Multifusion Categories and Finite Semisimple 2-Categories
aa r X i v : . [ m a t h . QA ] D ec On the Algebraic Theory of Fusion 2-Categories I
Thibault D. D´ecoppetJanuary 1, 2021
Abstract
The 3-categories of semisimple 2-categories and of multifusion cate-gories are shown to be equivalent. A weakening of the notion of fusion2-category is introduced. Multifusion 2-categories are defined, and thefusion rule of the fusion 2-categories associated to certain pointed braidedfusion categories is described.
Contents
A The Adjoints Monoidal 2-Functor 21B Completions 23
B.1 Completions of 1-Categories . . . . . . . . . . . . . . . . . . . . . 23B.2 Completions of 2-Categories . . . . . . . . . . . . . . . . . . . . . 25
References 29 ntroduction
After Lurie’s sketch of proof of the Cobordism Hypothesis of Baez and Dolan (see[Lur10]), there began a quest for interesting fully-dualizable objects in symmet-ric monoidal higher categories. Examples are given by fusion categories (overan algebraically closed fielf of characteristic zero), which are fully-dualizableobjects of symmetric monoidal 3-category of (multi)fusion categories, bimod-ules categories, bimodule functors, and bimodule natural transformations withmonoidal structure given by the Deligne tensor product (see [DSPS13]). Bydirect analogy, it is conjectured that fusion 2-categories (introduced in [DR18])are 4-dualizable objects of a suitable symmetric monoidal 4-category. In par-ticular, one case is already known; Namely, [BJS18] have proven that braidedfusion categories are 4-dualizable. In the language of fusion 2-categories, theyhave proven that connected fusion 2-categories are fully dualizable dualizable.We believe that without a thorough understanding of the properties of fusion 2-categories, it will not be possible to prove that they are 4-dualizable in general.In the present work, we initiate this program by introducing a weak versionof the notion of fusion 2-category and study its elementary properties. Let usalso remark that related objects called separable fusion 2-categories have beenintroduced in [JF20], and that the aforementioned conjecture is equivalent tothe statement that every fusion 2-category is separable (over an algebraicallyclosed fielf of characteristic zero).We now describe the content of the different sections of the present article.Section 1 contains most of the (monoidal) 2-category theory that we will need.We begin by reviewing adjoints and duals in monoidal 2-categories, and describehow adjoints can be assembled to give a monoidal 2-functor. Some properties ofmonoidal 2-categories that have right adjoints and right duals are given. Then,we recall the notion of 2-condensation monad in a 2-category introduced in[GJF19], and give a definition of the Karoubi envelope of a 2-category via a“3-universal property”.Finite semisimple 2-categories and fusion 2-categories as defined in [DR18]were required to be strict 2-categories. In section 2, we generalize their defini-tion by allowing the underlying 2-category to be a (weak) 2-category. Further,we show that every finite semisimple 2-category, respectively fusion 2-category,(as defined in the present text) is equivalent to a finite semisimple 2-category,respectively fusion 2-category, as defined by Douglas-Reutter. We also providea proof of a result conjectured in subsection 1.4.3 of [DR18].
Theorem (Thm. 2.2.2) . There is an equivalence of 3-categories between the3-category of multifusion categories and the 3-category of finite semisimple 2-categories.
The notion of a multifusion 2-category, which generalizes the notion of fusion2-category by not imposing the requirement that the unit object be simple, isintroduced. We show that connected fusion 2-categories correspond precisely tobraided fusion categories. Finally, we explain how to compute the fusion ruleof the connected fusion 2-category associated to certain pointed braided fusion1ategory, generalizing a result of [ENO02]. We use this to compute the fusionrules of some fusion 2-categories.I would like to thank Christopher Douglas, and David Reutter for helpfulconversations related to the content of this article.
We begin by reviewing the notions of adjoint for a 1-morphism in a (weak) 2-category and that of dual for an object of a monoidal 2-category. We explainhow the existence of adjoints in a monoidal 2-category give rise to a monoidal2-functor. Then, we examine how adjoints and duals interact in a monoidal 2-category. We go on to recall the definition of a 2-condensation, and some resultsfrom [GJF19]. Finally, we state the universal property of the Karoubi envelopeof a 2-category. Note that everything said also applies to linear monoidal 2-categories upon making the obvious changes.
The following definition is well-known.
Definition 1.1.1.
Let C be a 2-category and f : A → B a 1-morphism in C .A right-adjoint for f is a 1-morphism f ∗ : B → A together with 2-morphisms ǫ : f ◦ f ∗ ⇒ Id B and η : Id A ⇒ f ∗ ◦ f satisfying the snake equations( ǫ ◦ f ) · ( f ◦ η ) = Id f , ( f ∗ ◦ ǫ ) · ( η ◦ f ∗ ) = Id f ∗ , in which we have omitted the relevant coherence 2-isomorphisms. One definesleft-adjoints dually. Remark . It is well-known that a right-adjoint for a 1-morphism f is uniqueup to unique 2-isomorphism. Notation 1.1.3.
Let C be a 2-category. We denote by C op the 2-categoryobtained from C by reversing the direction of the 2-morphisms. Analogously,we denote by C op the 2-category obtained by reversing the direction of the1-morphisms. If C is monoidal, then it is clear that so is C op . The 2-category C op can also be given a monoidal structure, but this construction is slightlytrickier, as one need to “invert” 1-morphisms. Using the algebraic definition ofmonoidal 2-category given in [SP11], this poses no problem. Lemma 1.1.4.
Let C be a 2-category with right-adjoints. There is a 2-functor ( − ) ∗ : C → C op ;2 op that is the identity on objects and sends a 1-morphism f to the 1-morphismunderlying a chosen right adjoint. Dually, if C has left-adjoints, there is a 2-functor denoted by ∗ ( − ) that sends 1-morphisms to their left adjoints. Further,if C has left and right-adjoints, ∗ ( − ) is a pseudo-inverse for ( − ) ∗ . More precisely, { ∗ ( − ) } op ;2 op is a pseudo-inverse. roof. The is well-known, for instance, see [Gho10], or appendix A below.If we assume in addition that C is monoidal, then the 2-functor ( − ) ∗ can bemade monoidal. Lemma 1.1.5.
Let C be a monoidal 2-category with right-adjoints. The 2-functor ( − ) ∗ of lemma 1.1.4 can be made monoidal. If, in addition, C hasleft-adjoints, then the monoidal 2-functors ∗ ( − ) is a monoidal pseudo-inversefor ( − ) ∗ .Proof. See appendix A below.
Some care has to be taken with respect to what one calls a dual in a monoidal2-category C . For us, a right dual for an object A of C consists of an object A ♯ ,together with two 1-morphisms i A : I → A ♯ (cid:3) A and e A : A (cid:3) A ♯ → I satisyingthe snake equations up to 2-isomorphisms. Similarly, one can give a definitionof a left dual for A . These definitions have the advantage of being concise andeasy to check, but they are not convenient to use in constructions. That is whywe shall also need to consider the refinement defined in [Pst14] called a coherentright dual. Definition 1.2.1.
Let A be an object of a monoidal 2-category C . A coherentright dual for A consists of an object A ♯ in C , 1-morphisms i A : I → A ♯ (cid:3) A and e A : A (cid:3) A ♯ → I , and 2-isomorphisms C A : ( e A (cid:3) A ) ◦ a • A,A ♯ ,A ◦ ( A (cid:3) i A ) ⇒ Id A ,D A : Id A ♯ ⇒ ( A ♯ ◦ e A ) ◦ a A ♯ ,A,A ♯ ◦ ( i A (cid:3) A ♯ ) , satisfying the two swallowtail equations (depicted in figures 3 and 4 of [Pst14]).We will also say that the data ( A, A ♯ , i A , e A , C A , D A ) is a coherent dual pair, orthat ( A, A ♯ , i A , e A , C A , D A ) is a coherent left dual for A ♯ . Remark . In the notation of definition 1.2.1, if we assume that C is a strictcubical monoidal 2-category, the swallowtail equations simplify to (cid:2) e A ◦ ( C A (cid:3) A ♯ ) (cid:3) · h φ ( e A ,Id I ) , ( Id A (cid:3) A♯ ,e A ) ◦ ( A (cid:3) i A (cid:3) A ♯ ) i · [ e A ◦ ( A (cid:3) D A )] = Id e A , (cid:2) ( A ♯ (cid:3) C A ) ◦ i A (cid:3) · h ( A ♯ (cid:3) e A (cid:3) A ) ◦ φ ( i A ,Id I ) , ( Id A♯ (cid:3) A ,i A ) i · [( D A (cid:3) A ) ◦ i A ] = Id i A . It is clear that every coherent right dual is, in particular, a right dual. How-ever, it is natural to ask whether every right dual can be made into a coherentright dual. This question was solved in [Pst14]. The notation a • A,A ♯ ,A is used in [SP11] to refer to the chosen inverse adjoint equivalenceof the 1-morphism a A,A ♯ ,A . orollary 1.2.3. ([Pst14] corollary 2.8) Every right dual can be made coherent,and every left dual can be made coherent. Definition 1.2.4.
Let C be a monoidal 2-category. We say that C has right/leftduals if every object has a right/left dual. We say that C is rigid if it has rightand left duals.Using the above result from [Pst14], we believe that one can construct aright dual 2-functor on any monoidal 2-category with right duals. Notation 1.2.5.
Given C a monoidal 2-category with monoidal product (cid:3) , wedenote by C (cid:3) op the monoidal 2-category with the opposite monoidal product. Conjecture 1.2.6.
Let C be a monoidal 2-category with right duals. Thereexists a monoidal 2-functor ( − ) ♯ : C → C (cid:3) op ;1 op that sends an object A to the object underlying a right dual for A . If, in addition, C has left duals, there is a monoidal 2-functor, which we denote by ♯ ( − ) thatsends an object to its left dual. Further, ♯ ( − ) is a monoidal pseudo-inverse for ( − ) ♯ .Remark . Most of the proof of conjecture 1.2.6 is straightforward. However,many interchangers appear in the course of the construction. Checking thecoherence axioms involves making sure that big composites of interchangersagree, and we have not found a satisfactory way to deal with these.The decategorified version of the next lemma is well-known.
Lemma 1.2.8.
Let C be a monoidal 2-category, and A , B and C objects of C .1. If C has a right dual, there are natural equivalences Hom C ( A, B (cid:3) C ) ≃ Hom C ( A (cid:3) C ♯ , B ) ,Hom C ( C (cid:3) A, B ) ≃ Hom C ( A, C ♯ (cid:3) B ) .
2. If C has a left dual, there are natural equivalences Hom C ( A (cid:3) C, B ) ≃ Hom C ( A, B (cid:3) ♯ C ) ,Hom C ( A, C (cid:3) B ) ≃ Hom C ( ♯ C (cid:3) A, B ) . Proof.
Without loss of generality, we may assume that C is strict cubical (thisfollows from the coherence theorem of [GPS95]). Let ( C, C ♯ , i C , e C , C C , D C ) bea coherent dual pair. Then, the functors More precisely, { ∗ ( − ) } op ;2 op is a pseudo-inverse. om C ( A, B (cid:3) C ) ⇄ Hom C ( A (cid:3) C ♯ , B ) f ( B (cid:3) e C ) ◦ ( f (cid:3) C ♯ )( g (cid:3) C ) ◦ ( A (cid:3) i C ) ← [ g form an adjoint equivalence with counit[( A (cid:3) C C ) ◦ f ] · [( B (cid:3) e C (cid:3) C ) ◦ φ ( f,Id ) , ( Id,i C ) ]and unit [ φ ( g,Id ) , ( Id,e C ) ◦ ( A (cid:3) i C (cid:3) C ♯ )] · [ g ◦ ( B (cid:3) D C )] . The triangle identites follow from the swallowtail equations. The naturality in A and B is clear from the definition. Remark . The above lemma can be reformulated by saying that certain 2-functors form a 2-adjunction. For instance, if C has a right dual, then ( − ) (cid:3) C ♯ is left 2-adjoint to ( − ) (cid:3) C . The monoidal 2-categories we will consider have both adjoints and duals. Thatis why we now examine the properties of such monoidal 2-categories.
Lemma 1.3.1.
Let A ♯ be a right dual for A in C , a monoidal 2-category withright adjoints, with unit i A and counit e A . Then A ♯ has a right dual.Proof. Using the monoidal 2-functor ( − ) ∗ constructed in lemma 1.1.5, one getthat the image of i A and e A under ( − ) ∗ witness that A ♯ is a right dual for A in C op ;2 op . This means that A is a right dual for A ♯ in C . Remark . Lemma 1.3.1 can be generalized to n -categories, see [Ara17, Lem.4.1.2]. Corollary 1.3.3.
Let C be a monoidal 2-category with right adjoints and rightduals. For any object A of C , A ♯♯ is equivalent to A . Corollary 1.3.4.
Let C be a monoidal 2-category with right adjoints. If C hasright duals, then it also has left duals. The notion of a n -condensation monad in an n -category was introduced in[GJF19]. These objects categorify the notion of an idempotent to an n -category.In particular, a 1-condensation monad is an idempotent in a (1-)category. Below,we recall the definition in the case n = 2. Definition 1.4.1.
Let C be a 2-category. A 2-condensation monad in C consistsof: 1. An object A ; 5. A 1-morphism e : A → A ;3. And two 2-morphisms µ : e ◦ e ⇒ e and δ : e ⇒ e ◦ e ,such that1. The pair ( e, µ ) is a (non-unital) associative algebra;2. The pair ( e, δ ) is a (non-counital) coassciative algebra;3. The Frobenius relations hold, i.e. δ is an ( e, e )-bimodule map;4. We have: µ · δ = Id e . Remark . More concisely, a 2-condensation monad on A in C is a non-unitalnon-counital special Frobenius algebra in the monoidal category End C ( A ).Generalizing the notion of a split surjection, [GJF19] also introduced n -condensations. Definition 1.4.3.
Let C be a 2-category. A 2-condensation in C consists of:1. Two objects A and B ;2. Two 1-morphism f : A → B and g : B → A ;3. And two 2-morphisms φ : f ◦ g ⇒ Id B and γ : Id B ⇒ f ◦ g such that1. We have: φ · γ = Id Id B .Given a 2-condensation as above, observe that the induced structure on theobject A alone is precisely that of a 2-condensation monad. Definition 1.4.4.
Given a 2-condensation monad (
A, e, µ, δ ) in C , an exten-sion of this 2-condensation monad to a 2-condensation is a 2-condensation( A, B, f, g, φ, γ ) together with a 2-isomorphism θ : g ◦ f ∼ = e such that µ = θ · ( g ◦ φ ◦ f ) · ( θ − ◦ θ − ) ,δ = ( θ ◦ θ ) · ( g ◦ γ ◦ f ) · θ − . Remark . Extensions of a 2-condensation monad to a 2-condensation arepreserved by 2-functors (upon insertion of the relevant coherence 2-isomorphisms).Categorifying the notion of being idempotent complete [GJF19] makes thefollowing definition.
Definition 1.4.6.
A 2-category C has all condensates if it is locally idempotentcomplete and every 2-condensation monad can be extended to a 2-condensation.6 emark . Theorem 2.3.2 of [GJF19] states that if C is a locally idempotentcomplete 2-category, then the 2-category of extensions of any 2-condensationmonad to a 2-condensation is either empty or contractible. (This is the cate-gorified version of the statement that if an idempotent splits, it does so uniquelyup to unique isomorphism.)Given a locally idempotent complete 2-category C , [GJF19] constructs aKaroubi envelope for C , i.e. a 2-category Kar ( C ) that has all condensates andcomes with a canonical fully faithful inclusion ι C : C ֒ → Kar ( C ). This is given bya certain 2-category with objects 2-condensation monads, 1-morphism bimod-ules and 2-morphisms bimodule maps. For our purposes, it will be importantto have a universal characterization for Kar ( C ) and ι C . Definition 1.4.8.
Let C be a locally idempotent complete 2-category. AKaroubi envelope is a 2-functor ι C : C → Kar ( C ) satisfying the following 3-universal property:0. The locally idempotent complete 2-category Kar ( C ) has all condensates.1. For any locally idempotent complete 2-category A that has all condensatesand every 2-functor F : C → A , there exists a 2-functor F ′ : Kar ( C ) → A and a 2-natural equivalence φ : F ′ ◦ ι C ⇒ F .2. For every 2-functors G, H : Kar ( C ) → A and 2-natural transformation λ : G ◦ ι C ⇒ H ◦ ι C , there exists a 2-natural transformation ξ : G ⇒ H and an invertible modification Γ : ξ ◦ ι C ⇛ λ .3. Furthermore, for any 2-natural transformations π, ρ : G ⇒ H and modifi-cation ∆ : π ◦ ι C ⇒ ρ ◦ ι C , there exists a unique modification Ξ : π ⇛ ρ such that Ξ ◦ ι C = ∆. Remark . By its very definition, a Karoubi envelope, if it exists, is uniquein a precise way; Namely, the 3-category of Karoubi envelopes is a contractible3-groupoid.In appendix B.2, we shall prove the following proposition:
Proposition 1.4.10.
Every locally idempotent complete 2-category has a Karoubienvelope.
In the context of R -linear 2-categories, where R is a fixed commutative ring,the natural notion of completion is that of Cauchy completion. Definition 1.4.11. An R -linear 2-category is Cauchy complete if it has allcondensates, is locally additive, and has all direct sums (of objects).In the linear context, one can formulate an analogue of definition 1.4.8, whereKaroubi envelope is replaced by Cauchy complete, and the 2-functors appearingare assumed to be R -linear. We prove the following result in appendix B.2. Proposition 1.4.12.
Let C be a locally additive and locally idempotent complete R -linear 2-category, then there exists an R -linear 2-functor C → Cau ( C ) that isa Cauchy completion. Fusion 2-Categories
We set up the analogues of the definitions of semisimple 2-category, fusion 2-category, and pivotal fusion 2-category of [DR18] in the setting of (weak) 2-categories. We prove that our definition of fusion 2-category is, up to stricti-fication, equivalent to the definition of fusion category intorduced by Douglas-Reutter. We also prove that the 3-category of multifusion categories as definedin [Sch13] is 3-equivalent to the 3-category of semisimple 2-categories, a resultthat was conjectured in [DR18]. Throughout, we work over a fixed algebraicallyclosed field k of characteristic zero. Semisimple 2-categories were originally defined in [DR18]; In order to avoidconfusion, we shall call these objects strict semisimple 2-categories. We givea (categorically) weaker definition compatible with the existing strict one. As2-categories can always be strictified, both definitions are equivalent.
Definition 2.1.1.
A semisimple 2-category is a k -linear 2-category that is lo-cally semisimple, has adjoints for 1-morphisms, and is Cauchy complete. Definition 2.1.2.
Let C be a semisimple 2-category. An object C of C is saidto be simple if End C ( C ) has simple monoidal unit. Definition 2.1.3.
A finite semisimple 2-category is a semisimple 2-categorythat is locally finite semisimple, and has only finitely many equivalence classesof simple objects.
Lemma 2.1.4.
Let C be a strict 2-category. It is (finite) semisimple in thesense of definition 2.1.1 if and only if it is (finite) semisimple in the sense of[DR18].Proof. It is enough to observe that all the defining properties of a semisimple2-category (as in 2.1.1) do not depend on any underlying strictness hypothesis.The result follows from the fact that for a locally idempotent complete andlocally additive linear 2-category, being Cauchy complete is equivalent to beingadditive and idempotent complete (in the sense of [DR18]) by theorem 3.3.3 of[GJF19].
Lemma 2.1.5.
Given a (finite) semisimple 2-category C (as in definition 2.1.1),there exists a linearly 2-equivalent (finite) semisimple strict 2-category.Proof. By a linear version of the coherence theorem for 2-categories, there existsa linear 2-equivalence F : C → D , where D is a strict linear 2-category. Thus,it is enough to check that all the properties in the definition of a semisimple2-categories are invariant under equivalences of linear 2-categories. This is clearfor all of them. For instance, one can see that being Cauchy complete is invariantusing the 3-universal property of the Cauchy completion.8 emark . In particular, all the results of [DR18] on strict (finite) semisimple2-categories apply to (finite) semisimple 2-categories.The following result gives many examples of finite semisimple 2-categories.
Theorem 2.1.7. [DR18, Thm 1.4.8] Let C be a multifusion 2-category. Then M od ( C ) , the 2-category of finite semisimple right C -module categories, right C -module functors, and right C -module natural transformations is a finite semisim-ple 2-category.Remark . The 2-category
M od ( C ) can equivalently be described as the 2-category of separable algebras in C , bimodules, and bimodule maps. This isthe content of proposition 1.3.13 of [DR18]. We shall alternate between thesetwo descriptions of the finite semisimple 2-category M od ( C ) depending on thesituation. Alternatively, we can also think of M od ( C ) as the 2-category of 2-condensation monads in C . Definition 2.1.9.
Let C be a finite semisimple 2-category. We say that anobject C ∈ C is a generator if the canonical inclusion BEnd C ( C ) → C of the one object 2-category with endomorphism category End C ( C ) into C is aCauchy completion.One of the most important results on finite semisimple 2-categories is thatthey are precisely the 2-categories of finite semisimple module categories overa multifusion category, as was shown in [DR18]. We recall this result in ourlanguage now. Lemma 2.1.10.
Every finite semisimple 2-category C has a generator.Proof. Without loss of generality, we may assume that C is strict. By the-orem 1.4.9 of [DR18], there exists an object C of C such that the inclusion BEnd C ( C ) → C is a direct sum completion followed by an idempotent comple-tion, whence it is a Cauchy completion (by theorem 3.3.3 [GJF19]).Let us examine an example of a finite semisimple 2-category in details. Example 2.1.11.
Let p be a prime, and Vect Z /p Z denote the category of Z /p Z -graded vector spaces with its standard monoidal structure. We shall denoteits simple objects by k i with i ∈ Z /p Z . By theorem 2.1.7, M od ( Vect Z /p Z )is a finite semisimple 2-category. Using theorem 1.1 of [Nat16], we see thatthis finite semisimple 2-category has two equivalence classes of simple objects,which we denote by M (0 , triv ) and M ( Z /p Z , triv ), where triv denotes thetrivial cohomology class (of the appropriate cohomology group). In this case,it is easy to give explicit Vect Z /p Z -module categories representative: We haveequivalences M (0 , triv ) ≃ Vect Z /p Z and M ( Z /p Z , triv ) ≃ Vect . Using these9epresentatives, one can describe the finite semisimple hom-categories betweenthe two simple objects. We obtain the following schematic description: M (0 , triv ) M ( Z /p Z , triv ) VectVect Z /p Z Vect Vect Z /p Z It should be noted that we have written the endomorphism category of thesimple object represented by M ( Z /p Z , triv ) as Vect Z /p Z , whereas the canonicaldescription of this hom-category is Rep ( Z /p Z ). As Z /p Z is abelian, these twofusion categories are equivalent.Interpreted correctly, the diagram above contains some information aboutthe composition of 1-morphisms. Viewing Vect Z /p Z as a monoidal category,and Vect as a
Vect Z /p Z -bimodule categories in the standard way describeshow most of the 1-morphisms compose. The remaining compositions are givenby Vect × Vect → Vect Z /p Z ( k , k ) M i ∈ Z /p Z k i , which is a Vect Z /p Z -balanced functor. Let us observe that finite semisimple 2-categories form a 3-category. Namely, itis the full sub-3-category of the linear 3-category 2
Cat k of k -linear 2-categorieson the finite semisimple 2-categories. We denote the 3-category of finite semisim-ple 2-categories by SS2C .In the thesis [Sch13], it has been proven that fusion categories, bimodulecategories, bimodule functors and bimodule natural transformations form a 3-category, which is denoted by
Bimod . However, the proof supplied there doesnot use all the properties of fusion categories; Inspection shows that the following(slightly) stronger result was proven.
Proposition 2.2.1. [Sch13] Multifusion categories, bimodule categories, bi-module functors and bimodule natural transformations form a linear 3-category,which we denote by
Mult . We will now proceed to prove that these two 3-categories are equivalent,solving a conjecture of [DR18].
Theorem 2.2.2.
There is a linear 3-functor
M od : Mult → SS2C that sendsa multifusion category C to the associated finite semisimple 2-category of finitesemisimple right C -module categories. Moreover, this 3-functor is an equiva-lence. roof. Let us consider the linear 3-functor
Hom
Mult ( Vect , − ) : Mult → k . By definition of the linear 3-category
Mult , this 3-functor sends a multifusioncategory C to the associated 2-category of finite semisimple ( Vect , C )-bimodulecategories. But, as a left Vect -module category structure on a finite semisim-ple category is essentially unique, we may identify this 2-category with the2-category of finite semisimple right C -module categories. As linear 2-categoriesof this form are finite and semisimple by 2.1.7, the 3-functor Hom
Mult ( Vect , − )factors through the full sub-3-category SS2C . Let us denote this 3-functor by
M od ( − ) : Mult → SS2C . It remains to prove that
M od is an equivalence of linear 3-categories. Accord-ing to the definition of a 3-equivalence given in [GPS95], it is enough to showthat it is essentially surjective and induces equivalences on hom-2-categories.Lemma 2.1.10 shows that
M od is essentially surjective. Now, let C and D bemultifusion categories. We wish to prove that the linear 2-functor Hom
Mult ( C , D ) = Bimod ( C , D ) → F un k ( M od ( C ) , M od ( D ))induced by M od is a equivalence. By construction, it sends a finite semisimple( C , D )-bimodule category M to the 2-functor M od ( C ) → M od ( D ) . N 7→ N ⊠ C M Firstly, we have to show that every linear 2-functor F : M od ( C ) → M od ( D )can be obtained in this way. Observe that the image of C ∈
M od ( C ) under F inherits a ( C , D )-bimodule structure from the left C -module structure of C .Moreover, we have the following commutative diagram: BEnd
Mod ( C ) ( C ) F ′ ' ' PPPPPPPPPPP κ (cid:15) (cid:15) M od ( C ) F / / M od ( D ) , where κ is a Cauchy completion. In particular, F is characterized uniquelyup to 2-natural equivalence by the linear 2-functor F ′ : BEnd
Mod ( C ) ( C ) → M od ( D ) through the 3-universal property of the Cauchy completion. Note that F ′ ( C ) = F ( C ) ≃ C ⊠ C F ( C ) . Said differently, there is a 2-natural equivalence κ ⊠ C F ( C ) ≃ F ′ . Thus, there isa 2-natural equivalence F ≃ ( − ) ⊠ C F ( C ).Secondly, let M , and M be two ( C , D )-bimodules, and α : ( − ) ⊠ C M ⇒ ( − ) ⊠ C M be a 2-natural transformation. Using the 3-universal property of11he Cauchy completion, we find that α is uniquely determined up to invertiblemodification by α C : M → M , which inherits a ( C , D )-bimodule functorstructure from the left C -module structure on C . More precisely, there exists aninvertible modification α ∼ = ( − ) ⊠ C α C .Thirdly, let f , f : M → M be two ( C , D )-bimodule functors, and let Γ :( − ) ⊠ C f ⇛ ( − ) ⊠ C f be a modification. The 3-universal property of the Cauchycompletions shows that Γ is uniquely determined by Γ C : f ⇒ f . Further, wefind that Γ C is a ( C , D )-bimodule natural transformation. Moreover, any ( C , D )-bimodule natural transformation f ⇒ f determines a unique modification.Finally, it follows from the 3-universal property of the Cauchy completion thatthese two operations are inverses to one another. Remark . It is known that every equivalence of linear 3-categories has aninverse. We believe that the linear 3-functor
End : SS2C → Mult given on objects by sending a finite semisimple category C to End C ( C ), where C ∈ C is an arbitrary generator, is such an inverse to M od .Fusion categories are the most important examples of multifusion categories.Therefore, it seems sensible to consider the full sub-3-category of
Mult on thefusion categories. However, this definition is problematic, as being fusion isnot a Morita invariant property. Said differently, the full sub-3-category on thefusion categories is not replete.
Definition 2.2.4.
A multifusion category C is connected if it is Morita equiva-lent to a fusion category. Remark . Let C be a multifusion category. It is known that we can write C as a matrix of finite semisimple categories whose diagonal entries are non-zero fusion categories (see [EGNO15, Section 4.3]). It is immediate that C isconnected if and only if all entries of the matrix are non-zero finite semisimplecategories.The folding and unfolding constructions of [DR18] suggest that the corre-sponding property for finite semisimple 2-categories is also a form of connected-ness. Definition 2.2.6.
Let C be a finite semisimple 2-category. We say that twosimple objects A, B of C are connected if the category Hom C ( A, B ) is non-zero.A semisimple 2-category C is said to be connected if all of its simple objects areconnected. Example 2.2.7.
The finite semisimple 2-category of example 2.1.11 is con-nected.
Proposition 2.2.8.
The linear 3-equivalence
M od of theorem 2.2.2 restrictsto an equivalence between the full sub-3-categories on the connected multifusioncategories and on the connected semisimple 2-categories. roof. Observe that it is enough to show the following two things: Firstly, givena fusion category C , then M od ( C ) is connected. Secondly, if C is a connectedfinite semisimple 2-category, then there exists a connected multifusion category D such that M od ( D ) ≃ C .We begin by proving the first point. Given a fusion category C , everyfinite semisimple indecomposable right C -module category M admits a non-zero right C -module functor C → M (Pick a non-zero object in M ). Thus, Hom
Mod ( C ) ( C , M ) is non-zero for every M as above. As finite semisimple inde-composable C -module categories are precisely the simple objects of M od ( C ), andthat C is indecomposable, we are done by the existence of adjoints for C -modulefunctors (see [DSPS19]) and proposition 1.2.19 of [DR18].Let C be a connected finite semisimple 2-category. Let A i be representativesfor the finitely many equivalence classes of simple objects of C . Let us define C := End C ( ⊞ i A i ) . By definition, we have that C is connected. Further, the proof of theorem 1.4.9 of[DR18] proves that the canonical inclusion M od ( C ) → C is an equivalence. Recall that multifusion categories are defined as finite semisimple categoriesequipped with a rigid monoidal structure. From a purely category theoreticstandpoint, multifusion 2-categories ought to be defined similarly. Namely, asa finite semisimple 2-category with a rigid monoidal structure. Further, fu-sion categories are those multifusion categories whose monoidal unit is simple.By analogy, we define fusion 2-categories as the multifusion 2-categories whosemonoidal unit is simple.
Definition 2.3.1.
A multifusion 2-category C is a finite semisimple 2-categoryequipped with a rigid k -linear monoidal structure (see [SP11]). In particular, itcomes equipped with a bilinear 2-functor (cid:3) : C × C → C , and a monoidal unit I . A fusion 2-category is a multifusion 2-category whosemonoidal unit is simple. Remark . By their very definition, all the standard results of monoidal2-category theory (up to linearization) apply to multifusion 2-categories. Forinstance, by [SP11], every multifusion 2-category is equivalent to a skeletal mul-tifusion 2-category.It is well-known that the monoidal unit of a multifusion category splits as adirect sum of non-isomorphic simple objects (see [EGNO15] section 4.3). Usingthe fact that every object of a finite semisimple 2-categories decomposes into adirect sum of simple objects (see proposition 1.4.5 of [DR18]), a similar resultholds for multifusion 2-categories. 13 emma 2.3.3.
Let C be a multifusion 2-category. Let X i , i = 1 , ..n be thefinitely many simple objects appearing in the decomposition of the monoidalunit as a direct sum of simple objects, i.e. I ≃ ⊞ ni =1 X i . Then,
Hom C ( X i , X j ) is non-zero if and only if i = j . Moreover, we have that X i (cid:3) X j is equivalent to X i if i = j and to otherwise.Proof. For any i , we have X i (cid:3) I ≃ X i . As X i is simple, there exists preciselyone j such that X i (cid:3) X j is non-zero. Together with the reverse argument on j ,this shows that X i ≃ X i (cid:3) X j ≃ X j . If i = j , then X i (cid:3) I would have ( X i (cid:3) X i ) ⊞ ( X i (cid:3) X j ) as a summand, whencewould not be simple. Thus, we must have i = j and X i (cid:3) X i ≃ X i . Moreover,this shows that X i is both a left and a right dual for X i .Let i , j be arbitrary. Then, we have: Hom C ( X i , X j ) ≃ Hom C ( X i (cid:3) ( ♯ X j ) , I ) ≃ Hom C ( δ ij X i , I ) . The last term is non-zero precisely when i = j . This finishes the proof. Remark . Lemma 2.3.3 can be seen as a generalization of the fact thatbraided multifusion categories have no non-zero entries away from the diagonal(see lemma 5.3 of [BDSPV14]).Let C be a multifusion 2-category. We write i C j for the semisimple 2-category X i (cid:3) C (cid:3) X j . The following result is a direct analogue of the usual decompositionof a multifusion category. Lemma 2.3.5.
The semisimple 2-categories i C i are fusion 2-categories, and thefinite semisimple 2-categories i C j are ( i C i , j C j ) -bimodule 2-categories. Finally,the following matrix C · · · C n ... . . . ... n C · · · n C n represents the fusion rule of the multifusion 2-category C . Example 2.3.6.
The 2-category of representations of a finite 2-groupoid G issemisimple and finite by [DR18]. It inherits a (symmetric) monoidal structurefrom the symmetric monoidal structure on 2 Vect . The monoidal unit is givenby the constant 2-functor
G → Vect with value
Vect . This 2-representationsplits as the direct sum of the simple 2-representations that are constant withvalue
Vect on exactly one component of G and 0 on the others.14 .4 Comparison with strict Fusion 2-Categories In the preceding section, we have defined fusion 2-categories. In particular, theunderlying monoidal 2-category is allowed to be weak. However, Douglas andReutter have used in [DR18] the term fusion 2-category to refer to certain Graymonoids (objects that we call strict fusion 2-categories). We now prove thatthese two definitions are in some sense equivalent. More precisely, strict fusion2-categories correspond to a strict (in the categorical sense) version of fusion2-categories. In particular, one can switch relatively harmlessly between thetwo definitions.
Lemma 2.4.1.
Let C be a multifusion 2-category. There exists a multifusion2-category D , whose underlying monoidal 2-category is a strict cubical k -linearmonoidal 2-category, that is linearly equivalent to C . Moreover, if C is fusion,so is D .Proof. Using a k -linear version of the coherence theorem of [GPS95], we ob-tain a strict cubical k -linear monoidal 2-category D that is linearly monoidallyequivalent to C . Observe that the underlying linear equivalence of 2-categorieswitnesses that the 2-category D is a finite semisimple 2-category. Moreover,rigidity is preserved by monoidal equivalences of 2-categories. This proves thefirst part of the result. The last part follows from the Whitehead theorem formonoidal 2-categories (see [SP11]). Lemma 2.4.2.
There is a bijection between weak fusion 2-categories, whose un-derlying monoidal 2-category is a strict cubical (or opcubical) k -linear monoidal2-category and strict fusion 2-categories. Proof.
The k -linear version of [BMS12] lemma 2.16 shows that there is a bi-jection between strict cubical k -linear monoidal 2-categories and k -linear Graymonoids with one object. Moreover, the equivalence does not affect the under-lying 2-categories. Thus, the only thing we have to prove is that this bijectionrespects the existence of duals. This property follows from the fact that themonoidal product with a fixed 1-morphism is invariant under this bijection byconstruction. Remark . In particular, we may invoke all the results that [DR18] haveproven for strict fusion 2-categories, and apply them to fusion 2-categories.
We give some consequences of the definition of a multifusion 2-category. Mostare analogues of standard results on fusion categories. We begin by comparingleft and right duals in multifusion 2-categories. Then, we prove a serie of resultsthat explain the behaviour of simple objects under the operation of taking rightduals, multiplication with a simple object, and arbitrary monoidal 2-functors. This statement can be made rigorous using a set-theoretic argument. For instance, onecould use a bigger universe. emma 2.5.1. Let C be a multifusion 2-category, and let A ♯ be a right dual for A in C , then A is a right dual for A ♯ .Proof. This is an immediate consequence of lemma 1.3.1.On the one hand, the decategorified analogue of the next result is well-known:it says that left and right duals in a fusion category agree. The proof reliescrucially on the category being semisimple. On the other hand, in the contextof fusion 2-categories, the proof has a very distinct flavour; it uses corollary1.3.3, which applies in great generality.
Corollary 2.5.2.
Let C be a multifusion 2-category, and A an object of C . Then, A ♯♯ is equivalent to A . Corollary 2.5.3.
Let C be a monoidal finite semisimple 2-category. If C hasright duals, then it also has left duals, i.e it is multifusion. As is the case in any multifusion category, the right dual of a simple objectis again a simple object.
Lemma 2.5.4.
In any multifusion 2-category, the left and right duals of asimple object are simple.Proof.
Let A be a simple object with right dual A ♯ . Observe that the right dualof a non-zero object has to be non-zero. Further, a right dual for a direct sum isgiven by the direct sum of the right duals. Thus, if A ♯ were not simple, i.e. hadtwo non-zero summands, then A ♯♯ would have two non-zero summands. Thiscontradicts the fact that A ≃ A ♯♯ is simple.Lemma 2.5.4 implies that the operation of taking the right dual inducesa bijection on the set of equivalence classes of simple objects. Now, observethat lemma 1.2.8 also applies to multifusion 2-categories, yielding the followingresults: Corollary 2.5.5.
Let C be a multifusion 2-category, and A , B two simple ob-jects such that Hom C ( A, B ) is non-trivial. Then, Hom C ( ♯ A, ♯ B ) is non trivial.Proof. Note that it is enough to prove that
Hom C ( ♯ B, ♯ A ) is non-trivial. Namely,the 2-functor ( − ) ∗ of lemma 1.1.4 provides us with a linear equivalence: Hom C ( ♯ A, ♯ B ) ≃ Hom C ( ♯ B, ♯ A ) . Now, using lemma 1.2.8, there are linear equivalences:
Hom C ( ♯ B, ♯ A ) ≃ Hom C ( ♯ B (cid:3) A, I ) ≃ Hom C ( A, B ) . This concludes the proof.
Corollary 2.5.6.
Let C be a fusion 2-category, and A , J two simple objectssuch that J is in the component of the monoidal unit (i.e. Hom C ( I, J ) is non-zero), then A (cid:3) J is a direct sum of simple objects in the connected componentof A . roof. By contradiction, let us assume that A (cid:3) J has a summand B not in theconnected component of A . In particular, there exists a non-trivial 1-morphismbetween A (cid:3) J and B . Thence, there exists a non-trivial 1-morphism between ♯ A (cid:3) B and J . Thus, there exists a non-trivial 1-morphism between ♯ A (cid:3) B and I ,which is equivalent to saying that A and B are in the same connected component.We examine the behaviour of simple objects under the monoidal productand arbitrary 2-functors. Lemma 2.5.7.
Let C be a fusion category and C , D two non-zero objects. Then C (cid:3) D is non-zero.Proof. By rigidity, C has a left dual, which we denote by ♯ C . In particular,the decomposition of ♯ C (cid:3) C into simple objects contains a copy of J , a simpleobject in the connected component of I . By definition, there exists a non-zero1-morphism f : I → J . Thus, we get a map I (cid:3) D f (cid:3) D / / J (cid:3) D (cid:31) (cid:127) / / ♯ C (cid:3) C (cid:3) D. On one hand, if C (cid:3) D were equal to zero, then J (cid:3) D ≃
0, whence we wouldhave f (cid:3) D ≃
0. On the other hand, f has a left adjoint ∗ f , and the 2-functor( − ) (cid:3) D preserves adjunctions. As Id I is a direct summand of ∗ f ◦ f , we findthat f (cid:3) D = 0. Consequently, C (cid:3) D must be non-zero. Proposition 2.5.8.
Let F : C → D be a monoidal 2-functor between two fusion2-categories. For any non-zero object C of C , we have that F ( C ) is non-zero.Proof. The proof is analogous to the proof of the lemma 2.5.7. Namely, if F ( C )were equivalent to 0, then we would have0 ≃ (cid:3) F ( ♯ C ) ≃ F ( C ) (cid:3) F ( ♯ C ) ≃ F ( C (cid:3) ♯ C ) . This shows that F ( I ) ≃
0, which contradicts the assumption that F is amonoidal 2-functor. The goal of this section is to study a special class of fusion 2-categories: con-nected fusion 2-categories. They are ubiquitous both because the 2-categoryof finite semisimple module categories associated to a braided fusion categoryis connected, and because every fusion 2-category has a connected fusion 2-category as a full sub-2-category. This is similar to the fact that every topo-logical monoid admits a connected submonoid corresponding to the connectedcomponent of the identity.
Definition 2.6.1.
A connected fusion 2-category is a fusion 2-category whoseunderlying finite semisimple 2-category is connected.17 emark . By proposition 1.2.19 of [DR18], in order to show that a fusion2-category is connected, it is enough to check that the hom-categories from themonoidal unit to any simple object is non-trivial.
Proposition 2.6.3. [DR18, Construction 2.1.19] Let C be a braided fusioncategory. Then, M od ( C ) is a connected fusion 2-category, with monoidal productgiven by ⊠ C the balanced Deligne tensor product.Proof. Note that the monoidal 2-category B C is a rigid. Then, through theproof of theorem 4.1.1 of [GJF19], we find that its Cauchy completion, givenby M od ( C ), is a multifusion 2-category with monoidal product (cid:3) . Explicitly,the monoidal structure is as follows: Given two separable algebras A , B in C , representing two right C -module categories M od C ( A ) and M od C ( B ), theirproduct A ⊗ B is again a separable algebra in C . The separable algebra A ⊗ B represents an object M od C ( A ⊗ B ) in M od ( C ), which is, by construction,the monoidal product of M od C ( A ) and M od C ( B ). The result follows from theequivalence of right C -module categories: M od C ( A ) ⊠ C M od C ( B ) ≃ M od C ( A ) ⊠ C RM od C ( B op ) ≃ Bimod C ( A, B op ) ≃ M od C ( A ⊗ B ) = M od C ( A ) (cid:3) M od C ( B ) , where we have used the equivalence of right C -module categories M od C ( B ) ≃ RM od C ( B op )between the category of left B -modules and the category of right B op -modules. Definition 2.6.4.
Let C be a fusion 2-category with monoidal unit I . Wedenote by C the connected component of the identity, i.e. the full additivesub-2-category on the simple objects that admit a non-zero morphism from I . Proposition 2.6.5.
The 2-category C is a fusion sub-2-category of C that isconnected.Proof. We begin by proving that C is finite semisimple. The only propertywhich is not obvious is that C has all condensates. Let ( A, ... ) be a 2-condensationmonad in C , and let ( A, B, f, g, ... ) be an extension of (
A, ... ) to a 2-condensation.Observe that for every simple summand C of B , the composite of f with theprojection B → C is a non-zero 1-morphism A → C . (If this 1-moprhism waszero, ( A, B, f, g, ... ) would not be a 2-condensation.) The claim thus followsfrom proposition 1.2.19 of [DR18]. Further, by definition, we have that C isconnected.By corollary 2.5.6, the monoidal product of C restricts to give C a monoidalstructure. Finally, as I ♯ ≃ I , we find by corollary 2.5.5 that this monoidalstructure is rigid, and C is clearly fusion.18 orollary 2.6.6. Let C be a fusion 2-category. Then, there is an equivalenceof monoidal 2-categories: M od ( End C ( I )) ≃ C . Proof.
As a consequence of the proof of the above proposition, we find that
BEnd C ( I ) ֒ → C is a Cauchy completion. Further, this inclusion is monoidal, whence, by the3-universal property of the Cauchy completion, we get the desired result.Corollary 2.6.6 shows that the behavior of the monoidal product on theconnected component of the identity is completely determined by the braidingon the fusion category End C ( I ). Proposition 2.6.7.
There is an equivalence between the category of connectedfusion 2-categories and equivalence classes of monoidal linear 2-functors, andthe category of braided fusion categories and equivalence classes of braided tensorfunctors.Proof.
Let us denote by A the category of connected fusion 2-categories andequivalence classes of monoidal linear 2-functors, and by B the category ofbraided fusion categories and equivalence classes of braided tensor functors. Tak-ing the endomorphism category of the monoidal unit yields a functor End ( − ) ( I ) : A → B , and taking the Cauchy completion of the delooped braided fusion category givesa functor M od ( − ) = Cau ( B ( − )) : B → A . Using the 3-universal property of the Cauchy completion, one can show thatthese functors are pseudo-inverses for one another.
Let C be a pointed braided fusion category. By results of [EGNO15], weknow that this corresponds equivalently to the data of a finite abelian group A equipped with an abelian 3-cocycle ( ω, β ). Further, we will assume that ω is trivial . We denote the braided fusion category associated to this data by Vect βA . Finite semisimple indecomposable right module categories over Vect βA correspond to pairs ( E, φ ), where E is a subgroup of A and φ is 2-cocycle on E with value in k × (considered up to 2-coboundary). We denote the correspondingright module category by M ( E, φ ). Proposition 3.16 of [ENO09], explains howto compute the relative Deligne tensor product of two right C -modules when β is trivial. We now generalize this result. If A has odd order, then this can always be done. E, φ ), and (
F, ψ ) be two pairs consisting of a subgroup of A , and anappropriate 2-cocycle. Let Alt ( φ ) : E × E → k × and Alt ( ψ ) : F × F → k × bethe corresponding skew-symmetric bilinear forms, i.e. Alt ( φ )( e , e ) := φ ( e , e ) /φ ( e , e ) , and similarly for Alt ( ψ ). We define a skew-symmetric bicharacter b on E ⊕ F by b (( e , f ) , ( e , f )) := Alt ( φ )( e , e ) Alt ( ψ )( f , f ) β ( f , e ) /β ( f , e ) . The group E ∩ F embeds in E ⊕ F via e ( e, − e ), thus we can consider itsorthogonal complement ( E ∩ F ) ⊥ under the bicharacter b . Now, let H be theimage of ( E ∩ F ) ⊥ under the canonical map E ⊕ F → A . The restriction of b to( E ∩ F ) ⊥ descends to a skew-symmetric bilinear form b ′ on H , which correspondsto an element of H ( H, k × ) represented by a chosen 2-cocycle ρ (see [Tam00]). Proposition 2.7.1.
We have M ( E, φ ) ⊠ C M ( F, ψ ) ≃ ⊞ mi =1 M ( H, ρ ) , where m = | ( E ∩ F ) ⊥ || ( E ∩ F ) || E || F | . Proof.
Let A ( E, φ ) be the algebra in C whose underlying object is k E and whosemultiplication is given by φ , and similarly for A ( F, ψ ). By definition, M ( E, φ ) =
M od C ( A ( E, φ )), and M ( F, ψ ) =
M od C ( A ( F, ψ )). The relative Deligne tensorproduct is given by the category of left modules over the algebra A ( E, φ ) ⊗ A ( F, ψ ) in C . Note that the multiplication of this algebra is twisted by thebraiding β of C . More precisely, it is given by the 2-cocycle τ on E ⊕ F definedby (( e , f ) , ( e , f )) φ ( e , e ) ψ ( f , f ) β ( f , e ) . In particular, the corresponding skew-symmetric bicharacter is b as definedabove. Using proposition 2.11 of [ENO09], we obtain the desired result. Remark . It should be possible to generalize proposition 2.7.1 to the casewere ω is not assumed to be trivial. However, the above proof does not immedi-ately generalize because if ω is not trivial, then φ and ψ may not be 2-cocycles,and thus Alt ( φ ) and Alt ( ψ ) may not be skew-symmetric bicharacters. Example 2.7.3.
Recall the notations of example 2.1.11. It is well-known thatthe fusion category
Vect Z /p Z admits p distinct braided structures (up to braidedmonoidal automorphism of Vect Z /p Z that is the identity on objects). A braiding b on Vect Z /p Z is determined by its value b k , k = e πikp , for any 0 ≤ k < p . Ifwe allow arbitrary braided monoidal automorphisms, there are two if p = 2, andthree otherwise. The symmetric or trivial braiding is specified by k = 0, andwe denote the corresponding braided fusion category by Vect triv Z /p Z . The other20nes correspond to the cases where 0 < k < p is a quadratic residue or not. Forsimplicity, we only treat the case where k is a quadratic residue, for which wemay assume k = 1, and denote the corresponding braided fusion category by Vect β Z /p Z . The remaining case is entirely analogous.Let us begin by examining the monoidal product on the finite semisimple2-category of finite semisimple right modules categories over C := Vect triv Z /p Z . As C is the monoidal unit for the induced monoidal structure on M od ( C ), we onlyhave to determine Vect ⊠ C Vect . Also recall that as right
Vect Z /p Z -modulecategories, we have Vect ≃ M ( Z /p Z , triv ) in the notations used above. Now,a straightforward computation using proposition 2.7.1 shows: Vect ⊠ C Vect ≃ ⊞ pi =1 Vect , as right C -module categories.We now turn our attention to the case D := Vect β Z /p Z . As above, D is themonoidal unit of the induced monoidal structure on M od ( D ), whence we onlyhave to determine Vect ⊠ D Vect . In order to use proposition 2.7.1, we computethat h (1 , − i ⊥ ⊆ Z /p Z ⊕ Z /p Z is precisely h (1 , − i . This gives m = 1, andthus Vect ⊠ D Vect ≃ M (0 , triv ) ≃ D . This examples shows that the braiding we put on a fusion category can have abig impact on the fusion rule of the associated fusion 2-category.
A The Adjoints Monoidal 2-Functor
Notation A.0.1.
Let C be a (monoidal) 2-category. Given f : A → B a 1-morphism in C , we denote by { f } op ;2 op : B → A the corresponding 1-morphismin C op ;2 op . We write ◦ op for the composition of 1-morphisms in C op ;2 op . Given α : f ⇒ g a 2-morphism in C , we denote by { α } op ;2 op : { g } op ;2 op ⇒ { f } op ;2 op the corresponding 2-morphism in C op ;2 op . Proof. (Lem. 1.1.4) This is well-known (for instance, see [Gho10]), but letus indicate briefly how to proceed. Thanks to the coherence theorem for 2-categories, we can omit the coherence 2-isomorphisms for C . The 2-functor( − ) ∗ is defined as follows:It sends the obejct C of C to itself. Given a 1-morphism f , we set f ∗ := { f ′ } op ;2 op , where f ′ is a fixed right adjoint for f with unit η f and counit ǫ f .Given a 2-morphism α : f ⇒ g , we define α ∗ := { ( f ′ ◦ ǫ g ) · ( f ′ ◦ α ◦ g ′ ) · ( η f ◦ g ′ ) } op ;2 op . Given two 1-morphisms f : A → B, g : B → C , the structure 2-isomorphism g ∗ ◦ op f ∗ ⇒ ( g ◦ f ) ∗ in C op ;2 op witnessing that ( − ) ∗ respects the compositionof 1-morphisms is given by the 2-isomorphisms { ( f ′ ◦ g ′ ◦ ǫ g ◦ f ) · ( f ′ ◦ η g ◦ f ◦ ( g ◦ f ) ′ ) · ( η f ◦ ( f ◦ g ) ′ ) } op ;2 op . { η − Id C } op ;2 op : Id ∗ C ⇒ Id C for every object C of C . This data specifies the 2-functor ( − ) ∗ . The associativity of compositionfollows from the triangle identities, and the unitality of composition is clear.Further, if C also has left-adjoints, we can dually define a 2-functor ∗ ( − ). Inparticular, for any given 1-morphism f , we fix a left adjoint ′ f with unit ξ f andcounit κ f . It is not hard to see that ∗ ( − ) is a pseudo-inverse for ( − ) ∗ (see thefootnote on page 2). For instance, a 2-natural equivalence θ : ( ∗ ( − )) ∗ ⇒ Id isgiven by the identity 1-morphism on objects, and on a 1-morphism f by the2-isomorphism ( ǫ ( ′ f ) ◦ f ) · (( ′ f ) ′ ◦ ξ f ) . Lemma A.0.2.
Let F : C → D be a 2-functor. There is a 2-natural equivalence e that fits into the following diagram: C F / / ( − ) ∗ (cid:15) (cid:15) D ( − ) ∗ (cid:15) (cid:15) C op ;2 op e < qqqqqqqqqq qqqqqqqqqq F op ;2 op / / D op ;2 op . Proof.
As above, we omit the coherence 2-isomorphisms of C , and D . Given anobject C of C , we set e C := Id F ( C ) . On the 1-morphism f : C → D , we define e f as the following 2-isomorphism: { ( η F ( f ) ◦ F ( f ′ )) · ( F ( f ) ′ ◦ F − f ′ ,f ) · ( F ( f ) ′ ◦ F ( ǫ f )) } op ;2 op , where F f ′ ,f : F ( f ′ ) ◦ F ( f ) ⇒ F ( f ′ ◦ f ) is the coherence 2-isomorphismsupplied by F . It is not hard to check that this defines a 2-natural equivalence. Proof. (Lem. 1.1.5) Thanks to the coherence theorem for monoidal 2-categories,there is an equivalence of monoidal 2-categories F : C → D such that D is strictcubical. Below, we will endow the adjoint 2-functor ( − ) ∗ of D with a monoidalstructure. Through the natural 2-equivalence of lemma A.0.2, this shows thatthe adjoint 2-functor ( − ) ∗ of C is equivalent as a 2-functor to a monoidal one.Hence, it is monoidal itself.In order to specify a monoidal structure on the 2-functor ( − ) ∗ on D , we needto give some data. To make this more digestible, we use the notations of [SP11].We begin by defining the 2-natural equivalence χ : ( − ) ∗ (cid:3) ( − ) ∗ ⇒ (( − ) (cid:3) ( − )) ∗ . Given two objects A , B we let χ A,B := Id A (cid:3) B . Given two 1-morphisms f : A → B and g : C → D we let the 2-isomorphism χ f,g : f ∗ (cid:3) g ∗ ⇒ ( f (cid:3) g ) ∗ begiven by { (cid:0) ( f ′ (cid:3) g ′ ) ◦ ǫ f (cid:3) g (cid:1) · (cid:0) φ − f ′ ,g ′ ) , ( f,g ) ◦ ( f (cid:3) g ) ′ (cid:1) · (cid:0) ( η f (cid:3) η g ) ◦ ( f (cid:3) g ) ′ (cid:1) } op ;2 op .
22t is not hard to see that χ is a 2-natural transformation. Further, it isclearly an isomorphism; and we pick χ • to be its inverse.We choose the 1-equivalence ι to be the identity 1-morphism on the monoidalunit. The modifications ω , γ , δ are uniquely specified by the universal propertyof right adjoints. Namely, we let γ C be the 2-isomorphism { η Id C } op ;2 op in C and δ C be the identity 2-morphism of { Id C } op ;2 op , for every object C of C . Further,for every A, B, C in C , the modification ω A,B,C is given by the 2-isomorphism { ǫ Id A (cid:3) B (cid:3) C } op ;2 op .The commutativity of the coherence diagrams can be checked using theuniqueness up to unique isomorphism of right adjoints. Similarly, one can endowthe 2-functor ∗ ( − ) with a monoidal structure.Finally, we need to construct monoidal 2-natural equivalences witnessingthat ( − ) ∗ and ∗ ( − ) are pseudo-inverse monoidal 2-functors. We construct themonoidal 2-natural equivalence ( ∗ ( − )) ∗ ⇒ Id , the other one can be constructedanalogously. Observe that using the argument at the beginning of this proof, it isenough to construct this monoidal 2-natural equivalence on D . As its underlying2-natural equivalence we take the 2-natural equivalence θ defined in the proofof lemma 1.1.4 above. Then, in the notation of [SP11], the 2-isomorphism M is given by the identity 2-morphism on Id I , and the modification Π is specifiedon A, B in C by the 2-isomorphism { η Id A (cid:3) B } op ;2 op . B Completions
B.1 Completions of 1-Categories
We recall some facts about the idempotent completion of a 1-category and itsadditive counterpart, the Cauchy completion. Moreover, we give 2-universalproperties characterizing these constructions.
Definition B.1.1.
Let C be a category. A Karoubi envelope, also called idempo-tent completion, is a functor ι C : C →
Kar ( C ) satisfying the following 2-universalproperty:0. The category Kar ( C ) is idempotent complete.1. For every idempotent complete category A , and every functor F : C → A ,there exists a functor F ′ : Kar ( C ) → A and a natural isomorphism φ : F ′ ◦ ι C ⇒ F .2. For every functors G, H : Kar ( C ) → A and natural transformation λ : G ◦ ι C ⇒ H ◦ ι C , there exists a unique natural transformation ξ : G ⇒ H such that ξ ◦ ι C = λ . Remark
B.1.2 . In the terminology of [Fio06, Def. 9.4.], the 1-morphism ι C isbiuniversal with respect to the forgetful 2-functor U : i.c. → fromthe 2-category of (small) idempotent complete categories to the 2-category of(small) categories. However, we have used the terminology 2-universal more23roadly than [Fio06], as he works exclusively with strict 2-categories. Giventhat every 2-category is 2-naturally equivalent to a strict one, this is not a causefor concern. Lemma B.1.3.
Given any category C , its Karoubi envelope exists.Proof. The existence of a functor ι C : C →
Kar ( C ) is standard. Verifyingthe 2-universal property follows from the universal property of splittings ofidempotents, and the fact that splittings of idempotents are preserved by allfunctors.The following result is part of the folklore. Proposition B.1.4.
The Karoubi envelope defines a 2-functor
Kar : → , and a 2-natural transformation ι : Id ⇒ Kar.
Further, if C is an idempotent complete category, ι C is an equivalence of cate-gories.Remark B.1.5 . To be precise, the definitions of
Kar and ι involve choices. Letus now explain how to give a universal characterization for the pair ( Kar, ι ).Recall that we write U : i.c. → for the forgetful 2-functor from the2-category of idempotent complete categories to the 2-category of categories.Using remark B.1.2, and theorem 9.16 of [Fio06], we have: Proposition B.1.6.
The 2-functor
Kar : → i.c. is a left 2-adjointfor the 2-functor U , with unit given by ι . It is useful to know when the Karoubi envelope of a functor is an equivalence.The following characterization is well-known.
Lemma B.1.7.
Let F : C → D . Then,
Kar ( F ) is an equivalence if and only if F is fully faithful, and every object of D can be expressed as the splitting of anidempotent supported on an object in the image of F . Let us presently indicate the changes that are necessary to accommodate R -linear categories, where R is a fixed commutative ring. In this context, theappropriate notion of completion is called the Cauchy completion, and involvesboth the splitting of idempotents and the existence of finite direct sums. TheCauchy completion is given by first taking the direct sum completion, and thenthe Karoubi envelope. Given a R -linear category C , we denote its Cauchy com-pletion by κ C : C →
Cau ( C ). The R -linear functor κ C has a 2-universal propertysimilar to the one of the Karoubi envelope. Lemma B.1.8.
Every R -linear category has a Cauchy completion. roof. The existence of the direct sum completion is standard. Combining thisobservation with lemma B.1.3, and the fact that the Karoubi envelope of an R -linear category that has finite direct sums is R -linear and has finite directsums yields the result.Let us denote by R the R -linear 2-category of R -linear categories, andby cR the full sub-2-category on the Cauchy complete R -linear categories.Using the the 2-universal property of the Cauchy completion, and [Fio06, Thm.9.16], we have: Proposition B.1.9.
Cauchy completion defines a R -linear 2-functor Cau : R → cR that is left 2-adjoint to the forgetful 2-functor V : cR → R , with unit κ . It is not hard to deduce a characterization of those R -linear functors thatbecome equivalences upon application of Cau from lemma B.1.7.
Lemma B.1.10.
Let F : C → D be a R -linear functor of R -linear categories.Then, Cau ( F ) is an equivalence if and only if F is fully faithful and every objectof D is the splitting of an idempotent supported on a finite direct sum of objectsin the image of F . B.2 Completions of 2-Categories
Let us now turn our attention to the 2-categorical story. From now on, weshall assume that all 2-categories under consideration are locally idempotentcomplete, i.e. their
Hom -categories are idempotent complete. The goal of thissection is to prove proposition 1.4.10. Before doing so, we need to recall adefinition from [GJF19].
Definition B.2.1.
Let C be a 2-category, ( A , e , µ , δ ) and ( A , e , µ , δ ) be2-condensation monads. An ( A , A ) bimodule consists of:1. A 1-morphism b : A → A ;2. Two 2-morphisms ν r : b ◦ e ⇒ b and β r : b ⇒ b ◦ e ;3. Two 2-morphisms ν l : e ◦ b ⇒ b and β l : b ⇒ e ◦ b such that1. The pair ( f, ν r , ν l ) is an associative ( e , e )-bimodule;2. The pair ( f, β r , β l ) is a coassociative ( e , e )-bicomodule;3. The 2-morphisms ν r and β l commute, and the 2-moprhisms ν l and β r commute;4. The Frobenius relations for modules hold for ( b, ν r , β r ) and ( b, ν l , β l );5. We have β r · ν r = Id b and β l · ν l = Id b .25 roof. (Prop. 1.4.10) Without loss of generality, we may assume that C is a strict2-category. The condensate complete 2-category Kar ( C ) and the 2-functor ι C are defined in theorem 2.3.10 of [GJF19]. In particular, recall that Kar ( C ) is the2-category of 2-condensation monads in C , with 1-morphisms given by bimodulesand 2-morphisms by bimodule maps. The 2-functor ι C sends an object C of C to the trivial 2-condensation monad on C .Let F : C → A be a 2-functor to a condensation complete 2-category, whichwe may assume is strict. Given 1-morphisms f : A → B , and g : B → C in C , wewrite the coherence 2-isomorphism witnessing that F respects the compositionby F g,f : F ( g ) ◦ F ( f ) ∼ = F ( g ◦ f ). On an object C of C , we denote the unitor by F C : Id F ( C ) ∼ = F ( Id C ).The 2-functor F ′ is defined on objects using theorem 2.3.1 of [GJF19],which says that the 2-category of extensions to a 2-condensation of a given2-condensation monad is either empty or contractible. In our case, as A has allcondensates, they are all contractible. Explicitly, given a 2-condensation monad( A i , e i , µ i , ν i ) in C , we fix a 2-condensation ( F ( A i ) , B i , f i , g i , φ i , γ i ) together witha 2-isomorphism θ i : g i ◦ f i ∼ = e i providing an extension of F ( A i , e i , µ i , ν i ) in A to a 2-condensation, and we set F ′ ( A i ) := B i .In order to define F ′ on 1-morphisms, we use the fact that A is locallyidempotent complete. Let ( b , ν r , β r , ν l , β l ) be an ( A , A ) bimodule. The 1-morphism f ◦ F ( b ) ◦ g supports the idempotent 2-morphism( φ ◦ f ◦ ν r ◦ g ) · ( f ◦ θ − ◦ b ◦ θ ◦ g ) · ( f ◦ β l ◦ g ◦ γ ) . The 1-morphism F ′ ( b j ) is given by a choice of splitting for this idempotent in Hom A ( B , B ). The value of F ′ on 2-morphisms is defined similarly using theuniversal property of splittings of idempotents.It remains to be proven that F ′ is a 2-functor and that there exists a 2-natural equivalence F ′ ◦ ι C ⇒ F . We begin by tackling the first point. Let b be an ( A , A ) bimodule and b be an ( A , A ) bimodule. Recall that theircomposite b ⊗ A b in Kar ( C ) is given by a choice of splitting for the idempotent( ν r ◦ b ) · ( b ◦ β l )supported on b ◦ b (see [GJF19] for details). In particular, both F ′ ( b ) ◦ F ′ ( b )and F ′ ( b ⊗ A b ) are defined as splittings of idempotents. Using the fact thatsplittings of idempotents are universal, we find that F ′ ( b ) ◦ F ′ ( b ) is a splittingof the idempotent supported on f ◦ F ( b ) ◦ e ◦ F ( b ) ◦ g given by:( φ ◦ f ◦ ν r ◦ µ ◦ ν r ◦ g ) · ( f ◦ θ − ◦ b ◦ e ◦ e ◦ e ◦ b ◦ θ ◦ g ) · ( f ◦ β l ◦ δ ◦ β l ◦ g ◦ γ ) . Likewise, one can see that F ′ ( b ⊗ A b ) is a splitting of the idempotent supportedon f ◦ F ( b ) ◦ F ( b ) ◦ g specified by:( φ ◦ f ◦ ν r ◦ ν r ◦ g ) · ( f ◦ θ − ◦ b ◦ b ◦ θ ◦ g ) · ( f ◦ β l ◦ β l ◦ g ◦ γ ) . φ ◦ f ◦ ν r ◦ e ◦ ν r ◦ g ) · ( f ◦ θ − ◦ b ◦ δ ◦ b ◦ θ ◦ g ) · ( f ◦ β l ◦ β l ◦ g ◦ γ )from f ◦ F ( b ) ◦ F ( b ) ◦ g to f ◦ F ( b ) ◦ e ◦ F ( b ) ◦ g , and( φ ◦ f ◦ ν r ◦ ν r ◦ g ) · ( f ◦ θ − ◦ b ◦ µ ◦ b ◦ θ ◦ g ) · ( f ◦ β l ◦ e ◦ β l ◦ g ◦ γ )from f ◦ F ( b ) ◦ e ◦ F ( b ) ◦ g to f ◦ F ( b ) ◦ F ( b ) ◦ g exhibit an equivalenceof idempotents. Thus, there exists a unique 2-isomorphism F ′ ( b ) ◦ F ′ ( b ) ∼ = F ′ ( b ⊗ A b ). A similar argument can be used to show that the 2-isomorphismsconstructed above are compatible, meaning that F ′ respects the composition of1-morphisms. Unitality can be dealt with analogously.Finally, we have to construct a 2-natural equivalence F ′ ◦ ι C ⇒ F . Ob-serve that on an object C i of C , F ′ ι C ( C i ) is given by an extension of the im-age under F of the 2-condensation monad ( C i , Id C i , Id Id Ci , Id Id Ci ). Note thatthe 2-condensation ( F ( C i ) , F ( C i ) , Id F ( C i ) , F ( Id C i ) , F C i , F − C i ) is such an exten-sion. (On the nose!) But, we have already picked an extension above, namely( F ( C i ) , B i , f i , g i , φ i , γ i ). Now, observe the proof theorem 2.3.2 of [GJF19] showsthat g i : B i → F ( A i ) is a 1-equivalence F ′ ι C ( A i ) → F ( A i ) in A . Given a 1-morphism c : C → C in C , there is a trivial bimodule structure on F ( c ). Then,one can see that g ◦ ( F ′ ι C )( c ) is a splitting of the idempotent supported on g ◦ f ◦ F ( c ) ◦ g given by( g ◦ φ ◦ F ( c ) ◦ F − C ◦ g ) · ( g ◦ f ◦ F C ◦ F ( c ) ◦ g ◦ γ ) . The 2-morphisms( F ( Id C ) ◦ F ( c ) ◦ F − C ◦ g ) · ( F C ◦ F ( c ) ◦ γ ◦ g ) : F ( c ) ◦ g ⇒ g ◦ f ◦ F ( c ) ◦ g ( F − C ◦ F ( c ) ◦ F − C ◦ g ) · ( F ( Id C ) ◦ F ( c ) ◦ γ ◦ g ) : g ◦ f ◦ F ( c ) ◦ g ⇒ F ( c ) ◦ g exhibit F ( c ) ◦ g as a splitting of this idempotent. Thus, there is a unique2-isomorphism g ◦ ( F ′ ι C )( c ) ∼ = F ( c ) ◦ g . Using a similar argument, it is nothard to prove that these assemble to produce the desired 2-natural equivalence F ′ ◦ ι C ⇒ F .The second and third point of definition 1.4.8 follow using analogous argu-ments.It will be important to have characterized those 2-functors that become 2-equivalences upon taking the Karoubi envelope. Lemma B.2.2.
Let F : C → D be a 2-functor. Then, Kar ( F ) is an equivalenceof 2-categories if and only if F is fully faithful (it induces a equivalences on Hom -categories), and every object of D is the splitting of a condensation monad onan object in the image of F . roof. The backward implication is straightforward (using the fact that the 2-functors ι C and ι D are fully faithful as stated in theorem 2.3.10 of [GJF19]).Let us assume that F has the above properties. Then, it follows from theconstruction, and the fact that F is fully faithful, that Kar ( F ) is fully faithful.It remains to prove essential surjectivity. In spirit, the argument is similar tothe proof that the Karoubi envelope of a 2-category is condensation complete(see proposition A.5.3 of [DR18]). Let ( D, e, µ, δ ) be a 2-condensation monad in D . By hypothesis, there exists a 2-condensation( F ( C ) , D, f, g, φ, γ )in D for some C in C . We can compose the 2-condensation monad on D withthis 2-condensation, and produce a 2-condensation monad( F ( C ) , g ◦ e ◦ f, ( f ◦ µ ◦ g ) · ( g ◦ e ◦ φ ◦ e ◦ f ) , ( g ◦ e ◦ γ ◦ e ◦ f ) · ( f ◦ δ ◦ g )) . Now, note that these 2-condensation monads are equivalent as objects of
Kar ( D )via the canonical 2-condensation bimodules given by f , and g . As F is fullyfaithful, the 2-condensation monad ( F ( C ) , g ◦ e ◦ f ) comes from a 2-condensationmonad in C , which proves the result.Let R be a fixed commutative ring. When working with R -linear locallyadditive ( Hom -categories have finite direct sums) 2-categories and R -linear 2-functors, the natural notion of completion is the Cauchy completion, meaningbeing condensation complete and having finite direct sums by theorem 4.3.5 of[GJF19].Further, the Cauchy completion of a locally additive R -linear 2-category C can be constructed as follows: First, take the direct sum completion of C ,denoted by M at ( C ). This is the locally additive 2-category whose objects arefinite vectors with values in C , and whose morphisms are finite matrices ofmorphisms in C . Observe that the canonical 2-functor C → M at ( C ) has anobvious 3-universal property. Second, take the Karoubi envelope. We get an R -linear 2-functor C → Kar ( M at ( C )). Observing that the Karoubi envelope ofa locally additive R -linear 2-category with finite direct sums has finite directsums shows that the target 2-category is indeed Cauchy complete.Given a locally additive R -linear 2-category C , we denote the Cauchy com-pletion constructed above by κ C : C → Cau ( C ). It enjoys a 3-universal propertyanalogous to the one of the Karoubi envelope. Namely, it is enough to mod-ify definition 1.4.8 by replacing ι C by κ C : C → Cau ( C ), requiring that all2-categories and 2-functors be R -linear, and that A be additive as well as con-densation complete.The next lemma is a consequence of lemma B.2.2. Lemma B.2.3.
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