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Dual Fusion 2-Categories

Given a fusion 2-category and a suitable module 2-category, the dual tensor 2-category is the associated 2-category of module 2-endofunctors. In order to study the properties of this 2-category, we begin by proving that the relative tensor product of modules over a separable algebra in a fusion 2-category exists. We use this result to construct the Morita 3-category of separable algebras in a fusion 2-category. Then, we explain how module 2-categories form a 3-category, and we prove that, over a fusion 2-category, the 2-adjoint of a left module 2-functor carries a canonical left module structure. We define separable module 2-categories over a fusion 2-category, and prove that the Morita 3-category of separable algebras is equivalent to the 3-category of separable module 2-categories. Finally, we show that the dual tensor 2-category with respect to a separable module 2-category is a multifusion 2-category.


Introduction
The definition of the dual tensor 1-category to a fusion 1-category with respect to a suitable module 1-category originated in [Ost03], where it was used to define and study Morita equivalences between fusion 1-categories. This notion of Morita equivalence has many classical applications in the theory of fusion 1-categories (for instance, see [ENO05] and [ENO09]), but also in the theory of subfactors (see [Müg03]), and in conformal field theory (see [FRS02]). In particular, it is important to understand when the dual tensor 1-category to a fusion 1-category is again a fusion 1-category. This is guaranteed by requiring that the module 1-categories under consideration be separable in the sense of [DSPS21], restriction which features implicitly in the earlier work of [Müg03] and [FRS02].
In the present article, we examine the categorified version of this last question. That is we study separable module 2-categories, and dual tensor 2-categories to a fusion 2-category with respect to such module 2-categories. The need to examine these objects has multiple origins. Firstly, as recalled above, this constitutes a necessary first steps towards defining and studying Morita equivalences between fusion 2-categories, point to which we will return in [Déc]. Secondly, it was conjectured in [DR18] that fusion 2-categories are the object of a symmetric monoidal 4-category with duals (in the sense of [Lur10]). Proving this conjecture undoubtedly requires a thorough understanding of this 4-category. But, by analogy with the decategorified setting studied in [DSPS21], the 1-morphisms of the aforementioned 4-category are separable bimodule 2-categories. Thirdly, at the moment, the only construction available to produce new fusion 2-categories out of the ones that are already known is the 2-Deligne tensor product introduced in [Déc21a]. Taking the dual tensor 2-category to a fusion 2-category with respect to a separable module 2-category provides a new method to build interesting fusion 2-categories. In a somewhat different direction, we will see that separable module 2-categories are intimately linked to the Morita theory of separable algebras in fusion 2-categories. Further, it was shown in [Déc22b] that many familiar objects in the theory of fusion 1-categories such as G-graded fusion 1-categories and G-crossed fusion 1-categories over a finite group G are separable algebras in certain fusion 2-categories. Thus, as a byproduct of our investigations, we recover the equivariant Morita theory of G-graded fusion 1categories introduced in [GJS21], but also obtain appropriate versions of Morita theory for G-crossed fusion 1-categories and various other flavours of fusion 1-categories.
Let us now recall the notion of the dual tensor 1-category to a fusion 1category in detail. Let C be a multifusion 1-category (over an algebraically closed field of characteristic zero), and let M be a finite semisimple left C-module 1-category. Following [Ost03], the dual tensor 1-category to C with respect to M is the rigid monoidal 1-category End C (M) of left C-module endofunctors of M, also denoted by C * M . This 1-category admits another description. Namely, it was shown in [Ost03] that there exists an algebra A in C such that M M od C (A). Thus, there is a monoidal equivalence between End C (M) and Bimod C (A) ⊗ A op , the monoidal 1-category of A-A-bimodules in C. Following [DSPS21], let us call the module 1-category M separable if the the algebra A is separable. Provided M is separable, we have that End C (M) is a multifusion 1-category, so that the dual of a multifusion 1-category with respect to a separable module 1-category is again a multifusion 1-category. In fact, this property characterizes separable module 1-categories precisely. Let us also remark that the above discussion remains sensible over an field. Further, every finite semisimple module 1-category is separable over a field of characteristic zero, but that this property does not hold over fields of positive characteristic (see [DSPS21]).
Our goal is to categorify the results recalled in the last paragraph. More precisely, for now, let us work over an algebraically closed field of characteristic zero, and recall from [DR18] that a multifusion 2-category is a finite semisimple rigid monoidal 2-category, and that a fusion 2-category is a multifusion 2category whose monoidal unit is a simple object. We fix a multifusion 2-category C together with an algebra A in C that is separable, which implies that the 2-category Bimod C (A) of A-A-bimodules in C is finite semisimple (see [Déc22b]). We now wish to endow Bimod C (A) with a monoidal structure. As expected, the desired monoidal structure is given by the relative tensor product of modules over the separable algebra A, which generalizes the relative tensor product of finite semisimple module 1-categories over a fusion 1-category introduced in [ENO09]. We establish more generally the existence of the relative tensor product of modules over a separable algebra in any monoidal 2-category D that is Karoubi complete in the sense of [GJF19].
Theorem 3.1.6. Let B be a separable algebra in a Karoubi complete monoidal 2-category D. Then, the relative tensor product of any right B-module M , and left B-module N in D exists.
In fact, elaborating on the above result, we construct the Morita 3-category Mor sep (D) of separable algebras, bimodules, and their morphisms in D. Related 3-categories have previously been considered in [Hau17] and [GJF19].
We then turn our attention towards the 2-category End C (M) of left C-module 2-endofunctors on the left C-module 2-category M. We show that this 2-category has a canonical monoidal structure given by composition. More generally, for any fixed monoidal 2-category D, we will construct a 3-category LMod(D) of left D-module 2-categories, left D-module 2-functors, left D-module 2-natural transformations, and left D-module modification, by promoting the 3-category of 2-categories considered in [Gur13]. Further, if D is rigid, we will show that if a left D-module 2-functor has a 2-adjoint as a plain 2-functor, it has a 2-adjoint as a D-module 2-functor. In particular, for any left D-module 2-category N, the monoidal 2-category End C (N) is a rigid if every (plain) 2-endofunctor on N has a 2-adjoint. Now, it was shown in [Déc21c] that the 2-category Mod C (A) of right Amodule in C admits a canonical left C-module structure. By analogy with the decategorified setting, we wish to compare the monoidal 2-categories Bimod C (A) and End C (Mod C (A)). We will do so in more generality by working over an arbitrary field, and letting C be a compact semisimple tensor 2-category C in the sense of [Déc21b]. Over an algebraically closed field of characteristic zero, this recovers precisely the notion of a multifusion 2-category. Under these hypotheses, we say that a left C-module 2-category is separable if it is equivalent to the 2-category of modules over a separable algebra, and write LMod sep (C) for the full sub-3-category of LMod(C) on the separable module 2-categories. We then prove the following twice categorified version of the classical Eilenberg-Watts theorem.
Theorem 5.1.2. Let C be a compact semisimple 2-category. There is a contravariant linear 3-functor that sends a separable algebra in C to the associated separable left C-module 2-category of right modules. Moreover, this 3-functor is an equivalence.
Let us mention that various particular cases of the above theorem have already appeared in the literature (see theorem 3.2.2 of [Déc22a], the finite semisimple case of theorem 4.16 of [GJS21], and corollary 3.1.5 of [Déc21b]). Finally, under a mild assumption on the compact semisimple 2-category tensor C, we can bring together the various results of this article in order to obtain the following theorem.
Theorem 5.3.2. Let k be a perfect field, and A a separable algebra in a locally separable compact semisimple tensor 2-category C. Then, is a compact semisimple tensor 2-category.
In particular, given a separable module 2-category M, we call End C (M) the dual tensor 2-category to C with respect to M, which we denote by C * M . Specializing the above theorem to the case of algebraically closed fields of characteristic zero, we obtain the following sought-after result, which will be used in the subsequent article [Déc] to set up the Morita theory of fusion 2-categories.
Corollary 5.3.5. Let k be an algebraically closed field of characteristic zero, C a multifusion 2-category, and M a separable left C-module 2-category. Then, C * M , the dual tensor 2-category to C with respect to M, is a multifusion 2-category.

Acknowledgments
I am indepted to Christopher Douglas for his value feedback on the contents of the present article.

Graphical Conventions
The main objects of study of the present article are (weak) 2-categories with additional structures. In this context, it is convenient to use the graphical calculus originally developed in [GS16], and subsequently modified in [Déc21c]. More precisely, we use string diagrams, in which regions correspond to objects, strings to 1-morphisms, and coupons to 2-morphisms. Our diagrams are to be read from top to bottom, which gives the composition of 1-morphisms, and from left to right, which gives the composition of 2-morphisms. We use the symbol 1 to denote the identity 1-morphism on an object, but will omit it from the notations if it is not necessary. To illustrate our conventions, let C be a 2-category, and let f : A → B, and g, h : B → C be 1-morphisms. Given a 2-morphism γ : g ⇒ h, the composite 2-morphism γ • f is represented in our graphical calculus by the following diagram: . Throughout, we will work with a monoidal 2-category C in the sense of [SP11]. In particular, we write : C × C → C for the monoidal product of C and I for its monoidal unit. In fact, we will most often assume that C is strict cubical, which is not a loss of generality thanks to [Gur13]. More precisely, a strict cubical monoidal 2-category is a strict 2-category C, such that the monoidal product is strictly associative and the unit I is strict. We will therefore systematically omit I from the notations in this case. In addition, the 2-functor is strict in either variable separately. In general, the 2-functor is not strict though. In detail, given pairs of composable 1-morphisms f 1 , f 2 and g 1 , g 2 in C, the 2-isomorphism witnessing that preserves the composition of 1-morphisms, called the interchanger, is not trivial. Nevertheless, the strict cubical hypothesis guarantees that φ (f2,g2),(f1,g1) is trivial when either f 2 = 1 or g 1 = 1. Given f and g two 1-morphisms in C, the 2-isomorphism will be depicted using the diagram below on the left, and its inverse using the diagram on the right: , .
In particular, note that we have omitted the symbol . We will systemically do so in order to improve the readability of our diagrams.
In section 4, we will also consider 2-functors and 2-natural transformations, and we now recall from [Déc21c] how to extend the above graphical calculus to these objects. In detail, given F : for the 2-isomorphism witnessing that F preserves the composition of the two composable 1-morphisms g and f in A. These 2-isomorphisms satisfy well-known compatibility conditions. Now, given any 2-morphism υ : g • f ⇒ k • h in A, we set: We extend this convention in the obvious way to the image under a 2-functor of a general 2-morphism, and note that it is well-defined thanks to the coherence axioms for a 2-functor. Now, let F, G : A → B be two 2-functors, and let τ : F ⇒ G be 2-natural transformation. This means that, for every object A in A, we have a 1-morphism τ A : F (A) → G(A), and for every 1-morphism f : The collection of these 2-isomorphisms has to satisfy the obvious coherence relations. In our graphical language, we will depict the 2-isomorphism τ f using the following diagram on the left, and its inverse using the diagram on the right: , .

2-Condensation Monads
We review the definition of a 2-condensation monad introduced in [GJF19] as a categorification of the notions of an idempotent (see also [DR18]). More precisely, we recall the unpacked version of this definition given in section 1.1 of [Déc22a].
Definition 1.2.1. A 2-condensation monad in a 2-category C is an object A of C equipped with a 1-morphism e : A → A and two 2-morphisms µ : e • e ⇒ e and δ : e • e ⇒ e such that µ is associative, δ is coassociative, the Frobenius relations hold (i.e. δ is a 2-morphism of e-e-bimodules) and µ · δ = Id e .
Categorifying the notion of split surjection, [GJF19] gave the definition of a 2-condensation, which we recall below. Further, we also review the definition of the splitting of a 2-condensation monad by a 2-condensation, which is spelled out in [Déc22a].
Definition 1.2.2. A 2-condensation in a 2-category C is a pair of objects A, B in C together with two 1-morphisms f : A → B and g : B → A and two 2-morphisms φ : f • g ⇒ Id B and γ : Definition 1.2.3. Let C be a 2-category, and (A, e, µ e , δ e ) a 2-condensation monad in C. A splitting of C is a 2-condensation (A, B, f, g, φ, γ) together with a 2-isomorphism θ : g • f ∼ = e such that Remark 1.2.4. Let C be a 2-category whose Hom-categories are idempotent complete. It was shown in theorem 2.3.2 of [GJF19] that the 2-category of splittings of a fixed 2-condensation monads in C is either empty or a contractible 2-groupoid. Following [GJF19], we will call a 2-category locally idempotent complete if its Hom-categories are idempotent complete, that is idempotents splits. Further, when working over a fixed field k, we will call a k-linear 2-category locally Cauchy complete if its Hom-categories are Cauchy complete, that is they have direct sums and idempotents splits.
Definition 1.2.5. A locally idempotent complete 2-category is Karoubi complete if every 2-condensation monad splits. A locally Cauchy complete k-linear 2category is Cauchy complete if it is Karoubi complete and has direct sums for objects.
Remark 1.2.6. It is always possible to Karoubi complete an arbitrary locally idempotent complete 2-category (see [DR18] and [GJF19]). Further, this process satisfies a precise 3-universal property as explained in [Déc22a].

Compact Semisimple 2-Categories
Let k be a field. We now review the definition of a semisimple 2-category, given in [DR18] over algebraically closed field of characteristic zero. We will then recall the notion of a compact semisimple 2-category introduced in [Déc21b]. Definition 1.3.1. A k-linear 2-category is semisimple if it is locally semisimple, has right and left adjoints for 1-morphisms, and is Cauchy complete.
An object C of a semisimple 2-category C is called simple if the identity 1-morphism Id C is a simple object of the 1-category End C (C). We say that two simple object C, D of C are in the same connected component if there exists a non-zero 1-morphism between them. As explained in section 1 of [Déc21b], this defines an equivalence relation on the set of simple object, whose equivalence classes are called the connected components of C. Definition 1.3.2. A semisimple k-linear 2-category is compact if it is locally finite semisimple and has finitely many connected component.
As was shown in [Déc21b], the notion of compact semisimple 2-category is the appropriate categorification of the definition of a finite semisimple 1-category. Namely, following [DR18], a finite semisimple 2-category is a semisimple 2categories which is locally finite semisimple and has finitely many equivalence classes of simple objects. However, it was proven in [Déc21b] that, over a general field, there does not exist any finite semisimple 2-category, but there always exists compact semisimple 2-categories. Let us note that, over algebraically closed fields or real closed fields, they do show that every compact semisimple 2-category is in fact finite.
Finally, we recall the definitions of a tensor 2-category and of a fusion 2category, as introduced in [DR18] over algebraically closed fields of characteristic zero. We proceed to give some examples. Definition 1.3.3. A tensor 2-category is a rigid monoidal k-linear 2-category. A fusion 2-category is finite semisimple tensor 2-category, whose monoidal unit is simple. Example 1.3.4. A perfect (k-linear) 1-category is a finite semisimple (k-linear) 1-category, for which the algebra of endomorphisms of any object is separable. Note that if k is algebraically closed or has characteristic zero, then every finite semisimple 1-category is perfect. We write 2Vect for the 2-category of perfect finite semisimple 1-categories, also called perfect 2-vector spaces. The Deligne tensor product endows 2Vect with the structure of a fusion 2-category. Example 1.3.5. Let G be a finite group. We use 2Vect G to denote the compact semisimple 2-category of G-graded perfect 2-vector spaces. The convolution product turns 2Vect G into a compact semisimple tensor 2-category. Furthermore, given a 4-cocycle π for G with coefficients in k × , we can form the fusion 2category 2Vect π G by twisting the structure 2-isomorphisms of 2Vect G using π (see construction 2.1.16 of [DR18] or [Del21]). Example 1.3.6. Let us fix C a finite semisimple tensor 1-category (over k). Following [DSPS21], we say that a finite semisimple right C-module 1-category is separable if it is equivalent to the 1-category of left modules over a separable algebra in C. If k has characteristic zero, every finite semisimple C-module 1category is separable. We write Mod(C) for the compact semisimple 2-category of separable right C-module 1-categories. If B is a braided finite semisimple tensor 1-category, then the relative Deligne tensor product over B endows the 2category Mod(B) with a rigid monoidal structure, so that Mod(B) is a compact semisimple tensor 2-category (see [Déc21b]). Example 1.3.7. Let G be a finite group whose order is coprime to char(k). We write BG for the 2-category with one object * , and End BG ( * ) = G. We may consider the compact semisimple 2-category Fun(BG, 2Vect) of (finite perfect) 2-representations of G, denoted by 2Rep(G). Said differently, the objects of 2Rep(G) are perfect 2-vector spaces equipped with a G-action. The symmetric monoidal structure of 2Vect endows 2Rep(G) with the structure of a symmetric compact semisimple 2-category. More precisely, given V and W two 2-vector spaces with a G-action, their monoidal product is given by the Deligne tensor product V W endowed with the diagonal G-action. The compact semisimple 2-category 2Rep(G) is fact rigid as can be seen either directly or from lemma 1.3.8 below.
The next lemma gives an alternative description of the symmetric monoidal 2-category 2Rep(G) of perfect 2-representations of a finite group G. To this end, let us write Rep(G) for the symmetric fusion 1-category of finite dimensional representations of G. Lemma 1.3.8. Let G be a finite group whose order is coprime to char(k). The symmetric monoidal compact semisimple 2-categories Mod(Rep(G)) and 2Rep(G) are equivalent. In particular, 2Rep(G) is rigid.
Proof. This follows from a slight elaboration on theorem 8.5 of [Gre10]. For completeness, we give a proof using the theory of compact semisimple tensor 2-categories. By construction, the monoidal unit I of 2Rep(G) is Vect, the 1category of finite k-vector spaces, equipped with the trivial G-action. Now, note that the underlying 2-category of 2Rep(G) is exactly Mod(Vect G op ). Namely, a G-action on a perfect 2-vector is precisely the data of a right Vect G op -module structure. In particular, 2Rep(G) is a connected compact semisimple 2-category. It then follows from theorem 3.14 of [Déc21b] and the fact that Vect induces a Morita equivalence between the separable finite semisimple tensor 1-categories Vect G op and Rep(G) that the 2-functor is an equivalence of 2-categories. But, we have End 2Rep(G) (Vect) Rep(G) as symmetric finite semisimple tensor 1-categories. Therefore, it follows from proposition 3.3.4 of [Déc21b] (see also proposition 2.4.7 of [Déc22c]) that the 2-functor Hom 2Rep(G) (Vect, −) induces an equivalence of symmetric monoidal 2-categories. This finishes the proof.

Algebras & Modules
We review some key definitions using our graphical calculus. More precisely, we begin recalling the definition of an algebra in a (strict cubical) monoidal 2-category. We go on to review the definitions of right and left modules as well as that of bimodules. We end this section by recollecting the definitions of rigid and separable algebras, and giving plenty of examples in fusion 2-categories.

Algebras
Throughout, we work with a fixed strict cubical monoidal 2-category C. We begin by recalling the definition of an algebra (also called pseudo-monoid in [DS97]) in C in the form of definition 1.2.1 of [Déc22b]. For the definition of an algebra in an arbitrary monoidal 2-category expressed using our graphical language, we refer the reader to definition 3.1.1 of [Déc21c]. (2) We will make use of the following coherence results for algebras derived in section 6.3 of [Hou07]. We will also use the analogue of equation (3) for ρ, which follows from lemma 2.2.2 below.

Modules
Let us fix an algebra A in the strict cubical monoidal 2-category C. We now recall the notion of a right A-module in C given in definition 1.2.3 of [Déc22b]. We invite the reader to consult definition 3.2.1 of [Déc21c] for a version of this definition in a general monoidal 2-category. b. We have: = .
(6) For later use, let us recall the following coherence result established in lemma 1.2.8 of [Déc22b].
Lemma 2.2.2. Given any right A-module M , we have the following equality: = . (7) Finally, let us recall definitions 3.2.6 and 3.2.7 of [Déc21c].
subject to the coherence relations: a. We have: b. We have: = . (9) Definition 2.2.4. Let M and N be two right A-modules, and f, g : M → M two right A-module 1-morphisms. A right A-module 2-morphism f ⇒ g is a 2-morphism γ : f ⇒ g in C that satisfies the following equality: The above structures can be assembled into a 2-category as was proven in lemma 3.2.10 of [Déc21c]. In fact, as we have assumed that C is strict cubical, this 2-category is strict.
Lemma 2.2.5. Right A-modules, right A-module 1-morphisms, and right Amodule 2-morphisms in C form a strict 2-category, which we denote by Mod C (A).

Let us now recall the definition of left
Definition 2.2.6. A left A-module in C consists of: 1. An object M of C; A slight variant of the proof of lemma 3.2.10 of [Déc21c] shows that left A-modules and their morphisms can be assembled into a 2-category. As C is strict cubical, this 2-category is strict.
Lemma 2.2.9. Left A-modules, left A-module 1-morphisms, and left A-module 2-morphisms in C form a strict 2-category, which we denote by LMod C (A).

Bimodules
be algebras in the strict cubical monoidal 2-category C. We now review the notion of an A-B-bimodule in C.
Definition 2.3.1. An A-B-bimodule in C consists of: 1. An object P of C; 2. The data (P, l P , λ P , κ P ) of a left A-module structure on P ; 3. The data (P, n P , ν P , ρ P ) of a right B-module structure on P ; a. We have: b. We have: = . (15) (16) Definition 2.3.3. Let P and Q be two A-B-bimodules, and f, g : which is both a left A-module 2-morphism and a right B-module 2-morphism.
A slight elaboration on the proof of lemma 3.2.10 of [Déc21c] proves that A-B-bimodules in C and their morphisms can be assembled into a 2-category. Further, as C is strict cubical, this 2-category is in fact strict.

Rigid and Separable Algebras
Let C be a strict cubical monoidal 2-category. A rigid algebra in C is an algebra A whose multiplication 1-morphism m : A A → A has a right adjoint as an A-A-bimodule 1-morphism. In particular, we wish to emphasize that this is a property of an algebra, and not additional structure. Let us also remark that this notion was first introduced in [Gai12], and was first considered in the study of fusion 2-categories in [JFR21]. Before giving examples of this notion in the next section, we review the unpacked version of this definition given in section 2.1 of [Déc22b].
c. The structures of left and right A-module 1-morphisms on m * constructed above are compatible, i.e. they turn m * into an A-A-bimodule 1-morphism: d. The 2-morphism m , depicted below as a cap, is an A-A-bimodule 2morphism: e. The 2-morphism η m , depicted below as a cup, is an A-A-bimodule 2morphism: = . (25) as in the above definition has a section as an A-A-bimodule 2-morphism. Let us again highlight that being separable is a property of an algebra. We now recall the detailed definition of a separable algebra given in definition 2.1.2 of [Déc22b].
(27) Let k be a field, and let assume that C is a monoidal compact semisimple k-linear 2-category. The properties of rigid and separable algebras in C have been investigated in details in [Déc22b]. In particular, theorem 3.1.6 of [Déc22b] shows that if A is a rigid algebra in C, then A is separable if and only if Bimod C (A) is compact semisimple. Further, if either of these conditions is satisfied, both Mod C (A), and LMod C (A) are compact semisimple 2-categories.

Examples
Let k be a field. Following [Déc22b], we examine rigid and separable algebras in some of the examples of compact semisimple tensor 2-categories given in section 1.3. We emphasize that these compact semisimple tensor 2-categories are not strict cubical monoidal 2-category, so that we really have to use the fully weak definition of an algebra in a monoidal 2-category.
Example 2.5.1. Algebras in 2Vect are precisely perfect monoidal (k-linear) 1categories, and rigid algebras are precisely perfect tensor 1-categories, i.e. perfect monoidal 1-categories whose objects have right and left duals. Corollary 3.1.7 of [Déc22b] shows that a perfect tensor 1-category C yields a separable algebra in 2Vect if and only if its Drinfel'd center Z(C) is a finite semisimple 1-category. If k has characteristic zero, it follows from corollary 2.6.8 of [DSPS21] that finite semisimple tensor 1-categories give all separable algebras in 2Vect.
Example 2.5.2. Let G be a finite group. Algebras in 2Vect G are precisely perfect G-graded monoidal 1-categories, and rigid algebras are exactly perfect G-graded tensor 1-categories. If k has characteristic zero, it is straightforward to check that finite semisimple G-graded tensor categories yield all separable algebras in 2Vect G . More generally, given a 4-cocycle for G with coefficients in k × , algebras in 2Vect π G should be thought of as perfect π-twisted G-graded monoidal 1-categories. If H ⊆ G is a subgroup, and γ is a 3-cochain for H such that dγ = π| H , we can consider the algebra Vect γ H in 2Vect π G . It follows from corollary 3.3.7 of [Déc22b] that Vect γ H yields a rigid algebra in 2Vect π G , which is separable if and only if the characteristic of k does not divide the order of H.
Example 2.5.3. Let B be a braided finite semisimple tensor 1-category. In the terminology of [BJS21], a B-central monoidal 1-category is a monoidal 1-category C equipped with a braided monoidal functor F : B → Z(C) to the Drinfel'd center of C. Note that this induces in particular a right B-module structure on C. This notion has also appeared under different names in [DGNO10], [HPT16] and [MPP18]. It follows from proposition 3.2 of [BJS21] that algebras in Mod(B) correspond exactly to finite semisimple B-central monoidal 1-categories, which are separable as right B-module 1-categories. By lemma 2.1.4 of [Déc22b], every B-central finite semisimple tensor 1-category, which is separable as right Bmodule 1-category, is a rigid algebra in Mod(B). If k has characteristic zero, one may check that separable algebras in Mod(B) are precisely B-central finite semisimple tensor 1-categories.
Example 2.5.4. Let G be a finite group of order coprime to char(k). Algebras in 2Rep(G) are given exactly by perfect monoidal 1-categories with a G-action. Further, rigid algebras 2Rep(G) are precisely perfect tensor 1-categories with a G-action, and it follows from lemma 3.3.5 of [Déc22b] that such a rigid algebra is separable if and only if the underlying perfect tensor 1-category is separable.
Remark 2.5.5. Lemma 1.3.8 has one particularly noteworthy consequence, which we now explain. As Mod(Rep(G)) and 2Rep(G) are equivalent as symmetric monoidal 2-categories, the associated (symmetric monoidal) 2-categories of algebra, 1-morphisms of algebras and 2-morphisms of algebras are equivalent. In particular, this induces an equivalence between the full sub-2-categories on the rigid algebras. If we assume that k is an algebraically closed field of characteristic zero, we therefore get an equivalence between the 2-category of multifusion 1categories with a G-action and Rep(G)-central multifusion 1-categories. In the theory of fusion 1-categories, this is a well-known result (see theorem 4.18 of [DGNO10]). In addition, we also get an equivalence between the (symmetric monoidal) 2-categories of braided rigid algebras. That is there is an equivalence between the 2-category of braided multifusion 1-categories with a braided Gaction and braided multifusion 1-categories equipped with a braided functor from Rep(G). This is also a classical result (see proposition 4.22 of [DGNO10]).

The Relative Tensor Product over Separable Algebras
Throughout this section, we work with a fixed monoidal 2-category C, which we assume to be strict cubical without loss of generality. Our first goal is to explain the 2-universal property of the relative tensor product of a right and a left module over an arbitrary algebra A. We then prove that if C is Karoubi complete and A is separable, then the relative tensor product over A always exists. Using this fact, we construct the Morita 3-category of separable algebras, bimodules, and their morphisms in C.

Definition & Existence
Let A be an algebra in C. We fix M a right A-module in C, N a left A-module in C. We begin by defining A-balanced 1-morphisms and 2-morphisms out of the pair (M, N ).
Definition 3.1.1. Let C be an object of C. An A-balanced 1-morphism (M, N ) → C consists of: a. We have: b. We have: = . (29) Definition 3.1.2. Let C be an object of C, and f, g : Definition 3.1.3. The relative tensor product of M and N over A, if it exists, is an object M A N of C together with an A-balanced 1-morphism t A : (M, N ) → M A N satisfying the following 2-universal property: Remark 3.1.4. Observe that, for any object C in C, A-balanced 1-morphisms and 2-morphisms out of (M, N ) form a 1-category, which we denote by Bal A (M, N ; C). Furthermore, this assignment is functorial in M , N , and C. Definition 3.1.3 may be rephrased as asserting that precomposition with t A induces an equivalence of 1-categories which is natural in the object C in C. Let us also note that it follows readily from the definition that the 2-category of relative tensor products M A N is either empty or a contractible 2-groupoid.
Remark 3.1.5. Over an algebraically closed field, with C = 2Vect, and C a multifusion 1-category, then definition 3.1.3 recovers the relative tensor product over C as in definition 3.3 of [ENO09]. As C is automatically separable in this case, theorem 3.1.6 below recovers the well-known statement that the relative tensor product of finite semisimple module 1-categories over C exists and is a finite semisimple 1-category. Other particular cases of definition 3.1.3 have already appeared as definition 3.2 [DSPS19] and definition 3.3 of [BZBJ18].
Theorem 3.1.6. Let A be a separable algebra in a Karoubi complete 2-category monoidal 2-category C. Then, the relative tensor product of any right A-module M and any left A-module N in C exists.
Proof. Let us consider the 2-condensation monad (M N, e, µ, δ) in C given by Clearly, µ · δ = Id e . We now prove that µ is associative using diagrams depicted in section A.1. Figure 1 depicts the composite µ · (e • µ). We begin by moving the two indicated coupons labeled 1κ N and ν M −1 11 to the left along the corresponding arrows, which brings us to figure 2. We then using equation (10) on the blue coupons and equation (5) on the green coupons to arrive at figure 3. We go on by moving the coupon labeled 11κ N up, as well as the coupons labeled ν M −1 1111, ν M −1 111, and 1µ −1 111 to the left along the green arrow. Having arrived at figure 4, we move the coupon labeled ν M −1 11111 up, and that labelled ν M −1 1111 down. Further, we also move the left most cap along the red arrow, and in doing so, use equations (23) and (22), which brings us to figure 5. Now, we use equations (21) on the blue coupons to get to figure 6. We then move the coupon labeled 1µ −1 1111 to the right, as well as the coupon labeled 11µ1 up in order to apply equation (19) to the green coupons, and use equation (17) on the red coupons, bringing us to figure 7. We can then make use of equation (3) on the blue coupons, and cancel the green coupons to arrive at figure 8. Finally, reorganising the diagram along the depicted arrows leads us to figure 9, which represents µ · (µ • e). Thence, we have established the associativity of µ as desired. The coassociativity of δ can be proven similarly.
Let us now move on to proving that (µ • e) · (e • δ) = δ · µ using diagrams depicted in section A.1. Figure 10 depicts the left hand-side of this equality. By moving the coupons labeled 1κ N −1 and ν M 11 to the right, we arrive at figure 11. Then, applying equation (10) to the blue coupons, and equation (5) to the green ones, we get to contemplate figure 12. We proceed to move some coupons along the depicted arrows, and use equation (19) on the blue coupons, and equation (17) on the green coupons, which brings us to figure 13. Using equation (3) on the blue coupons, and moving the coupons labeled 1ψ r −1 1 and 1κ N −1 to the right yields the diagram given in figure 14. Then, we first apply equation (21) to the blue coupons, and then equation (1) on the green coupon together with the coupon labeled 1µ1, which was just created. This brings us to figure 15. Finally, using in succession equation (23) on the blue coupons, equation (22) on the green coupons, and equation (3) on the red coupons, leads us to figure 16, which depicts δ · µ. This proves the desired equality. The equality (e • µ) · (δ • e) = δ · µ can be proven using a similar argument.
In order to prove that the relative tensor product of M and N over A exists, we will use the reformulation given in remark 3.1.4. To this end, recall that 2-condensation monads are preserved by all 2-functors, so that applying The fact that this defines an A-balanced structure can be seen as follows. Let us start with the right hand-side of equation (28) for β f •e . We begin by applying equation (21) after having moved some coupons, then we use equation (2). We continue by appealing to equations (5) and (10), followed by (19) and (17). At last, we can use equations (3) and (7) for A as well as reorganise the string diagram to get to the left hand-side of (28). Equation (29) N, e, µ, δ). To see this, it is enough to prove that for every 1-morphism f : M N → C, we have p f •e = f · µ and s f •e = f · δ. The first equality follows by first applying equations (5), (17), followed by equation (7) for A, and then by using successively equations (6), (18), and (4). The second equality is obtained in a similar fashion.
Finally, as C is Karoubi complete, the 2-condensation monad (M N, e, µ, δ) admits a splitting in C, which we denote by M A N . Now, the splitting of a 2-condensation monad is preserved by any 2-functor, so that Hom C (M A N, −) is also a splitting of the 2-condensation monad on Hom C (M N, −) induced by (M N, e, µ, δ). But, the 2-category of splittings of a 2-condensation monad is a contractible 2-groupoid, so that we get the desired equivalence. Thanks to the definition of the relative tensor product using a 2-universal property, the following result is an immediate consequence of the above theorem.
Corollary 3.1.8. If C is a Karoubi complete 2-category, and A is a separable algebra, the relative tensor product over A defines a 2-functor Remark 3.1.9. For completeness, let us note that if C is a linear monoidal 2category, then it follows from the 2-universal property of A and the fact that is a bilinear 2-functor that A is a bilinear 2-functor.

The Morita 3-Category
Our goal is now to explain how to construct the Morita 3-category of separable algebras in a Karoubi complete monoidal 2-category. In order to do so, we need to generalize the setup of the previous section to bimodules.   . We now prove a unitality property of the relative tensor product that will play a crucial role later on.
Lemma 3.2.5. Let A and B be arbitrary algebras in C. There is a 2-natural adjoint equivalence l M P : A A P P for any A-B-bimodule P in C.
Proof. Let P , Q be two A-B-bimodule in C. Observe that l P : A P → P is an A-balanced A-B-bimodule 1-morphism via β l P := κ P . We claim that this 1-morphism satisfies the 2-universal property defining the relative tensor product. Namely, given f : A P → Q an A-balanced A-B-bimodule 1-morphism, we define g as the composite right B-module 1-morphism In addition, the 2-isomorphism χ g := endows g with a compatible left A-module structure. Further, it follows from the definitions that the 2-isomorphism ξ : g • l P ∼ = f given by ξ := is an A-balanced A-B-bimodule 2-morphism as desired. Now, let g, h : P → Q be two A-B-bimodule 1-morphisms, and γ : g • l P ⇒ h • l P be an A-balanced A-B-bimodule 2-morphisms, then it is not hard to check that ζ := γ • (i P ) is an A-B-bimodule 2-morphism satisfying ζ • l p = γ. This finishes the proof of the claim. Finally, using the 2-universal property of the relative tensor product, one can readily construct the desired adjoint 2-natural equivalence l M .
For our purposes, it is also necessary to examine the relative tensor product of multiple bimodules.
But, using the 2-universal property of M t C again together with the fact that γ is B-balanced, we find that ζ is necessarily B-balanced. Thence, by the 2-universal property of t B , there exists an A-D-bimodule 2morphism ζ : g ⇒ h such that ζ • t B = ζ . Putting everything together, we find that ζ • t B • (M t C ) = γ as desired. This proves that M B (N C P ) is 2-universal with respect to (B, C)-balanced A-D-bimodule morphisms.

One proceeds analogously to show that the (B, C)-balanced A-D-bimodule
is 2-universal with respect to (B, C)-balanced A-D-bimodule morphisms. The second part of the statement then follows readily by appealing to the 2-universal property.
We are now ready to explain the main construction of this section.

Proof. Let
of corollary 3.2.4 provides us with the necessary composition 2-functor. Further, the identity 1-morphism on the algebra A is given by the canonical A-A-bimodule A. It remains to prove that these operations can be made suitably coherent in the sense of definition 4.1 of [Gur13]. Firstly, note that lemma 3.2.5 provides us with an adjoint 2-natural equivalence l M . Using a similar argument, one can construct a 2-natural equivalence r M given on the A-B-bimodule P by r M P : P B B P . Moreover, lemma 3.2.7 provides us with an adjoints 2-natural equivalence α M witnessing associativity of the composition of 1-morphisms.
Secondly, we have to supply invertible modifications λ M , µ M , ρ M , and π M between specific composites of l M , r M , and α M . Let us explain how to construct λ M . Let M be an A-B-bimodule and N a B-C-bimodule in C, and consider the diagram Analogously to the above arguments, π M is constructed using this 2-universal property.
Finally, one has to check that the equation between these invertible modifications given in definition 4.1 of [Gur13] are satisfied. All of them follow readily from the 2-universal property of the relative tensor product over either three or four algebras.
Remark 3.2.9. Over a perfect field, the 3-category Mor sep (2Vect) constructed above is the underlying 3-category of the symmetric monoidal 3-category TC sep of separable multifusion 1-categories considered in [DSPS21]. Over an algebraically closed field of characteristic zero, and given B a braided fusion 1-category, the 3-category Mor sep (Mod(B)) corresponds to the Hom-3-category from B to Vect in the symmetric monoidal 4-category BrFus of braided fusion 1-categories considered in [BJS21].
Remark 3.2.10. Let C be a Karoubi complete monoidal 2-category. In [GJF19], the authors outlined the construction of a 3-category Kar(C) of 3-condensation monads, condensation bimodules, condensation bimodule 1-morphisms, and condensation bimodule 2-morphisms. Using variants of the results proven in section 3 of [GJF19], we expect that one can prove that the 3-category Mor sep (C) considered above is equivalent to Kar(C).
Remark 3.2.11. Our proof of theorem 3.2.8 also applies to other setups. Namely, given any monoidal 2-category C and any set A of algebras in C such that for any algebras A, B, and C in A the relative tensor product over B of any A-B-bimodule and B-C-bimodule exists. The above proof constructs a 3-category Mor A (C) of algebras in A , bimodules between them, and their bimodule morphisms. In particular, if every codescent diagram admits a pseudocoequalizer in C, and that commutes with them, then it follows from remark 3.1.7 that the relative tensor product over any algebra in C exists. In this case, we can therefore consider the 3-category Mor(C) of all algebras in C, bimodules and their bimodule morphims. We note that this last example has already been thoroughly examined in [Hau17] in an ∞-categorical context.

Module 2-Categories
We recall the definitions of a module 2-category, module 2-functor, module 2-natural transformation, and module modification and show that, over a fixed monoidal 2-category, these objects assemble into a 3-category. We then review the definition of a 2-adjunction between two 2-functors, and explain how this concepts interacts with that of a module 2-functor over a rigid monoidal 2-category. Theses results are quite technical in nature, but will play a determining role in the last part of the present article.

The 3-Category of Module 2-Categories
Let C be a cubical monoidal 2-category. Our goal is to construct a 3-category whose objects are left C-module 2-categories in the sense of definition 2.1.3 of [Déc21c]. Now, it follows from proposition 2.2.8 of [Déc21c] that every pair (C, M) consisting of a monoidal 2-category C and a left C-module 2-category M is equivalent to a pair in which both C and M are strict cubical (see definition 4.1.1 below). Thus, there is no loss of generality in assuming that C and M are strict cubical. In fact, by remark 2.2.9 of [Déc21c], this strictification procedure holds for any set of module 2-categories.      Fixing two strict cubical left C-module 2-categories M and N, it was shown in proposition 2.2.1 of [Déc21c] that left C-module 2-functors, left C-module 2-natural transformation, and left C-module modifications form a 2-category, which we denote by F un C (M, N). In particular, given θ, θ, θ : F ⇒ G left Cmodule 2-natural transformations, and two left C-module modifications Ξ : θ θ, Z : θ θ, the vertical composite Z • Ξ is a left C-module modification. Further, given two left C-module 2-natural transformations θ : F ⇒ G and τ : G ⇒ H, their composite is the 2-natural transformation τ · θ equipped with the invertible modification Thanks to our strictness hypotheses, the above composition of left C-module 2-natural transformations is in fact strictly associative and unital. Thence, Fun C (M, N) is in fact a strict 2-category. For later use, we need to assemble all of these 2-categories together. Proof. In section 5.1 of [Gur13], the author constructs a 3-category of 2-categories. Our proof is precisely a left C-module version of this argument. In order to do this, it is enough to upgrade the structures defined in section 5.1 of [Gur13] with suitable left C-module actions. Furthermore, thanks to proposition 2.2.8 and remark 2.2.9 of [Déc21c], we may assume without loss of generality that C and every left C-module 2-category is strict cubical. Let M, N, and P be strict cubical left C-module 2-categories. We begin by constructing the 2-functor providing us with the composition of left C-module 2-functors. Given two left C-module 2-functors F : M → N and G : N → P, we endow their composite G • F with a left C-module structure using the following assignments. We define the adjoint 2-natural equivalence k G•F by for every A in C and M in M, and the two invertible modifications ω G•F , and γ G•F by It is not hard to show that the above data satisfies the axioms of definition 4.1.4. Then, given a left C-module 2-natural transformation θ : F 1 ⇒ F 2 between two left C-module 2-functors F 1 , F 2 : M → N, we endow the 2-natural transformation G • θ with a left C-module structure by setting Likewise, given a left C-module 2-natural transformation τ : G 1 ⇒ G 2 between two left C-module 2-functors G 1 , G 2 : N → P, we may similarly define a left C-module structure on the 2-natural transformation τ • F by Π τ •F := . Now, recall from the proof of proposition 5.1 of [Gur13] that τ •θ = (G 2 •θ)·(τ •F 1 ), so that the 2-natural transformation τ • θ inherits a C-module structure. These assignments can be straightforwardly extended to left C-module modifications, so that we obtain a functor between the 1-categories of left C-module 2-natural transformations. The additional structure constraints needed to define a 2-functor are the invertible modifications given in proposition 5.1 of [Gur13], and one checks easily that they respect the relevant left C-module structures defined above. Thus, we obtain the desired 2-functor • : Analogously, the adjoint 2-natural equivalences l, and r constructed in proposition 5.5 of [Gur13], witnessing that composition of 2-functors is unital, can be promoted to left C-module adjoint 2-natural equivalences. Namely, as we are working with strict 2-categories, these adjoint 2-natural equivalences are in fact both given by the identity 2-natural adjoint equivalence. Thus, given a C-module 2-functor F : M → N between strict cubical left C-module 2categories, the 2-natural transformations l F and r F can canonically be upgraded to left C-module adjoint 2-natural equivalences. With these additional pieces of data, l and r define adjoint 2-natural equivalence witnessing the unitality of the composition of left C-module 2-functors. The proof is then completed by checking that the invertible modification π, µ, λ, and ρ given in proposition 5.6 of [Gur13] are compatible with the left C-module structures we have defined. This is immediate as it follows from our strictness assumptions that π, µ, λ, and ρ are all identity modifications.
Remark 4.1.8. It follows immediately from our proof of theorem 4.1.7 that there is a forgetful 3-functor LMod(C) → 2Cat to the 3-category of 2-categories. We end this first section on module 2-categories with the following proposition, which will constitute a key ingredient in our study of the Morita theory of fusion 2-categories.

Module 2-Functors and 2-Adjunctions
The goal of this section is study the interaction between the notion of a module 2-functor recalled above, and that of a 2-adjunction between 2-functors, which we now recall. Let M and N be two 2-categories, and F : M → N and G : N → M be two 2-functors.
We also say that F is a left 2-adjoint to G, or that G is a right 2-adjoint to F .
Let C be a rigid monoidal 2-category, and assume that both M and N are left C-module 2-categories. The categorified version of corollary 2.13 of [DSPS19] holds, as we show in the next two propositions. In fact, our proof also establishes the categorifications of their lemmas 2.10 and 2.11. Proposition 4.2.2. Let C be a rigid monoidal 2-category, and let F : M → N be a left C-module 2-functor between left C-module 2-categories. If F has a right 2-adjoint G, then G is canonically a left C-module 2-functor.
Proof. Thank to proposition 2.2.8 and remark 2.2.9 of [Déc21c], we may assume that C is strict cubical, and that both M and N are strict cubical left C-module 2categories. We begin by proving that G can be endowed with an lax left C-module structure. Given A in C, and M in M, we let the 2-natural transformation k G be given by where (k F ) • is the pseudo-inverse of k F provided in the data of a left C-module 2-functor. The invertible modifications ω G and γ G are given by Using the axioms of definition 4.1.4 for F , it is easy to check that ω G and γ G satisfy the axioms of 4.1.4.
It remains to show that k G can be upgraded to an adjoint 2-natural equivalence. As every 2-natural equivalence can be upgraded to an adjoint 2-natural equivalence (see section 1 of [Gur12]), it is enough to exhibit for every A in C and M in M, a pseudo-inverse (k G ) • A,M for the 1-morphism k G A,M . Let A be a left dual for A in C with unit 1-morphism i A : I → A A, counit 1-morphism e A : A A → I and 2-isomorphisms where (k F ) • denotes the canonical pseudo-inverse of k F supplied by the definition of a module 2-functor. Let us denote by (ω F −1 ) • and (γ F −1 ) • the 2-isomorphisms given by where the cups and the caps denote the unit and counit 2-isomorphisms witnessing that k F and (k F ) • form an adjoint 2-natural equivalence. Finally, the two 2isomorphisms , , witness that (k G ) • A,M is a pseudo-inverse for k G A,M as desired. Proof. It is enough to upgrade u F and c F to left C-module 2-natural transformations, and show that Φ F and Ψ F define invertible left C-module modifications.
For the first part, we endow u F and c F with left C-module structures using the modifications Π u F and Π c F specified by It is easy to check that Π u F and Π c F satisfy the axioms of definition 4.1.5. Finally, using naturality together with axioms a and b of definition 4.1.4 for G, one can readily check that Φ F and Ψ F are compatible with the left C-module structure on u F and c F defined above, which finishes the proof of the result.
Proposition 4.2.2 above shows that the rightt 2-adjoint of a left C-module 2functor has a canonical left C-module structure. This result has a clear analogue for left adjoints, which we spell out below. Proof. We write C 1op for the monoidal 2-category obtained from C be reversing the direction of the 1-morphisms. Note that if C is strict cubical, then C 1op is strict opcubical. It is clear that M 1op and N 1op are left C 1op -module 2-categories. Further, G 1op : N 1op → M 1op is a left C 1op -module 2-functor, and our hypothesis guarantees that G 1op has a right adjoint given by F 1op . Then, by proposition 4.2.2, F 1op is a left C 1op -module 2-functor, so that F is a left C-module 2-functor as desired.
Similarly, one derives the following corollary.

Dual Tensor 2-Categories
In general, it is difficult to work with arbitrary compact semisimple module 2categories over a fixed compact semisimple tensor 2-category C. Motivated by the decategorified situation studied in [DSPS21], we therefore restrict our attention to a particularly nice class of compact semisimple module 2-categories called separable module 2-categories. We prove that the 3-category of separable algebras in C is equivalent to the 3-category of separable left C-module 2-categories. Under a mild assumption on C, we then show that the monoidal 2-category of bimodules over a separable algebra in C is a compact semisimple tensor 2-category. This allows us to define the dual tensor 2-category of C with respect to a separable module 2-category. Throughout, we work over a fixed field k, meaning that all categories and functors under consideration are k-linear.

Separable Module 2-Categories
Let us fix C a compact semisimple tensor 2-category, and M a compact semisimple left C-module 2-category. In theorem 4.1.7, we have proven that left C-module 2-categories form a 3-category, which we denote by LMod(C). We will write LMod sep (C) for the full sub-3-category whose objects are the separable module 2-categories. We are now ready to state our main theorem, which was conjectured in remark 5.4.9 of [Déc21c]. Let us also mention that when k is algebraically closed of characteristic zero and C = 2Vect, then we recover the main result of [Déc22a] as every multifusion 1-category is separable. With k algebraically closed of characteristic zero and C = 2Vect G for some finite group G, we also obtain the G-graded multifusion 1-category case of theorem 4.16 of [GJS21]. Further, if k is perfect and C = 2Vect, the theorem below also recovers corollary 3.1.5 of [Déc21b] thanks to proposition 2.5.10 of [DSPS21].
Theorem 5.1.2. Let C be a compact semisimple 2-category. There is a contravariant linear 3-functor that sends a separable algebra in C to the associated separable left C-module 2-category of right modules. Moreover, this 3-functor is an equivalence.
Proof. Without loss of generality, we may assume that C is strict cubical. The monoidal unit I of C is canonically a separable algebra in C. Thanks to theorem 3.2.8, this yields a contravariant linear 3-functor It remains to prove that Mod C is an equivalence of 3-categories, i.e. that it is essentially surjective and induces equivalences on Hom-2-categories. Essential surjectivty follows immediately from the definition of a separable left C-module 2-category. Therefore, it only remains to prove that for every separable algebras A, B in C, the 2-functor induced by Mod C is an equivalence of 2-categories. In order to exhibit a pseudo-inverse to F, consider the following 2-functor where Using this 2-universal property, we readily find that F • B(F ) F . Finally, the definition of the A-balanced right B-module 1-morphism f F is 2-natural in F , so that we get F • B Id from the 2-universal property. This finishes proving that F and B are pseudo-inverses.
We now prove an alternative characterization of separability for a compact semisimple left C-module 2-category M under mild assumptions on the underlying 2-category of C. More precisely, following [Déc21b], if k is perfect, we say that C is locally separable if for every simple object C of C, the finite semisimple tensor 1-category End C (C) is separable in the sense of [DSPS21]. Proof. The forward direction follows by combining theorem 5.1.2 above with proposition 3.1.3 of [Déc22b]. Conversely, let us assume that End C (M) is compact semisimple. Thanks to theorem 5.1.5 and remark 5.1.11 of [Déc21c], there exists an algebra A in C such that Mod C (A) M as left C-module 2-categories. We will use the 2-functor End C (M) has right adjoints for 1-morphisms by hypothesis, so that n has a right adjoint n * with counit n . As right adjoint are preserved by 2-functors, B(n * ) is a right adjoint for m as an A-A-bimodule 1-morphism with counit B( n ). This implies that A is rigid. Thirdly, note that the 2-morphism n : n · n * ⇒ Id is surjective. Namely, for every simple object M of M Mod C (A), the 2-morphism n M is surjective as n M : M A → M is a non-zero 1-morphism. But End C (M) is compact semisimple, so that n has a section γ n as a left C-module modification. As sections of 2-morphisms are preserved by arbitrary 2-functors, we find that B(γ n ) is a section of B( n ) as an A-A-bimodule 2-morphism. Consequently, A is separable, and the proof is complete.
The proof of the above proposition also establishes the following result (over any field k and compact semisimple tensor 2-category C).
Corollary 5.1.4. Assume that M is a separable left C-module 2-category, and let B be any algebra such that M Mod C (B) as left C-module 2-categories, then B is separable.
Remark 5.1.5. Let us call two algebras A, and B in C Morita equivalent if Mod C (A) Mod C (B) as left C-module 2-categories. Note that this is clearly an equivalence relation. The above corollary may then be succinctly reformulated as the statement that separability is a Morita invariant property. Further, rigidity is also a Morita invariant property. On one hand, if A is a rigid algebra, then Mod C (A) has right adjoints by theorem 2.2.8 of [Déc22b], so that End C (Mod C (A)) has right adjoints. On the other hand, the proof of proposition 5.1.3 above, proves that if End C (Mod C (A)) has right adjoints, then A is rigid.
We end this section by examining an example.
Example 5.1.6. Let G be a finite group whose order is coprime to char(k). Then, the monoidal forgetful 2-functor 2Vect G → 2Vect provides 2Vect with a canonical structure of a left 2Vect G -module 2-category. We claim that End 2Vect G (2Vect) 2Rep(G) as 2-categories. Namely, let F : 2Vect → 2Vect be a left 2Vect G -module 2-functor. As 2Vect is generated by Vect under direct sums and splittings of 2-condensation monads, the underlying linear 2-functor F is determined by V := F (Vect), a perfect 1-category. Further, unfolding the definition, we find that the left 2Vect G -module structure on F yields a G-action on V . But 2Vect G is the Cauchy completion of the monoidal 2-category G × 2Vect, so that this G-action on V characterizes F completely up to equivalence. A similar argument deals with 2Vect G -module 2-natural transformations and 2Vect G -module modifications, establishing the desired equivalence End 2Vect G (2Vect) 2Rep(G) of 2-categories. An immediate consequence of the above equivalence is that Vect is a separable 2Vect G -module 2-category. Over an algebraically closed field of characteristic zero, this equivalence was first observed in section 3.2 of [Del21].
We now wish to upgrade this to an equivalence of monoidal 2-categories. Observe that the identity 2Vect G -module 2-endofunctor of 2Vect corresponds to the object I = Vect of 2Rep(G) under the above equivalence. It follows from the proof of lemma 1.3.8 that 2Rep(G) is a connected compact semisimple 2-category. By proposition 3.3.4 of [Déc21b], the monoidal structure on 2Rep(G) is therefore completely determined by the braiding β on the finite semisimple tensor 1-category End 2Rep(G) (I) Rep(G). This means that, if we write 2Rep β (G) := Mod(Rep β (G)), then we have End 2Vect G (2Vect) 2Rep β (G) as monoidal 2-categories. But, by proposition 4.1.10, 2Vect is a left 2Rep β (G)module 2-category. Thus, by definition, there exists a braided monoidal functor Rep β (G) → Z(Vect) = Vect. As this functor is necessarily faithful, this forces the braiding β to be the trivial one, so that End 2Vect G (2Vect) 2Rep(G) as monoidal 2-categories.

Indecomposable Module 2-Categories
It is useful to know when the compact semisimple monoidal 2-category End C (M) has simple monoidal unit. We now explain when this is the case. Proof. The first assertion is immediate. Let us assume that M is separable, and M n i=1 M i be a decomposition of M as a direct sum of indecomposable compact semisimple left C-module 2-categories. We wish to prove that the projection P i : M M i → M is a simple object of End C (M). To this end, let Q, R : M → M be two C-module 2-functor such that P i = Q R. Let us additionally assume that Q(M ) is no-zero for some M (necessarily in M i ). Then, it follows from the proof of lemma 5.2.3 that given any simple object N of M i , there exists an object C of C and a non-zero 1-morphism C N → M . In particular, M is the splitting of a 2-condensation monad supported on C N . But splittings of 2-condensation monads are preserved by all 2-functors, so that M is the splitting of a 2-condensation monad on Q(C N ). As M is non-zero, so must be Q(C N ). Now, Q is a left C-module 2-functor, so that Q(C N ) C Q(N ), which implies that Q(N ) is non-zero. Finally, we have N = P i (N ) = Q(N ) R(N ), and N is simple, so that R(N ) = 0 by proposition 1.1.7 of [Déc21b]. As N was arbitrary, we find that R = 0, which finishes the proof of the lemma. Given the equivalence of 3-categories established in theorem 5.1.2, it is only natural to examine what property of a separable algebra corresponds to indecomposability of the associated module 2-category.
Definition 5.2.6. Let A be a separable algebra. We say that A is indecomposable if A is simple as an A-A-bimodule.
Corollary 5.2.7. A separable algebra A is indecomposable if and only if Mod C (A) is indecomposable.
Remark 5.2.8. In particular, this shows that being indecomposable is a Morita invariant property of separable algebras in C. Further, it follows from lemma 5.2.6 that any separable algebra A may be split into a direct sum of indecomposable separable algebras. A direct proof of this fact is given in the proof of theorem 3.1.6 of [Déc22b].

Dual Tensor 2-Categories
In this section, we assume throughout that k is perfect. The following technical result is needed to prove our main theorem over fields of positive characteristic. Before stating it, we need to recall some terminology from [Déc21b]. We say that a compact semisimple 2-category C is locally separable if for every simple object C of C, the finite semisimple tensor 1-category End C (C) is separable in the sense of [DSPS21]. We remark that, over fields of characteristic zero, they show that every finite semisimple tensor 1-category is separable, so that every compact semisimple 2-category is locally separable in this case.
Proposition 5.3.1. Let C be a locally separable compact semisimple monoidal 2-category, and A a separable algebra in C. Then, Mod C (A) is locally separable.
Proof. By theorem 1.4.7 of [Déc21b], there exists a separable finite semisimple tensor 1-category C such that Mod(C) C as 2-categories. It follows from theorem 5.1.2 that End(C) Bimod(C) C op .
Further, observe that algebras in Bimod(C) are precisely given by finite semisimple monoidal 1-categories D equipped with a monoidal functor C → D. Moreover, a variant of the proof of lemma 2.1.4 of [Déc22b] shows that such an algebra D is rigid if and only D is a tensor 1-category. Also note that the 2-category D-D-bimodules in Bimod(C) is canonically equivalent to Bimod(D), so that D is separable as an algebra in Bimod(C) if and only if it is separable as a finite semisimple tensor 1-category. Further, observe that the separable algebra A in C yields a separable algebra A in End(C) Bimod(C) C op via C → C A. It follows from the definition that Combining the theorem 5.3.2 above with corollary 2.2.4 of [Déc21b], we obtain the following result, which is the starting point of the study of Morita equivalences of fusion 2-category that we will develop in [Déc].
Corollary 5.3.5. Let k be an algebraically closed field of characteristic zero, C a multifusion 2-category, and M a separable left C-module 2-category. Then, C * M , the dual tensor 2-category to C with respect to M, is a multifusion 2-category.
The following corollary now follows from the discussion given in example 2.5.2.
Corollary 5.3.6. Let G be a finite group, and π a 4-cocycle for G with coefficients in k × . Further, let H ⊆ G be a subgroup of order coprime to char(k) and γ a 3-cochain for H with coefficients in k × . Then, Bimod 2Vect π G (Vect γ H ) is a compact semisimple tensor 2-category.
We end this section by examining some examples of dual tensor 2-categories.
Example 5.3.7. Let k be an algebraically closed field, and let C be a separable fusion 1-category. The compact semisimple 2-category Mod(C) admits a canonical left 2Vect-module structure. We claim that 2Vect *

Mod(C)
Mod(Z(C) βop ) as monoidal 2-categories. For the reader's convenience, we supply a different proof. Note that theorem 5.1.2 provides us with an equivalence of monoidal 2categories 2Vect *
Alternatively, as C is a fusion 1-category, C ⊗op C is also a fusion 1-category, so that the compact semisimple 2-category Bimod(C) Mod(C ⊗op C) is connected. Thus, by proposition 3.3.4 of [Déc21b], in order to determine the monoidal structure on Bimod(C), it is enough to understand the braiding on the finite semisimple tensor 1-category End C−C (C) = Z(C) of endomorphism the monoidal unit. Given that the monoidal structure of Bimod(C) is given by the relative Deligne tensor product C , the braiding on Z(C) is the canonical one.
Example 5.3.8. We now discuss a generalization of example 5.3.7. For simplicity, we will assume that k is an algebraically closed field of characteristic zero. Let B be a non-degenerate braided fusion 1-category, and C a B-central fusion 1-category. We claim that there is an equivalence of monoidal 2-categories Mod(B) *

Mod(C)
Mod(A βop ), where A is the centralizer of B in Z(C), which is non-degenerate by theorem 3.13 of [DGNO10]. By corollary 5.9 of [DMNO13], this implies further that B and A are Witt equivalent non-degenerate braided fusion 1-categories.
In order to prove the claim, observe that, as C is a separable algebra in Mod(B), corollary 5.3.5 establishes that Mod(B) *

Mod(C)
Bimod Mod(B) (C) C op is a fusion 2-category. Moreover, after having unfolded the definitions, we find Bimod Mod(B) (C) Mod(C ⊗op B C) as finite semisimple 2-categories. As B is non-degenerate, it follows from theorems 2.26 and 3.20 of [BJSS21] that there is an equivalence C ⊗op B C End A (C) of multifusion 1-categories. In particular, Bimod Mod(B) (C) Mod(C ⊗op B C) is a connected finite semisimple 2-category. The above equivalence of multifusion 1-categories implies that A End C ⊗op B C (C) as fusion 1-categories, so that the endomorphism fusion 1-category of C in Bimod Mod(B) (C) is given by A. But C is the monoidal unit of Bimod Mod(B) (C). Thence, appealing to proposition 2.4.7 of [Déc22c], it is enough to understand the braiding on the fusion 1-category A of endomorphisms of C in Bimod Mod(B) (C).
To this end, note that the forgetful 2-functor Bimod Mod(B) (C) → Bimod(C) induces the canonical inclusion of fusion 1-categories A → Z(C). But the forgetful monoidal 2-functor End Mod(B) (Mod(C)) → End(Mod(C)) is identified via theorem 5.1.2 to Bimod Mod(B) (C) → Bimod(C). This shows that the monoidal inclusion A → Z(C) is in fact braided, thereby completing the proof of the claim.
Remark 5.3.9. As one can readily observe from example 5.1.6, the non-degeneracy condition in example 5.3.8 above can not be omitted in general. We will return to this point in detail in [Déc].