THE MONOIDAL CENTRE FOR GROUP-GRADED CATEGORIES BRANKO NIKOLIC´ AND ROSS STREET Abstract. Let G be a group and k be a commutative ring. Our aim is to ameliorate the G-graded categorical structures considered by Turaev and Virelizier by fitting them into the monoidal bicategory context. We explain how these structures are monoidales in the monoidal centre of the monoidal bicategory of k-linear categories on which G acts. This provides a useful example of a higher version of Davydov’s full centre of an algebra. Introduction As well as being a mathematical discipline with its own structures, calculations and significant theorems, category theory provides a framework for understanding analogies between concepts in different disciplines as examples of the same concept interpreted in different categories. Bicategories do the same for categorical concepts, and so on, iterating the microcosm principle1. An example of this unification process is provided in Section 4.8 of Street [31] where the full centre of an algebra, as defined and constructed by Davydov [9], is shown to be a monoidal centre in the sense of Street [29] in the monoidal bicategory of pointed categories. We show here how structures and constructions defined by Turaev and Virelizier [33] (also see [32, 20, 2]) for G-graded categories, where G is a group, are precisely the usual monoidal structures taken in an appropriate monoidal bicategory. To some extent, we replace the monoidal category of k-modules by the monoidal bicategory Modop whose morphisms are two-sided modules on which categories act. There are several benefits gained by working at the bicategorical level. While k- modules only have monoidal duals if they are finitely generated and projective, all objects of Modop have duals. While k-modules only have finite direct sums, all small direct sums exist in Modop. While the monoidal centre of the category of all k-linear representations of a group G is not tortile (also called “ribbon”), the monoidal centre arXiv:2010.10656v1 [math.CT] 20 Oct 2020 of the bicategory of Modop-representations of G is tortile. Moreover, a higher version of the Davydov full centre is involved in the key example mentioned in [33]. At this stage we are not saying anything about the main theorem of [33] which is a G-graded version of a theorem of M¨uger [24]. However, note that the invertibility of the S-matrix in the G-graded case uses M¨uger’s result. Date: October 22, 2020. The second gratefully acknowledges the support of Australian Research Council Discovery Grant DP190102432. 1so named by Baez and Dolan; see footnote 2 on page 100 of [11] 1 2 BRANKONIKOLIC´ AND ROSS STREET 1. Basics on bicategories Bicategories as defined by B´enabou [3] are “monoidal categories with several ob- jects” in the sense that additive categories are “rings with several objects” [23]. For bi- categories A and B, we write PspA , Bq for the bicategory of pseudofunctors, pseudo- natural 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 struc- ture 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 B´enabou [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-Conduch´e[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 .