Invertible braided tensor categories
aa r X i v : . [ m a t h . QA ] M a r Invertible braided tensor categories
Adrien Brochier, David Jordan, Pavel Safronov, Noah SnyderApril 1, 2020
Abstract
We prove that a finite braided tensor category A is invertible in the Morita -category BrTens of braided tensor categories if, and only if, it is non-degenerate. Thisincludes the case of semisimple modular tensor categories, but also non-semisimpleexamples such as categories of representations of the small quantum group at goodroots of unity. Via the cobordism hypothesis, we obtain new invertible -dimensionalframed topological field theories, which we regard as a non-semisimple framed version ofthe Crane-Yetter-Kauffman invariants, after Freed–Teleman and Walker’s constructionin the semisimple case. More generally, we characterize invertibility for E - and E -algebras in an arbitrary symmetric monoidal ∞ -category, and we conjecture a similarcharacterization of invertible E n -algebras for any n . Finally, we propose the Picardgroup of BrTens as a generalization of the Witt group of non-degenerate braidedfusion categories, and pose a number of open questions about it.
Contents E n -algebras . . . . . . . . . . . . . . . . . . . . . . 41.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 E - and E -algebras 5 E n -algebras and higher Morita categories . . . . . . . . . . . . . . . . . . . . 52.2 Dualizability for E - and E -algebras . . . . . . . . . . . . . . . . . . . . . . 92.3 Invertibility for E - and E -algebras . . . . . . . . . . . . . . . . . . . . . . . 11 BrTens and the Witt group 22 Introduction
In the paper [BJS18], we introduced a symmetric monoidal 4-category
BrTens whose objectsare braided tensor categories, and whose morphisms encode their higher Morita theory,following [Hau17, JFS17], and gave sufficient conditions for 3-dualizability (“cp-rigidity”)and 4-dualizability (fusion) in
BrTens . In this paper we consider the related question ofinvertibility in
BrTens . We also treat both dualizability and invertibility in the more generalsetting of E -algebras in an arbitrary background symmetric monoidal 2-category S . Finite braided tensor categories are linear and abelian braided monoidal categories satisfyingstrong finiteness and rigidity conditions (see Section 3 for more details). Such an A is called non-degenerate if the Müger center of A is trivial, i.e. if for every non-trivial object X ∈ A ,there exists an object Y ∈ A such that the double-braiding σ Y,X ◦ σ X,Y on X ⊗ Y is not theidentity. The main result of this paper is: Theorem 1.1.
A finite braided tensor category A is an invertible object of BrTens if, andonly if, A is non-degenerate. We note that Theorem 1.1 includes modular tensor categories, whose invertibility isknown to experts through unpublished theorems of Freed–Teleman and Walker (c.f. [Fre12a,Fre12b] and [Wal]), but we emphasize that we require neither semisimplicity nor a ribbonstructure. In particular, our results include representation categories of small quantumgroups for good roots of unity, which are non-degenerate but not semisimple.Theorem 1.1 has an application to the construction of topological field theories viathe cobordism hypothesis [BD95, Lur09, Ber11, Fre13, AF17]. Namely, we obtain a -dimensional fully extended framed TFT attached to every non-degenerate finite braidedtensor category, which we may regard as a framed and non-semisimple analog of the fullyextended Crane-Yetter-Kauffman [CY93, CKY97, WW11, BB18, KT20] topological fieldtheory envisioned by Freed–Teleman and Walker. These TFTs are invertible in the senseof [Fre14, SP17, DGP19]. We believe that this aspect of the work will be important forapplications, as it was not generally expected that non-semisimple braided tensor categorieswould give rise to -dimensional TFTs due to the heavy reliance on semisimplicity in thetraditional state-sum approach to Crane-Yetter-Kauffman TFT’s. In future work we plan tostudy SO (4) -fixed point structures (hence the associated oriented TFT’s), and to give a re-formulation of non-semisimple Witten-Reshetikhin-Turaev theories of [CGPM14, RGG + BrTens . In Section 4,we discuss this Picard group, its relation to the Witt group, and a number of natural openquestions related to it. 2raided tensor categories define E -algebras in the symmetric monoidal -category S = Pr of locally presentable linear categories. Our approach in this paper is to work as much aspossible in the more general ( ∞ , -category Alg ( S ) of E -algebras in an arbitrary ambientclosed symmetric monoidal ( ∞ , -category S .In this generality we have analogs Z ( A ) , Z ( A ) , Z ( A ) , respectively, of the endo-functors,the Drinfeld center, and the Müger center of A (see Section 2.1 for detailed definitions). Wewill also use the Harish-Chandra category HC( A ) (a variant of the monoidal Hochschildhomology with the annulus/bounding framing rather than cylinder/product framing), andwe will denote by A ⊗ op and A σ op the E -algebras obtained as reflections of the E -structurethrough the x - and y -axes (see Section 2.1). We first prove the following result: Theorem 1.2. An E -algebra A ∈ Alg ( S ) is 3-dualizable if, and only if, A is dualizable asan object of S , as an A e -module, and as an HC( A ) -module. Turning next to invertibility we prove:
Theorem 1.3.
A 2-dualizable E -algebra C ∈ Alg ( S ) is invertible if, and only if:1. It is central : the natural morphism S → Z ( C ) is an equivalence.2. It is Azumaya : the natural morphism C ⊠ C ⊗ op → End( C ) is an equivalence. In the case S is the category of R -modules for R a commutative ring, Theorem 1.3 reducesto a characterization of Azumaya algebras, see Example 2.29. Our main general result is thefollowing characterization of invertible E -algebras: Theorem 1.4.
A 3-dualizable E -algebra A ∈ Alg ( S ) is invertible if, and only if:1. It is non-degenerate : the natural morphism S → Z ( A ) is an equivalence.2. It is factorizable : the natural morphism A ⊠ A σ op → Z ( A ) is an equivalence.3. It is cofactorizable : the natural morphism HC( A ) → End( A ) is an equivalence. Our proof of Theorem 1.1 is an application of Theorem 1.4 in the case
BrTens =Alg ( Pr ) . For this we require two additional claims: that finite braided tensor categoriesare -dualizable, and that for finite braided tensor categories, non-degeneracy, factorizabilityand cofactorizability are mutually equivalent. The first claim follows from the main resultof [BJS18] since finite tensor categories are cp-rigid. For the second claim, we have thefollowing two theorems, the second of which we prove in Section 3 (see Theorem 3.20 for thecomplete statement). Theorem 1.5 ([Shi19b]) . A finite braided tensor category is factorizable if and only if it isnon-degenerate.
Theorem 1.6.
A finite braided tensor category is factorizable if, and only if, it is co-factorizable. emark 1.7. Outside of finite braided tensor categories, non-degeneracy does not implyfactorizability, nor co-factorizability. For example, the category of integrable representationsof a semisimple quantum group at generic quantization parameter is non-degenerate, but isclearly not factorizable. We do not know whether either factorizability or co-factorizabilityimply finiteness (see Question 4.6).Besides clarifying proofs, the generality of Theorem 1.4 makes it possible to consider in-vertibility of E -algebras in contexts other than finite braided tensor categories. For instance,the 3-dimensional Rozansky–Witten TFT [RW97] associated to a holomorphic symplecticmanifold X gives rise to a ribbon braided tensor structure on the bounded derived category ofcoherent sheaves D bcoh ( X ) [RW10]. We expect that this braided tensor category is invertiblein the suitably derived version of BrTens , and hence defines an invertible 4D topological fieldtheory. In addition, we expect that one may formulate the Rozansky–Witten TFT as a fieldtheory relative to the resulting 4D theory, in precisely the way that the Witten-Reshetikhin–Turaev 3D TFT [Wit89, RT91] is constructed relative to the 4D Crane–Yetter–Kauffmantheory. We hope to return to this in future work. E n -algebras Let us now collect two conjectures generalizing our results for dualizability and invertibilityof E and E -algebras to E n -algebras. An important theorem of [GS18] (c.f. [Lur09, Claim4.1.14]) states that every E n algebra A ∈ Alg n ( S ) is n -dualizable. The following character-ization for ( n + 1) -dualizability of E n -algebras was formulated in [Lur09, Remark 4.1.27],where it is remarked that it would follow from an unpacking of the proof of the cobordismhypothesis. It is formulated in terms of the factorization homology (or topological chiralhomology) R M A of a framed n -manifold M , with coefficients in an an E n -algebra A (seeDefinition 2.2 for a brief recollection, or [AF15, Lur09] for complete definitions).Let us fix the framing on S k − × R which bounds an k -disk, and its product framing on S k − × R n − k +1 with the trivial framing on R n − k . This induces an action of the E n − k +1 -algebra R S k − × R n − k +1 A on R R n A ∼ = A . Conjecture 1.8. An E n -algebra A ∈ Alg n ( S ) is ( n + 1) -dualizable if, and only if, it isdualizable over the factorization homologies R S k − × R n − k +1 A for k = 0 , . . . , n . We note that the forward implication is clear. The case n = 1 of the conjecture is provedin [Lur09], and recalled in Theorem 2.15. We prove the case n = 2 relevant to the presentpaper in Theorem 2.22.Recall that the E k -center Z k ( A ) of an E n -algebra A , for k ≤ n , is: Z k ( A ) := End R Sk − × R n − k +1 A ( A ) . The pattern of Theorem 1.4 leads us to make the following analog of Conjecture 1.8:4 onjecture 1.9. An E n -algebra A is invertible if, and only if, it is ( n + 1) -dualizable andthe canonical maps Z S k − × R n − k +1 A → Z n − k ( A ) are equivalences for k = 0 , . . . , n . Again the forward implication is clear. Theorems 2.27 and 2.30 confirm the conjecturein the cases n = 1 and n = 2 , and the same techniques should work in the general caseprovided one has a sufficiently general calculus of mates (see Remark 2.28). We thereforeexpect Conjecture 1.9 to be significantly easier to prove than Conjecture 1.8. In particular,in contrast to Conjecture 1.8, we expect it can be proved independently from the proof oreven the statement of the cobordism hypothesis. In order to apply this conjecture to anyexamples, however, one must verify ( n + 1) -dualizability, which involves the previous moredifficult conjecture. DJ is supported by European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement no. 637618). PS is supported bythe NCCR SwissMAP grant of the Swiss National Science Foundation. NS is supported byNSF grant DMS-1454767. DJ and NS were partially supported by NSF grant DMS-0932078,administered by the Mathematical Sciences Research Institute while they were in residenceat MSRI during the Quantum Symmetries program in Spring 2020. The authors are gratefulalso to the ICMS RIGS program for hospitality during a research visit to Edinburgh.We would like to thank Dan Freed and Dmitri Nikshych helpful questions and commentsconcerning the Picard group of
BrTens . NS would like to thank Sean Sanford for excellentexpositional talks on Shimizu’s work. E - and E -algebras In this section we recall some fundamental definitions and results about the Morita categoryof E n -algebras, and we prove our main general results about dualizability and invertibilityfor E -algebras. To clarify notation, we will keep a running example S = Pr , but we stressthat the results in this section are all general, and that any results which are specific tobraided tensor categories are delayed until the next section. E n -algebras and higher Morita categories We begin by briefly recalling the notions of an E n -algebra in a closed symmetric monoidal ( ∞ , -category S which admits geometric realizations. We denote its symmetric monoidalstructure by ⊠ . We refer to [Lur, Chapter 5] and [AF15] for more details on the followingdefinitions. 5 efinition 2.1. Let
Mfld frn denote the topological category whose objects are framed n -manifolds, and whose space of morphisms between M and N is the topological space Emb fr ( M, N ) of framed embeddings . We equip Mfld frn with the structure of a symmetric monoidal cat-egory under disjoint union. We denote by
Disk frn the full subcategory consisting of finitedisjoint unions of the standard open disk (0 , n Definition 2.2. An E n -algebra in S a symmetric monoidal functor A : Disk frn → S . The factorization homology with coefficients in A is the left Kan extension of the functor A along the inclusion of Disk frn into
Mfld frn , and is denoted by M R M A . Example 2.3.
In the familiar example S = Pr (see Section 3), an E -algebra is a locallypresentable category tensor category with a colimit preserving tensor product. Henceforthwe will call this a ‘tensor category’, without a further requirement of rigidity, and will makeexplicit any notion of rigidity (e.g., compact-rigid or cp-rigid) we assume. Similarly, an E -algebra in Pr is a braided tensor category, and an E k -algebra, for k ≥ , is a symmetrictensor category.When we identify E -algebras in Pr with braided tensor categories, we use the follow-ing conventions, following [BJS18]: the tensor multiplication is in the x -direction, and thebraiding is given by counterclockwise rotation.Let us fix some more notation for later use. An E -algebra C has an opposite E -algebra C ⊗ op coming from precomposing by reflection. The notation ⊗ op is intended to emphasizethat we are taking the dual in the multiplication direction, and not taking a dual in theunderlying category S . An E -algebra A has two a priori distinct notions of opposite: A ⊗ op where we reflect in the x -direction and A σ op where we reflect in the y -direction. We havetwo canonical equivalences of E -algebras A ⊗ op ≃ A σ op , (1)given by a 180 degree rotation in the clockwise and counter-clockwise directions. Example 2.4.
Consider the case S = Pr and suppose A ∈ Alg ( S ) is a braided tensorcategory. The braided tensor category A ⊗ op has the same underlying category, the tensorproduct x ⊗ op y = y ⊗ x and the braiding x ⊗ op y = y ⊗ x σ − x,y −−→ x ⊗ y = y ⊗ op x. The braided tensor category A σ op has the same underlying tensor category and the braiding σ − y,x : x ⊗ y → y ⊗ x . The two canonical equivalences in this case are each given by equippingthe identity functor id : A ⊗ op → A σ op with a braided tensor structure given by the braiding,and its inverse. Recall that a framed embedding of M into N is an embedding of topological manifolds, together with anisotopy γ between the framing on M and the framing pulled back from N . In particular, a framed embeddingis not required to preserve the framing strictly. efinition 2.5 (See [Gin15, Section 7]) . Let A be an E n -algebra. Its enveloping algebra is the E -algebra U E n A = Z S n − × R A , where S n − × R carries the framing bounding the n -ball.The enveloping algebra U E n A has a natural left module action on R R n A ∼ = A , coming fromthe sphere bounding the ball. Definition 2.6.
Let A be an E n -algebra. Its E n -center is the object Z n ( A ) = End U En A ( A ) ∈ S . By [Fra13, Proposition 3.16] we have an equivalence of ∞ -categories LMod U En A ∼ = Mod E n A between the ∞ -category of left U E n A -modules and the ∞ -category of E n - A -modules. More-over, the notion of a center introduced in [Lur, Definition 5.3.1.6] is shown in [Lur, Theorem5.3.1.30] to coincide with End
Mod En A ( A ) . So, Z n ( A ) is indeed the center of the E n -algebra inthe sense of Lurie.In addition, by [Lur, Theorem 3.3.3.9] Mod E n A is an E n -monoidal ∞ -category with A ∈ Mod E n A the unit object. So, by the Dunn–Lurie additivity theorem [Lur, Theorem 5.1.2.2](cf. [Dun88, FV15]) Z n ( A ) is an E n +1 -algebra. Example 2.7.
Suppose C is an E -algebra. We denote C e = U E C ∼ = C ⊠ C ⊗ op . In the case S = Pr , it is shown in Proposition 3.6 that the E -center Z ( C ) coincides withthe Drinfeld center of the tensor category C . Example 2.8.
Suppose A is an E -algebra. We denote HC( A ) = U E A ∼ = A ⊠ A ⊠ A σ op A ⊗ op , where we regard A (resp., A ⊗ op ) as an E -algebra in right (resp., left) A ⊠ A σ op -modules,hence the relative tensor product inherits again an E -structure. In the case S = Pr , it isshown in Proposition 3.7 that the E -center Z ( A ) of coincides with the Müger center of thebraided tensor category A .The collection of E n -algebras in a fixed S carries the structure of a symmetric monoidal ( ∞ , n + 2) -category Alg n ( S ) . We refer to the papers [JFS17, Hau17, Sch14, CS] for a rigorousconstruction of Alg n ( S ) in the model of iterated complete Segal spaces, and to [GS18] for adigestable exposition, and a treatment of dualizability. Let us remark for experts that we7ill require Haugseng’s “unpointed” model in order to treat questions of higher dualizabilityand invertibility (see [GS18, §1.4] and the discussion there).While the works cited above are required in order to make rigorous sense of the compo-sition laws of higher morphisms, and their compatibilities, it is possible nevertheless to givean informal description of objects and morphisms themselves, in Alg ( S ) and Alg ( S ) : Definition 2.9 (Sketch) . The ( ∞ , -category Alg ( S ) has:• As objects E -algebras A , B , . . .• As 1-morphisms the ( A , B ) -bimodules M , N , . . . • As 2-morphisms the bimodule 1-morphisms• As 3-morphisms the bimodule 2-morphismsThe symmetric monoidal structure, as well as the composition of 2- and 3-morphisms, arethose inherited from S . Composition of -morphisms is the relative tensor product of bimod-ules. Definition 2.10 (Sketch) . The ( ∞ , -category Alg ( S ) has:• As objects E -algebras A , B , . . .• As 1-morphisms the E -algebra objects C , D , . . . in ( A , B ) -bimodules,• As 2-morphisms the ( C , D ) -bimodule objects M , N , . . . in ( A , B ) -bimodules• As 3-morphisms the bimodule 1-morphisms• As 4-morphisms the bimodule 2-morphismsThe symmetric monoidal structure, as well as the composition of 3- and 4-morphisms, arecompatible with those in S . For example, to compose -morphisms we endow the compo-sition of the underlying -morphisms with the structure of a bimodule -morphism in anappropriate way. Composition of - and -morphisms is given by the relative tensor prod-uct of bimodules, equipping the resulting composition in the case of -morphisms with acanonical E -algebra structure.There is a potential ambiguity in the notion of ( A , B ) -bimodule appearing above, since A and B each admit multiplications in both the x - and y - directions. Hence let us fix thefollowing conventions, following [BJS18]:• The ( A , B ) -bimodule structure on a -morphism is with respect to multiplication inthe y -direction for both A and B , whereas (by Example 2.3) the underlying E -algebraassociated to an E -algebra is in the x -direction. This means that an ( A , B ) -bimoduleis an A ⊠ B -module (rather than an A ⊠ B op -module).• Hence the data of an E -algebra in ( A , B ) -bimodules is equivalent to the data of an E -algebra C in S , together with a morphism A ⊠ B σ op → Z ( C ) . We call such data a ( A , B ) -central algebra .• Given E -algebras A , B and ( A , B ) -central algebras C , D a ( C , D ) -bimodule object M in the monoidal category of ( A , B ) -bimodules is an ( A , B ) -centered ( C , D ) -bimodule .8 .2 Dualizability for E - and E -algebras Recall from [Lur09, Definition 2.3.16] that a symmetric monoidal ( ∞ , k ) -category has duals if every object has a dual, and every i -morphism has a left and right adjoint for ≤ i < k . Anobject in an ( ∞ , n ) -category is called k -dualizable if it belongs to a full sub ( ∞ , k ) -categorywhich has duals.In this section we discuss dualizability and adjointability in the Morita categories Alg ( S ) and Alg ( S ) . The case of Alg ( S ) is well-known, but the results for Alg ( S ) are new. Webegin by looking at dualizability in Alg ( S ) . Proposition 2.11.
Every E -algebra C ∈ Alg ( S ) is 1-dualizable with dual C ∨ = C ⊗ op , andwith evaluation and co-evaluation given by versions of the regular bimodule:• ev is C as a ( C ⊗ op ⊠ C , S ) -bimodule.• coev is C as a ( S , C ⊠ C ⊗ op ) -bimodule. It turns out that a bimodule having a left adjoint or a right adjoint depends only onthe left action or the right action respectively, as explained in the following definition andproposition.
Definition 2.12.
Let C be an E -algebra in S . A right (resp. left) C -module M is called dualizable if it has a right (resp. left) adjoint as a ( C , S ) bimodule (resp. ( S , C ) bimodule). Proposition 2.13 ([Lur, Proposition 4.6.2.13]) . A 1-morphism M : C → D in Alg ( S ) hasa right (resp. left) adjoint if, and only if, M is dualizable as a D -module (resp. C -module). In order to give a complete characterization of 2-dualizable objects in
Alg ( S ) , we firstrecall the following result (see [Lur09, Proposition 4.2.3] and [Pst14, Theorem 3.9]) whichreduces 2-dualizability to a finite number of conditions. Theorem 2.14.
A 1-dualizable object X of a symmetric monoidal 2-category is -dualizableif, and only if, the evaluation and coevaluation maps each admit a right adjoint. We may finally state the following well-known characterization of 2-dualizable objects in
Alg ( S ) . Theorem 2.15. An E -algebra C ∈ Alg ( S ) is 2-dualizable if, and only if, it is dualizable asan object of S , and as a C e -module.Proof. We recall from Proposition 2.11 the 1-dualizability data C ∨ , ev , coev . By Theo-rem 2.14, C is 2-dualizable if, and only if, ev and coev both admit right adjoints. Then byProposition 2.13, these right adjoints exist if, and only if, C is dualizable as a S -module (i.e.as an object of S ), and as a C e -module.Now we turn to 2- and 3-dualizability of E -algebras. In particular, we prove the n = 2 case of Conjecture 1.8). 9 emma 2.16. If C is an E -algebra, there is a canonical equivalence of E -algebras Z ( C ⊗ op ) ∼ = Z ( C ) ⊗ op . (2) Theorem 2.17 (See [GS18, Section 4]) . Let C : A → B be a 1-morphism in Alg ( S ) givenby an E -algebra C equipped with an ( A , B ) -central structure A ⊠ B σ op → Z ( C ) . Its rightadjoint is C ⊗ op equipped with a ( B , A ) -central structure via the composite B ⊠ A σ op −→ Z ( C ) σ op ∼ = Z ( C ) ⊗ op ∼ = Z ( C ⊗ op ) , where the penultimate equivalence is given by a -degree clockwise rotation (1) , and thelast equivalence is given by (2) . The unit of the adjunction is C viewed as ( A , A ) -centered ( A , C ⊠ B C ⊗ op ) -bimodule. The counit of the adjunction is C viewed as a ( B , B ) -centered ( C ⊗ op ⊠ A C , B ) -bimodule. Remark 2.18.
The right adjoint constructed above used clockwise rotation, the left adjointwould use counterclockwise rotation.
Theorem 2.19 (See [GS18, Section 4]) . Every E -algebra A ∈ Alg ( S ) is 2-dualizable withdual A ∨ = A σ op , and with evaluation and coevalation given by the regular central algebra,• ev is A as a ( A σ op ⊠ A , S ) -central algebra,• coev is A as a ( S , A ⊠ A σ op ) -central algebra.The right adjoints to evaluation and coevaluation are given by• ev R is A ⊗ op as a ( S , A σ op ⊠ A ) -central algebra,• coev R is A ⊗ op as a ( A ⊠ A σ op , S ) -central algebra,as in Theorem 2.17. Their unit and counit morphisms are given by:• η coev is A as a ( S , HC( A )) -bimodule.• ǫ coev is A as a ( A ⊠ A σ op , A ⊠ A σ op ) -centered ( A ⊗ op ⊠ A , A ⊠ A ) -bimodule.• η ev is A as a ( A σ op ⊠ A , A σ op ⊠ A ) -centered ( A ⊠ A , A ⊠ A ⊗ op ) -bimodule.• ǫ ev is A as an (HC( A ) , S ) -bimodule. Next, we have the following analog of Proposition 2.13 establishing dualizability for -morphisms in Alg ( S ) . Proposition 2.20 ([BJS18, Proposition 5.17]) . Let C , D : A → B be -morphisms in Alg ( S ) ,i.e. ( A , B ) -central algebras, and let M : C → D be a 2-morphism in Alg ( S ) , i.e. an ( A , B ) -central ( C , D ) -bimodule. Then M has a right (resp. left) adjoint in Alg ( S ) if, and only if, ithas a right (resp. left) adjoint in Alg ( S ) , when regarded as a ( C , D ) -bimodule. The adjointsin Alg ( S ) are given by equipping the adjoints in Alg ( S ) with canonical central structures. We recall the following analog of Theorem 2.14, which reduces 3-dualizability to a finitelist of conditions: 10 heorem 2.21 ([Ara17, Proposition 1.2.1]) . Let X be an object in a symmetric monoidal 3-category C . Suppose that X has a dual and that the evaluation and coevaluation -morphismshave right adjoints. Then X is 3-dualizable if, and only if, the unit and counit 2-morphismswitnessing each of these two adjunctions (four maps in total) have right adjoints. We may finally prove a complete characterization of 3-dualizable objects in
Alg ( S ) . Theorem 2.22. An E -algebra A ∈ Alg ( S ) is 3-dualizable if, and only if, A is dualizableas an object of S , as an A e -module, and as an HC( A ) -module.Proof. We recall from Theorem 2.19 the 2-dualizability data A ∨ , ev , coev , ev R , coev R ,together with their units and counits η ev , η coev , ǫ ev , ǫ coev . By Theorem 2.21, A is 3-dualizableif, and only if, these last four morphisms have right adjoints. By Proposition 2.20 these existif, and only if, the underlying bimodules have right adjoints which can be analyzed usingProposition 2.13:• η ev has a right adjoint if, and only if, A is dualizable as a A e -module.• η coev has a right adjoint if, and only if, A is dualizable as a HC( A ) -module.• ǫ ev has a right adjoint if, and only if, A is dualizable as an object of S .• ǫ coev has a right adjoint if, and only if A is dualizable as a A ⊠ A -module via the right A ⊠ A -action on A . Using the braiding we may identify A ⊠ A ∼ = A ⊠ A ⊗ op as monoidalcategories, so that under this identification the A ⊠ A -action on A goes to the canonical A e -action on A . Remark 2.23.
Recall that the main result of [BJS18] was the construction of a 3-dualizablesubcategory of
BrTens based on the notion of cp-rigidity, showing in particular that everycp-rigid braided tensor category A is 3-dualizable. As a byproduct, we gave sufficient condi-tion for dualizability of higher morphisms, not just those appearing as dualizing data for A .On the other hand, there was no proof that cp-rigidity was necessary for 3-dualizability of abraided tensor category, just that it was sufficient. Indeed, since cp-rigidity is not a priori aMorita invariant, we do not expect it is a necessary condition.By contrast, Theorem 1.2 gives a complete characterization of 3-dualizability, it is Moritainvariant, and the characterization holds for a general S . However, in the case S = Pr , letus underscore that it remains an open question to characterize necessary conditions for 1-dualizability in Pr [BCJF15, Remark 3.6], let alone as A e - and HC( A ) -modules, so thatin practice one must still appeal to the method of [BJS18] to establish the conditions inTheorem 1.2. E - and E -algebras The goal of this section is to give a complete characterization of invertible objects in
Alg ( S ) and Alg ( S ) . We begin with an elementary lemma.11 emma 2.24. Suppose C is a bicategory and f : x → y a 1-morphism. It is invertible if, andonly if, it is right-adjointable and the unit and counit of the adjunction are isomorphisms. By an iterated application of this lemma, we may give a straightforward characterizationof invertible objects in
Alg n ( S ) . We begin with a characterization of invertible 1-morphismsin Alg ( S ) . Notation 2.25.
Give C ∈ Alg ( S ) , we denote C ! = Hom C e ( C , C e ) , and C ∗ = Hom S ( C , S ) . Theorem 2.26.
Suppose B ∈ Alg ( S ) is an E -algebra and C an B -central algebra viewedas a 1-morphism B → S . Then C is invertible if, and only if, C ∈ Alg ( S ) is 2-dualizableand the following maps are equivalences:1. The evaluation map Hom C e ( C , C e ) ⊠ B C → C e .2. The map B → Z ( C ) given by the B -central structure on C .3. The evaluation map Hom( C , S ) ⊠ C ⊠ B C ⊗ op C → S .4. The map C ⊗ op ⊠ B C → Hom( C , C ) given by the left and right action of C on itself.Proof. By Theorem 2.17 C admits a right adjoint C ⊗ op : S → B . The unit of the adjunction η is C viewed as an ( B , B ) -centered ( B , C ⊠ C ⊗ op ) -bimodule. The counit of the adjunction ǫ is C viewed as a ( C ⊗ op ⊠ B C , S ) -bimodule.By Lemma 2.24 the 1-morphism C : B → is invertible if, and only if, η and ǫ areinvertible. We will now analyze the invertibility of these 2-morphisms separately:• By Lemma 2.24 the unit η is invertible if, and only if, it is right-adjointable withthe unit and counit being isomorphisms. By Propositions 2.13 and 2.20 η is right-adjointable if, and only if, C is dualizable as a C e -module, with dual C ! . The unit ofthis adjunction is B → C ! ⊠ C e C ∼ = Z ( C ) . The counit of this adjunction is C ! ⊠ B C → C e .• By Lemma 2.24 the counit ǫ is invertible if, and only if, it is right-adjointable withthe unit and counit being isomorphisms. By Proposition 2.13 ǫ is right-adjointableif, and only if, C is dualizable as an object of S , with dual C ∗ . The unit of thisadjunction is C ⊗ op ⊠ B C → Hom( C , S ) ⊠ C ∼ = Hom( C , C ) . The counit of this adjunctionis C ∗ ⊠ C ⊠ B C ⊗ op C → S .The previous theorem recovers a well-known characterization of invertible objects in Alg ( S ) (see [Par76, Vit96, Joh14] for related characterizations). Theorem 2.27. An E -algebra C ∈ Alg ( S ) is invertible if, and only if, it is 2-dualizableand the following maps are isomorphisms:1. C ⊗ op ⊠ C → Hom( C , C ) given by the left and right action of C on itself.2. The inclusion of the unit S → Z ( C ) . roof. We have an equivalence
Alg ( S ) ∼ = Hom Alg ( S ) ( S , S ) of monoidal ( ∞ , -categories.Hence, C ∈ Alg ( S ) is invertible if, and only if, it is invertible when viewed as a 1-morphism S → S in Alg ( S ) . By Theorem 2.26 it is equivalent to C ∈ Alg ( S ) being 2-dualizable andsatisfying the 4 conditions of the theorem. Let us analyze them in pairs:• The evaluation map C ! ⊠ C → C e is a map of right C e -modules, where on the lefthandside C e acts on C ! . Since C ∈ S is dualizable and C ! is dualizable as a C e -module, C ! ⊠ C is dualizable as a C e -module. In particular, the evaluation map is an isomorphism if,and only if, its C e -linear dual map is an isomorphism. But the dual map is C e → C ∗ ⊠ C ∼ = Hom( C , C ) , which is the map in the fourth condition of Theorem 2.26.• In the map S → Z ( C ) ∼ = C ! ⊠ C e C both sides are dualizable in S : the dual of therighthand side is Hom( C ! ⊠ C e C , S ) ∼ = Hom C e ( C ! , C ∗ ) ∼ = C ∗ ⊠ C e C . In particular, the dual of this map is the map C ∗ ⊠ C e C → S in the third condition ofTheorem 2.26. Remark 2.28.
In the preceding proof an essential role was played by the dual -morphisms.Recall that the dual of a morphism f : x → y between dualizable objects is the composite y ∨ id ⊗ coev −−−−−→ y ∨ ⊗ x ⊗ x ∨ id ⊗ φ ⊗ id −−−−−→ y ∨ ⊗ y ⊗ x ∨ ev ⊗ id −−−→ x ∨ . In fact, if we view modules as 1-morphisms in the ( ∞ , -category Alg ( S ) , morphisms ofmodules are 2-morphisms. The dual module is the adjoint 1-morphism and the dual of amorphism of modules is called a “mate” of the 2-morphism. Example 2.29.
Let R be a commutative algebra and S = Mod R the symmetric monoidalcategory of R -modules. Theorem 2.27 characterizes invertible objects in Alg (Mod R ) as R -algebras A satisfying the following four conditions:1. A is dualizable as an R -module (i.e., A is finitely generated and projective as an R -module).2. A is dualizable as an A e -module (i.e., it is separable).3. The natural morphism A op ⊗ A → Hom R ( A, A ) given by the left and right action is anisomorphism.4. The morphism R → Z( A ) is an isomorphism (i.e., A is a central R -algebra).In this case one may prove a stronger claim that some of these conditions are equivalent toeach other. Concretely, invertible objects in Alg (Mod R ) are R -algebras A satisfying eitherof the following equivalent conditions (see [DI71, Theorem II.3.4]):• ( Azumaya ) A is a faithful dualizable R -module and the morphism A op ⊗ A → Hom R ( A, A ) is an isomorphism. 13 ( Central separable ) A is a dualizable A e -module and the morphism R → Z( A ) is anisomorphism.Finally, we come to the main result of this section: Theorem 2.30. An E -algebra A ∈ Alg ( S ) is invertible if, and only if, it is 3-dualizableand the following maps are isomorphisms:1. (cofactorizability) HC( A ) → Hom( A , A ) .2. (factorizability) A ⊠ A σ op → Z ( A ) .3. (nondegeneracy) The inclusion of the unit → Z ( A ) .Proof. Clearly, an invertible E -algebra is 3-dualizable, so by Theorem 2.22 dualizability of A as an HC( A ) -module is necessary. From now on we make this assumption. Recall that A ∈ Alg ( S ) is 1-dualizable with the dual given by A σ op . The evaluation map ev associatedwith this duality is A viewed as a A ⊠ A σ op -central algebra. Therefore, A ∈ Alg ( S ) isinvertible if, and only if, ev : A ⊠ A σ op → S is an isomorphism. By Theorem 2.26, this isequivalent to the 4 conditions listed there.Since A is assumed to be dualizable over A e , condition (1) is equivalent to the condition(2) after taking the dual over A e . Condition (2) is precisely the condition that A ⊠ A σ op → Z ( A ) is an isomorphism. Since A is dualizable over HC( A ) , condition (3) is equivalent afterapplying Hom( − , S ) to the condition that S → Hom
HC( A ) ( A , A ) = Z ( A ) is an isomor-phism. Finally, the condition (4) is the condition that the map HC( A ) → Hom( A , A ) is anisomorphism. In this section we show that for finite braided tensor categories (in the sense of [EO04]), thethree conditions for invertibility given in Theorem 1.4 are already mutually equivalent.
We begin by recalling some categorical background. All categories we consider are k -linear(i.e, enriched and tensored over k ), where k is an algebraically closed field of characteristiczero, and all functors are k -linear functors.We will consider the symmetric monoidal 2-category Pr of locally presentable categories,their colimit preserving functors, and their natural transformations. By the special adjointfunctor theorem, a functor between locally presentable categories is colimit preserving if, andonly if, it is a left adjoint: to emphasize this, we will use the notation Fun L ( C , D ) in placeof Hom Pr ( C , D ) . The symmetric monoidal structure is given by the so-called Deligne-Kellytensor product. For recollections about the notion of local presentability see [BJS18]. Themost important class of locally presentable categories for us are those which have enoughcompact projectives. 14 efinition 3.1. An object X of a presentable category C is compact-projective if the func-tor Hom( X, − ) : C → Vect is colimit preserving. The category C has enough compactprojectives if every object of C can be expressed as a colimit of compact-projective objects.Equivalently, a category C has enough compact projectives if, and only if, it is equivalentto the free cocompletion Fun(˜ C op , Vect) of a small category ˜ C , i.e. if it is a presheaf category. Remark 3.2.
It is shown in [BCJF15] that categories with enough compact-projectives are1-dualizable as objects of Pr , and conjectured there that these are the only 1-dualizableobjects. Definition 3.3.
A category C has a compact-projective generator if the category ˜ C may be taken to have a single object, and is finite if the endomorphism algebra of thatobject is finite-dimensional.Equivalently, a category with a compact-projective generator is one which is equivalentto A -mod for some associative algebra A , and a finite category is one for which A may betaken to be finite-dimensional. The following proposition gives a characterization of thesenotions internally to Pr . The proof is straightforward. Proposition 3.4.
Suppose that C ∈ Pr has enough compact projectives.1. The identity endofunctor id C ∈ End Pr ( C ) is a compact object if, and only if, C admitsa compact-projective generator.2. The identity endofunctor id C ∈ End Pr ( C ) is a projective object if, and only if, C issemisimple.3. Under assumption (1) (resp, (1) and (2)), C is finite (resp, finite semisimple) if, andonly if, Hom spaces between compact objects are finite-dimensional. Remark 3.5.
It follows easily from Proposition 3.4 that amongst categories with enoughcompact projectives the 2-dualizable categories are precisely the finite semisimple categories.This also follows from [BDSV15, Appendix A] and [Til98], because their category
Bim (ofCauchy complete categories, bimodules, and bimodule maps) is equivalent – via taking thefree cocompletion of categories – to the full subcategory of Pr whose objects are those withenough compact projectives.We will use the term tensor category to mean an E -algebra in Pr , and the term braided tensor category to mean an E -algebra in Pr . In particular, we will always assumethat the underlying category in each case is locally presentable, and that the tensor productbifunctor A × A → A preserves colimits in each variable, so that it defines a morphism A ⊠ A → A in Pr . We introduce the notations Tens = Alg ( Pr ) and BrTens = Alg ( Pr ) .Let us begin by relating the Drinfeld and Müger centers to the more general notionsintroduced in Section 2.1. The following statement is proved in [EGNO15, Proposition7.13.8]. 15 roposition 3.6. Let C ∈ Alg ( Pr ) be a tensor category. Then Z ( C ) is equivalent to theDrinfeld center: the category of pairs ( x, γ ) , where x ∈ C and γ : ( − ) ⊗ x ∼ −→ x ⊗ ( − ) is anassociative natural isomorphism. We may also analyze the E -center of a braided tensor category. Proposition 3.7.
Let A ∈ Alg ( Pr ) be a braided tensor category. Then Z ( A ) is equivalentto the Müger center: the full subcategory of A consisting of objects x ∈ A such that σ y,x ◦ σ x,y : x ⊗ y → x ⊗ y is the identity for every y ∈ A .Proof. Suppose B ∈ Alg ( Pr ) and A is a B -central algebra. The central structure boils downto the data of a tensor functor T : B → A together with a natural isomorphism τ : T ( z ) ⊗ x → x ⊗ T ( z ) for every z ∈ B and x ∈ A . By [Lau18, Proposition 3.34] Hom A ⊠ B A ⊗ op ( A , A ) is a fullsubcategory of the Drinfeld center Z ( A ) consisting of objects ( x, γ ) , where γ T ( z ) : T ( z ) ⊗ x → x ⊗ T ( z ) coincides with τ for every z ∈ B .The E -center Z ( A ) is given by this construction with B = A ⊠ A σ op . The B -centralstructure on A sends z ⊠ ∈ B to z ∈ A with τ given by σ z,x : z ⊗ x → x ⊗ z and ⊠ z ∈ B to z ∈ A with τ given by σ − x,z : z ⊗ x → x ⊗ z . Therefore, Z ( A ) ⊂ Z ( A ) is a full subcategoryconsisting of objects ( x, γ ) , where γ z = σ x,z = σ − z,x , i.e. of objects lying in the Müger center.Let us recall some standard rigidity and finiteness assumptions on tensor categories. Definition 3.8.
Suppose a tensor category A has enough compact projectives.• We say A is cp-rigid if all compact projective objects of A are dualizable.• We say A is compact-rigid if all compact objects of A are dualizable.• We say A is a finite tensor category if it is compact-rigid, and its underlying categoryis finite.• We say A is fusion if it is a finite tensor category and the underlying category issemisimple.• We say A is a finite braided tensor category (resp., braided fusion category ) ifit is braided, and its underlying tensor category is a finite tensor category (resp., fusioncategory).These definitions are compatible with the most standard definition of [EO04] in thefollowing sense: a tensor (resp., braided tensor) category is finite in the above sense if,and only if, it is the ind-completion of a finite tensor category (resp., finite braided tensorcategory) in the sense of [EO04] and [Shi19b]. The next proposition is proved in [BJS18],by verifying closure under composition. Both parts of this proposition require characteristic zero, otherwise one needs to restrict to fusioncategories and braided fusion categories of nonzero global dimension. roposition 3.9. We have higher subcategories of
Tens and
BrTens , defined as follows:• Fusion categories, semisimple bimodule categories, compact-preserving cocontinuous bi-module functors, and natural transformations form a subcategory
Fus of Tens .• Braided fusion categories, fusion categories equipped with central structures, finitesemisimple bimodule categories, compact-preserving cocontinuous bimodule functors,and bimodule natural transformations form a subcategory
BrFus of BrTens . Remark 3.10.
We warn the reader that although there is a -category of finite tensorcategories, finite bimodule categories, compact-preserving cocontinuous bimodule functors,and bimodule natural transformations which was the main object of study in [DSPS13], wedo not know of a similar -category whose objects are finite braided tensor categories. Theissue is that the relative tensor product of finite tensor categories over a finite braided tensorcategory will not again be finite (because it will only be cp-rigid and not compact rigid). Remark 3.11.
It is shown in [DSPS13] and [BJS18], respectively, that finite tensor cat-egories and cp-rigid tensor categories are 2-dualizable in
Tens . It is shown in [BJS18]that cp-rigid braided tensor categories are 3-dualizable in
BrTens . Finally, it is shown in[DSPS13] and [BJS18] that fusion categories are 3-dualizable in
Tens and braided fusioncategories are 4-dualizable in
BrFus . We expect that a finite braided tensor category is4-dualizable in
BrTens if, and only if, its Müger center is semisimple, but we do not knowa proof.
Remark 3.12.
Consider a non-semisimple and non-degenerate braided tensor category A ,such as the category of representations for the small quantum group at a primitive ℓ -th rootof unity, where ℓ is odd, not divisible by the lacing number and coprime to the determinantof the Cartan matrix [Ros93], [Tur10, Chapter XI.6.3], [LO17]. Note that this example isnot braided fusion, and so its -dualizability does not follow from [BJS18]. Furthermore,its underlying category of A is not -dualizable by Remark 3.5 However, simply becauseinvertibility implies full dualizability, we may conclude in particular that A is fully-dualizable.In particular, we see that an E -algebra in S can be -dualizable even when the underlyingobject is not -dualizable in S .We end this section with a result identifying HC( A ) and Z ( A ) as plain categories. Proposition 3.13.
Let A be a cp-rigid braided tensor category. Then there is an equivalenceof categories HC( A ) ∼ = Z ( A ) .Proof. Consider the monoidal equivalence L : A → A ⊗ op which sends every compact pro-jective object x to the left dual ∨ x . By [BZN09, Proposition 3.13] and [DSPS13, Theorem3.2.4] we may identify Z ( A ) = Hom A e ( A , A ) ∼ = A ⊗ A e id A LL , where id A LL is the ( A , A ) -bimodule A which has a regular left A -action, but whose right A -action is given by LL . 17ow consider the identity functor id : A → A equipped with the monoidal structure σ : x ⊗ y σ x,y −−→ y ⊗ x σ y,x −−→ x ⊗ y . By [BZBJ18, Lemma 3.9] we may identify HC( A ) ∼ = A ⊗ A e id A (id ,σ ) , where id A (id ,σ ) is the ( A , A ) -bimodule A which has a regular left A -action, but whose right A -action is given by the monoidal functor (id , σ ) .But by [EGNO15, Proposition 8.9.3] we have a natural monoidal isomorphism (id , σ ) ⇒ LL which identifies the two bimodules.Note that in the cp-rigid case the natural monoidal structures on HC( A ) and Z ( A ) are nevertheless different as illustrated in Example 3.14. In the symmetric fusion case thecompatibility between the two tensor structures on HC( A ) ∼ = Z ( A ) is studied in [Was20].From a TFT perspective, these monoidal structures may be understood as follows: themonoidal structure on HC( A ) is obtained by embedding annuli inside one another (see[BZBJ18, Figure 1]), while the monoidal structure on Z ( A ) comes from the embeddingof the two incoming and one outgoing annuli as the boundary of the pair of pants cobor-dism. From this point of view it becomes clear that the latter tensor product is braidedmonoidal, while the former is only monoidal in general. Example 3.14.
Suppose G is a finite group and let A = Rep ( G ) be the category of G -representations. Then HC( A ) ∼ = Z ( A ) ∼ = QCoh (cid:0) GG (cid:1) is the category of adjoint-equivariantquasi-coherent sheaves on G . The symmetric tensor structure coming from HC( A ) corre-sponds to the pointwise tensor product of quasi-coherent sheaves, while the braided tensorstructure coming from Z ( A ) corresponds to the convolution tensor structure. Let A be a cp-rigid braided tensor category and denote by A cp ⊂ A the full subcategory ofcompact projective objects. The tensor product functor T : A ⊠ A → A admits a colimit-preserving right adjoint T R : A → A ⊠ A (see e.g. [BJS18, Section 5.3]), so that we have acoend formula T R ( A ) = Z x ∈ A cp x ∨ ⊠ x ∈ A ⊠ A . Definition 3.15.
The canonical coend is the object F ∈ A defined as F = T T R ( A ) = Z x ∈ A cp x ∨ ⊗ x. We denote by π x : x ∨ ⊗ x → F the natural projection. The canonical coend F admitsa natural structure of a braided Hopf algebra in A illustrated in Fig. 1. These have beenstudied extensively, see e.g. [LM94, Lyu95, BV08, Shi19b].Moreover, F is equipped with the following additional algebraic structures. There is aHopf pairing ω : F ⊗ F → A and an isomorphism τ V : F ⊗ V → V ⊗ F for every V ∈ A ∨ x y ∨ y x ∨ xx x ∨ x ∨ x x ∨∨ x ∨ x ∨ x Figure 1: Multiplication m , coproduct ∆ , counit ǫ and antipode S on F .illustrated in Fig. 2. The isomorphism τ V allows one to identify left F -modules with right F -modules, so the category Mod F ( A ) inherits a monoidal structure given by the relativetensor product over F . We denote by triv r : A −→ Mod F ( A ) the functor which sends an object V ∈ A to the trivial right F -module. The following isproved in [BZBJ18, Section 4] for compact-rigid categories, and can be extended to cp-rigidcategories using [BJS18, Proposition 5.10]. Proposition 3.16.
We have an equivalence of monoidal categories
HC( A ) ∼ = Mod F ( A ) .Under this equivalence the HC( A ) -module structure on A is given by M, V M ⊗ F triv r ( V ) .x ∨ x y ∨ y Figure 2: Hopf self-pairing ω and the isomorphism τ V .Dually, we may also consider the canonical end. Recall that Fun L ( A , A ) denotes the cat-egory of colimit-preserving functors A → A . Consider the tensor product functor tens : A → Fun L ( A , A ) given by x x ⊗ ( − ) . It admits a right adjoint [JK01, Shi19a] tens R ( F ) = Z x ∈ A cp F ( x ) ⊗ x ∨ . efinition 3.17. The canonical end is the object E ∈ A defined as E = tens R (id) = Z x ∈ A cp x ⊗ x ∨ . The object E is naturally an algebra via the lax tensor structure on tens R . We will usethe following result. Proposition 3.18.
Let A be a finite compact-rigid braided tensor category. Then tens R : Fun L ( A , A ) → A is monadic and it identifies Fun L ( A , A ) ∼ = Mod E ( A ) . Proof.
Let A c ⊂ A be the full subcategory of compact objects. Since A is locally finitelypresentable, we may identify Fun L ( A , A ) ∼ = Ind
Fun rex ( A c , A c ) , where Fun rex ( − , − ) is the category of right exact functors.Since A is compact-rigid, A c is rigid. So, A c is an exact A c -module category in the senseof [EO04, Definition 3.1]. Clearly, it is also indecomposable. According to [Shi19a, Theorem3.4], tens R restricts to an exact and faithful functor Fun rex ( A c , A c ) → A c . So, tens R : Fun( A , A ) → A is cocontinuous.Since A is compact-rigid, tens R carries an A -module structure, so that the composition tens R ◦ tens is canonically isomorphic to the endofunctor of A given by tensoring with E . Theresult then follows from the standard monadic argument (see [BZBJ18, Section 4.1]). There is a canonical
Drinfeld map
Dr : F → E shown in Fig. 3.In this section we establish that in the finite setting cofactorizability is equivalent toinvertibility of the Drinfeld map. Proposition 3.19.
Let A be a finite compact-rigid braided tensor category. It is cofactoriz-able if, and only if, the Drinfeld map Dr : F → E is an isomorphism.Proof. Let free : A → HC( A ) be the functor x x ⊗ F sending x ∈ A to the free right F -module; its right adjoint is the forgetful functor forget : HC( A ) → A .20 ∨ xy y ∨ Figure 3: Drinfeld map
Dr : F → E .Consider the commutative diagram HC( A ) / / Hom( A , A ) A free c c ❋❋❋❋❋❋❋❋❋ tens ssssssssss Passing to right adjoints of vertical functors by Proposition 3.16 and Proposition 3.18we get monadic functors. Therefore, the functor
HC( A ) → Hom( A , A ) is an equivalence if,and only if, the associated functor of monads forget ◦ free ⇒ tens R ◦ tens is an equivalence.Since both monads are given by tensor product with an algebra, it is enough to show thatthe value of the above functor on A is an isomorphism, i.e. that the map (forget ◦ free)( A ) = F −→ (tens R ◦ tens)( A ) = E is an isomorphism.For V ∈ A we have a commutative diagram Hom A ( V, F ) / / ∼ (cid:15) (cid:15) Hom A ( V, E ) ∼ (cid:15) (cid:15) Hom
HC( A ) ( V ⊗ F , F ) / / Hom
Hom( A , A ) ( V ⊗ ( − ) , id) where the map at the bottom sends f : V ⊗ F → F to the bottom map in the commutativediagram ( V ⊗ F ) ⊗ F y f ⊗ id / / ∼ (cid:15) (cid:15) F ⊗ F y ∼ (cid:15) (cid:15) V ⊗ y / / y Taking V = F equipped with the identity map F → F the induced map F ⊗ y → y is givenby treating y ∈ A as a left F -module via triv r . The map F → y ⊗ y ∨ is therefore given bythe y ⊗ y ∨ -component of the Drinfeld map Dr : F → E , see Fig. 4.21 ∨ x y x ∨ xy y ∨ Figure 4: The action map F ⊗ triv r ( y ) → triv r ( y ) and the corresponding map F → y ⊗ y ∨ .We can now collect all results about invertibility of finite braided tensor categories in thefollowing statement. Theorem 3.20.
Let A be a compact-rigid finite braided tensor category. The followingconditions are equivalent:1. A is invertible.2. A is non-degenerate : the natural functor Vect → Z ( A ) is an equivalence.3. A is factorizable : the natural functor A ⊠ A σ op → Z ( A ) is an equivalence.4. A is cofactorizable : the natural functor HC( A ) → Hom( A , A ) is an equivalence.5. The Hopf pairing ω : F ⊗ F → A is non-degenerate.6. The Drinfeld map Dr : F → E is an isomorphism.Proof. [Shi19b, Theorem 1.1] establishes an equivalence between conditions (2) , (3) and (5) . [FGR19, Proposition 4.11] establishes an equivalence between conditions (5) and (6) .Proposition 3.19 establishes an equivalence between conditions (4) and (6) .Finally, Theorem 2.30 asserts that condition (1) is equivalent to a combination of condi-tions (2) , (3) and (4) which finishes the proof. BrTens and the Witt group
In this section we recall the Witt group of non-degenerate braided fusion categories [DMNO13],and identify it with the Picard group of
BrFus . We then state a number of questions con-cerning the Picard group of
BrTens , which we regard as a non-fusion generalization of theWitt group.Recall that non-degenerate braided fusion categories form a monoid under the Deligne-Kelly tensor product. Non-degenerate braided fusion categories A and B are Witt equiva-lent if there exist fusion categories C and D with A ⊠ Z ( C ) ≃ B ⊠ Z ( D ) as braided tensorcategories. That is, Witt equivalence is the equivalence relation generated by equating cat-egories which are braided tensor equivalent, and setting Drinfeld centers to be trivial. Definition 4.1 ([DMNO13]) . The
Witt group of non-degenerate braided fusion cat-egories is the quotient monoid of non-degenerate braided fusion categories, by Witt equiv-22lence. The inverse operation is [ A ] − = [ A σ op ] , due to the factorizability property Z ( A ) ≃ A ⊠ A σ op of non-degenerate braided fusion categories.Recall that the Picard group Pic( T ) of a symmetric monoidal n -category T is the groupwhose elements are equivalence classes of invertible objects and whose composition is givenby tensor product. The main results of this paper give a concrete description of the elementsof the Picard group of BrTens and its subcategory
BrFus . In the latter case, we have:
Theorem 4.2.
The Picard group of
BrFus is naturally isomorphic to the Witt group ofnon-degenerate braided fusion categories.Proof.
By Theorem 3.20, the non-degenerate braided fusion categories are exactly the in-vertible objects of
BrFus , so it only remains to show that the a A -central fusion category C gives an equivalence between A and Vect if and only the natural map A → Z ( C ) is anequivalence. This was already proved in [JMPP19, Theorem 2.23].We also give an alternate proof using our techniques. Let us apply Theorem 2.26 in thecase S = Pr and B = A . Note that C is dualizable as an A - and as a C ⊗ op ⊠ A C -module (e.g.by [BJS18, Theorem 5.16]). Then we obtain that an A -central fusion category C gives anequivalence between A and Vect if and only if the following four functors define equivalencesof categories:1. C e → Hom A ( C , C ) given by the left and right action of C on itself.2. A → Z ( C ) given by the A -central structure on C .3. Vect → Hom C ⊗ op ⊠ A C ( C , C ) given by the inclusion of the identity.4. C ⊗ op ⊠ A C → Hom( C , C ) given by the left and right action of C on itself.Condition (2) holds by assumption. Using (2), condition (3) reduces to the triviality of Hom C ⊗ op ⊠ Z1( C ) C ( C , C ) . As in the proof of Proposition 3.7, we use [Lau18, Proposition 3.34]to rewrite Hom C ⊗ op ⊠ Z1( C ) C ( C , C ) as the full subcategory of Z ( C ) consisting of objects ( x, γ ) ,where for every other pair ( y, γ ′ ) we have γ y = ( γ ′ x ) − as maps x ⊗ y → y ⊗ x . Equivalently, γ ′ x ◦ γ y = id x ⊗ y , i.e. ( x, γ ) lies in the Müger center. Thus, condition (3) becomes the trivialityof Z (Z ( C )) , which automatically holds for finite tensor categories, see [ENO04, Proposition4.4].We then have that (2) implies (1) and (3) implies (4) by the double commutant theorem[EGNO15, Theorem 7.12.11].According to Theorem 4.2, we may regard Pic(
BrTens ) as a natural generalization ofthe Witt group, without the finite and semisimple assumptions. The inclusion of BrFus into
BrTens induces a group homomorphism ρ : Pic( BrFus ) → Pic(
BrTens ) . This observation leads to a number of interesting and apparently non-trivial questions. Letus stress that we are not venturing conjectural answers to any of these questions.23 uestion 4.3.
Is the homomorphism ρ injective? In other words, can it happen that twonon-degenerate braided fusion categories are equivalent in BrTens , via a central algebrawhich is not itself fusion, hence not a -morphism in BrFus ? Question 4.4.
If a finite tensor category is trivial in the Witt group, must it be the centerof a finite tensor category?
Question 4.5.
Is the homomorphism ρ surjective? In other words, is every invertible braidedtensor category in fact equivalent in BrTens to a non-degenerate braided fusion category?
Question 4.6.
Is every invertible braided tensor category equivalent in
BrTens to a finite braided tensor category?
Question 4.7.
Is the Drinfeld center of any finite tensor category trivial in
Pic(
BrTens ) ?We note that Drinfeld centers of infinite tensor categories are not typically invertible, letalone trivial in Pic(
BrTens ) . We note also that condition (2) in Theorem 2.26 establishesthe reverse implication, that trivial elements of Pic(
BrTens ) necessarily represent Drinfeldcenters. However, the argument from Theorem 4.2 showing that centers of fusion categoriesare trivial does not apply, a priori . This is because the relative tensor product of finite tensorcategories over a centrally acting braided tensor category might not be finite (in particular,might not be compact-rigid) and so the double-commutant theorem does not apply.Recall that the higher Picard groupoid Pic( T ) of a symmetric monoidal n -category isthe subgroupoid of invertible objects and invertible higher morphisms in T . By definition,we have π (Pic( T )) = Pic( T ) but it is then interesting to study higher homotopy groups.It is known that π , π , and π of Pic(
BrFus ) vanish (the proof of this uses the notion ofFP-dimensions), and that π = k × . Question 4.8.
What is the Postnikov k-invariant relating the π and π of Pic(
BrFus ) ? Question 4.9.
What is the homotopy type of
Pic(
BrTens ) ?A symmetric tensor category may be regarded as an E k -algebra in Pr for any k ≥ .Moreover, as observed in [Hau17, Section 1.2], one has ΩPic(Alg k ( Pr )) ∼ = Pic(Alg k − ( Pr )) .So, the collection of symmetric monoidal ∞ -groupoids Pic(Alg k ( Pr )) forms a spectrum. Question 4.10.
What are the homotopy groups of this spectrum?
References [AF15]
D. Ayala, J. Francis . Factorization homology of topological manifolds.
Jour-nal of Topology ( ). 8(4):1045–1084. arXiv:1206.5522 .[AF17] D. Ayala, J. Francis . The cobordism hypothesis. arXiv preprint ( ). arXiv:1705.02240 . 24Ara17] M. Araújo . Coherence for 3-dualizable ob-jects . Ph.D. thesis, University of Oxford ( ). https://ora.ox.ac.uk/objects/uuid:a4b8f8de-a8e3-48c3-a742-82316a7bd8eb .[BB18] M. Bärenz, J. Barrett . Dichromatic state sum models for four-manifolds from pivotal functors.
Comm. Math. Phys. ( ). 360(2):663–714. arXiv:1601.03580 .[BCJF15] M. Brandenburg, A. Chirvasitu, T. Johnson-Freyd . Reflexivity anddualizability in categorified linear algebra.
Theory and Applications of Categories ( ). 30(23):808–835. arXiv:1409.5934 .[BD95] J. C. Baez, J. Dolan . Higher-dimensional algebra and topological quantumfield theory.
J. Math. Phys. ( ). 36(11):6073–6105. arXiv:q-alg/9503002 .[BDSV15] B. Bartlett, C. L. Douglas, C. J. Schommer-Pries, J. Vicary . Modularcategories as representations of the 3-dimensional bordism 2-category. arXivpreprint ( ). arXiv:1509.06811 .[Ber11] J. E. Bergner . Models for ( ∞ , n ) -categories and the cobordism hypothesis. In Mathematical foundations of quantum field theory and perturbative string theory ,vol. 83 of
Proc. Sympos. Pure Math. , pp. 17–30 (Amer. Math. Soc., Providence,RI, ). arXiv:1011.0110 .[BJS18] A. Brochier, D. Jordan, N. Snyder . On dualizability of braided tensorcategories. arXiv preprint ( ). arXiv:1804.07538 .[BV08] A. Bruguières, A. Virelizier . The double of a Hopf monad. arXiv preprint ( ). arXiv:0812.2443 .[BZBJ18] D. Ben-Zvi, A. Brochier, D. Jordan . Quantum character varieties andbraided module categories.
Selecta Mathematica ( ). 24(5):4711–4748. arXiv:1606.04769 .[BZN09] D. Ben-Zvi, D. Nadler . The character theory of a complex group. arXivpreprint ( ). arXiv:0904.1247 .[CGPM14] F. Costantino, N. Geer, B. Patureau-Mirand . Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories.
Journalof Topology ( ). 7(4):1005–1053. arXiv:1202.3553 .[CKY97] L. Crane, L. H. Kauffman, D. N. Yetter . State-sum invariantsof -manifolds. J. Knot Theory Ramifications ( ). 6(2):177–234. arXiv:hep-th/9409167 .[CS] D. Calaque, C. Scheimbauer . Factorization homology as a fully extendedtopological field theory. (In preparation) .25CY93]
L. Crane, D. Yetter . A categorical construction of 4d topological quan-tum field theories. In
Quantum topology , pp. 120–130 (World Scientific, ). arXiv:hep-th/9301062 .[DGP19] A. Debray, S. Galatius, M. Palmer . Lectures on invertible field theories. arXiv preprint ( ). arXiv:1912.08706 .[DI71] F. DeMeyer, E. Ingraham . Separable algebras over commutative rings . Lec-ture Notes in Mathematics, Vol. 181 (Springer-Verlag, Berlin-New York, ).[DMNO13]
A. Davydov, M. Müger, D. Nikshych, V. Ostrik . The Witt group of non-degenerate braided fusion categories.
J. Reine Angew. Math. ( ). 677:135–177. arXiv:1009.2117 .[DSPS13] C. L. Douglas, C. Schommer-Pries, N. Snyder . Dualizable tensor cate-gories. arXiv preprint ( ). arXiv:1312.7188 .[Dun88] G. Dunn . Tensor product of operads and iterated loop spaces.
J. Pure Appl.Algebra ( ). 50(3):237–258.[EGNO15] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik . Ten-sor categories , vol. 205 of
Mathematical Surveys and Mono-graphs (American Mathematical Society, Providence, RI, ). .[ENO04] P. Etingof, D. Nikshych, V. Ostrik . An analogue of Radford’s S formulafor finite tensor categories. Int. Math. Res. Not. ( ). 2004(54):2915–2933. arXiv:math/0404504 .[EO04] P. Etingof, V. Ostrik . Finite tensor categories.
Mosc. Math. J. ( ).4(3):627–654. arXiv:math/0301027 .[FGR19] V. Farsad, A. Gainutdinov, I. Runkel . SL (2 , Z ) -action for ribbon quasi-Hopf algebras. J. Algebra ( ). 522:243–308. arXiv:1702.01086 .[Fra13] J. Francis . The tangent complex and Hochschild cohomology of E n -rings. Compositio Mathematica ( ). 149(3):430–480. arXiv:1104.0181 .[Fre12a] D. Freed . 3-dimensional TQFTs through the lens ofthe cobordism hypothesis ( ). Slides available at .[Fre12b]
D. S. Freed . 4-3-2-8-7-6.
Aspects of topology conference talk slides. Availableat ( ).[Fre13] D. S. Freed . The cobordism hypothesis.
Bull. Amer. Math. Soc. (N.S.) ( ).50(1):57–92. arXiv:1210.5100 . 26Fre14] D. S. Freed . Anomalies and invertible field theories. In
String-Math 2013 ,vol. 88 of
Proc. Sympos. Pure Math. , pp. 25–45 (Amer. Math. Soc., Providence,RI, ). arXiv:1404.7224 .[FV15] Z. Fiedorowicz, R. M. Vogt . An additivity theorem for the interchange of E n structures. Adv. Math. ( ). 273:421–484. arXiv:1102.1311 .[Gin15] G. Ginot . Notes on factorization algebras, factorization homology and applica-tions. In
Mathematical aspects of quantum field theories , pp. 429–552 (Springer, ).[GS18]
O. Gwilliam, C. Scheimbauer . Duals and adjoints in the factorization higherMorita category. arXiv preprint ( ). arXiv:1804.10924 .[Hau17] R. Haugseng . The higher Morita category of E n –algebras. Geometry & Topol-ogy ( ). 21(3):1631–1730. arXiv:1412.8459 .[JFS17] T. Johnson-Freyd, C. Scheimbauer . (Op) lax natural transformations,twisted quantum field theories, and “even higher” Morita categories.
Advancesin Mathematics ( ). 307:147–223. arXiv:1502.06526 .[JK01] G. Janelidze, G. M. Kelly . A note on actions of amonoidal category.
Theory Appl. Categ ( ). 9(61-91):02. .[JMPP19] C. Jones, S. Morrison, D. Penneys, J. Plavnik . Extension theory forbraided-enriched fusion categories. arXiv preprint ( ). arXiv:1910.03178 .[Joh14] N. Johnson . Azumaya objects in triangulated bicategories.
J. Homotopy Relat.Struct. ( ). 9(2):465–493. arXiv:1005.4878 .[KT20] A. Kirillov, Jr., Y. H. Tham . Factorization homology and 4d TQFT. arXivpreprint ( ). arXiv:2002.08571 .[Lau18] R. Laugwitz . The relative monoidal center and tensor products of monoidalcategories. arXiv preprint ( ). arXiv:1803.04403 .[LM94] V. Lyubashenko, S. Majid . Braided groups and quantum Fourier transform.
Journal of Algebra ( ). 166(3):506 – 528.[LO17] S. Lentner, T. Ohrmann . Factorizable r -matrices for small quantumgroups. SIGMA Symmetry Integrability Geom. Methods Appl. ( ). 13:076. arXiv:1612.07960 .[Lur] J. Lurie . Higher algebra (Preprint, Available at ).27Lur09]
J. Lurie . On the classification of topological field theories.
Current develop-ments in mathematics ( ). 2008:129–280. arXiv:0905.0465 .[Lyu95] V. Lyubashenko . Modular transformations for tensor categories.
Journal ofPure and Applied Algebra ( ). 98(3):279–327.[Par76] B. Pareigis . Non-additive ring and module theory. IV. The Brauer group ofa symmetric monoidal category. In
Brauer groups (Proc. Conf., NorthwesternUniv., Evanston, Ill., 1975) ( ) pp. 112–133. Lecture Notes in Math., Vol.549.[Pst14] P. Pstrągowski . On dualizable objects in monoidal bicategories, framed sur-faces and the cobordism hypothesis. arXiv preprint ( ). arXiv:1411.6691 .[RGG + M. D. Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand,I. Runkel . 3-dimensional TQFTs from non-semisimple modular categories. arXiv preprint ( ). arXiv:1912.02063 .[Ros93] M. Rosso . Quantum groups at a root of 1 and tangle invariants.
Internat. J.Modern Phys. B ( ). 7(20-21):3715–3726. Yang-Baxter equations in Paris(1992).[RT91] N. Reshetikhin, V. G. Turaev . Invariants of -manifolds via link polyno-mials and quantum groups. Invent. Math. ( ). 103(3):547–597.[RW97] L. Rozansky, E. Witten . Hyper-Kähler geometry and invariants of three-manifolds.
Selecta Math. (N.S.) ( ). 3(3):401–458. arXiv:hep-th/9612216 .[RW10] J. Roberts, S. Willerton . On the Rozansky-Witten weight systems.
Algebr.Geom. Topol. ( ). 10(3):1455–1519. arXiv:math/0602653 .[Sch14] C. I. Scheimbauer . Factorization homology as a fully ex-tended topological field theory . Ph.D. thesis, ETH Zurich ( ). .[Shi19a] K. Shimizu . Further results on the structure of (co)ends in finite tensor cate-gories.
Applied Categorical Structures ( ). arXiv:1801.02493 .[Shi19b] K. Shimizu . Non-degeneracy conditions for braided finite tensor categories.
Advances in Mathematics ( ). 355:106778. arXiv:1602.06534 .[SP17] C. Schommer-Pries . Invertible topological field theories. arXiv preprint ( ). arXiv:1712.08029 .[Til98] U. Tillmann . S -structures for k -linear categories and the definition of a mod-ular functor. J. London Math. Soc. (2) ( ). 58(1):208–228.28Tur10] V. G. Turaev . Quantum invariants of knots and 3-manifolds , vol. 18 of deGruyter Studies in Mathematics (Walter de Gruyter & Co., Berlin, ), re-vised ed.[Vit96]
E. M. Vitale . The Brauer and Brauer-Taylor groups of a symmetric monoidalcategory.
Cahiers Topologie Géom. Différentielle Catég. ( ). 37(2):91–122.[Wal] K. Walker . Note on topological field theories (Available at http://canyon23.net/math/tc.pdf ).[Was20]
T. A. Wasserman . The Drinfeld centre of a symmetric fusion category is 2-foldmonoidal.
Adv. Math. ( ). 366:107090. arXiv:1711.03394 .[Wit89] E. Witten . Quantum field theory and the Jones polynomial.
Comm. Math.Phys. ( ). 121(3):351–399.[WW11] K. Walker, Z. Wang . ( )-TQFTs and topological insulators.
Frontiers ofPhysics ( ). 7(2):150–159. arXiv:1104.2632arXiv:1104.2632