morphism $\gamma:f\Rightarrow g$ in $\mathfrak{C}$ such that, for every object C in $\mathfrak{C}$ , we have

in $Hom_{\mathfrak{C}}(YC, CZ)$ .

The identity 1-morphism on the object $(Z,b_Z,R_Z)$ of $\mathscr{Z}(\mathfrak{C})$ is given by $(Id_Z, Id_{b_Z})$ . Given two 1-morphisms $(f,R_f)$ from $(X,b_X,R_X)$ to $(Y,b_Y,R_Y)$ and $(g,R_g)$ from $(Y,b_Y,R_Y)$ to $(Z,b_Z,R_Z)$ in $\mathscr{Z}(\mathfrak{C})$ their composite is the 1-morphism $(g\circ f, (R_g\circ f1)\cdot (1g\circ R_f))$ . This endows $\mathscr{Z}(\mathfrak{C})$ with the structure of a strict 2-category.

The monoidal structure of $\mathscr{Z}(\mathfrak{C})$ is given as follows. The monoidal unit is I, the monoidal unit of $\mathfrak{C}$ , equipped with the identity adjoint 2-natural equivalence, and identity modification. The monoidal product of two objects $(Y,b_Y,R_Y)$ and $(Z,b_Z,R_Z)$ of $\mathscr{Z}(\mathfrak{C})$ is given by

\[(Y,b_Y,R_Y)\Box (Z,b_Z,R_Z) = (Y\Box Z, (b_Y\Box 1) \circ (1\Box b_Z), R_{YZ}),\]

where $R_{YZ}$ is the invertible modification given on C, D in $\mathfrak{C}$ by

in $Hom_{\mathfrak{C}}(YZCD,CDYZ)$ . The monoidal product of two 1-morphisms $(f,R_f)$ and $(g,R_g)$ of $\mathscr{Z}(\mathfrak{C})$ is given by $(f,R_f)\Box (g,R_g) = (f\Box g,(R_f\Box g)\cdot (f\Box R_g)).$ Finally, the monoidal product of two 2-morphisms $\gamma$ and $\delta$ in $\mathscr{Z}(\mathfrak{C})$ is given by $\gamma\Box \delta$ . Using the interchange $\phi^{\Box}$ of the monoidal product of $\mathfrak{C}$ , we can upgrade the above assignment to a 2-functor that defines a strict cubical monoidal structure on $\mathscr{Z}(\mathfrak{C})$ as in [Reference CransCra98]. Furthermore, it follows from the definitions that the forgetful 2-functor $\mathscr{Z}(\mathfrak{C})\rightarrow \mathfrak{C}$ is monoidal.

Finally, in the notation of [Reference Schommer-PriesSch11], the braiding on $\mathscr{Z}(\mathfrak{C})$ is given as follows. For any two objects $(Y,b_Y,R_Y)$ and $(Z,b_Z,R_Z)$ of $\mathscr{Z}(\mathfrak{C})$ , we define the braiding b by $b_{Y,Z}:= (b_Y)_Z$ . Furthermore, b is promoted to an adjoint 2-natural equivalence in an obvious way using the definitions of the objects and 1-morphisms in $\mathscr{Z}(\mathfrak{C})$ . For any three objects $(X,b_X,R_X)$ , $(Y,b_Y,R_Y)$ , and $(Z,b_Z,R_Z)$ of $\mathscr{Z}(\mathfrak{C})$ , we also define invertible modifications $R_{X,Y,Z}:= (R_X)_{Y,Z}$ and $S_{X,Y,Z}:= Id_{XYZ}$ . It follows from [Reference CransCra98] that these assignments define a braiding on $\mathscr{Z}(\mathfrak{C})$ .

3.2 Elementary properties

As $\mathfrak{C}$ is a monoidal 2-category, it has the canonical structure of a $\mathfrak{C}$ - $\mathfrak{C}$ -bimodule 2-category (see Definition 2.1.1 of [Reference DécoppetDéc21]). We begin with the following familiar observation, which will be generalized in Proposition 3.6 below.

Using $\mathfrak{C}^{mop}$ to denote $\mathfrak{C}$ with the opposite monoidal structure, we may equivalently think of $\mathfrak{C}$ as a $\mathfrak{C}$ - $\mathfrak{C}$ -bimodule 2-category or as a left $\mathfrak{C}\times\mathfrak{C}^{mop}$ -module 2-category. By combining the above lemma with Propositions 4.2.3 and 4.2.4 of [Reference DécoppetDéc23b], we therefore obtain the following result.

We also record the following lemma, where $\mathscr{Z}(\mathfrak{C})^{rev}$ denotes the braided monoidal 2-category obtained from $\mathscr{Z}(\mathfrak{C})$ by using the opposite braiding. More precisely, given $(Y,b_Y,R_Y)$ and $(Z,b_Z,R_Z)$ in $\mathscr{Z}(\mathfrak{C})^{rev}$ , the braiding on the objects is given by $b^{\bullet}_{Y,Z}:Y\Box Z\rightarrow Z\Box Y$ , the prescribed adjoint equivalence of $b_{Y,Z}$ .

From now on, we will work over a fixed field $\mathbb{k}$ . In particular, we will assume that $\mathfrak{C}$ is a monoidal $\mathbb{k}$ -linear 2-category.

Let $\mathfrak{C}$ and $\mathfrak{D}$ be two Cauchy complete monoidal linear 2-categories. It follows from Theorem 5.2 of [Reference DécoppetDéc24] that the completed tensor product $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ inherits a monoidal structure. It is therefore natural to attempt to compare its Drinfeld center with that of $\mathfrak{C}$ and $\mathfrak{D}$ . To this end, we prove the following technical result.

Proof. We claim that there is a bilinear 2-functor

\[\begin{array}{ccc}\mathscr{Z}(\mathfrak{C})\times\mathscr{Z}(\mathfrak{D}) & \rightarrow &\mathscr{Z}(\mathfrak{C}\widehat{\otimes}\mathfrak{D})\\(Y,Z) & \mapsto & Y\widehat{\otimes}Z\end{array}\]

Namely, given Y in $\mathscr{Z}(\mathfrak{C})$ and Z in $\mathscr{Z}(\mathfrak{D})$ , we can equip the object $Y\widehat{\otimes} Z$ of $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ with the adjoint 2-natural equivalence $b_Y\widehat{\otimes} b_Z$ , which is defined for all objects of $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ of the form $C\widehat{\otimes} D$ . But, by the construction given in the proof of Proposition 3.4 of [Reference DécoppetDéc24], $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ is the Cauchy completion of its full sub-2-category on the objects of the form $C\widehat{\otimes} D$ with C in $\mathfrak{C}$ and D in $\mathfrak{D}$ . Thus, thanks to the 3-universal property of the Cauchy completion (see [Reference DécoppetDéc22a]), $b_Y\widehat{\otimes} b_Z$ can be canonically extended to an adjoint 2-natural equivalence defined on all the objects of $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ . An analogous argument shows that the invertible modification $R_Y\widehat{\otimes} R_Z$ can be canonically extended to all of $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ . Furthermore, it follows from its construction that this extension satisfies equation (1), so that $Y\widehat{\otimes} Z$ is indeed an object of $\mathscr{Z}(\mathfrak{C}\widehat{\otimes}\mathfrak{D})$ . This assignment is extended to morphisms using the same techniques.

Thanks to Lemma 3.4 above and the 3-universal property of the completed tensor product obtained in Proposition 3.4 of [Reference DécoppetDéc24], the bilinear 2-functor above induces a linear 2-functor $\mathscr{Z}(\mathfrak{C})\widehat{\otimes}\mathscr{Z}(\mathfrak{D})\rightarrow\mathscr{Z}(\mathfrak{C}\widehat{\otimes}\mathfrak{D})$ . By construction, this is a braided monoidal 2-functor, and it is compatible with the forgetful monoidal 2-functors.

3.3 Morita invariance

Let us now fix a multifusion 2-category $\mathfrak{C}$ over an algebraically closed field $\mathbb{k}$ of characteristic zero, and a separable left $\mathfrak{C}$ -module 2-category $\mathfrak{M}$ . Thanks to Proposition 2.2.8 of [Reference DécoppetDéc21], we may assume without loss of generality that $\mathfrak{C}$ and $\mathfrak{M}$ are strict cubical in the sense of Definition 2.2.7 of [Reference DécoppetDéc21]. Now, it is shown in Proposition 4.1.10 of [Reference DécoppetDéc23b] that $\mathfrak{M}$ can be canonically viewed as a left $\mathfrak{C}\times\mathfrak{C}_{\mathfrak{M}}^*$ -module 2-category. Furthermore, this left action is multilinear, so that we may view $\mathfrak{M}$ as a left $\mathfrak{C}\boxtimes\mathfrak{C}_{\mathfrak{M}}^*$ -module 2-category.

Proposition 3.6. Let $\mathfrak{M}$ be a faithful separable left $\mathfrak{C}$ -module 2-category. Then there is an equivalence

\[(\mathfrak{C}\boxtimes\mathfrak{C}_{\mathfrak{M}}^*)_{\mathfrak{M}}^*\simeq\mathscr{Z}(\mathfrak{C})\]

of monoidal 2-categories.

Given $(Z,b_Z,R_Z)$ in $\mathscr{Z}(\mathfrak{C})$ , we can consider the 2-endofunctor $\mathbf{J}(Z)$ on $\mathfrak{M}$ given by $M\mapsto Z\Box M$ . For any F in $\mathfrak{C}_{\mathfrak{M}}^*$ , the left $\mathfrak{C}$ -module structure on F yields an adjoint 2-natural equivalence $(\chi^F_{Z,M})^{\bullet}:F(Z\Box M)\simeq Z\Box F(M)$ and invertible modification, which endow $\mathbf{J}(Z)$ with a canonical left $\mathfrak{C}_{\mathfrak{M}}^*$ -module structure. Furthermore, $\mathbf{J}(Z)$ carries a compatible left $\mathfrak{C}$ -module structure given by $b^{\bullet}_{Z,C}\Box M: C\Box Z\Box M\simeq Z\Box C\Box M$ . This assignment can be extended to morphisms in the obvious way, so that we have a 2-functor $\mathbf{J}:\mathscr{Z}(\mathfrak{C})\rightarrow\mathbf{End}_{\mathfrak{C}\times\mathfrak{C}_{\mathfrak{M}}^*}(\mathfrak{M})$ . Furthermore, it follows by inspection that $\mathbf{J}$ is monoidal.

Now, Corollary 5.4.4 of [Reference DécoppetDéc23b] implies that the canonical monoidal 2-functor $\mathfrak{C}\rightarrow \mathbf{End}_{\mathfrak{C}_{\mathfrak{M}}^*}(\mathfrak{M})$ given by $C\mapsto \{M\mapsto C\Box M\}$ is an equivalence. In particular, objects of $\mathbf{End}_{\mathfrak{C}\times\mathfrak{C}_{\mathfrak{M}}^*}(\mathfrak{M})$ are identified with 2-endofunctors $M\mapsto Z\Box M$ for some Z in $\mathfrak{C}$ equipped with a left action by $\mathfrak{C}$ . More precisely, for every C in $\mathfrak{C}$ , we have a left $\mathfrak{C}_{\mathfrak{M}}^*$ -module adjoint 2-natural equivalence $C\Box Z\Box M \simeq Z\Box C\Box M$ . By the above equivalence of monoidal 2-categories, this corresponds exactly to an adjoint 2-natural equivalence $b_{Z,C}^{\bullet}:C\Box Z \simeq Z\Box C$ . An analogous argument shows that the invertible modification witnessing the coherence of the left $\mathfrak{C}$ -action corresponds precisely to the data of an invertible modification $R_Z$ satisfying equation (1). This shows that $\mathbf{J}$ is essentially surjective on objects. One proceeds similarly to show that $\mathbf{J}$ is essentially surjective on 1-morphisms and fully faithful on 2-morphisms. This concludes the proof.

Were now ready to prove the main theorem of this section.

Now, given any object Z in $\mathscr{Z}(\mathfrak{C})$ , the left $\mathfrak{C}$ -module 2-endofunctor $M\mapsto Z\Box M$ can be canonically upgraded to an object of $\mathscr{Z}(\mathfrak{C}_{\mathfrak{M}}^*)$ . Namely, for every object F of $\mathfrak{C}_{\mathfrak{M}}^*$ , the left $\mathfrak{C}$ -module structure on F provides us with an adjoint 2-natural equivalence $\chi^F_{Z,M}:Z\Box F(M)\simeq F(Z\Box M)$ , and an invertible modification satisfying equation (1). This defines a half-braiding on $M\mapsto Z\Box M$ , and yields a monoidal 2-functor $\mathbf{L}:\mathscr{Z}(\mathfrak{C})\rightarrow\mathscr{Z}(\mathfrak{C}_{\mathfrak{M}}^*)$ . Furthermore, observe that the following diagram of monoidal 2-functors is weakly commutative.

It follows from the first part of the proof that both $\mathbf{J}$ and $\mathbf{K}$ are equivalences, so that $\mathbf{L}$ is an equivalence of monoidal 2-categories.

Finally, let $(Y,b_Y,R_Y)$ and $(Z,b_Z,R_Z)$ be objects of $\mathscr{Z}(\mathfrak{C})$ . By the definition recalled in § 3.1 above, the braiding on $\mathbf{L}(Y)$ and $\mathbf{L}(Z)$ in $\mathscr{Z}(\mathfrak{C}_{\mathfrak{M}}^*)$ is given by

\[b_{\mathbf{L}(Y),\mathbf{L}(Z)}(M) = \chi^{\mathbf{L}(Z)}_{Y,M}= b^{\bullet}_{Z,Y}\Box M: Y\Box Z\Box M\rightarrow Z\Box Y\Box M.\]

Thus, $\mathbf{L}$ yields an equivalence $\mathscr{Z}(\mathfrak{C})\rightarrow\mathscr{Z}(\mathfrak{C}_{\mathfrak{M}}^*)^{rev}$ of braided monoidal 2-categories. This concludes the proof.

Corollary 3.9. There is an equivalence of braided fusion 2-categories

\[\mathscr{Z}(\mathbf{2Vect}_G)\simeq\mathscr{Z}(\mathbf{2Rep}(G)).\]

In addition, we also get an equivalence between the 2-categories of braided rigid algebras. More precisely, we get an equivalence between the 2-category of G-crossed braided multifusion 1-categories (see Example 2.22) and that of braided multifusion 1-categories equipped with a braided monoidal functor from $\mathbf{Rep}(G)$ (see Table 1 of [Reference Davydov and NikshychDN21]). This is a well-known result in the theory of fusion 1-categories, which dates back to [Reference KirillovKir01, Reference MügerMüg04].

4.1 Comparison with Witt equivalence

We begin by introducing a notion of Witt equivalence between arbitrary braided fusion 1-categories.

Given a braided fusion 1-category $\mathcal{B}$ with braiding $\beta$ , we use $\mathcal{Z}_{(2)}(\mathcal{B})$ to denote the symmetric center of $\mathcal{B}$ , that is, the full fusion sub-1-category of $\mathcal{B}$ on those objects B for which

\[\beta_{C,B}\circ \beta_{B,C} = Id_{B\otimes C}\]

for every object C of $\mathcal{B}$ . It follows from the definition that $\mathcal{Z}_{(2)}(\mathcal{B})$ is a symmetric fusion 1-category.

Let $\mathcal{C}$ be a $\mathcal{B}$ -central fusion 1-category $\mathcal{C}$ . We write $\beta$ for the braiding on $\mathcal{Z}(\mathcal{C})$ , and $F:\mathcal{B}\rightarrow \mathcal{Z}(\mathcal{C})$ for the braided monoidal functor supplying $\mathcal{C}$ with its central structure. Then we use $\mathcal{Z}(\mathcal{C},\mathcal{B})$ to denote the centralizer of the image of $\mathcal{B}$ in $\mathcal{Z}(\mathcal{C})$ , that is, the full fusion sub-1-category of $\mathcal{Z}(\mathcal{C})$ on those objects Z for which

\[\beta_{F(B),Z}\circ \beta_{Z,F(B)} = Id_{Z\otimes F(B)}\]

for every object B of $\mathcal{B}$ .

Let $\mathcal{E}$ be a symmetric fusion 1-category. A braided fusion 1-category over $\mathcal{E}$ , also known as an $\mathcal{E}$ -central braided fusion 1-category, is a pair $(\mathcal{B},F)$ consisting of a braided fusion 1-category $\mathcal{B}$ and a braided embedding $F:\mathcal{E}\hookrightarrow\mathcal{Z}_{(2)}(\mathcal{B})$ . We say that $(\mathcal{B},F)$ , a braided fusion 1-category over $\mathcal{E}$ , is non-degenerate if F induces an equivalence $\mathcal{E}\simeq\mathcal{Z}_{(2)}(\mathcal{E})$ . In [Reference Davydov, Nikshych and OstrikDNO13], the authors introduced a notion of Witt equivalence over $\mathcal{E}$ between non-degenerate braided fusion 1-categories over $\mathcal{E}$ . More precisely, $(\mathcal{B}_1,F_1)$ and $(\mathcal{B}_2,F_2)$ are Witt equivalent over $\mathcal{E}$ if there exists a fusion 1-category $\mathcal{C}$ , together with a fully faithful braided embedding $\mathcal{B}_1\hookrightarrow \mathcal{Z}(\mathcal{C})$ and a braided monoidal equivalence $\mathcal{B}_2\simeq \mathcal{Z}(\mathcal{C},\mathcal{B}_1)^{rev}$ , such that the two braided functors $\mathcal{E}\rightarrow\mathcal{Z}(\mathcal{C})$ are equivalent. We now explain how this is related to the notion of Witt equivalence we have introduced above.

Proof. Let us fix an identification $F_1:\mathcal{E}\simeq \mathcal{Z}_{(2)}(\mathcal{B}_1)$ . If there exists a fusion 1-category $\mathcal{C}$ , together with a fully faithful braided embedding $\mathcal{B}_1\hookrightarrow \mathcal{Z}(\mathcal{C})$ and a braided monoidal equivalence $\mathcal{B}_2\simeq \mathcal{Z}(\mathcal{C},\mathcal{B}_1)^{rev}$ , then there is a canonical identification $F_2:\mathcal{E}\simeq \mathcal{Z}_{(2)}(\mathcal{B}_2)$ such that

\[\mathcal{B}_1\boxtimes_{\mathcal{E}}\mathcal{B}_2^{rev}\simeq\mathcal{B}_1\boxtimes_{\mathcal{E}}\mathcal{Z}(\mathcal{C},\mathcal{B}_1)\simeq\mathcal{Z}(\mathcal{C},\mathcal{E})\]

by Proposition 4.3 of [Reference Davydov, Nikshych and OstrikDNO13]. This shows that $(\mathcal{B}_1,F_1)$ and $(\mathcal{B}_2^{rev},F_2^{rev})$ define opposite elements in the Witt group $\mathcal{W}(\mathcal{E})$ in the sense of [Reference Davydov, Nikshych and OstrikDNO13], so that $(\mathcal{B}_1,F_1)$ and $(\mathcal{B}_2,F_2)$ are Witt equivalent over $\mathcal{E}$ in the sense of Definition 5.1 of [Reference Davydov, Nikshych and OstrikDNO13]. The converse follows by running the argument in reverse.

We can now state the main theorem of this section, which completely characterizes which connected fusion 2-categories are Morita equivalent. In particular, it generalizes Example 5.4.6 of [Reference DécoppetDéc23b]. The proof relies on the technical results derived in § 4.2 below.

Proof. Assume that $\mathbf{Mod}(\mathcal{B}_1)$ and $\mathbf{Mod}(\mathcal{B}_2)$ are Morita equivalent fusion 2-categories. Then it follows from Theorem 3.7 that

\[\mathscr{Z}(\mathbf{Mod}(\mathcal{B}_1))\simeq \mathscr{Z}(\mathbf{Mod}(\mathcal{B}_2))\]

as braided fusion 2-categories. Thanks to Lemma 2.16 of [Reference Johnson-Freyd and ReutterJR24] or Corollary 4.13 below, we find that

\[\mathcal{Z}_{(2)}(\mathcal{B}_1)\simeq\Omega\mathscr{Z}(\mathbf{Mod}(\mathcal{B}_1))\simeq\Omega\mathscr{Z}(\mathbf{Mod}(\mathcal{B}_2))\simeq \mathcal{Z}_{(2)}(\mathcal{B}_2)\]

as symmetric fusion 1-categories. We denote this symmetric fusion 1-category by $\mathcal{E}$ . Now, as $\mathbf{Mod}(\mathcal{B}_1)$ and $\mathbf{Mod}(\mathcal{B}_2)$ are Morita equivalent fusion 2-categories, there exist a fusion 1-category $\mathcal{C}$ equipped with a braided monoidal functor $F:\mathcal{B}_1\rightarrow \mathcal{Z}(\mathcal{C})$ , and a monoidal equivalence $\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B}_1)}(\mathcal{C})\simeq \mathbf{Mod}(\mathcal{B}_2)$ of fusion 2-categories. Let us write $\mathcal{A}$ for the image of $\mathcal{B}_1$ in $\mathcal{Z}(\mathcal{C})$ , which is a braided fusion 1-category by definition. Then it follows from Proposition 4.3 of [Reference Davydov, Nikshych and OstrikDNO13] and from the definitions that

\[\mathcal{Z}_{(2)}(\mathcal{A})\simeq \mathcal{Z}_{(2)}(\mathcal{Z}(\mathcal{C},\mathcal{A}))\simeq\mathcal{Z}_{(2)}(\mathcal{Z}(\mathcal{C},\mathcal{B}_1))\]

as symmetric fusion 1-categories. On the other hand, by Lemma 4.11 below, we find that $\mathcal{Z}(\mathcal{C},\mathcal{B}_1)^{rev}\simeq \mathcal{B}_2$ as braided fusion 1-categories. This implies that $\mathcal{Z}_{(2)}(\mathcal{A})\simeq \mathcal{E}$ . Thence, it follows from Corollary 3.24 of [Reference Davydov, Müger, Nikshych and OstrikDMNO13] that F is fully faithful, so that $\mathcal{B}_1$ and $\mathcal{B}_2$ are Witt equivalent.

On the other hand, if $\mathcal{B}_1$ and $\mathcal{B}_2$ are Witt equivalent, then, by definition, there exist a $\mathcal{B}_1$ -central fusion 1-category $\mathcal{C}$ such that $\mathcal{B}_1\rightarrow \mathcal{Z}(\mathcal{C})$ is a fully faithful embedding, and an equivalence $\mathcal{Z}(\mathcal{C},\mathcal{B}_1)^{rev}\simeq \mathcal{B}_2$ of braided fusion 1-categories. By Corollary 4.16 below, the $\mathcal{B}_1$ -central fusion 1-category $\mathcal{C}$ witnesses the desired Morita equivalence.

The technical results used in the proof of the above theorem will be established below. But first, we record the following corollary. Let us fix $\mathcal{E}$ a symmetric fusion 1-category. A non-degenerate braided fusion 1-category $\mathcal{B}$ equipped with a fully faithful embedding of $\mathcal{E}$ is called minimal if the centralizer of $\mathcal{E}$ in $\mathcal{B}$ is exactly $\mathcal{E}$ . The set of minimal non-degenerate extensions of $\mathcal{E}$ forms a group (see [Reference Lan, Kong and WenLKW17]), which is denoted by $Mext(\mathcal{E})$ . We say that a minimal non-degenerate extension $\mathcal{B}$ of $\mathcal{E}$ is Witt trivial if the class of the non-degenerate braided fusion 1-category $\mathcal{B}$ is trivial in the Witt group $\mathcal{W}$ . This is equivalent to requiring that $\mathcal{B}$ is braided equivalent to the Drinfeld center of a fusion 1-category. We note that the set of Witt trivial minimal non-degenerate extensions of $\mathcal{E}$ is a subgroup of $Mext(\mathcal{E})$ thanks to Theorem 3.6.20 of [Reference SunSun22].

Proof. It follows readily from Corollary 4.5 that every Morita autoequivalence of $\mathbf{Mod}(\mathcal{E})$ is given by an $\mathcal{E}$ -central fusion 1-category $\mathcal{C}$ such that $\mathcal{Z}(\mathcal{C},\mathcal{E})=\mathcal{E}$ together with a braided autoequivalence of $\mathcal{E}$ . On the other hand, any two $\mathcal{E}$ -central fusion 1-categories $\mathcal{C}_1$ and $\mathcal{C}_2$ with $\mathcal{Z}(\mathcal{C}_1,\mathcal{E})=\mathcal{E}=\mathcal{Z}(\mathcal{C}_2,\mathcal{E})$ can induce the same Morita equivalence only if $\mathbf{Mod}(\mathcal{C}_1)$ and $\mathbf{Mod}(\mathcal{C}_2)$ are equivalent as left $\mathbf{Mod}(\mathcal{E})$ -module 2-categories. This implies that $\mathcal{C}_1$ and $\mathcal{C}_2$ are Morita equivalent fusion 1-categories so that we have an equivalence $\mathcal{Z}(\mathcal{C}_1)\simeq\mathcal{Z}(\mathcal{C}_2)$ of braided fusion 1-categories (see Theorem 1.1 of [Reference Etingof, Nikshych and OstrikENO09]). Finally, $\mathbf{Mod}(\mathcal{C}_1)$ and $\mathbf{Mod}(\mathcal{C}_2)$ are equivalent as left $\mathbf{Mod}(\mathcal{E})$ -module 2-categories if and only if the diagram of braided monoidal functors

commutes up to braided natural equivalence. This is the case if and only if $\mathcal{Z}(\mathcal{C}_1)$ and $\mathcal{Z}(\mathcal{C}_2)$ are equivalent minimal non-degenerate extensions of $\mathcal{E}$ . This concludes the proof.

On the other hand, the above corollary shows that $\mathrm{BrPic}(\mathbf{2SVect})\simeq 0$ . Namely, the group of minimal non-degenerate extensions of $\mathbf{SVect}$ is identified with $\mathbb{Z}/16\mathbb{Z}$ by Example 2.2 of [Reference NikshychNik22], and only one of the 16 minimal non-degenerate extensions of $\mathbf{SVect}$ is Witt trivial (see Appendix A.3.2 of [Reference Drinfeld, Gelaki, Nikshych and OstrikDGNO10]). But every minimal non-degenerate extension $\mathcal{C}$ of $\mathbf{SVect}$ induces a braided autoequivalence of $\mathscr{Z}(\mathbf{2SVect})$ . In detail, the left $\mathbf{2SVect}$ -module 2-category $\mathbf{Mod}(\mathcal{C})$ induces a braided monoidal 2-functor $\mathscr{Z}(\mathbf{2SVect})\simeq \mathscr{Z}(\mathbf{2SVect}\boxtimes\mathbf{Mod}(\mathcal{C}))$ , and, thanks to Corollary 6.17 below, the right-hand side is canonically equivalent to $\mathscr{Z}(\mathbf{2SVect})\boxtimes\mathscr{Z}(\mathbf{Mod}(\mathcal{C}))\simeq\mathscr{Z}(\mathbf{2SVect})$ as a braided monoidal 2-category. In fact, it was sketched in § 2.3 of [Reference Johnson-FreydJoh21] that this construction induces a bijection of sets $Aut^{br}(\mathscr{Z}(\mathbf{2SVect}))\cong \mathbb{Z}/16\mathbb{Z}$ . We therefore expect that the map $\Phi:\mathrm{BrPic}(\mathbf{2SVect})\rightarrow Aut^{br}(\mathscr{Z}(\mathbf{2SVect}))$ is not an isomorphism, so that the full version of Theorem 1.1 of [Reference Etingof, Nikshych and OstrikENO09] cannot be naïvely categorified.

Let $\mathcal{E}$ be a symmetric fusion 1-category. The group $\mathcal{W}(\mathcal{E})$ of classes of non-degenerate braided fusion 1-categories over $\mathcal{E}$ up to Witt equivalence over $\mathcal{E}$ was introduced in [Reference Davydov, Nikshych and OstrikDNO13]. The group $Aut^{br}(\mathcal{E})$ acts on $\mathcal{W}(\mathcal{E})$ . Namely, given $E:\mathcal{E}\simeq\mathcal{E}$ an equivalence of symmetric fusion 1-categories, and $(\mathcal{B},F)$ a non-degenerate braided fusion 1-category over $\mathcal{E}$ , we can consider $(\mathcal{B},F\circ E)$ . Furthermore, it is clear that this assignment is compatible with Witt equivalence over $\mathcal{E}$ . We write $\widehat{\mathcal{W}}(\mathcal{E})$ for the quotient of $\mathcal{W}(\mathcal{E})$ under the action of $Aut^{br}(\mathcal{E})$ . The next result follows by combing Lemma 4.2 with Theorem 4.4.

Remark 4.10. Let us fix a symmetric fusion 1-category $\mathcal{E}$ . A $\mathbf{Mod}(\mathcal{E})$ -central fusion 2-category, or a fusion 2-category over $\mathbf{Mod}(\mathcal{E})$ , is a fusion 2-category $\mathfrak{C}$ equipped with a braided monoidal 2-functor $\mathbf{F}:\mathbf{Mod}(\mathcal{E})\rightarrow \mathscr{Z}(\mathfrak{C})$ . A Morita equivalence between two $\mathbf{Mod}(\mathcal{E})$ -central fusion 2-categories $(\mathfrak{C}_1,\mathbf{F}_1)$ and $(\mathfrak{C}_2,\mathbf{F}_1)$ is a Morita equivalence between the underlying fusion 2-categories $\mathfrak{C}_1$ and $\mathfrak{C}_2$ such that the induced diagram of braided monoidal 2-functors

commutes up to braided monoidal 2-natural equivalence. Now, the data of a $\mathbf{Mod}(\mathcal{E})$ -central structure for a connected fusion 2-category $\mathbf{Mod}(\mathcal{B})$ corresponds exactly to a symmetric monoidal functor $F:\mathcal{E}\rightarrow \mathcal{Z}_{(2)}(\mathcal{B})$ . Namely, we have $\Omega\mathscr{Z}(\mathbf{Mod}(\mathcal{B}))\simeq \mathbf{Mod}(\mathcal{Z}_{(2)}(\mathcal{B}))$ , and any monoidal 2-functor $\mathbf{F}:\mathbf{Mod}(\mathcal{E})\rightarrow\mathscr{Z}(\mathbf{Mod}(\mathcal{B}))$ is completely determined by the braided monoidal functor $\Omega\mathbf{F}$ . In particular, if $\mathcal{B}_1$ and $\mathcal{B}_2$ are braided fusion 1-categories, and $F_1:\mathcal{E}\simeq\mathcal{Z}_{(2)}(\mathcal{B}_1)$ and $F_2:\mathcal{E}\simeq\mathcal{Z}_{(2)}(\mathcal{B}_2)$ are equivalences of symmetric fusion 1-categories, it follows from Theorem 4.4 that the $\mathbf{Mod}(\mathcal{E})$ -central fusion 2-categories $(\mathbf{Mod}(\mathcal{B}_1),\mathbf{Mod}(F_1))$ and $(\mathbf{Mod}(\mathcal{B}_2),\mathbf{Mod}(F_1))$ are Morita equivalent if and only if $(\mathcal{B}_1, F_1)$ and $(\mathcal{B}_2, F_2)$ are Witt equivalent over $\mathcal{E}$ in the sense of [Reference Davydov, Nikshych and OstrikDNO13].

4.2 Technical results

We establish the technical lemmas that were used in the proof of Theorem 4.4.

Lemma 4.11. Let $\mathcal{B}$ be a braided fusion 1-category, and $\mathcal{C}$ a $\mathcal{B}$ -central fusion 1-category. There is an equivalence

\[\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})^0\simeq\mathbf{Mod}(\mathcal{Z}(\mathcal{C},\mathcal{B})^{rev})\]

of monoidal 2-categories.

Proof. We have seen in Example 2.20 above that there is an equivalence of 2-categories

\[\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})\simeq\mathbf{Mod}(\mathcal{C}^{mop}\boxtimes_{\mathcal{B}}\mathcal{C}).\]

Furthermore, as observed in Remark 2.21, the 2-category on the left-hand side is the 2-category of finite semisimple $\mathcal{B}$ -balanced $\mathcal{C}$ - $\mathcal{C}$ -bimodule 1-categories as considered in § 3.4 of [Reference LaugwitzLau20]. Now, $\mathcal{C}$ with its canonical $\mathcal{B}$ -balanced $\mathcal{C}$ - $\mathcal{C}$ -bimodule structure is the monoidal unit of $\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})$ . Let us write $End^{\mathcal{B}}_{\mathcal{C}\textrm{-}\mathcal{C}}(\mathcal{C})$ for the braided fusion 1-category of endomorphisms of $\mathcal{C}$ in $\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})$ . Now, observe that the forgetful 2-functor

\[\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})\rightarrow\mathbf{Bimod}(\mathcal{C})\]

is monoidal. On the monoidal unit, this 2-functor induces a braided monoidal functor $End^{\mathcal{B}}_{\mathcal{C}\textrm{-}\mathcal{C}}(\mathcal{C})\rightarrow End_{\mathcal{C}\textrm{-}\mathcal{C}}(\mathcal{C})$ . But, by Example 5.3.7 of [Reference DécoppetDéc23b], we know that there is an equivalence $End_{\mathcal{C}\textrm{-}\mathcal{C}}(\mathcal{C})\simeq\mathcal{Z}(\mathcal{C})^{rev}$ of braided fusion 1-categories. Moreover, it follows from Definition 3.24 of [Reference LaugwitzLau20] that $End^{\mathcal{B}}_{\mathcal{C}\textrm{-}\mathcal{C}}(\mathcal{C})$ is the full fusion sub-1-category of $End_{\mathcal{C}\textrm{-}\mathcal{C}}(\mathcal{C})\simeq\mathcal{Z}(\mathcal{C})^{rev}$ on those objects centralizing the image of $\mathcal{B}^{rev}$ in $\mathcal{Z}(\mathcal{C})^{rev}$ . Thus, we find that $End^{\mathcal{B}}_{\mathcal{C}\textrm{-}\mathcal{C}}(\mathcal{C})\simeq\mathcal{Z}(\mathcal{C},\mathcal{B})^{rev}$ as braided fusion 1-categories. Finally, the proof is concluded by appealing to Proposition 2.4.7 of [Reference DécoppetDéc22b].

We recover Lemma 2.16 of [Reference Johnson-Freyd and ReutterJR24].

Corollary 4.13 Let $\mathcal{B}$ be a braided fusion 1-category. Then we have

\[\Omega\mathscr{Z}(\mathbf{Mod}(\mathcal{B}))\simeq \mathcal{Z}_{(2)}(\mathcal{B}).\]

We now give a sufficient criterion for the fusion 2-category $\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})$ to be connected.

We need the following slightly more general version of the previous lemma.

Proof. Let us consider the fusion 2-category $\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{Z}(\mathcal{C}))$ , where $\mathcal{Z}(\mathcal{C})$ is endowed with its canonical $\mathcal{B}$ -central structure. Thanks to Lemma 4.14 above, it is connected. On the other hand, by Theorem 2.23 above, there is an equivalence of fusion 2-categories

\[\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{Z}(\mathcal{C}))\simeq\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C}^{mop}\boxtimes\mathcal{C}).\]

More precisely, $\mathcal{C}^{mop}\boxtimes\mathcal{C}$ is equipped with the braided monoidal functor

\[\mathcal{B}\rightarrow{F}\mathcal{Z}(\mathcal{C})\hookrightarrow\mathcal{Z}(\mathcal{C}^{mop})\boxtimes\mathcal{Z}(\mathcal{C})\simeq\mathcal{Z}(\mathcal{C}^{mop}\boxtimes\mathcal{C}).\]

Then, as $\mathcal{Z}(\mathcal{C})$ and $\mathcal{C}^{mop}\boxtimes\mathcal{C}$ are Morita equivalent fusion 1-categories, $\mathbf{Mod}(\mathcal{Z}(\mathcal{C}))$ and $\mathbf{Mod}(\mathcal{C}^{mop}\boxtimes\mathcal{C})$ are equivalent finite semisimple 2-categories. Furthermore, this Morita equivalence of fusion 1-categories is compatible with the $\mathcal{B}$ -central structures as the following diagram of braided monoidal functors commutes.

But, as the $\mathcal{B}$ -central structure on $\mathcal{C}^{mop}\boxtimes\mathcal{C}$ factors through $\mathcal{C}$ , we have that

\[\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C}^{mop}\boxtimes\mathcal{C})\simeq\mathbf{Bimod}(\mathcal{C}^{mop})\boxtimes \mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})\]

as fusion 2-categories. Finally, the fusion 2-categories $\mathbf{Bimod}(\mathcal{C}^{mop})$ and $\mathbf{Mod}(\mathcal{Z}(\mathcal{C}^{mop}))$ are equivalent (see Example 5.3.7 of [Reference DécoppetDéc23b]). But, thanks to Lemma 4.3 of [Reference DécoppetDéc24], the set of connected components of a 2-Deligne tensor product is the product of the set of connected components, so that $\mathbf{Bimod}_{\mathbf{Mod}(\mathcal{B})}(\mathcal{C})$ is indeed connected.

5.1 Strongly fusion and invertible

We begin by introducing invertible multifusion 2-categories, and we characterize them explicitly.

Before stating the first theorem of this section, we will establish a technical lemma. In order to do so, it is convenient to generalize the terminology of Definition 2.10.

Proof. Let $\mathfrak{C}$ be a bosonic fusion 1-category. We write $\mathcal{E}:=\mathcal{Z}_{(2)}(\Omega\mathfrak{C})$ . Thanks to Deligne’s theorem (see [Reference DeligneDel02]), there exists a symmetric monoidal functor $\mathcal{E}\rightarrow\mathbf{Vect}$ . We define a braided fusion 1-category $\mathcal{D}:=\Omega\mathfrak{C}\boxtimes_{\mathcal{E}}\mathbf{Vect}$ . By Proposition 4.30 of [Reference Drinfeld, Gelaki, Nikshych and OstrikDGNO10], $\mathcal{D}$ is non-degenerate. Furthermore, we view $\mathcal{D}$ as a connected rigid algebra in $\mathbf{Mod}(\Omega\mathfrak{C})\simeq \mathfrak{C}^0$ using the canonical braided monoidal functor $\Omega\mathfrak{C}\rightarrow \mathcal{D}\rightarrow\mathcal{Z}(\mathcal{D})$ . It follows from Example 2.20 that $\mathcal{D}$ is a separable algebra. Thus, by Theorem 5.4.3 of [Reference DécoppetDéc23b], the multifusion 2-category $\mathbf{Bimod}_{\mathfrak{C}}(\mathcal{D})$ is Morita equivalent to $\mathfrak{C}$ . Furthermore, as $\mathcal{D}$ is connected, it is necessarily indecomposable, so that it follows from § 5.2 of [Reference DécoppetDéc23b] that $\mathbf{Bimod}_{\mathfrak{C}}(\mathcal{D})$ is in fact a fusion 2-category, and we set $\mathfrak{D}:=\mathbf{Bimod}_{\mathfrak{C}}(\mathcal{D})$ . It remains to analyze $\mathfrak{D}^0$ . As $\mathcal{D}$ is an algebra in $\mathfrak{C}^0$ , Corollary 2.3.6 of [Reference DécoppetDéc22b] shows that the underlying object of any simple $\mathcal{D}$ - $\mathcal{D}$ -bimodule in $\mathfrak{C}$ is contained in a single connected component of $\mathfrak{C}$ . In particular, we have an equivalence

\[\mathfrak{D}^0\simeq (\mathbf{Bimod}_{\mathfrak{C}^0}(\mathcal{D}))^0\simeq(\mathbf{Bimod}_{\mathbf{Mod}(\Omega\mathfrak{C})}(\mathcal{D}))^0\]

of fusion 2-categories. Moreover, it was shown in Lemma 4.11 above that

\[(\mathbf{Bimod}_{\mathbf{Mod}(\Omega\mathfrak{C})}(\mathcal{D}))^0\simeq\mathbf{Mod}(\mathcal{Z}(\mathcal{D},\Omega\mathfrak{C}))\]

as fusion 2-categories. Finally, as $\mathcal{D}$ is non-degenerate, we have that $\mathcal{Z}(\mathcal{D})\simeq \mathcal{D}\boxtimes\mathcal{D}^{rev}$ by Proposition 3.7 of [Reference Drinfeld, Gelaki, Nikshych and OstrikDGNO10]. Furthermore, the canonical braided monoidal functor $\Omega\mathfrak{C}\rightarrow\mathcal{D}$ is surjective (in the sense of Definition 2.1 of [Reference Drinfeld, Gelaki, Nikshych and OstrikDGNO10]), so that $\mathcal{Z}(\mathcal{D},\Omega\mathfrak{C})\simeq \mathcal{D}^{rev}$ as braided fusion 1-categories. This concludes the proof in this case.

If $\mathfrak{C}$ is fermionic, then it follows from [Reference DeligneDel02] that there exists a dominant symmetric monoidal functor $\mathcal{E}\rightarrow\mathbf{SVect}$ . We then define a braided fusion 1-category $\mathcal{D}:=\Omega\mathfrak{C}\boxtimes_{\mathcal{E}}\mathbf{SVect}$ . By Proposition 4.30 of [Reference Drinfeld, Gelaki, Nikshych and OstrikDGNO10], $\mathcal{D}$ is slightly degenerate. Then the argument used above shows that $\mathfrak{D}:=\mathbf{Bimod}_{\mathbf{Mod}(\Omega\mathfrak{C})}(\mathcal{D})$ is a fusion 2-category with $\mathfrak{D}^0\simeq\mathbf{Mod}(\mathcal{Z}(\mathcal{D},\Omega\mathfrak{C}))$ as fusion 2-categories. It follows from Proposition 4.3 of [Reference Davydov, Nikshych and OstrikDNO13] that $\mathcal{Z}(\mathcal{D},\Omega\mathfrak{C})$ is slightly degenerate as desired, thereby finishing the proof.

Case 1. We assume that the fusion 2-category $\mathfrak{C}$ is bosonic. Then it follows from Lemma 5.5 that there exists a fusion 2-category $\mathfrak{D}$ that is Morita equivalent to $\mathfrak{C}$ , and such that $\mathcal{B}:=\Omega\mathfrak{D}$ is a non-degenerate braided fusion 1-category. Let us now define $\mathfrak{F}:= \mathfrak{D}\boxtimes \mathbf{Mod}(\mathcal{B}^{rev})$ , so that $\Omega\mathfrak{F}\simeq \mathcal{B}\boxtimes \mathcal{B}^{rev}$ by Proposition 5.1 of [Reference DécoppetDéc24]. In particular, equipping $\mathcal{B}$ with the canonical braided monoidal functor $\mathcal{B}\boxtimes \mathcal{B}^{rev}\rightarrow \mathcal{Z}(\mathcal{B})$ , we can view $\mathcal{B}$ as a separable algebra in the fusion 2-category $\mathbf{Mod}(\mathcal{B}\boxtimes\mathcal{B}^{rev})\simeq\mathfrak{F}^0$ . Let us write $\mathfrak{G}$ for the fusion 2-category given by $\mathbf{Bimod}_{\mathfrak{F}}(\mathcal{B})$ . By Example 5.3.7 of [Reference DécoppetDéc23b] (see also Lemma 4.15 above), we find that

\[\mathfrak{G}^0=\mathbf{Bimod}_{\mathfrak{F}^0}(\mathcal{B})\simeq \mathbf{2Vect}.\]

This shows that $\mathfrak{G}$ is a bosonic strongly fusion 2-category.

Finally, it follows from Lemma 2.26 that Morita equivalence is compatible with the 2-Deligne tensor product. In particular, as $\mathbf{2Vect}$ is Morita equivalent to $\mathbf{Mod}(\mathcal{B}^{rev})\boxtimes\mathbf{Mod}(\mathcal{B})$ , we have that $\mathfrak{C}$ is Morita equivalent to the fusion 2-category $\mathfrak{C}\boxtimes\mathbf{Mod}(\mathcal{B}^{rev})\boxtimes\mathbf{Mod}(\mathcal{B})$ . Furthermore, the above discussion shows that $\mathfrak{C}\boxtimes\mathbf{Mod}(\mathcal{B}^{rev})\boxtimes\mathbf{Mod}(\mathcal{B})$ is Morita equivalent to $\mathfrak{G}\boxtimes \mathbf{Mod}(\mathcal{B})$ . But Morita equivalence is an equivalence relation by Theorem 2.24, so that we get the desired result in this case.

Case 2. We assume that the fusion 2-category $\mathfrak{C}$ is fermionic. Then it follows from Lemma 5.5 that there exists a fusion 2-category $\mathfrak{D}$ that is Morita equivalent to $\mathfrak{C}$ , and such that $\mathcal{B}:=\Omega\mathfrak{D}$ is a slightly degenerate braided fusion 1-category. Now, thanks to the main theorem of [Reference Johnson-Freyd and ReutterJR24], there exists a non-degenerate braided fusion 1-category $\mathcal{C}$ together with a braided embedding $\mathcal{B}\subseteq \mathcal{C}$ , such that the centralizer of $\mathcal{B}$ in $\mathcal{C}$ is exactly $\mathbf{SVect}$ . Let us write $\mathfrak{E}$ for the fusion 2-category $\mathbf{Bimod}_{\mathfrak{D}}(\mathcal{C})$ . By Lemma 4.15 above, we find that

\[\mathfrak{E}^0=\mathbf{Bimod}_{\mathfrak{D}^0}(\mathcal{C})\simeq\mathbf{Mod}(\mathbf{SVect}\boxtimes\mathcal{C}^{rev}).\]

Namely, as $\mathcal{C}$ is a non-degenerate braided fusion 1-category, we have that $\mathcal{Z}(\mathcal{C})\simeq \mathcal{C}\boxtimes\mathcal{C}^{rev}$ by Proposition 3.7 of [Reference Drinfeld, Gelaki, Nikshych and OstrikDGNO10], so that $\mathcal{Z}(\mathcal{C},\mathcal{B})\simeq \mathbf{SVect}\boxtimes\mathcal{C}^{rev}$ as braided fusion 1-categories. Then we consider the fusion 2-category $\mathfrak{F}:=\mathfrak{E}\boxtimes \mathbf{Mod}(\mathcal{C})$ . Using the same argument as that used in the proof of the first case, we find that $\mathfrak{G}:=\mathbf{Bimod}_{\mathfrak{F}}(\mathcal{C})$ is a fermionic strongly fusion 2-category.

Finally, as $\mathbf{2Vect}$ is Morita equivalent to $\mathbf{Mod}(\mathcal{C})\boxtimes\mathbf{Mod}(\mathcal{C}^{rev})$ , we have that $\mathfrak{C}$ is Morita equivalent to the fusion 2-category $\mathfrak{C}\boxtimes\mathbf{Mod}(\mathcal{C})\boxtimes\mathbf{Mod}(\mathcal{C}^{rev})$ . Furthermore, the above discussion implies that $\mathfrak{C}\boxtimes\mathbf{Mod}(\mathcal{C})\boxtimes\mathbf{Mod}(\mathcal{C}^{rev})$ is Morita equivalent to $\mathfrak{G}\boxtimes \mathbf{Mod}(\mathcal{C}^{rev})$ . This concludes the proof.

5.2 Connected

We say that an algebra A in a fusion 2-category is called strongly connected if its unit $i:I\rightarrow A$ is the inclusion of a simple summand. We begin by proving the following lemma.

Proof. As A is connected, it is a fortiori indecomposable, which ensures that $\mathbf{Bimod}_{\mathfrak{C}}(A)$ is a fusion 2-category by Corollary 5.2.5 of [Reference DécoppetDéc23b]. Let P be any simple A-A-bimodule. There exist a simple object C of $\mathfrak{C}$ and a non-zero 1-morphism $A\Box C\Box A\rightarrow P$ of A-A-bimodules. For instance, pick any summand of P viewed as an object of $\mathfrak{C}$ , and consider the induced map of A-A-bimodules. We claim that $A\Box C\Box A$ is a simple A-A-bimodule. Namely, under the adjunction

\[Hom_{A\textrm{-}A}(A\Box C\Box A,A\Box C\Box A)\simeq Hom_{\mathfrak{C}}(C,A\Box C\Box A),\]

the identity A-A-bimodule 1-morphism on P corresponds to the 1-morphism $i\Box Id_C\Box i$ in $\mathfrak{C}$ . But, thanks to our assumptions, the 1-morphism $i:I\hookrightarrow A$ is the inclusion of the monoidal unit. It therefore follows that $i\Box Id_C\Box i$ is a simple 1-morphism in $\mathfrak{C}$ , which establishes that $A\Box C\Box A$ is a simple A-A-bimodule. Now, A has a summand in every connected component of $\mathfrak{C}$ , so that there exists a non-zero 1-morphism $f:C\rightarrow A$ . Furthermore, as A is connected, the composite A-A-bimodule 1-morphism $A\Box C\Box A\rightarrow A\Box A\Box A\rightarrow A\Box A$ is non-zero. Thence, every simple object of $\mathbf{Bimod}_{\mathfrak{C}}(A)$ is in the connected component of the simple object $A\Box A$ , concluding the proof of the lemma.

We now establish the second main theorem of this section.

We fix $\mathfrak{C}$ a strongly fusion 2-category, and write $\mathfrak{C}^{\times}$ for the monoidal sub-2-category of $\mathfrak{C}$ on the invertible objects and the invertible morphisms. As $\mathfrak{C}$ is a monoidal $\mathbb{k}$ -linear 2-category, there is an equivalence of monoidal 2-categories $\mathfrak{C}^{\times}\simeq \mathcal{G}\times^{\pi}\mathrm{B}^2\mathbb{k}^{\times}$ for some finite 2-group $\mathcal{G}$ and 4-cocycle $\pi$ for $\mathcal{G}$ with coefficients in $\mathbb{k}^{\times}$ . We consider the fusion 2-category $\mathbf{2Vect}_{\mathcal{G}}^{\pi}$ from Example 2.13. Then the monoidal inclusion $\mathfrak{C}^{\times}\hookrightarrow \mathfrak{C}$ induces a monoidal 2-functor $\mathbf{J}:\mathbf{2Vect}_{\mathcal{G}}^{\pi}\rightarrow \mathfrak{C}$ . This follows from the fact that $\mathbf{2Vect}_{\mathcal{G}}^{\pi}$ is constructed from $\mathfrak{C}^{\times}$ through linearization, local Cauchy completion, and Cauchy completion, which are all operations satisfying sufficient universal properties. Moreover, because $\mathfrak{C}$ is a strongly fusion 2-category, the main results of [Reference Johnson-Freyd and YuJY21] yield that every simple object of $\mathfrak{C}$ in invertible. As a consequence, we find that $\mathbf{J}$ is essentially surjective on objects.

Now, it follows from Lemma 5.11 below that there exist a finite group H and a monoidal 2-functor $H\times\mathrm{B}^2\mathbb{k}^{\times}\rightarrow\mathcal{G}\times^{\pi}\mathrm{B}^2\mathbb{k}^{\times}\simeq\mathfrak{C}^{\times}$ , which is essentially surjective on objects. After linearization, local Cauchy completion, and Cauchy completion, this yields a monoidal 2-functor $\mathbf{K}:\mathbf{2Vect}_H\rightarrow\mathbf{2Vect}_{\mathcal{G}}^{\pi}$ between fusion 2-categories such that the composite $\mathbf{J}\circ\mathbf{K}$ is essentially surjective on objects. The fusion 1-category $\mathbf{Vect}_H$ equipped with the non-trivial grading by H defines a strongly connected rigid algebra in $\mathbf{2Vect}_H$ , which is separable thanks to Corollary 3.3.7 of [Reference DécoppetDéc23d]. We write A for the separable algebra $\mathbf{J}(\mathbf{K}(\mathbf{Vect}_H))$ in $\mathfrak{C}$ . We note that A is strongly connected as $\mathbf{J}\circ\mathbf{K}$ is a monoidal 2-functor between fusion 2-categories. By construction, we also have that the underlying object of A contains every simple object of $\mathfrak{C}$ as a direct summand. Thus, the hypotheses of Lemma 5.8 are satisfied, and we get that $\mathbf{Bimod}_{\mathfrak{C}}(A)$ is a connected fusion 2-category.

We will need the following technical lemma generalizing Theorem 1 of [Reference OpolkaOpo94] to cohomology with twisted coefficients.

Proof. We begin by recording the following fact, which is an immediate consequence of Corollary 6.2.23 of [Reference WeibelWei95], and which we will use repeatedly. For any free group F, abelian group B with an action by F, and $m\geq 2$ , we have $H^m(F;B)=0$ . Let us now pick a surjective group homomorphism $f:F\twoheadrightarrow G$ with F a finitely generated free group. In particular, it is clear that $f^*\omega$ is a coboundary. For our purposes, it is useful to recast this observation through a Lyndon–Hochschild–Serre spectral sequence. More precisely, let us write N for the kernel of f, and note that, thanks to the Nielsen–Schreier theorem, N is again a finitely generated free group, and that it acts trivially on A. We can consider the Lyndon–Hochschild–Serre spectral sequence [Reference Hochschild and SerreHS53]

\[E_2^{p,q}=H^p(G;H^q(N;A))\Rightarrow H^{p+q}(F;A)\]

associated to the short exact sequence of groups $N\hookrightarrow F\twoheadrightarrow G$ . By the previous observations, the $E_2$ page of this spectral sequence is zero unless $q=0,1$ , and its $E_{\infty}$ page is zero unless $p+q\leq 1$ . In particular, $\omega$ viewed as an element of $E^{0,n}_2=H^n(G;A)$ is in the image of the $d_2$ differential

\[d_2:H^1(N;H^{n-2}(G;A))=Hom(N,H^{n-2}(G;A))\rightarrow H^0(N;H^n(G;A))=H^n(G;A).\]

Let us write $\psi:N\rightarrow H^{n-2}(G;A)$ for a homomorphism representing a class in $H^1(N;H^{n-2}(G;A))$ such that $d_2(\psi) = \omega$ . Now, if k denotes the exponent of the image of $\psi$ in the torsion abelian group $H^{n-2}(G;A)$ , we have that $\psi$ factors through the quotient $N/K$ with K the subgroup of N generated by commutators and kth powers. Moreover, the subgroup K of N is characteristic, that is, it is preserved by any automorphism of N, so that K is also a normal subgroup of F. In particular, we may consider the quotient group $\widetilde{G}:=F/K$ , which is finite as $N/K$ is finite. Furthermore, there is a surjective group homomorphism $p:\widetilde{G}\rightarrow G$ . We can therefore consider the following homomorphism of short exact sequences of groups.

By naturality of the Lyndon–Hochschild–Serre spectral sequence, there is a map of spectral sequence $\widetilde{E}^{p,q}_2\rightarrow E^{p,q}_2$ . Moreover, as $\psi$ factors through $N/K$ , it can be lifted to $\widetilde{\psi}$ in $\widetilde{E}^{1,n-2}_2=Hom(N/K;H^{n-2}(G;A))$ , and it follows that $\widetilde{d}_2(\widetilde{\psi})=\omega$ . To put it another way, the class represented by $\omega$ in $\widetilde{E}^{0,n}_2$ is killed by $\widetilde{d}_2$ , and therefore does not survive to $\widetilde{E}^{0,n}_{\infty}$ . This precisely says that the pullback $p^*\omega$ is a coboundary as the subgroup $\widetilde{E}^{0,n}_{\infty}$ of $H^n(\widetilde{G};A)$ is the image of the pullback map $p^*$ .

Proof. Let us write $G:= \pi_1(\mathcal{G})$ for the finite group of equivalence classes of objects of $\mathcal{G}$ , and $A:=\pi_2(\mathcal{G})$ for the finite abelian group of automorphisms of the monoidal unit of $\mathcal{G}$ . Furthermore, G acts on A by conjugation. It was shown in [Reference SínhSín75] that the finite 2-group $\mathcal{G}$ is uniquely determined by the above data together with $\omega$ , a 3-cocycle for G with coefficients in A. Thanks to Lemma 5.10, there exist a finite group $\widetilde{G}$ , and a surjective group homomorphism $p:\widetilde{G}\twoheadrightarrow G$ such that $p^*\omega$ is a coboundary. In particular, the data of p together with a cochain trivializing $p^*\omega$ induces an essentially surjective monoidal functor $p:\widetilde{G}\rightarrow \mathcal{G}$ . Then $p^*\pi$ , the pullback of $\pi$ along p, is a 4-cocycle for $\widetilde{G}$ with coefficients in $\mathbb{k}^{\times}$ . By Lemma 5.10 above, there exist a finite group H and a surjective group homomorphism $q:H\twoheadrightarrow \widetilde{G}$ such that $q^*p^*\pi$ is a coboundary. Taking $K:=p\circ q$ concludes the proof of the lemma.

Theorem 5.9 has the following interesting consequence.

6.1 Rigid algebras

Thanks to Theorem 5.9, we have extremely good control over the Morita equivalence classes of fusion 2-categories. We use this to prove the following deep result on the structure of rigid algebras in fusion 2-categories, which internalizes Corollary 2.6.8 of [Reference Douglas, Schommer-Pries and SnyderDSS20], and positively answers a question raised in [Reference Johnson-Freyd and ReutterJR24].

By combining the last corollary with Theorem 5.3.4 of [Reference DécoppetDéc21] we get the following result.

Remark 6.5. Let $\mathfrak{C}$ be a multifusion 2-category. As a consequence of the above results, one can show that the relative 2-Deligne tensor product of any finite semisimple $\mathfrak{C}$ -module 2-categories exists (see [Reference Douglas, Schommer-Pries and SnyderDSS20] for a version of this statement that applies to fusion 1-categories). In particular, it then follows from [Reference Johnson-Freyd and ScheimbauerJS17] that multifusion 2-categories, finite semisimple bimodule 2-categories, and bimodule morphisms form a symmetric monoidal 4-category, which we denote by $\mathbf{F2C}$ . Let us also consider $\mathbf{BrFus}$ , the Morita 4-category of braided multifusion 1-categories studied in [Reference Brochier, Jordan and SnyderBJS21]. We expect that the assignment

\[\mathbf{Mod}(-):\mathbf{BrFus}\rightarrow \mathbf{F2C},\]

given by sending a braided multifusion 1-category $\mathcal{B}$ to the associated multifusion 2-category $\mathbf{Mod}(\mathcal{B})$ , extends to a symmetric monoidal 4-functor. In fact, let us note that, under the assumptions of [Reference Gaiotto and Johnson-FreydGJ19a], the construction of this 4-functor was sketched in [Reference Johnson-FreydJoh22b]. Then Theorem 5.9 proves that the 4-functor $\mathbf{Mod}$ is essentially surjective on objects. Moreover, it follows from Theorem 5.1.2 of [Reference DécoppetDéc23b] and Lemma 2.2.6 of [Reference DécoppetDéc21] that $\mathbf{Mod}$ induces equivalence between Hom-3-categories. Thus, provided it exists, the symmetric monoidal 4-functor $\mathbf{Mod}$ is an equivalence. As braided multifusion 1-categories are fully dualizable objects of $\mathbf{BrFus}$ thanks to [Reference Brochier, Jordan and SnyderBJS21], it would then follow that multifusion 2-categories are fully dualizable objects of $\mathbf{F2C}$ . We give another approach to establishing the full dualizability of multifusion 2-categories in Remark 6.9 below.

6.2 Fusion 2-categories

We work with a fixed multifusion 2-category $\mathfrak{C}$ . The following definition categorifies Definition 2.5.8 of [Reference Douglas, Schommer-Pries and SnyderDSS20].

We now associate a canonical rigid algebra to any multifusion 2-category. This construction should be compared with that of § 2.6 of [Reference Douglas, Schommer-Pries and SnyderDSS20], in which a canonical Frobenius algebra is associated to any fusion 1-category.

Construction 6.7. As $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ is a multifusion 2-category acting on the finite semisimple 2-category $\mathfrak{C}$ , it follows from Theorem 4.2.2 of [Reference DécoppetDéc21] that $\mathfrak{C}$ can be given the structure of a $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ -enriched 2-category. More precisely, for any C,D in $\mathfrak{C}$ , the Hom-object $\underline{Hom}(C,D)$ in $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ is characterized by the adjoint 2-natural equivalence

\[Hom_{\mathfrak{C}}(H\Box C, D)\simeq Hom_{\mathfrak{C}\boxtimes\mathfrak{C}^{mop}}(H, \underline{Hom}(C,D)),\]

for every H in $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ . It was established in Proposition 4.1.1 of [Reference DécoppetDéc21] that such an object $\underline{Hom}(C,D)$ of $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ always exists. We write $\mathcal{R}_{\mathfrak{C}}$ for the algebra $\underline{End}(I)$ in $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ , and observe that $\mathcal{R}_{\mathfrak{C}}$ is rigid by Theorem 5.2.7 of [Reference DécoppetDéc21]. Moreover, the monoidal unit I of $\mathfrak{C}$ generates $\mathfrak{C}$ under the action of $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ in the sense of Definition 5.3.1 of [Reference DécoppetDéc21]. Thus, by Theorem 5.3.4 of [Reference DécoppetDéc21], there is an equivalence

\[\mathfrak{C}\simeq \mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{C}^{mop}}(\mathcal{R}_{\mathfrak{C}})\]

of left $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ -module 2-categories. Let us also record for later use that, if we assume that $\mathfrak{C}$ is a fusion 2-category, then $\mathcal{R}_{\mathfrak{C}}$ is connected. Namely, its unit 1-morphism corresponds to the identity 1-morphism $I\rightarrow I$ under the adjunction above.

Thanks to Theorem 6.1 and Corollary 6.2, the rigid algebra $\mathcal{R}_{\mathfrak{C}}$ is in fact separable. This yields the following result, which categorifies Corollary 2.6.8 of [Reference Douglas, Schommer-Pries and SnyderDSS20].

We now argue that this is possible given that the multifusion 2-category $\mathfrak{C}$ is separable. Through the equivalence of left $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ -module 2-categories of Construction 6.2, the 2-functor T is identified with the left $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ -module 2-functor $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}\rightarrow\mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{C}^{mop}}(\mathcal{R}_{\mathfrak{C}})$ given by $H\mapsto H\Box \mathcal{R}_{\mathfrak{C}}$ . We let $\Delta$ be the forgetful 2-functor $\mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{C}^{mop}}(\mathcal{R}_{\mathfrak{C}})\rightarrow\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ . Then the composite $T\circ\Delta:\mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{C}^{mop}}(\mathcal{R}_{\mathfrak{C}})\rightarrow\mathbf{Mod}_{\mathfrak{C}\boxtimes\mathfrak{C}^{mop}}(\mathcal{R}_{\mathfrak{C}})$ is identified with the 2-functor $H\mapsto H\Box \mathcal{R}_{\mathfrak{C}}$ . But Theorem 5.1.2 of [Reference DécoppetDéc23b] exhibits an equivalence between the Morita 3-category of separable algebras in $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ and the 3-category of separable left $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ -module 2-categories. Under this equivalence, the composite left $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ -module 2-functor $T\circ\Delta$ corresponds to the $\mathcal{R}_{\mathfrak{C}}$ - $\mathcal{R}_{\mathfrak{C}}$ -bimodule $\mathcal{R}_{\mathfrak{C}}\Box \mathcal{R}_{\mathfrak{C}}$ . Conversely, the identity left $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ -module 2-functor $Id_\mathfrak{C}$ corresponds to the $\mathcal{R}_{\mathfrak{C}}$ - $\mathcal{R}_{\mathfrak{C}}$ -bimodule $\mathcal{R}_{\mathfrak{C}}$ . Finally, as $\mathfrak{C}$ is separable, $\mathcal{R}_{\mathfrak{C}}$ is a separable algebra, so that the $\mathcal{R}_{\mathfrak{C}}$ - $\mathcal{R}_{\mathfrak{C}}$ -bimodule $\mathcal{R}_{\mathfrak{C}}\Box \mathcal{R}_{\mathfrak{C}}$ can be extended to a 3-condensation with splitting $\mathcal{R}_{\mathfrak{C}}$ . We therefore find that $T\circ\Delta$ can be extended to a 3-condensation with splitting $Id_{\mathfrak{C}}$ , which proves the desired result.

We explain how to categorify the notion of the global (also called categorical) dimension of a fusion 1-category. If $\mathfrak{C}$ is a fusion 2-category, we have obtained in Construction 6.2 a connected rigid algebra $\mathcal{R}_{\mathfrak{C}}$ in $\mathfrak{C}\boxtimes\mathfrak{C}^{mop}$ . Then, via the construction given in _§ 3.2 of [Reference DécoppetDéc23d], one can associate to $\mathcal{R}_{\mathfrak{C}}$ a well-defined scalar $\mathrm{Dim}_{\mathfrak{C}\boxtimes\mathfrak{C}^{mop}}(\mathcal{R}_{\mathfrak{C}})$ , called its dimension.

Definition 6.10. The dimension of a fusion 2-category $\mathfrak{C}$ is

\[\mathrm{Dim}(\mathfrak{C}):=\mathrm{Dim}_{\mathfrak{C}\boxtimes\mathfrak{C}^{mop}}(\mathcal{R}_{\mathfrak{C}}).\]

Thanks to Theorem 6.1, the rigid algebra $\mathcal{R}_{\mathfrak{C}}$ is actually separable. Combining this observation with Theorem 3.2.4 of [Reference DécoppetDéc23d] yields the following result, which categorifies Theorem 2.3 of [Reference Etingof, Nikshych and OstrikENO05].