The Monoidal Centre for Group-Graded Categories 3

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1. Basics on bicategories Bicategories as defined by Bénabou [3] are “monoidal categories with several objects” in the sense that additive categories are “rings with several objects” [23]. For bicategories A and B, we write PspA , Bq for the bicategory of pseudofunctors, pseudonatural transformations, and modifications (in the terminology of Kelly-Street [19]). We will use monoidal bicategory terminology from Day-Street [11] except that we now use “monoidale” in preference to “pseudomonoid”. A good example of an autonomous symmetric monoidal bicategory is Mod. The objects are categories. The homcategories are defined by ModpA , Bq“rBop ˆ A , Sets ; objects of these homs are called modules. Composition ModpB, C qˆ ModpA , Bq ÝÑ˝ ModpA , C q is defined by B pN ˝ MqpC, Aq“ MpB, Aqˆ NpC, Bq . ż The identity module A Ñ A is the hom functor of A . Tensor product is finite cartesian product of categories; it is not the product in Mod. Coproduct in Mod is coproduct of categories; it is also product in Mod. We identify each functor F : A Ñ B with the module F˚ : A Ñ B defined by F˚pB, Aq “ BpB,FAq; this gives a faithful locally-full pseudofunctor p´q˚ : Cat Ñ Mod which is the identity on objects, strong monoidal, and coproduct preserving. The dual of A in Mod is the opposite category A op. Recall (for example from [11]) that monoidales in Modop are promonoidal categories A in the sense of Day [10]. The tensor product is a module P : A Ñ A ˆ A and the unit is a module J : A Ñ 1. The Day convolution monoidal structure on the functor category rA , Sets has unit J : A Ñ Set and tensor product F ‹ G defined by A,B pF ‹ GqC “ P pA, B; Cqˆ F A ˆ GB . ż Suppose V is a complete cocomplete closed symmetric monoidal category. We have a symmetric (weak) monoidal pseudofunctor r´, V s : Modop ÝÑ V -CAT (1.1) taking each category A to the V -enriched functor category rA , V s. Therefore, each promonoidal category is taken to a convolution monoidal V -category. In the case V “ Set, (1.1) is the contravariant representable pseudofunctor Modp´, 1q.

Let G be a group regarded as a one-object category. The object will be denoted by oG. The morphisms of this category will also be called elements of G; so a P G will mean, as usual, that a is an element. We also use the conjugation notation ca “ cac´1 and ac “ c´1ac. Like every category, G has a promonoidal structure for which the convolution structure on rG, Sets is pointwise cartesian product. In this case, the unit J is constant at the one-point set and the module P : G Ñ G ˆ G is deﬁned by

P poG,oG; oGq“tpx, yq : x, y P Gu , P pa, b; cqpx, yq“pcxa, cybq . (1.2) THE MONOIDAL CENTRE FOR GROUP-GRADED CATEGORIES 3

As we expound the theory, we will carry along an example based on Example 3.5 of [33] which, in turn, was based on an example in [22]. At this point it is just to show some naturally occurring lax functors and pseudofunctors. Example 1.1. A group morphism π : H Ñ G is a functor. According to Bénabou [4] (see [28]), any functor over a category G corresponds to a normal lax functor H : G Ñ Modop. The functor π is a fibration if and only if it is an opfibration if and only if it is Giraud-Conduché[16, 6] if and only if it is a surjective group morphism. In this case, the normal lax functor is actually a pseudofunctor. Let us now describe it. The fibre of the functor π over oG is the kernel K of the group morphism π; so HoG “ K as a one-object category. For a : oG Ñ oG in G, the module Ha : K Ñ K is defined by

pHaqpoK,oKq “ tx P H : πpxq“ au . b a In particular, H1 “ K is the identity module. For a composable pair oG ÝÑ oG ÝÑ oG in G, the composite module Hb ˝ Ha is seen to be the set

pHb ˝ HaqpoK,oKq“ tx P H : πpxq“ au ˆ ty P H : πpyq“ bu {K of orbits under the action of `K on the right in the ﬁrst factor of the product˘ and on the left in the second. The composition constraint is the bijection

pHb ˝ HaqpoK,oKq ÝÑ HpabqpoK,oKq , rx, ys ÞÑ xy . A cleavage for the fibration π : H Ñ G amounts to a splitting σ of π as a function on elements; that is, πσa “ a for all a P G. The chosen cartesian morphism over op a : oG Ñ oG is σpaq : oH Ñ oH . The pseudofunctor H : G Ñ Cat corresponding to σpaq this cloven fibration is defined by HoG “ K, pHaqk “ k for a P G and k P K, while the component at oK of the invertible composition constraint pKbqpKaq ñ Kpabq is equal to σpaqσpbqσpabq´1 P K. We use the same symbol for the pseudofunctors H since the first is equivalent to the composite of the second with the canonical pseudofunctor p´q˚.

The groupoid of automorphisms in G will be denoted by Gaut. The objects are elements a P G. A morphism f : a Ñ b is an element f P G such that fa “ bf. If Gconj denotes the G-set whose elements are those of G and whose G-action is by conjugation then there is an equivalence of categories aut rG, Sets{Gconj »rG , Sets . (1.3)

Since Gconj is a monoid in rG, Sets, there is a monoidal structure on the left hand side of (1.3) whose tensor product takes cartesian product of the morphisms over Gconj followed by the monoid multiplication. This monoidal structure is closed (on both sides) and so transports to a promonoidal structure on Gaut. Recall from the end of Section 4 of [13] that this promonoidal structure is deﬁned by 1 if c “ 1 P pa, b; cq“tpu, vq P G ˆ G : ua vb “ cu and Jc “ " ∅ if c ‰ 1 and that there is a braiding – u γa,b;c : P pa, b; cq ÝÑ P pb, a; cq , pu, vq ÞÑ p av,uq . 4 BRANKONIKOLIC´ AND ROSS STREET

It also has a twist τa “ a : a Ñ a ; compare Section 2 of [27]. The reader is invited to check the commutativity of (1.4) which is the main twist condition.

γa,b;c P pa, b; cq / P pb, a; cq

P p1,1;τcq P pτa,τb;1q (1.4)

γb,a;c P pa, b; cq o P pb, a; cq Furthermore, Gaut is a ˚-autonomous promonoidal category in the sense of [12]: we have the natural isomorphisms P pa, b; c´1q ÝÑ– P pb, c; a´1q , pu, vq ú pu´1v,u´1q ;

P pa, b; c´1q ÝÑ– P pb´1, a´1; cqop , pu, vq ú pv,uq . Proposition 1.2. [29, 13, 14] The braided monoidale Gaut is the monoidal centre of the monoidale G in Modop. The convolution braided monoidal category rGaut, Sets is braided monoidal equivalent to the monoidal centre ZrG, Sets2 of the cartesian monoidal category rG, Sets of G-sets. We note that the equivalence becomes balanced on transport of the convolution twist to the monoidal centre. Example 1.3. We return to our surjective group morphism π : H Ñ G. Since fibrations in Cat are preserved by 2-functors of the form rD, ´s, we have the fibration πaut : Haut Ñ Gaut. The corresponding pseudofunctor Haut : Gaut Ñ Modop is defined as follows. For each a P G, the category Hauta has objects those x P H with πpxq“ a; morphisms k : x Ñ x1 are those k P K with kx “ x1k; composition is that of K. For f : b Ñ a in Gaut, the module Hautf : Hauta Ñ Hautb is defined by pHautfqpy, xq “ th P Hautpy, xq : πphq“ fu . The splitting σ for π also gives a cleavage for πaut. The corresponding pseudofunctor Haut : Gaut Ñ Catop has the same value on objects as in the last paragraph. For aut aut aut aut aut f : b Ñ a in G , the functor H f : H a Ñ H b takes k : x Ñ x1 in H a σpfq σpfq σpfq aut aut to k : x Ñ x1 . The invertible composition constraint pH gqpH fq ñ Hautpfgq has component at x P Hauta equal to σpfqσpgqσpfgq´1.

2. Monoidales in convolution bicategories aut One virtue of the promonoidal groupoid G over the monoid Gconj is that we can obtain convolution balanced monoidal structures on functors from Gaut, not only into Set but, into any nice enough monoidal category; or even on pseudofunctors from Gaut into any nice enough monoidal bicategory. Let K be a monoidal bicategory with coproducts preserved by horizontal composition in each variable. The tensor product will be denoted by ´b´ : K ˆ K Ñ K

2We are using Z for the centre of a monoidal category since we like the fact that it is the ﬁrst letter of the German words for both centre and braid. THE MONOIDAL CENTRE FOR GROUP-GRADED CATEGORIES 5 with unit object I . Think of Gaut as a bicategory with only identity 2-cells. We will make explicit the convolution monoidal structure on the bicategory PspGaut, K q. Take S, T P PspGaut, K q. Put3 a,b pS ‹ Tqc “ Sa b Tb » P pa, b; cq ¨ Sa b Tb (2.5) ż abÿ“c ` ps ˘ aut and, for f : c Ñ c1 in G , deﬁne pS ‹ Tqf by commutativity in

in S T a,b S T a b b / ab“c a b b ř SfbTf pS‹Tqf (2.6) Sfa Tfb / Sa Tb . b a1b1“c1 1 b 1 infa,f b ř This deﬁnes the tensor product S ‹ T for a monoidal structure on PspGaut, K q with unit J : Gaut Ñ K deﬁned by I if c “ 1 Jc “ (2.7) " 0 if c ‰ 1 which becomes functorial on noting that, for f : c Ñ c1, if c “ 1 then c1 “ 1. We can now contemplate monoidales M in PspGaut, K q. Such a monoidale consists of a pseudofunctor M : Gaut Ñ K equipped with morphisms

I : I Ñ M1 and ˝a,b : Ma b Mb Ñ Mpabq in K and invertible 2-cells

1b˝b,c Ma b Mb b Mc / Ma b Mpbcq α ˝ a,b,c ˝ a,bb1 +3 a,bc (2.8) – Mpabqb Mc / Mpabcq ˝ab,c

Ma ❧❧ ❘❘❘ I ❧❧❧ ❘❘❘ I b1❧❧❧ ❘❘1❘b ❧❧❧ ❘❘❘ ❧❧❧ ❘❘ v λa– ρa– ( M1 b Ma +3 1 +3 Ma b M1 (2.9) ❘❘ ❧ ❘❘❘ ❧❧❧ ❘❘❘ ❧❧❧ ˝ ❘❘ ❧❧˝ 1,a ❘❘❘ ❧❧❧ a,1 ❘❘ ❧❧❧ ( Ma v all subject to pseudonaturality

˝a,b Ma b Mb / Mc

MfbMf – Mf ùñ (2.10) f f 1 M a b M b ˝ / Mc , f a,f b modiﬁcationality and coherence conditions.

3The coend here is in the pseudo-sense appropriate to bicategories. 6 BRANKONIKOLIC´ AND ROSS STREET

Example 2.1. The H of Example 1.1 is a monoidale in PspG, Modopq. The monoidal structure is provided by the modules

˝ : HoG Ñ HoG ˆ HoG and I : HoG Ñ 1 deﬁned by

˝poK,oK; oKq“tpu, vq : u, v P Ku , ˝pk,ℓ; mqpu, vq“pmuk, mvℓq , I “!˚

3. The Turaev-Virelizier structures Definition. [33] A G-graded category over k is a k-linear monoidal category C , with finite direct sums, endowed with a system of pairwise disjoint full k-linear subcate- gories Ca, a P G, with finite direct sums, such that

(a) each object X P C splits as a direct sum ‘aXa where Xa P Ca and a runs over a ﬁnite subset of G; (b) if X P Ca and Y P Cb then X b Y P Cab; (c) if X P Ca and Y P Cb with a ‰ b then C pX,Y q“ 0; (d) the tensor unit I of C is in C1.

Turaev-Virelizier call an object X of a G-graded category C homogeneous when there exists a (necessarily unique) a P G such that X P Ca; this a is denoted by |X|. They write G¯ for the discrete monoidal category of elements of G with the multiplication as tensor product. They write AutpC q for the monoidal category of monoidal endo-equivalences of the monoidal k-linear category C and monoidal natural isomorphisms; the tensor product is composition of functors.

Deﬁnition. [33] A G-crossed category C is a G-graded category over k equipped with a strong monoidal functor ϕ : G¯ Ñ AutpC q such that ϕapCbqĎ Ca´1ba for all a, b P G.

Proposition 3.1. Let V be the monoidal category of modules over a fixed commutative ring. Then monoidales in PspGaut, V -Catq are equivalent to the G-crossed categories of [33]. Proof. Take a monoidale M in PspGaut, V -Catq. Let C be the V -category obtained by taking the completion, with respect to finite direct sums, of the coproduct Chom “ M V C M ´1 a a in -Cat. Then is a G-graded category with ϕf “ f . řConversely, take a G-graded category C and define Ma to be the full sub-V -category Ca of C consisting of the objects homogeneous over a P G. Then M is a monoidale in PspGaut, V -Catq. Corollary 3.2. With V as in Proposition 3.1, any monoidale in PspGaut, Modopq delivers a G-crossed category on application of the pseudofunctor (1.1). Definition. [33] A G-braided category C is a G-crossed category pC ,ϕq equipped with a natural family of isomorphisms

γX,Y : X b Y ÝÑ Y b ϕ|Y |pXq , for X,Y P C and Y homogeneous, subject to three axioms. In the next section, we will see how this ﬁts into our theory of braided monoidales. THE MONOIDAL CENTRE FOR GROUP-GRADED CATEGORIES 7

4. Internal homs, biduals and braidings If K is left closed, it is straightforward to see that so too is PspGaut, K q: rT, Usa “ rTb, Upabqs . (4.11) źb Proposition 4.1. Suppose in K that direct sums indexed by the elements of G exist and that each Tb has a left bidual pTbq_. Then T has a left bidual T_a “rT, Jsa “pTa´1q_ (4.12) in PspGaut, K q. Proof. Taking (4.12) as the deﬁnition of T_, we need to prove that the canonical morphism S ‹ T_ ÝÑ rT, Ss is an equivalence for all S. The component of this canonical pseudonatural transformation at c is Sa b Tpb´1q_ » Spcdqb Tpdq_ » rTd, Spcdqs ÝÝÝÑcanon. rTd, Spcdqs abÿ“c ÿd ÿd źd in which the arrow is an equivalence because of our assumption about direct sums. Corollary 4.2. All biduals exist in PspGaut, Modopq; that is, the monoidal bicategory is autonomous (also called “compact” or “rigid”).

If K is equipped with a braiding γX,Y : X b Y Ñ Y b X then we obtain a braiding aut γS,T : S ‹ T Ñ T ‹ S on PspG , K q as deﬁned by the commutative pentagon (4.13).

γSa,Tb Tab1 Sa b Tb / Tb b Sa / Tab b Sa

ina,b inab,a (4.13) 1 1 Sa b Tb / 1 1 Tb b Sa ab“c γS,Tc b a “c ř ř

If K is balanced then so too is PspGaut, K q with twist

Sa θSa θS,a “ Sa ÝÑ Sa ÝÝÑ Sa . Recall that, if K is symmetric, we` choose its twist to˘ be the identity. We already have the example Gaut of a ˚-autonomous balanced monoidale in Modop. Proposition 4.3. The monoidal bicategory PspGaut, Modopq is tortile. Moreover, with PspGaut, K q a braided monoidal bicategory, according to [29], we can contemplate monoidal centres ZM for monoidales M therein. Since the centre is a limit, it is formed pointwise in K . From [29], we know that ZM is a braided monoidale in PspGaut, K q.

Example 4.4. The Haut of Example 1.3 is a balanced monoidale in PspGaut, Modopq. The monoidal structure is provided by the modules aut aut aut aut ˝a,b : H pabqÑ H a ˆ H b and I : H 1 Ñ 1 8 BRANKONIKOLIC´ AND ROSS STREET deﬁned by 1 if z “ 1 ˝ px, y; zq“tpu, vq : u, v P K,u x vy “ zu and Iz “ . a,b " ∅ if z ‰ 1 The braiding is

– u γx,y;z : ˝a,bpx, y; zq ÝÑ ˝b,apy, x; zq , pu, vq ÞÑ p xv,uq .

aut aut aut aut aut The twist τ : H Ñ H is given by τa “ H a : H a Ñ H a. We suspect this example is also tortile [25] (also called “ribbon”). Let us consider the case of K “ V -Cat where V is a complete cocomplete closed symmetric monoidal category. For S P PspGaut, V -Catq, f : a Ñ b in Gaut and A P S, we put fA “pSfqA P Sb. Let M be a monoidale in PspGaut, V -Catq. The tensor product consists of V - functors ˝a,b : Ma b Mb Ñ Mpabq. The unit is an object I of M1. The associativity constraint consists of a V -natural family

αa,b,c : pA ˝a,b Bq ˝ab,c C ÝÑ A ˝a,bc pB ˝b,c Cq . (4.14) A braiding for M consists of a V -natural family

γa,b : A ˝a,b B ÝÑ aB ˝ab,a A. (4.15) Proposition 4.5. In the setting of Proposition 3.1, the braided monoidales in PspGaut, V -Catq are equivalent to the G-braided categories of [33].

According to Section 3 of [29], since pseudolimits limits are formed pointwise, the monoidal centre ZM of a monoidale M in PspGaut, V -Catq is constructed as follows. The V -category pZMqa has objects pairs pA, υq where A is an object of Ma and υ is a half G-braiding for A consisting of a V -natural family of isomorphisms

υb : A ˝a,b B ÝÑ aB ˝ab,a A (4.16) such that υ1 : A ˝a,1 I ÝÑ I ˝1,a A transports the right unit constraint into the left unit constraint and the following hexagon commutes.

υ A ˝a,bc pB ˝b,c Cq / paB ˝ab,a c aCq ˝apbcq,a Aq ❦❦❦5 α ❦❦❦ ❦❦❦ α ❦❦❦❦ ❦❦ pA ˝a,b Bq ˝ab,c C aB ˝ab,ac paC ˝ac,a Aq ❚❚❚❚ O ❚❚❚❚ ❚❚❚ 1bυ υb1 ❚❚❚ ❚) ˝ ˝ ˝ ˝ paB ab,a Aq ab,c C α / aB ab,ac pA a,c Cq

f 1 1 For f : a Ñ a, we have ZMpA, υq“pfA,υ q where υb for B P Mb is the composite

fυ ´ ´1 f 1B ´1 f ˝f ˝ f´1 ˝a ´1 ˝ f fA a,b B – fpA a, b f Bq ÝÝÝÝÑ fpaf B pf bq,a Aq– aB ab,f a fA . THE MONOIDAL CENTRE FOR GROUP-GRADED CATEGORIES 9

5. Full centres Davydov [9] deﬁned the full centre of a monoid M in a (not necessarily braided) monoidal category E to be a commutative monoid zM in the (braided) monoidal centre ZE of E satisfying an appropriate universal property. Street [31] pointed out that the pair pZE , zMq is the monoidal centre of the monoidale pE , Mq in the monoidal bicategory of pointed categories. We can lift this concept of full centre from the monoidal category level to the monoidal bicategory level. The full centre of a monoidale M in a monoidal bicategory K is a braided monoidale zM in the monoidal centre ZK (see [1, 8, 21]) of K satisfying an appropriate universal property. Proposition 5.1. (i) The braided monoidal bicategory PspGaut, Modopq is the monoidal centre of the monoidal bicategory PspG, Modopq. (ii) The full centre of the monoidale H in PspG, Modopq (see Example 2.1) is the braided monoidale Haut (see Example 4.4) in PspGaut, Modopq.

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Department of Mathematics, Macquarie University, NSW 2109, Australia Email address: [email protected] Department of Mathematics, Macquarie University, NSW 2109, Australia Email address: [email protected]