Monoidal Ladder Categories

Home , Category of modules, Monoidal category, Monoidal functor

Mitchell Rowett

October 2019

A thesis submitted for the degree of Bachelor of Philosophy (Honours) of the Australian National University

Declaration

The work in this thesis is my own except where otherwise stated.

Mitchell Rowett

Acknowledgements

Firstly, I would like to thank my supervisor, Scott Morrison. Thank you for teach- ing me so much over the past few years, for your kindness and support, and for your incredible generosity with your time. I would also like to thank the lecturers at the MSI for their patience and enthusiasm, and their willingness to indulge all kinds of questions. Thanks should also go to the students and rest of the community at the MSI; it has been a privilege to have been a part of such a supportive and welcoming community. I would especially like to thank the other honours students, who have taken this journey with me. Last but not least, I would like to thank my friends and family for their continual support. Mum, Dad, Cooper, Brittany, and Nat, your encouragement is the reason I have made it this far. Chris and Huon, thank you for sticking by me this year, and for doing so much to help me when I needed it most. Natalie, I think you already know how much I have relied on your patience and positivity. I am so very proud of you.

Abstract

In this thesis, we construct a tensor product of module categories over a linear rigid monoidal category, which we call a ladder category. In the case of monoidal module categories over a braided category, we exhibit a monoidal structure on the ladder category. We then give two major examples. For the first, we show that given a fusion category C with a central functor from RepG, the de-equivariantisation of C can be realised as the idempotent completion of the ladder category of C with Vec over RepG. We also give a proof that the definition of de-equivariantisation by de-enrichment given in [MP19] is equivalent to the standard definition of de- equivariantisation. For the second example we give an explicit description of the ladder category of the two unital Ad E8 fusion categories over Fib, which appears to be a fusion category not previously studied. We also show that the ladder category construction is equivalent to the Deligne tensor product in the case of fusion categories.

vii

Contents

Acknowledgements v

Abstract vii

1 Introduction 1

2 Background 3 2.1 Monoidal Categories ...... 3 2.1.1 Deﬁnition of a Monoidal Category ...... 3 2.1.2 The Coherence Theorem ...... 4 2.1.3 String Diagrams ...... 6 2.1.4 Monoidal Functors and Monoidal Equivalences ...... 6 2.1.5 Braided Monoidal Categories ...... 8 2.1.6 Rigid Monoidal Categories ...... 10 2.2 Linear Categories ...... 11 2.2.1 Additive and Linear Categories ...... 11 2.2.2 Additive Completion ...... 13 2.2.3 Idempotent Complete Categories and the Karoubi Envelope 14 2.2.4 Semisimple Categories ...... 14 2.3 (Co)Algebras and (Co)Modules ...... 17 2.3.1 Monoids and Algebras ...... 17 2.3.2 Module Objects ...... 18 2.3.3 Coalgebras and Comodules ...... 18 2.3.4 Categories of Modules and Comodules ...... 19 2.3.5 Free Comodules ...... 21 2.4 Monoidal Module Categories ...... 22 2.4.1 Module Categories ...... 22 2.4.2 Monoidal Module Categories ...... 24

ix x CONTENTS

3 Ladder Categories 25 3.1 Ladder Categories ...... 25 3.1.1 Deﬁnition of Ladder Categories ...... 25 3.1.2 Ladder Categories Over Semisimple Categories ...... 28 3.2 Monoidal Ladder Categories ...... 28 3.2.1 Deﬁnition of Monoidal Ladder Categories ...... 28 3.2.2 Ladder Categories over Vec ...... 31 3.3 Properties of Ladder Categories ...... 32

4 De-equivariantisation 37 4.1 Deﬁnitions of de-equivariantisation ...... 37 4.1.1 Equivariantisation ...... 37 4.1.2 De-equivariantisation ...... 38 4.2 Frobenius Algebras ...... 39 4.2.1 Deﬁnition of Frobenius Algebras ...... 40

4.2.2 Equivalence Between Cpcomod-A and Ccomod-A ...... 42 4.3 Ladders and de-equivariantisation ...... 47 4.4 The Enrichment De-equivariantisation ...... 54 4.4.1 Enriched Categories ...... 54 4.4.2 Monoidally Enriched Categories ...... 56 4.4.3 Transporting Enrichment ...... 57 4.4.4 The Enrichment De-equivariantisation ...... 58 4.4.5 The Underlying Monoidal Categories ...... 60 4.4.6 Equivalence to De-equivariantisation ...... 63

5 Tensor Categories and the Deligne Product 69 5.1 Tensor Categories ...... 69 5.1.1 Fusion Categories as Tensor Categories ...... 71 5.2 The Deligne Product ...... 72 5.2.1 Monoidal Structure on the Deligne Product ...... 73 5.3 Ladder Categories and the Deligne Product ...... 74

− + 6 An Explicit Example: Ad E8 ⊗Fib Ad E8 77 6.1 Classiﬁcation of “Small” Fusion Categories ...... 77 6.1.1 Frobenius-Perron Dimension ...... 77 6.1.2 Principal Graphs ...... 78 6.1.3 The ADE Fusion Categories ...... 79 6.1.4 Planar Algebras and a Classiﬁcation ...... 80 CONTENTS xi

6.1.5 Fib and Ad E8 ...... 81 − + 6.2 Fusion Rules for Ad E8 ⊗Fib Ad E8 ...... 83 − + 6.2.1 Simple Objects of Ad E8 ⊗Fib Ad E8 ...... 83 − + 6.2.2 Calculating Fusion Rules of Ad E8 ⊗Fib Ad E8 ...... 87 − + 6.3 Ad E8 ⊗Fib Ad E8 as a Category of Modules ...... 90

A Adjunctions 93

Bibliography 94

1 Introduction

In this thesis, we give a construction of a tensor product of “monoidal module categories” M, N over a braided category C, which we call the ladder category

M ⊗C N . We give two major examples. The ﬁrst is of the de-equivariantisation of a monoidal module category over RepG, where G is a ﬁnite group.

Theorem A. Given a braided linear functor RepG → Z(C) for a fusion category C, the de-equivariantisation can be realised as the idempotent completion of a monoidal ladder category

CG ' Kar(C ⊗RepG Vec).

A third deﬁnition of de-equivariantisation was given without proof in [MP19]; we provide a proof that it is indeed equivalent to the standard deﬁnition of de- equivariantisation. The second example is of unitary ADE fusion categories over a common subcategory.

Theorem B. The adjoint subcategories of the two unitary E8 fusion categories are (right and left) monoidal module categories for Fib. Hence

− + Ad E8 ⊗Fib Ad E8 is a monoidal category, and if semisimple has principal graph

2 XR 5 3 . 1 X 2 Y τ

YR 2

We believe this to be a fusion category which has not previously been studied.

1 2 CHAPTER 1. INTRODUCTION

We also provide a comparison between monoidal ladder categories and the Deligne tensor product of tensor categories, and show that they are equivalent for fusion categories. One can think of the monoidal ladder category construction as an alternative to the Deligne tensor product which can be applied to a strictly larger class of categories, though unfortunately it does not always agree with the Deligne tensor product. However, monoidal ladder categories have an advantage as they give a way of taking tensor products of categories which are not necessarily abelian, which can be inaccessible for some constructions of linear monoidal categories. We do not assume any familiarity with most of the notions described above, but rather assume only basic knowledge of categories, functors, natural transformations, and adjunctions. Most of the definitions used above are defined in Chapter 2. The remainder of this thesis is structured as follows. In Chapter 2 we recall definitions from category theory which will be useful as we develop later theory. In Chapter 3 we define monoidal ladder categories and show some basic results about them. In Chapter 4 we give two alternative definitions of de- equivariantisation, firstly as a monoidal ladder category and secondly as a “de- enrichment” [MP19], and show that these are both equivalent to the standard definition of de-equivariantisation. In Chapter 5 we give an overview of the basic definitions of tensor categories and the Deligne tensor product, and show that the Deligne tensor product is equivalent to the monoidal ladder category in the case of fusion categories. Finally, in Chapter 6 we further describe the theory of the unitary − + ADE fusion categories, and given an explicit construction of Ad E8 ⊗Fib Ad E8 . 2 Background

In this chapter, we recall aspects of the theory of monoidal, linear, and semisimple categories which will be important to the developments in later chapters. We also give a brief overview of the theory of algebra and module objects, which will be especially useful in Chapter 4, and then progress to the categorical analogues by deﬁning module and monoidal module categories. We assume the reader is familiar with the deﬁnitions of categories, functors, natural transformations, and adjunctions (a brief summary of the last is given in the appendix). A discussion of this theory can be found in [Mac98]. Where examples are given, they are chosen primarily for use later in this thesis.

2.1 Monoidal Categories

2.1.1 Deﬁnition of a Monoidal Category

A monoidal category is a category with a functor ⊗ which acts as a kind of “tensor product”. For example, in the category of ﬁnite-dimensional vector spaces Vec we can take tensor products of both objects and morphisms using the standard tensor product. A category can often be made into a monoidal category in more than one way. For example, Vec can also be made into a monoidal category with the direct sum as the tensor product. Deﬁnition 2.1. A monoidal category is a category C with • a functor ⊗ : C × C → C called the tensor product, • a distinguished object 1 ∈ C called the unit object, • a natural isomorphism a with components of the form

∼ ax,y,z :(x ⊗ y) ⊗ z −→ x ⊗ (y ⊗ z) called the associator, • natural isomorphisms l and r with components of the form

∼ lx : 1 ⊗ x −→ x ∼ rx : x ⊗ 1 −→ x

3 4 CHAPTER 2. BACKGROUND

called the left unitor and right unitor respectively, such that the following two diagrams commute for all w, x, y, z ∈ C: • the triangle identity:

ax,1,y (x ⊗ 1) ⊗ y x ⊗ (1 ⊗ idy)

rx⊗idy idx ⊗ly x ⊗ y

• the pentagon identity:

((w ⊗ x) ⊗ y) ⊗ z

aw,x,y⊗idz aw⊗x,y,z

(w ⊗ (x ⊗ y)) ⊗ z (w ⊗ x) ⊗ (y ⊗ z)

aw,x⊗y,z aw,x,y⊗z w ⊗ ((x ⊗ y) ⊗ z) w ⊗ (x ⊗ (y ⊗ z)) idw ⊗ax,y,z

We will often suppress the ⊗ symbol, instead simply concatenating both objects and morphisms as xy and fg. We will therefore always write composition explicitly as g ◦ f.

Examples 2.2. i) As discussed above, the category Vec of ﬁnite-dimensional vector spaces with ⊗ given by the tensor product of vector spaces.

ii) The category RepG of ﬁnite-dimensional representations over a ﬁnite group

G with ⊗ given by the tensor product of representations; that is, if (V, ρV :

G → GL(V )) and (W, ρW : G → GL(W )) are representations, then their

tensor product is (V ⊗ W, ρV ⊗W ) where

ρV ⊗W (g) = ρV (g) ⊗ ρW (g).

The unit object is the trivial representation.

2.1.2 The Coherence Theorem

For any ordered sequence of objects x1, . . . , xn ∈ C, we can take the tensor product x1 . . . xn. However, we can take diﬀerent ways of parenthesising this product, and a priori these may be diﬀerent objects. For n = 3, the associator gives an ∼ isomorphism (x1x2)x3 −→ x1(x2x3), while for n ≥ 4 it is fairly simple to see that 2.1. MONOIDAL CATEGORIES 5

for any two ways P1,P2 of parenthesising there is a chain of associators (tensored with identity morphisms) which gives an isomorphism from P1 to P2. However, we run into a problem in that there may well be multiple ways of chaining these associators to obtain an isomorphism, giving potentially diﬀerent isomorphisms. If these isomorphisms are truly diﬀerent, then we cannot say that

P1 and P2 are canonically isomorphic, as there is no reason to prefer one particular chain of associators over another. The purpose of the pentagon diagram, then, is to assert that for n = 4 all sequences of associators from one parenthesisation to another are equal. The coherence theorem makes an even stronger claim for all n, which is that every (well-behaved1) diagram comprised of associators and unitors commutes. In other words, all isomorphisms formed from associators and unitors (potentially tensored with identities) between two parenthesisations are equal.

Theorem 2.3 (Coherence Theorem). Let x1, . . . , xn ∈ C, and let P1,P2 be two parenthesisations of the product x1 . . . xn with arbitrary insertions of the unit 1C.

Let f, g : P1 → P2 be two isomorphisms formed from the composition of associators, unitors and their inverses (potentially tensored with identity morphisms). Then f = g.

We will omit the proof of this theorem here. It can be found as Theorem 2.9.2 in [EGNO15]. This is one of the most fundamental results in monoidal category theory, and importantly it allows us to suppress associators and unitors from our morphisms. For example, if we have a morphism f : xx → x, we can write a diagram

f⊗idx xxx xx

idx ⊗f

The reader may insert their favourite isomorphism (xx)x −→∼ x(xx) where required to make this formally correct, and any possible composition of associators and unitors will in fact be the same isomorphism.

1We cannot say that every diagram commutes, as two formally diﬀerent vertices may be equal in a certain category in such a way that breaks the commutativity. However, so long as every vertex contains the same ordered sequence of objects (plus units) this issue does not arise. 6 CHAPTER 2. BACKGROUND

2.1.3 String Diagrams

String diagrams are a useful graphical tool for describing monoidal categories, and can help make certain arguments more intuitive. In this section, we let C be a monoidal category, and we will suppress associators and unitors as permitted by the coherence theorem. We denote a morphism f : x → y by a string with a box labelled f:

Note that we read the morphism from bottom to top. We then concatenate vertically to denote composition, concatenate horizontally to denote the tensor product, and use a plain line to denote the identity morphism:

x g

= g ◦ f f g = f ⊗ g idx = f x

We will often omit labels of objects (as above) when the object is either unam- biguous or arbitrary.

A general function f : x1 . . . xn → y1 . . . ym from and to a tensor product can be denoted by multiple lines going into the morphism:

y1 ...ym

x1 ... xn

We will denote the identity on 1C by the omission of a line, and so a morphism f : 1 → x is denoted as x

f .

2.1.4 Monoidal Functors and Monoidal Equivalences

Deﬁnition 2.4. Let C and D be monoidal categories. A monoidal functor from C to D is • a functor F : C → D, 2.1. MONOIDAL CATEGORIES 7

• an isomorphism ∼ : 1D −→ F (1C) called the unitor, and • a natural isomorphism µ with components of the form

∼ µx,y : F (x)F (y) −→ F (xy)

called the tensorator, satisfying the following conditions: • (Associativity) The following diagram commutes for all x, y, z ∈ C, where aC and aD are the associators in C and D:

aD (F (x)F (y))F (z) F (x),F (y),F (z) F (x)(F (y)F (z))

µx,y idF (z) idF (x) µx,y F (xy)F (z) F (x)F (yz)

µxy,z µx,yz

F ((xy)z) C F (x(yz)) F (ax,y,z)

• (Unitality) The following diagrams commutes for all x ∈ C, where lC, lD, rC, and rD are the left and right unitors in C and D:

D lF (x) 1DF (x) F (x)

C −1 idF (x) F (lx)

F (1C)F (x) F (1Cx) µ1C,x and D rF (x) F (x)1D F (x)

C −1 idF (x) F (rx )

F (x)F (1C) F (x1C) µx,1C We can weaken this definition slightly. A lax monoidal functor has the same definition as for a monoidal functor, except we do not require and µx,y to be isomorphisms. An oplax monoidal functor has instead a morphism : F (1C) → 1D and a natural transformation µx,y : F (xy) → F (x)F (y), satisfying appropriately modified associativity and unitality conditions. Mostly in this thesis we will discuss monoidal functors (with isomorphisms for the constraints), the exception being §4.4. 8 CHAPTER 2. BACKGROUND

Deﬁnition 2.5. A monoidal functor F is called a monoidal equivalence if it is an equivalence of ordinary categories.

2.1.5 Braided Monoidal Categories

A braided monoidal category can be thought of as a monoidal category where the tensor product is commutative; as always, we mean this only up to isomorphism.

Deﬁnition 2.6. A braided monoidal category is a monoidal category C with a natural isomorphism ∼ βx,y : xy −→ yx called the braiding, such that the following two hexagon diagrams commute for all x, y, z ∈ C:

β x(yz) x,yz (yz)x

ax,y,z ay,z,x

(xy)z y(zx)

βx,y idz idy βx,z (yx)z y(xz) ay,x,z and β (xy)z xy,z z(xy) −1 −1 ax,y,z az,x,y

x(yz) (zx)y

xβy,z βx,zy

x(zy) −1 (xz)y ax,z,y

As a string diagram, the braiding is denoted by crossed strings:

β = β−1 =

The use of the opposite crossing as the inverse braiding is intuitive, as it amounts to the assertion that

= = 2.1. MONOIDAL CATEGORIES 9

We can now describe the hexagon diagrams as string diagrams:

= =

Deﬁnition 2.7. Let C and D be braided categories with braidings βC and βD respectively. A monoidal functor (F, µ) is called braided if the following diagram commutes for all x, y ∈ C:

βD F (x)F (y) F (x),F (y) F (y)F (x)

µx,y µy,x

F (xy) C F (yx) F (βx,y)

Given any monoidal category C, we can construct a braided category from the objects in C which can be braided past any other object.

Deﬁnition 2.8. Let C be a monoidal category. The (Drinfeld) centre of C is the category Z(C) with objects (z, γ), where z ∈ C and

∼ γx : xz −→ zx is a natural isomorphism called a half-braiding, denoted in string diagrams by

γx = , x z such that the following holds for all x, y ∈ C:

= .

x y z x y z

0 0 0 A morphism (z, γ) → (z , γ ) is a morphism f ∈ C(z, z ) such that (f idx) ◦ γx = 0 γx ◦ (idx f) for all x ∈ C, which as string diagrams is

z0 x z0 x

f = . f

x z x z 10 CHAPTER 2. BACKGROUND

It is straightforward to verify that the Drinfeld centre of a monoidal category C is a braided monoidal category with the associativity morphism inherited from C, and braiding given by 0 β(z,γ),(z0,γ0) := γz. For any monoidal category D, there is a forgetful functor R : Z(D) → D which takes (z, γ) to z.

2.1.6 Rigid Monoidal Categories

Let C be a monoidal category, and let x ∈ C.

Deﬁnition 2.9. An object x∗ ∈ C is said to be a left dual of x if there exist ∗ ∗ morphisms evx : x ⊗ x → 1 and coevx : 1 → x ⊗ x , called evaluation and coevaluation respectively, such that the compositions

x coevx ⊗ idx x ⊗ x∗ ⊗ x idx ⊗ evx x

id ∗ ⊗ coev ev ⊗ id ∗ x∗ x x x∗ ⊗ x ⊗ x∗ x x x∗ are the identity morphisms.

Deﬁnition 2.10. An object ∗x ∈ C is said to be a right dual of x if there exist ∗ ∗ morphisms evx : x ⊗ x → 1 and coevx : 1 → x ⊗ x, called evaluation and coevaluation respectively, such that the compositions

x idx ⊗ coevx x ⊗ ∗x ⊗ x evx ⊗ idx x

coevx ⊗ id∗ id∗ ⊗ evx ∗x x x∗ ⊗ x ⊗ x∗ x ∗x are the identity morphisms.

It is clear that if x∗ is a left dual of x, then x is a right dual of x∗, and vice versa. It is fairly simple to show that if x has a left (resp. right) dual, then it is unique up to unique isomorphism. When working with duals, string diagrams are directed, with an upward arrow indicating an object and a downward arrow indicating its dual. The evaluation and coevaluation maps are written as directed cups and caps:

Left duals: ev = coev =

Right duals: ev = coev = 2.2. LINEAR CATEGORIES 11

Then the conditions on compositions amount to the assertion that we can “straighten”:

Left duals: = =

Right duals: = =

Deﬁnition 2.11. A monoidal category C is said to be rigid if every object in C has both left and right duals.

Deﬁnition 2.12. Since left duals are unique up to unique isomorphism, when C is rigid we can deﬁne a contravariant left duality functor (−)∗ : C → C which sends x to x∗ and sends f : x → y to

x∗

f ∗ := f .

y∗

This gives a monoidal functor (−)∗ : Cop → Cmop, where Cmop is the monoidal category with the same underlying category as C, but opposite tensor product.

That is, x ⊗Cmop y = y ⊗C x.

Deﬁnition 2.13. A pivotal category is a rigid category C together with a natural ∼ ∗∗ isomorphism ψ : idC −→ (−) .

2.2 Linear Categories

2.2.1 Additive and Linear Categories

Deﬁnition 2.14. A category C is called additive if • Every set C(x → y) has the structure of an abelian group, and composition is a bi-homomorphism with respect to this structure, • There exists a zero object 0 such that C(0 → 0) = 0, and

• All ﬁnite biproducts exist. That is, for all x1, x2 ∈ C there exists a p ∈ C and

morphisms p1 : p → x1, p2 : p → x2, i1 : x1 → p and i2 : x2 → p such that 12 CHAPTER 2. BACKGROUND

· p1 ◦ i1 = idx1 ,

· p2 ◦ i2 = idx2 ,

· p2 ◦ i1 = 0,

· p1 ◦ i2 = 0, and

· i1 ◦ p1 + i2 ◦ p2 = idp.

Such a biproduct is called a direct sum and denoted by x1 ⊕ x2. It is unique up to unique isomorphism. If C is monoidal, then we also require that ⊗ is a bi-homomorphism with respect to the abelian group structure on the sets C(x → y).

Let k be a field. Definition 2.15. An additive category C is called k-linear if every set C(x → y) has the structure of a vector space over k, and composition is bilinear with respect to this structure. If C is monoidal, then we also require that ⊗ is bilinear with respect to the vector space structure on the sets C(x → y). Definition 2.16. Let F : C → D be a functor between two additive categories. We say F is additive if the associated maps C(x → y) → D(F (x) → F (y)) are homomorphisms of abelian groups. If C and D are linear categories, then we say F is linear if the homomorphisms above are linear. Since the direct sums in an additive category are biproducts, the hom-sets in an additive category are distributive with respect to direct sums. That is, there is a canonical isomorphism C(x ⊕ y → z) −→C∼ (x → z) ⊕ C(y → z), and similarly if the direct sum is the target. It follows that if F : C → D is an additive functor, then there is a natural isomorphism F (x ⊕ y) −→∼ F (x) ⊕ F (y), corresponding to the image of idx⊕y in C(x ⊕ y → x ⊕ y) ∼= C(x ⊕ y → x) ⊕ C(x ⊕ y → y).

Example 2.17. Both Vec and RepG are linear braided monoidal categories, and are rigid with duals given by the usual duals of vector spaces and representations respectively. They also have a canonical pivotal structure coming from the natural isomorphism of vector spaces V −→∼ V ∗∗. 2.2. LINEAR CATEGORIES 13

2.2.2 Additive Completion

A category C is called pre-additive if every set C(x → y) has the structure of an abelian group, and composition is a bi-homomorphism with respect to this structure. Given such a category, we can formally add direct sums in order to construct an additive category Mat(C) such that there is a fully faithful functor C → Mat(C).

Deﬁnition 2.18. The additive completion Mat(C) of a pre-additive category C is the category with Ln • objects given by formal sums i=1 ci (including the empty sum), Ln Lm • for objects c = i=1 ci and d = j=1 dj, a morphism c → d is given by a m × n matrix M, where each entry Mji is a morphism Mji : ci → dj, • composition of morphism is given by “multiplication” of matrices, except using composition in C instead of multiplication of scalars: X (M ◦ N)ki = Mkj ◦ Nji, j • addition of morphisms is given by addition of matrices.

Remark 2.19. It is worth checking that taking the additive completion of a category in which some direct sums exist respects those direct sums. Indeed, if x1 ⊕ x2 exists in a pre-additive category C, then it is isomorphic to the formal sum of x1 and x by the maps 2 " # h i p1 i1 i2 and . p2 Remark 2.20. Let C be a pre-additive category, let D be an additive category, and let F : C → D be a functor which is a homomorphism on hom-groups (this is precisely the deﬁnition of an additive functor, except we do not require C to be additive). Then F extends to an additive functor Mat(C) → D, which takes the formal direct sum x⊕y to F (x)⊕F (y). The analogous result holds for constructing linear functors Mat(C) → D.

Let C and D be linear categories. We have two obvious (though somewhat naive) constructions.

Deﬁnition 2.21. The direct sum C ⊕ D is the additive completion of the category with objects given by pairs (c, d) for c ∈ C and d ∈ D, and hom-spaces

C ⊕ D((c1, d1) → (c2, d2)) := C(c1 → c2) ⊕ D(d1 → d2). 14 CHAPTER 2. BACKGROUND

Deﬁnition 2.22. The naive tensor product C ⊗D is the additive completion of the category with objects given by pairs (c, d) for c ∈ C and d ∈ D, and hom-spaces

C ⊗ D ((c, d) → (c0, d0)) := C(c → c0)D(d → d0).

2.2.3 Idempotent Complete Categories and the Karoubi Envelope

An idempotent in some category C is an endomorphism p : c → c such that p◦p = p. We would often like an idempotent to have something analogous to an image, on which the idempotent is the identity. Such an idempotent is called split.

Deﬁnition 2.23. An idempotent p : c → c is split if there exists r : c → b and s : b → c such that s ◦ r = p and r ◦ s = idb. The object b is denoted im(p). A category is said to be idempotent complete if every idempotent is a split idempotent.

For an arbitrary category C we can construct an idempotent complete category Kar(C) called the Karoubi envelope of C, for which there is a fully faithful functor I : C → Kar(C). This is also called the idempotent completion of C.

Deﬁnition 2.24. The Karoubi envelope Kar(C) of a category C has as objects pairs (c, p) where c ∈ C and p : c → c is an idempotent. The morphisms (c, p) → (c0, p0) are given by morphisms f : c → c0 such that f ◦ p = f = p0 ◦ f. Composition is given simply by the composition of morphisms in C, and the identity morphism on (c, p) is given by p. If C is monoidal, then Kar(C) is also monoidal, with

(c, p) ⊗ (c, p0) := (cc0, pp0).

The fully faithful functor I : C → Kar(C) is given by sending c to (c, idc), which is monoidal if C is monoidal. Thus we can consider C to be a full subcategory of Kar(C).

2.2.4 Semisimple Categories

The following is not the conventional deﬁnition of semisimplicity, which is deﬁned only for abelian categories, though it is equivalent in that case. We discuss this further in Chapter 5.

Deﬁnition 2.25. A k-linear monoidal category C is called (M¨uger)semisimple if: 2.2. LINEAR CATEGORIES 15

• all hom-spaces C(x → y) are ﬁnite-dimensional; •C is idempotent complete;

• there exist objects xi ∈ C indexed by a set I such that  k if i = j C(xi → xj) = 0 if i 6= j and such that for any pair a, b ∈ C, the composition law M C(xi → b) ⊗ C(a → xi) → C(a → b) i∈I

is an isomorphism. The objects xi are called simple objects. A semisimple category C is said to be ﬁnitely semisimple if it has ﬁnitely many simple objects.

We will often also call an object simple if it is isomorphic to an object xi, in which case the definition of finitely semisimple should say that it requires finitely many isomorphism classes of simple objects. Remark 2.26. This definition implies that every object in a semisimple category

C is a direct sum of simple objects. To see this, we note that C(xi → c) and

C(c → xi) are dual vector spaces for every simple xi and c ∈ C; the coevaluation map

C(c → xi)C(xi → c) → k is given by sending f ⊗ g to f ◦ g, noting that C(xi → xi) = k (that is to say, all morphisms xi → xi are scalar multiples of the identity idxi ). P Under the composition law isomorphism, idc corresponds to i∈I ei for ei ∈ C(xi → c)C(c → xi). The evaluation map

k → C(xi → c)C(c → xi) is given by sending idxi = 1 ∈ k to ei. This indeed satisﬁes the straightening relations, and so we see that C(c → xi) and C(xi → c) are dual vector spaces. ∗ Hence if we choose a basis {αi,j} of C(c → xi), we have a dual basis {αi,j} of

C(xi → c). When we consider the meaning of the coevaluation and evaluation maps, this means that  id if i = i and j = j ∗ xi1 1 2 1 2 αi1,j1 ◦ αi2,j2 = 0 otherwise

X ∗ αi,jαi,j = idc . i,j 16 CHAPTER 2. BACKGROUND

If we deﬁne

[c : xi] := dim(C(c → xi)) = dim(C(xi → c)), then this exactly means that  .  .   M ⊕[c:xi] αi,j : c → xi  .  . i∈I is an isomorphism with inverse h i ∗ M ⊕[c:xi] . . . αi,j ... : xi → c. i∈I

We can therefore think of [c : xi] as the number of copies of xi in c. Note that ∼ if c = d, then [c : xi] = [d : xi], and so this decomposition into a direct sum of simple objects is unique.

Remark 2.27. In particular, we can consider a semisimple category to be the additive completion of its full subcategory of simple objects, and so by Remark 2.20 to define a linear functor from a semisimple category it suffices to define it on the full subcategory of simple objects.

Proposition 2.28. A ﬁnitely semisimple category C is equivalent to Vec⊕n, where n is the number of simple objects in C.

⊕n Proof. Observe that Vec is clearly ﬁnitely semisimple, with simple objects ki which are a copy of k in the ith copy of Vec. We can deﬁne a linear functor ⊕n C → Vec which sends a simple object xi to ki. This is essentially surjective since every simple object in Vec⊕n is in the image of the functor. The hom- spaces between simple objects are either k or 0, and so it is fully faithful on simple objects since it must take idxi to idki . Hence it is fully faithful, since all hom-spaces decompose into the direct sum of hom-spaces between simple objects.

Definition 2.29. Let k be an algebraically closed field. A fusion category is a finitely semisimple rigid monoidal k-linear category C in which 1C is a simple object.

Examples 2.30. Let k be an algebraically closed ﬁeld. i) The category Vec is a fusion category a single simple object k. ii) For any ﬁnite group G, RepG is a fusion category with simple objects the irreducible representations. 2.3. (CO)ALGEBRAS AND (CO)MODULES 17 2.3 (Co)Algebras and (Co)Modules

2.3.1 Monoids and Algebras

We can generalise the notion of a monoid to describe a type of object in any monoidal category. Recall that a monoid is deﬁned to be a set with an associative multiplication and a unit for that multiplication. The following is a generalisation of this deﬁnition.

Deﬁnition 2.31. A monoid object (or simply monoid) in a monoidal category C is an object a together with • a morphism µ : aa → a called multiplication, and

• a morphism η : 1C → a called the unit such that the following associativity and unitality axioms hold:

a a a a a

µ µ µ µ = and = = µ µ η η

a a a a a a a a a

Examples 2.32. a) Monoid objects in Set with the Cartesian product are precisely monoids in the usual sense.

b) In Ab, the category of abelian groups with ⊗Z as the tensor product, the monoid objects are rings.

c) The monoid objects in Veck are the (ﬁnite-dimensional) k-algebras. For this reason, monoid objects in k-linear categories are more commonly referred to as algebra objects (or simply algebras).

d) Let G be a ﬁnite group. The object Fun G ∈ RepG of functions G → k (where the G-action is given by g · f(h) = f(gh)) is an algebra in RepG, with multiplication given by pointwise multiplication of functions and unit given by the map k → Fun G which sends c ∈ k to the constant function c(g) = c for all g.

We will develop this last example further over the next sections, as it is used extensively in Chapter 4.

Deﬁnition 2.33. An algebra (A, µ) in a braided category C is called commutative 18 CHAPTER 2. BACKGROUND

if µ = µ ◦ βA,A, where β is the braiding in C. That is, if

A A

µ = µ .

A A A A

2.3.2 Module Objects

Just as we can have modules over rings in the usual sense, we can also have modules over monoid objects in monoidal categories (recalling that a ring is a monoid object in Ab). We can think of a right module over a ring R as an abelian group M with a “right multiplication by R” M ⊗Z R → M which respects multiplication and addition in R. We formalise the generalisation of this in the deﬁnition below. Let C be a monoidal category and let a ∈ C be a monoid. Deﬁnition 2.34. A (right) module over a in C is an object m ∈ C, together with a morphism · : ma → m (called the action of a on m), such that the following associativity and unitality axioms hold: m m m m

· · · = and = µ · η

m a a m a a m m There is a corresponding notion of a left module, where the monoid acts from the left. From here we will simply refer to modules over a monoid, which should be taken to mean right modules unless otherwise speciﬁed. Example 2.35. In Ab, the monoid objects are rings. Correspondingly, the module objects over a ring R are precisely the modules over R.

2.3.3 Coalgebras and Comodules

There is also a notion dual to that of algebras and modules. We will assume C is linear to justify the use of the word coalgebra, though the deﬁnitions in this section do not require that assumption. Let C be a linear monoidal category. Deﬁnition 2.36. A coalgebra in C is an algebra in Cop (the category with the same objects as C but formally reversed morphisms). That is, a coalgebra is an object A ∈ C together with 2.3. (CO)ALGEBRAS AND (CO)MODULES 19

• a morphism ∆ : A → AA called comultiplication,

• a morphism ε : A → 1C called the counit, satisfying appropriate coassociativity and counitality conditions (which are dual to the associativity and unitality axioms for algebras).

A coalgebra (A, ∆) in a braided category C is called commutative if ∆ = βA,A ◦ ∆, where β is the braiding in C.

Let A be a coalgebra in C.

Deﬁnition 2.37. A (right) comodule for A is a right module for A as an algebra in Cop. That is, a right comodule for A is an object c ∈ C together with a morphism α : c → cA (called the coaction of A on c) satisfying appropriate coassociativity and counitality conditions (which are dual to the associativity and unitality axioms for right modules).

Example 2.38. For G a ﬁnite group, Fun G ∈ RepG is a coalgebra with comulti- P plication which sends f to g f(g)δg ⊗ δg, where δg is the indicator function on g. P The counit sends f to g f(g). This is not the usual coalgebra structure given to Fun G, but it gives Fun G the structure of a separable Frobenius algebra, which we will deﬁne and use in Chapter 4. The usual coalgebra structure on Fun G has comultiplication X δg 7→ δgk ⊗ δk−1 , k∈G and counit given by evaluation at the group identity; this gives Fun G the structure of a Hopf algebra (which we do not discuss in this thesis).

2.3.4 Categories of Modules and Comodules

Let C be a linear monoidal category.

Deﬁnition 2.39. Given an algebra A ∈ C, we deﬁne the category of right modules

Cmod-A to be the category with objects (m, p) (where p : mA → m is the action of A on m) which are right modules for A, and morphisms (m1, p1) → (m2, p2) are given by f ∈ C(m1 → m2) such that

f ◦ p1 = p2 ◦ (f idA).

We can similarly deﬁne the category A-modC of left modules for A. Given a coalgebra A ∈ C, we can deﬁne the category of right (respectively, left) comodules Ccomod-A (respectively, A-comodC) analogously. 20 CHAPTER 2. BACKGROUND

Remark 2.40. Suppose (A, µ, η) is an algebra in C. Then A∗ is a coalgebra, with comultiplication given by µ∗ and counit given by η∗ as defined in Definition 2.12. Since the left duality functor is a monoidal functor Cop → Cmop, it induces a monoidal functor ∗ Cmod-A → A -comodC, which sends (m, p) to (m∗, p∗). This in fact defines an equivalence, where the inverse functor is the right duality functor. Example 2.41. Recall that when G is a finite group, Fun G ∈ RepG is an algebra and a coalgebra with

• multiplication µ(f1 ⊗ f2)(g) = f1(g)f2(g), • unit η which sends c ∈ k to the constant function with value c, P • comultiplication ∆(f) = g f(g)δg ⊗ δg where δg is the indicator function on g, and P • counit ε(f) = g f(g). We note that Fun G is self-dual, and that if we dualise the algebra structure on Fun G we get the coalgebra structure on Fun G. Hence we have ∼ Cmod- Fun G = Fun G-comodC. We can also note that Fun G is commutative both as an algebra and as a coalgebra. Deﬁnition 2.42. An central algebra (respectively, coalgebra) A with half-braiding β is the image under the forgetful functor R : Z(C) → C of an algebra (A, β) ∈ Z(C). A central algebra (respectively, coalgebra) is commutative if it is commutative in Z(C). Given a central algebra A ∈ C with half-braiding β and a right A-module (m, p), we can view it as a left module with action

β p Am −→m mA −→ m.

Given a right A-module (m, p) and a left A-module (n, q), the tensor product of m and n over A is deﬁned to be the coequaliser of the diagram

p idn mAn mn m ⊗A n. idm q

When A is a central algebra, the category of right modules Cmod-A is monoidal, with the tensor product given by the tensor product over A, considering the right modules as left modules where needed. Once again, we can make analogous deﬁnitions for the categories of left modules, and of left/right comodules. 2.3. (CO)ALGEBRAS AND (CO)MODULES 21

Proposition 2.43. Let A ∈ C be an algebra in an idempotent complete category

C. Then Cmod-A is idempotent complete.

Proof. We need to exhibit a splitting of an arbitrary idempotent. An idempotent e ∈ Cmod-A((m, p) → (m, p)) is given by a idempotent e ∈ C(m → m) such that e ◦ p = p ◦ (e idA). Since C is idempotent complete, we can take im(e) and deﬁne the right action of A by im(e)

r p s

im(e) A where s : im(e) → m and r : m → im(e) are the maps which exhibit im(e) as a splitting of e. It is then straightforward to check that this deﬁnes a right action on im(e), and so e is a split idempotent.

The analogous result holds for Ccomod-A when A is a coalgebra in an idempotent complete category C.

2.3.5 Free Comodules

Deﬁnition 2.44. Let A ∈ C be an algebra. The category Cfcomod-A of free comodules has the same objects as C, and hom-spaces

Cfcomod-A(x → y) := C(xA → y).

For f ∈ Cfcomod-A(x → y) and g ∈ Cfcomod-A(y → z), their composition is given by

z g z y f g f ◦ = . ∆

y A x A x A

When A is a central algebra, Cfcomod-A is monoidal with tensor product on 22 CHAPTER 2. BACKGROUND objects inherited from the tensor product in C, and on morphisms given by

y1 y2

y1 y2 f1 f2

f1 f2 ⊗ = . ∆

x1 A x2 A x1 x2 A

Deﬁnition 2.45. We will write Cpcomod-A for the idempotent completion

Kar(Cfcomod-A).

We can make analogous deﬁnitions for the categories of left free comodules and left/right modules.

2.4 Monoidal Module Categories

2.4.1 Module Categories

Let C be a monoidal category.

Deﬁnition 2.46. A (right) module category over C is • a category M, • a functor · : M × C → M called the action of C on M, • a natural isomorphism α with components

∼ αm,x,y : m · (xy) −→ (m · x) · y

for all x, y ∈ C and m ∈ M, called the associator, • a natural isomorphism e with components

∼ em : m · 1C −→ m

for all m ∈ M, called the unitor, such that the following two diagrams commute: • the triangle identity:

αm,1,x m · (1Cx) (m · 1) · x

idm ·lx em·idx x · m 2.4. MONOIDAL MODULE CATEGORIES 23

• the pentagon identity: m · ((xy)z)

idm ·ax,y,z αm,xy,z

m · (x(yz)) (m · (xy)) · z

αm,x,yz αm,x,y·idz (m · x) · (yz) ((m · x) · y) · z αm·x,y,z Example 2.47. If C and M are monoidal categories, then a monoidal functor F : C → M gives M the structure of a right module over C where m · c = mF (c), and using the tensorator as the associator. It also gives M the structure of a left module over C where c · m = F (c)m. Example 2.48 (Linear categories as modules over Vec). Given any linear category

C, there is a monoidal linear functor Vec → C determined by sending k to 1C. Hence we have both left and right actions of Vec on C, and so can write V · c and c · V for c ∈ C,V ∈ Vec. Observe that since the functor Vec → C is linear, we have V · c ∼= c⊕ dim V ∼= c · V. Indeed, this is an example of a monoidal module category, as we will see in the next section. We can use the example above to deﬁne both a left and right adjoint to any linear functor between two fusion categories. We can state this in slightly more generality.

Lemma 2.49. Let k be an algebraically closed field, let C be a finitely semisimple category, and let D be a semisimple category. Then for any linear functor F : C → D, F has a right adjoint R defined on simple objects by M R(yj) := xi ·D(F (xi) → yj),

xi∈C simple and a left adjoint L deﬁned on simple objects by M L(yj) := xi ·D(yj → F (xi)).

xi∈C simple Proof. To see that this is a right adjoint, it suﬃces to check for simple objects in

C and D. Let xi ∈ C and yj ∈ D be simple objects. Then we have

C(xi → R(yj)) = C(xi → xi ·D(F (xi) → yj)) ∼ = C(xi → xi) ⊗ D(F (xi) → yj) ∼ = D(F (xi) → yj), 24 CHAPTER 2. BACKGROUND

and these are natural in xi and yj. The proof for the left adjoint is analogous.

2.4.2 Monoidal Module Categories

If C and M are monoidal categories, then a monoidal functor F : C → M gives M the structure of a right module over C where m · c = mF (c), and using the tensorator as the associator. However, if C is braided, then we can define a different right module structure on M, where m · c = F (c)m, and the associator is given by first using the braiding in C and then the tensorator. For these two structures to be equivalent, we need F (c)m and mF (c) to be isomorphic in a way which is compatible with the braiding in C. In other words, we need that F have the structure of a central functor.

Deﬁnition 2.50. A central functor is a functor F : C → D with a factorisation

Z F : C −−→ZF (D) −→DR where F Z is braided and R is the forgetful functor.

Deﬁnition 2.51. Let V be a braided monoidal category. A (right) monoidal module category over V is a monoidal category C with a central functor F : V → C. A left monoidal module category over V is a monoidal category C with a central functor F : Vrev → C. 3 Ladder Categories

In this chapter, all categories and functors are assumed to be linear. For C a rigid monoidal category, M a right C-module category, and N a left C-module category, we build the ladder category M ⊗C N . In the case of monoidal module categories over a braided rigid monoidal category, we describe an explicit construction of a monoidal structure on the ladder category. We then develop some basic results about monoidal ladder categories. Ladder categories can be considered to be a special case of the tensor product of n-categories construction as described in section 6 of [MW12]. An explicit definition in the case where the modules are over a fusion category is given in [BBJ19]. In this thesis we generalise this definition to simply require the category be rigid (and linear), and define a monoidal structure on the ladder category in the case of monoidal modules over a braided rigid monoidal category.

3.1 Ladder Categories

3.1.1 Deﬁnition of Ladder Categories

Let C be a rigid monoidal category, let M be a right C-module category, and let N be a left C-module category.

Deﬁnition 3.1. The ladder category M ⊗C N is the additive completion of the category with objects given by pairs (m, n) for m ∈ M and n ∈ N , and

!, 0 0 M 0 0 M ⊗C N ((m, n) → (m , n )) := M(m · c → m )N (n → c · n ) K, c∈C where K is a subspace we will describe shortly. For f ∈ M(m · c → m0) and g ∈ N (n, c · n0), we will write (f, c, g) for their tensor product in M(m · c → m0)N (n, c · n0), and call such a morphism pure (so that an arbitrary morphism is a sum of pure morphisms).

25 26 CHAPTER 3. LADDER CATEGORIES

We can depict a pure morphism (f, c, g):(m, n) → (m0, n0) as

m0 n0

f c , g

m n where we consider the left side of the ladder to be in M and the right side of the ladder to be in N . For morphisms, we require

m0 n0 m0 n0

f h f = h , g g

m n m n by letting K be the span of morphisms of the form

0 (f ◦ (idm ·h), c, g) − (f, c , (h · idn0 ) ◦ g) for all h ∈ C(c → c0), f ∈ M(m · c0 → m0), and g ∈ N (n → c · n0). Hence we can treat a morphism h ∈ C(c → c0) as not belonging to either side of the ladder, but rather to the middle:

m0 n0

f h . g

m n The reader can choose to interpret this diagram as having h associate to the left or to the right as preferred, and this requirement ensures that both interpretations are equal.

The identity on (m, n) is given by (idm, id1C , idn), which in string diagrams is m n

m n Composition of pure morphisms is given by

0 0 0 0 0 0 (f , c , g ) ◦ (f, c, g) = (f ◦ (f · idc0 ), cc , (idc ·g ) ◦ g), 3.1. LADDER CATEGORIES 27 which in string diagrams is simply vertical concatenation:

m00 n00 m0 n0 m00 n00

0 0 0 f c f c0 f c f . ◦ = 0 0 g g g c g

m0 n0 m n m n ∼ Remark 3.2. We can observe that (m1 ⊕ m2, n) = (m1, n) ⊕ (m2, n) (where this second direct sum is the formal sum introduced by taking the additive completion) by the maps " # h i (p1, 1C, idn) (i1, 1C, idn)(i2, 1C, idn) and , (p2, 1C, idn) where i1, i2, p1, p2 are the inclusion and projection maps as described in Deﬁnition ∼ 2.14. Similarly we have (m, n1 ⊕ n2) = (m, n1) ⊕ (m, n2). Proposition 3.3. Suppose f ∈ M(m · c → m0) and g ∈ N (n → c · n0) are 0 0 isomorphisms. Then (f, c, g):(m, n) → (m , n ) is an isomorphism in M ⊗C N , with inverse m n

f −1 (f,˜ c∗, g˜) := c , g−1

m0 n0 where we are using the existence of a right dual for c (as C is rigid). Proof. The composition (f,˜ c∗, g˜) ◦ (f, c, g) is given by m n m n m n m n

f −1 f c c g−1 = g−1 = g−1 = . c c g g g m n m n m n m n −1 The ﬁrst equality is simply that f ◦f = idm·c. Next, we use one of the “straighten-

= −1 ing” identities for right duals: . The ﬁnal equality is simply that g ◦g = idn. ˜ ∗ Hence (f, c , g˜) ◦ (f, c, g) = id(m,n). ˜ ∗ The proof that (f, c, g) ◦ (f, c , g˜) = id(m0,n0) is analogous. Corollary 3.4. There is a canonical isomorphism (m, c · n) −→∼ (m · c, n) for all m ∈ M, n ∈ N , and c ∈ C.

Proof. Take (idm·c, c, idc·n), which is an isomorphism by Proposition 3.3. 28 CHAPTER 3. LADDER CATEGORIES

3.1.2 Ladder Categories Over Semisimple Categories

Let V be a semisimple rigid category with simple objects {xi}i∈I , let M be a right C-module category, and let N be a left C-module category. 0 0 For any pure morphism (f, c, g): M ⊗C N ((m, n) → (m , n )), we have from Remark 2.26 X ∗ idc = αi,j ◦ αi,j, i,j ∗ where {αi,j}j are a basis for C(c → xi) and {αi,j}j are a dual basis for C(xi → c) under the evaluation and coevaluation maps deﬁned in Remark 2.26. Hence we can write

m0 n0 m0 n0

f c X f xk ∈ L M(m · x → m0)N (n, x · n0). = α∗ k k g i,j g k∈I i,j αi,j m n m n

Therefore in the deﬁnition of morphisms in M ⊗C N , taking the direct sum over simple objects is the same as taking the direct sum over all objects (after taking respective quotients), giving !, 0 0 M 0 0 ˜ M ⊗C N ((m, n) → (m , n )) = M(m · xk → m )N (n, xk · n ) K. k∈I

Here K˜ is the restriction of K to the direct sum over just simple objects, since all of the relations can also be split into maps with simple objects in the middle, and this decomposition gives morphisms in K˜ .

3.2 Monoidal Ladder Categories

3.2.1 Deﬁnition of Monoidal Ladder Categories

Let V be a braided rigid monoidal category, let F : V → C be the structure of a right monoidal V-module C, and let G : Vrev → D be the structure of a left monoidal V-module D. Recall that F is by definition a central functor, and so comes equipped with a braided monoidal functor F Z : V → Z(C). For any v ∈ V, we define σv to be the half-braiding given by F Z (v) = (F (v), σv). We do likewise for the central functor G, defining γv by GZ (v) = (G(v), γv). 3.2. MONOIDAL LADDER CATEGORIES 29

We will often write v rather than F (v) as an object in C, and likewise will write v ∈ D rather than G(v).

Proposition 3.5. The ladder category C ⊗V D has the structure of a monoidal category, with tensor product on objects given by (c, d) ⊗ (m, n) = (cm, dn), and on pure morphisms given by

c0 d0 m0 n0 c0 m0 d0 n0

f v h w w ⊗ = , g i v

c d m n c m d n where the crossings are deﬁned as

v w := σm := γd0 . m v d0 w

This is to say that

v w (f, v, g) ⊗ (h, w, i) := (fh ◦ idc σm idw, vw, idv γd0 idn0 ◦gi).

Proof. We need to show that the tensor product is functorial, which is to say that

((f 0, v0, g0) ◦ (f, v, g)) ⊗ ((h0, w0, i0) ◦ (h, w, i)) = ((f 0, v0, g0) ⊗ (h0, w0, i0)) ◦ ((f, v, g) ⊗ (h, w, i)).

Evaluating the left side, we obtain

c00 d00 m00 n00

f 0 v0 h0 w0 ⊗ c00 d00 m00 n00 g0 i0 f 0 h0 c0 d0 m0 n0 f h v0 w0 ◦ ◦ = ⊗ v w c0 d0 m0 n0 g0 i0 g i f v h w ⊗ c d m n g i

c d m n 30 CHAPTER 3. LADDER CATEGORIES

c00 m00 d00 n00

= .

c m d n

Evaluating the right side, we obtain

c00 d00 m00 n00 c00 m00 d00 n00

f 0 v0 h0 w0 w0 ⊗ g0 i0 v0

c0 d0 m0 n0 c0 m0 d0 n0

◦ ◦ = ◦

c0 d0 m0 n0 c0 m0 d0 n0

f v h w w ⊗ g i v

c d m n c m d n c00 m00 d00 n00

= .

c m d n

To show that these morphisms are equal, we need to show that we can swap the middle two rungs of the ladder:

c00 m0 d00 n0 c00 m0 d00 n0

↔ .

c0 m d0 n c0 m d0 n 3.2. MONOIDAL LADDER CATEGORIES 31

Indeed, we have

c00 m0 d00 n0 c00 m0 d00 n0 c00 m0 d00 n0 c00 m0 d00 n0

0 f h

w = = = . v0

0 g i c0 m d0 n c0 m d0 n c0 m d0 n c0 m d0 n

−1 The ﬁrst equality is a substitution idv0 idw = βw,v0 ◦ βw,v0 , where β is the braiding in V. Next, we consider v acting on C, and by the naturality of the half-braiding σv on F (v) we can swap the order of the braiding and h, which gives the second equality. Likewise, the third equality comes from considering the half-braiding γw0 on G(w0).

The unit object is given by (1C, 1D), while the associator

α(c1,d1),(c2,d2),(c3,d3) : ((c1, d1) ⊗ (c2, d2)) ⊗ (c3, d3) → (c1, d1) ⊗ ((c2, d2) ⊗ (c3, d3)) is given by taking the associator on both sides:

C D α(c1,d1),(c2,d2),(c3,d3) := αc1,c2,c3 , 1V , αd1,d2,d3 .

The left and right unitors are similarly given by taking the unitors on both sides.

Since all of these natural isomorphisms have 1V in the middle, the two sides of the ladder are not interacting. Hence the triangle and pentagon identities are immediate from the commutativity of the corresponding diagrams in both C and D.

3.2.2 Ladder Categories over Vec

For any linear monoidal category C, we have a linear functor Vec → Z(C) which determined by sending k to (1C, id1C ), and hence C is a monoidal Vec-module (both rev left and right, as Vec is braided equivalent to Vec ). Hence for any two k-linear monoidal categories C and D we can form C ⊗Vec D. Furthermore, Vec is semisimple with only simple object k, and so by § 3.1.2 we have that C ⊗Vec D has hom-objects

0 0 0 0 C ⊗Vec D((c, d) → (c , d )) = (C(c → c )D(d → d ))/ K, 32 CHAPTER 3. LADDER CATEGORIES where K is the subspace generated by

(f ◦ (idc ·h), k, g) − (f, k, (h · idd0 ) ◦ g) for h : k → k. However, h is simply a scaling, and so this is simply

(λf, k, g) − (f, k, λg) for some λ ∈ k, which is 0 in C(c → c0)D(d → d0). Therefore

0 0 0 0 C ⊗Vec D((c, d) → (c , d )) = C(c → c )D(d → d ), and so in fact C ⊗Vec D = C ⊗ D as described in section 2.2.2.

3.3 Properties of Ladder Categories

Let V be a braided rigid monoidal category, let (C,F : V → C) be a right monoidal V-module and let (D,G : V → D) be a left monoidal V-module. There is a canonical way to make V into a right monoidal V-module, with

V → Z(V)

v 7→ (v, β−,v) where β is the braiding on V. Alternatively, we can make V into a left monoidal V-module by

Vrev → Z(V)

v 7→ (v, βv,−)

Proposition 3.6. There is a monoidal equivalence between C and the monoidal ladder category C ⊗V V. Similarly, there is a monoidal equivalence between D and the monoidal ladder category V ⊗V D.

Proof. Deﬁne functors

L : C ⊗V V → C (c, v) 7→ cF (v)

c0 v0 c0 F (v0)

f w f F (w) 7→ (f idF (v0)) ◦ (idc F (g)) = g F (g)

c v c F (v) 3.3. PROPERTIES OF LADDER CATEGORIES 33 and

R : C → C ⊗V V

c 7→ (c, 1V )

0 c 1V f 7→ f

c 1V Note that in the definition of L we have suppressed the tensorator of F and written F (g): F (v) → F (w)F (v0). It is immediate that L ◦ R is naturally isomorphic to the identity functor on C, simply by using the unitor of F and the right unitor in C. For the other natural ∼ isomorphism, we need an isomorphism (c, v) −→ (cF (v), 1V ) which is natural in c and v. By Proposition 3.3, it suffices to find two isomorphisms in C(c · v → cF (v)) and V(v → v · 1V ), for which we can take idcF (v) and the inverse of the right unitor in V respectively. The naturality of this isomorphism follows immediately from the naturality of the right unitor in V. To see that this equivalence is monoidal, it suffices to show that R is a monoidal functor. However, we have

R(vw) = (vw, 1) R(v)R(w) = (v, 1)(w, 1) = (vw, 1), and hence the identity on C ⊗V V suﬃces as a tensorator, and trivially satisﬁes all of the required properties.

The proof in the case of V ⊗V D is analogous.

Proposition 3.7. If C and D are rigid, then C ⊗V D is rigid.

∗ ∗ Proof. The left dual of (c, d) ∈ C ⊗V D is given by (c , d ), with

ev(c,d) := (evc, 1V , evd)

coev(c,d) := (coevc, 1V , coevd).

The required “straightening” conditions follow immediately from the straightening conditions in both C and D. The deﬁnition of the right dual is analogous.

Deﬁnition 3.8. Let V and W be braided monoidal categories. A monoidal (V − W)-bimodule is a monoidal module category over Vrev ⊗ W. 34 CHAPTER 3. LADDER CATEGORIES

rev There is a canonical functor W → V ⊗ W which takes w to (1V , w). If C is a monoidal (V − W)-bimodule with central functor BZ :(Vrev ⊗ W) → Z(C), then composing these two functors gives C the structure of a monoidal W-module, as we would expect. Similarly, C is also a left monoidal V-module. Proposition 3.9. Let V, W, and U be braided rigid categories, let C be a (V −W)- bimodule, and let D be a monoidal (W −U)-bimodule. Then C ⊗W D is a monoidal (V − U)-bimodule. Proof. Let F : Vrev → C be the central functor inherited from the (V − W)- bimodule structure on C, and let G : U → D be the central functor inherited from the (W − U)-bimodule structure on C. Then there is a central functor

Z rev B :(V ⊗ U) → Z(C ⊗W D) v u (v, u) 7→ ((F (v),G(u)), (σ , 1V , γ ))

Z v Z u v u where F (v) = (v, σ ) and G (u) = (u, γ ). Note that (σ , 1V , γ ) is immediately a half-braiding on (F (v),G(u)) since σv and γu are half-braidings on v and u respectively. Proposition 3.10. Let V and W be braided rigid categories, let A be a right monoidal V-module, let B be a monoidal (V − W)-bimodule, and let C be a left monoidal W-module. Then there is a canonical monoidal equivalence

(A ⊗V B) ⊗W C ' A ⊗V (B ⊗W C) . Proof. We deﬁne a functor F which on objects is

F :(A ⊗V B) ⊗W C → A ⊗V (B ⊗W C) ((a, b), c) 7→ (a, (b, c)).

Every morphism ((a, b), c) → ((a0, b0), c0) is a sum of morphisms of the form

a0 b0 c0

f v w , g h

a b c and we send this morphism to

a0 b0 c0

f v w . g h

a b c 3.3. PROPERTIES OF LADDER CATEGORIES 35

There is an obvious functor in the opposite direction (which does precisely the reverse of this functor), and these two functors form an equivalence (using the identities as the required natural isomorphisms). Further, we have that

F (((a, b), c) ⊗ ((a0, b0), c0)) = F ((aa0, bb0), cc0) = (aa0, (bb0, cc0)) = F ((a, b), c) ⊗ F ((a0, b0), c0).

Hence F is monoidal (with the identity as the tensorator) and so this is a monoidal equivalence.

Let V be a braided subcategory of W. Then a monoidal W-module inherits the structure of a monoidal V-module by restricting the central functor. Proposition 3.11. Let V be a full braided subcategory of W, and let C and D be

(right and left respectively) monoidal W-modules. Then C ⊗V D is a subcategory of C ⊗W D, where C and D have the inherited monoidal V-module structures.

Proof. Both C ⊗V D and C ⊗W D have the same objects, given by ﬁnite sums L i(ci, di) of objects ci ∈ C and di ∈ D. Recall that !, 0 0 M 0 0 ˜ C ⊗V D((c, d) → (c , d )) := C(c · v → c )D(d → v · d ) K v∈V !, 0 0 M 0 0 C ⊗W D((c, d) → (c , d )) := C(c · w → c )D(d → w · d ) K w∈W The direct sum over V is a subspace of the direct sum over W, so it remains to show that K˜ is the intersection of K with the direct sum over W. Recall that K is the span of morphisms of the form

m0 n0 m0 n0

0 0 f w f w w − w h g h g

m n m n for every h ∈ W(w → w0). Such a morphism is in the direct sum over V if and only if w, w0 ∈ V, and K˜ is precisely the span of morphisms of the above form where w, w0 ∈ V. Therefore K˜ is the intersection of K with the direct sum over V. 0 0 0 0 Hence C ⊗V D((c, d) → (c , d )) is a subspace of C ⊗W D((c, d) → (c , d )), and so

C ⊗V D is a subcategory of C ⊗W D.

4 De-equivariantisation

Given a monoidal category C with an action of G, we can equivariantise by taking G G the category of equivariant objects C , and obtain a central functor RepG → Z(C ) (in the case of Vec with the trivial action, we obtain precisely RepG). Conversely, if D is a monoidal category with such a central functor, we can de-equivariantise to obtain a category DG with an action of G. In this chapter we ﬁrst show that the de-equivariantisation of a category D is precisely Kar(D ⊗RepG Vec). We then describe the “de-equivariantisation by de- enrichment” given without proof in [MP19], and show that this is equivalent to the standard deﬁnition of de-equivariantisation.

4.1 Deﬁnitions of de-equivariantisation

4.1.1 Equivariantisation

For a monoidal category C, we will write Aut⊗(C) for the monoidal category of monoidal autoequivalences of C with morphisms given by isomorphisms of functors, which has composition as the tensor product. For a group G, we will write G⊗ for the monoidal category with elements of the group as objects, only identity morphisms, and group multiplication as the tensor product.

Deﬁnition 4.1. An action of G on C is a monoidal functor

⊗ T : G → Aut⊗(C).

Given an action T , we will often write either Tg rather than T (g).

⊗ Deﬁnition 4.2. Let T : G → Aut⊗(C) be an action of G on C. The G- equivariantisation CG of C has ∼ • objects given by pairs (x, {ug : Tgx −→ x}g∈G), where x ∈ C and each ug is

37 38 CHAPTER 4. DE-EQUIVARIANTISATION

an isomorphism such that the diagram

Tg(uh) Tg(Th(x)) Tg(x)

µg,h ug T x gh ugh

commutes for all g, h ∈ G, where µ is the tensorator for T ,

• morphisms (x, {ug}) → (y, {vg}) given by maps f ∈ C(x → y) such that

Tg(f) Tg(x) Tg(y)

ug vg x y f

commutes for all g ∈ G. The equivariantisation CG inherits a monoidal structure from the monoidal structure on C, and the forgetful functor CG → C is monoidal. The objects of CG are called equivariant objects of C.

Example 4.3. Every group G has an action on Vec, namely the trivial action. G G Under that action, we have Vec = RepG. Indeed, an object in Vec is a vector space with a collection of linear automorphisms ug satisfying precisely the conditions required for g 7→ ug to be a representation, and the morphisms between two objects are precisely the intertwiners between representations.

4.1.2 De-equivariantisation

For the remainder of this chapter we will assume G is a ﬁnite group, and that k is an algebraically closed ﬁeld of characteristic 0. For any linear monoidal category C there is a canonical fully faithful linear ⊗ monoidal functor Vec → C which sends k to 1C. If C has a group action T : G → Aut⊗(C), this extends to a fully faithful monoidal functor

G G F : RepG = Vec → C .

Proposition 4.4. The embedding F canonically has the structure of a central functor.

Proof. Recall from Example 2.48 that for every linear category C, Vec is both a left and right C-module category via a functor which sends k to 1C, and then c · V = cF (V ) for c ∈ C and V ∈ Vec. 4.2. FROBENIUS ALGEBRAS 39

For any representation ρ : G → GL(V ), its image under F is given by (V ·

1C, {ug}) where

ρg·id1C ug : Tg(V · 1C) = V · 1C −−−−→ V · 1C. The unitors in C give a natural isomorphism

⊕ dim V −1 ⊕ dim V lx (rx ) γx :(V · 1C) ⊗ x −−−−→ x −−−−−−−→ x ⊗ (V · 1C), which is indeed a morphism in CG. Hence we obtain a braided monoidal functor

Z G F : RepG → Z(C )

(V, ρ) 7→ ((V, {ug}), γ). such that F is the composition of F Z with the forgetful functor Z(CG) → CG.

Let C be a fusion category and let F : RepG → C be a fully faithful central functor. Let A be the image under this functor of the algebra Fun(G) of functions G → k, as described in Example 2.41.

Deﬁnition 4.5. The de-equivariantisation of C is given by the category of right modules over A:

CG := Cmod-A.

Proposition 4.6. The de-equivariantisation CG is a fusion category, and there is G an action of G on CG such that C is monoidally equivalent to (CG) .

For more details on this definition and a proof of this equivalence, see Section 8.23 of [EGNO15]. We will make the following definition of the ladder de-equivariantisation, and will later justify this terminology by showing that it is equivalent to the standard definition of de-equivariantisation.

Deﬁnition 4.7. The ladder de-equivariantisation is given by Kar(C ⊗RepG Vec).

4.2 Frobenius Algebras

The goal of this chapter is to build an equivalence between Kar(C ⊗RepG Vec) and the standard deﬁnition of de-equivariantisation Cmod-A, where A = F (Fun G). It will prove more useful, given the deﬁnition of the ladder category, to deal with comodules rather than modules. We have the following result: 40 CHAPTER 4. DE-EQUIVARIANTISATION

Lemma 4.8. There is a monoidal equivalence

Cmod-A ' Ccomod-A, where A = F (Fun G) with the algebra and coalgebra structure as described in §2.3. Proof. We know that Fun G is self-dual, and that monoidal functors preserve duals. Therefore by 2.40 we have that

Cmod-A ' A-comodC. However, we also have that F is a central functor, so we have (A, β) ∈ Z(C) where ∼ βx : xA −→ Ax is a half-braiding. Let (c, α) be a right A-comodule. Then we can define a left A-comodule structure on c with coaction βc ◦ α. Similarly if (d, γ) is a left A-comodule we can define a right A-comodule structure on d with coaction −1 βd ◦ γ. It is then fairly simple to check that these define a monoidal equivalence

A-comodC ' Ccomod-A.

We will begin by showing that C ⊗RepG Vec ' Cfcomod-F (Fun G), and deduce from this

Kar(C ⊗RepG Vec) ' Kar(Cfcomod-F (Fun G)) = Cpcomod-F (Fun G).

So to show that Kar(C ⊗RepG Vec) is equivalent to the standard deﬁnition of de-equivariantisation, we need

Cpcomod-F (Fun G) ' Ccomod-F (Fun G). This is indeed the case, as F (Fun G) is a separable Frobenius algebra in C, as we will see in Example 4.11.

4.2.1 Deﬁnition of Frobenius Algebras

Deﬁnition 4.9. A Frobenius algebra in a monoidal category C is an object A with

• a multiplication µ : AA → A and unit η : 1C → A which make A into an algebra, and

• a comultiplication ∆ : A → AA and counit ε : A → 1C which make A into a coalgebra, such that the following Frobenius compatibility condition holds:

µ µ ∆ = = . ∆ µ ∆

A A A A A A 4.2. FROBENIUS ALGEBRAS 41

Given a Frobenius algebra A, we also have maps

ε µ := ∆ and := η which satisfy the standard straightening relations (as for duals)

= = = and = = = .

The ﬁrst equality in each diagram is by deﬁnition of the maps, the second by Frobenius compatibility, and the third by the axioms of algebras/coalgebras.

Deﬁnition 4.10. A Frobenius algebra is called separable if

µ = . ∆

Given a separable Frobenius algebra A, we have

======.

The first and last of these equalities are simply the definitions. The second equality is by Frobenius compatibility, the third is a coalgebra axiom, and the fourth is again by Frobenius compatibility. The fifth is the definition of a separable Frobenius algebra. Similarly, we have

= .

Example 4.11. Recall from §2.3 that when G is a ﬁnite group, Fun G ∈ RepG is an algebra and a coalgebra with

P • counit ε(f) = g f(g), and is commutative as both an algebra and a coalgebra. It is then straightforward to verify that Fun G is a separable Frobenius algebra with this structure. When F : RepG → C is a monoidal functor, it preserves algebra and coalgebra structure as well as identities, and so F (Fun G) is a separable Frobenius algebra in C.

4.2.2 Equivalence Between Cpcomod-A and Ccomod-A Proposition 4.12. Let A be a commutative central separable Frobenius algebra in an idempotent complete monoidal category C. Then there is a monoidal equivalence between Cpcomod-A and Ccomod-A.

Proof. Recall that Cpcomod-A is the idempotent completion Kar(Cfcomod-A) of the category of free right A-comodules. Given

f ∈ Cfcomod-A(x → y) = C(xA → y)

g ∈ Cfcomod-A(y → z) = C(yA → z) their composition g ◦ f in Cfcomod-A is given by y z z g f g ◦ = f ∆ x A y A y A For any idempotent

p ∈ Cfcomod-A(x → x) = C(xA → x), the map p∆ ∈ Ccomod-A(xA → xA) given by

x A

p A p∆ = ∆

x A is an idempotent. Indeed, we have

x A x A x A p p ∆ p p , p = = ∆ ∆ ∆ ∆ x A x A x A 4.2. FROBENIUS ALGEBRAS 43 where the ﬁrst equality comes from the associativity of ∆ and the second equality is because p is an idempotent in Cfcomod-A.

We know Ccomod-A is idempotent complete by Proposition 2.43, and so for every idempotent p : x → x there exists a splitting of p. That is to say, there exists and object im(p) and morphisms rp : x → im(p) and sp : im(p) → x such that sp ◦ rp = p and rp ◦ sp = idim(p).

We can deﬁne a functor T : Cpcomod-A → Ccomod-A, which takes an object

(x, p : x → x) to im(p∆). A morphism f :(x, p) → (y, q), which is a morphism f ∈ C(xA → y) such that

y y y q f = f = , f p ∆ ∆ x A x A x A

is sent to rq∆ ◦ f∆ ◦ sp∆ . Note that for f :(x, p) → (y, q) and g :(y, q) → (z, t), we have

T (g) ◦ T (f) = rt∆ ◦ g∆ ◦ sq∆ ◦ rq∆ ◦ f∆ ◦ sp∆

= rt∆ ◦ g∆ ◦ q∆ ◦ f∆ ◦ sp∆ , and

g g g q q ∆ g ◦ f f f g∆ ◦ q∆ ◦ f∆ = ∆ = = = = (g ◦ f)∆. f ∆ ∆ ∆ ∆ ∆ ∆ ∆

Hence T (g) ◦ T (f) = T (g ◦ f), and so T is indeed functorial. To see that T is an equivalence, we must show that T is essentially surjective and fully faithful. For the proof that T is essentially surjective, let (x, α) ∈ Ccomod-A. Deﬁne x p := . α x A

Then p : xA → x is an idempotent in Cfcomod-A(x → x), since x x x

α = = . ∆ ∆ α ∆ α α x A x A x A 44 CHAPTER 4. DE-EQUIVARIANTISATION

Here the ﬁrst equality is the associativity axiom for a coaction on a comodule, and the second equality follows because A is a separable Frobenius algebra.

Hence (x, p) ∈ Cpcomod-A(x → x). We claim that T (x, p) = im(p∆) is isomorphic to x via the maps

p ◦ sp∆ : im(p∆) → x

rp∆ ◦ α : x → im(p∆).

For one composition, we obtain

p ◦ sp∆ ◦ rp∆ ◦ α = p ◦ p ◦ α = p ◦ α, and we have

p ◦ α = = = = = , α α α α α α where the ﬁrst two equalities are deﬁnitional, the third equality is the associativity axiom for a coaction on a comodule, the fourth is because A is a separable Frobenius algebra, and the last is the unitality axiom for a coaction. For the other composition, we obtain

rp∆ ◦ α ◦ p ◦ sp∆ , and observe that

x A x A x A x A x A x A

α α ◦ p ======p∆. α α α α α α x A x A x A x A x A x A Here the second equality uses the associativity property of the action α and the definition of the cap map. The third equality is by Frobenius compatibility, the fourth and fifth are by the unitality of the coalgebra structure on A, the sixth is once again by Frobenius compatibility, and the last is the definition. So this composition is

rp∆ ◦ p∆ ◦ sp∆ = rp∆ ◦ sp∆ ◦ rp∆ ◦ sp∆ = idp∆ .

Hence the arbitrary (x, α) ∈ Ccomod-A is isomorphic to T (x, p), and so T is essentially surjective. 4.2. FROBENIUS ALGEBRAS 45

To see that T is full, let f : T (x, p) → T (y, q). This is precisely the statement that the map ˜ f := rq∆ ◦ f ◦ sp∆ : xA → yA satisﬁes the properties that ˜ ˜ ˜ f ◦ p∆ = f = q∆ ◦ f,

∆ and f˜ = f˜ . ∆

0 Then we claim the map f ∈ Cpcomod-A((x, p) → (y, q)) deﬁned by y

f 0 := f˜

x A maps to f. First we need to show that f 0 ◦ p = f 0 = q ◦ f 0. We have

f˜ 0 ˜ 0 f ◦ p = p = f = f ,

∆

˜ ˜ where this equality is the statement that f ◦ p∆ = f. Similarly

q q ∆ q ◦ f 0 = f˜ = f˜ = f˜ = f 0,

∆ where the first and last equalities are definitions, the second equality is the property that ∆ can be ‘moved through’ f˜, and the third equality is the property that ˜ ˜ q∆ ◦ f = f. 0 0 So we have that indeed f ∈ Cpcomod-A((x, p) → (y, q)). To see that T (f ) = f, 0 ˜ it suffices to prove that f∆ = f. We have

∆ 0 ˜ ˜ ˜ f∆ = f = f = f = f.˜ ∆

Here the ﬁrst and last equalities are deﬁnitions, the second equality is the property that ∆ can be ‘moved through’ f˜, and the third equality is the identity axiom of coalgebras. Hence our arbitrary f is the image of f 0, and so T is full. 46 CHAPTER 4. DE-EQUIVARIANTISATION

To see that T is faithful, suppose that f∆ = g∆ for f, g ∈ Cpcomod-A(x → y). Then we have

f = g , ∆ ∆ and therefore f = g by the identity axiom of coalgebras. To see that T is monoidal, we ﬁrst note that

x y A

p q T ((x, p) ⊗ (y, q)) = im ∆

x y A

For the tensor product in Ccomod-A, recall that the tensor product of right comodules (x, ν) and (y, χ) is given by the equaliser of

ν⊗idy x ⊗A y xy xAy, idx ⊗(βy◦χ) where βy : yA → Ay is the half-braiding on A. When A is Frobenius, we can realise this equaliser as the image of the idempotent

ν χ x y .

Hence

T (x, p) ⊗ T (y, q) = im . p q

x A y A We deﬁne the tensorator T (x, p) ⊗ T (y, q) → T ((x, p) ⊗ (y, q)) to be the map between images of idempotents induced by the xAyA → xyA deﬁned as

p q , p

x A y A 4.3. LADDERS AND DE-EQUIVARIANTISATION 47 which has inverse p

p q .

x y A By similar calculations to those above, one can verify that this indeed gives a tensorator for T .

4.3 Ladders and de-equivariantisation

By Proposition 4.12, and we see that

Cpcomod-F (Fun G) ' Ccomod-F (Fun G), and the category on the right is precisely the de-equivariantisation of C. Therefore, in order to show that Kar(C ⊗RepG Vec) is equivalent to the standard deﬁnition of de-equivariantisation, it suﬃces to show that

C ⊗RepG Vec ' Cfcomod-F (Fun G).

The following well-known result will be useful when manipulating the separable Frobenius algebra F (Fun G).

Lemma 4.13. There exists a decomposition as representations M Fun G ∼= W · W ∗, W ∈Irrep G where the sum is over irreducible representations of G, and W ∗ is the dual of the underlying vector space of W .

A proof of this lemma can be found as Proposition 10 in [Ser77].

Theorem 4.14. The ladder de-equivariantisation is monoidally equivalent to the de-equivariantisation.

Proof. As described above, it suﬃces to exhibit a monoidal equivalence

C ⊗RepG Vec ' Cfcomod-A, where A = F (Fun G). 48 CHAPTER 4. DE-EQUIVARIANTISATION

Since the ladder construction respects direct sums, we can write an object

(x, V ) ∈ C ⊗RepG Vec as a direct sum of objects (x, k). Hence we can consider C ⊗RepG Vec to be the additive completion of the full subcategory of objects of the form (x, k). Hence to define a linear functor from C ⊗RepG Vec it suffices to define a functor from this full subcategory, by Remark 2.20.

The hom-spaces between such objects in C ⊗RepG Vec are (formal sums of) M C ⊗RepG Vec((x, k) → (y, k)) = C(x · W → y)Vec(k → W · k), W ∈Irrep G where the sum is over the irreducible representations of G (the simple objects in

RepG). Morphisms in C(x · W → y)Vec(k → W · k) are sums of pure morphisms

y k

f W . g

x k

We can canonically consider a map g : k → W · k to be a linear map k → W ; any such map is precisely determined by choosing a vector w ∈ W to be the image of 1 ∈ k, and we will abuse notation slightly to write w : k → W for this map.

Recall that Cfcomod-A has the same objects as C, and hom-spaces

Cfcomod-A(x → y) := C(xA → y).

Note that for any representation W , a map of representations ρ : Fun G → W is determined by where it sends δe, as it must therefore send δg to g · ρ(δe). This gives an isomorphism W ∼= RepG(Fun G → W ).

Hence we can consider w ∈ W to be a map of representations w : Fun G → W , which sends δg to g · w.

We can now construct a linear functor T : C ⊗RepG Vec → Cfcomod-A, which on objects sends (x, k) to x, and on pure morphisms (f, W, w) ∈ C ⊗RepG Vec(x → y) is given by y k y

f W f 7→ , w F (w)

x k x A 4.3. LADDERS AND DE-EQUIVARIANTISATION 49 where F (w): A = F (Fun G) → F (W ), and so the morphism on the right is indeed a morphism in C(xA → y).

We now check that T is functorial. Composition in C ⊗RepG Vec is given by

z k y k z k f W f W f W 2 2 2 2 1 1 f ◦ = 1 . w2 w2 w1 W1 w1

y k x k x k

In order to understand where this is sent by T , we need to decompose this into morphisms which have an irreducible representation in the middle of the ladder.

Since RepG is semisimple, we can decompose idW1W2 (as described in Remark 2.26) to obtain

W1 W2 W1 W2

∗ αW,i = P P W , W ∈Irrep G i αW,i

W1 W2 W1 W2

∗ where {αW,i} is a basis for RepG(W1W2 → W ), and {αW,i} is the dual basis with respect to the evaluation and coevaluation maps described in Remark 2.26. Therefore we have

z k z y z k k k f 2 ∗ f W f αW,i f W f W 2 2 1 2 2 1 1 f P P ◦ = 1 = W . w2 W ∈Irrep G i w2 w1 W1 w1 αW,i w2 w1 y k x k x k x k

We will write

z W k f 2 ∗ w F (α ) wW,i = αW,i 2 , fW,i = f1 W,i and w1

x F (W ) k where fW,i is a diagram in C and wW,i is a diagram in Vec. Observe that wW,i is given by precisely αW,i(w1w2) as an element of W . 50 CHAPTER 4. DE-EQUIVARIANTISATION

Hence

fW,i P P F (W ) T ((f2,W2, w2) ◦ (f1,W1, w1)) = . W ∈Irrep G i F (wW,i)

x A

Alternatively, we have

y y

f1 f2 F (W1) F (W2) . T (f1,W1, w1) = and T (f2,W2, w2) = F (w1) F (w2)

x A x A

Composition in Cfcomod-A is given by

f2 z y f1 f2 f1 F (W2) ◦ F (W1) = F (w2) F (w1) F (w1) F (w2) y A x A F (µ)

x A z z f2

fW,i f1 F (α∗ ) = P P W,i = P P F (αW,i) W ∈Irrep G i W ∈Irrep G i F (αW,i) F (w1) F (w2) F (w1) F (w2) F (∆) F (∆) x A x A where ∆ is the comultiplication on Fun G. In these diagrams we have suppressed the tensorator of F for clarity. To show that T respects composition, it therefore suﬃces to show that for every 4.3. LADDERS AND DE-EQUIVARIANTISATION 51

W and i, W W

αW,i

w1 w2 = wW,i . ∆ Fun G Fun G

On the right, we have the map which sends a basis element δg ∈ Fun G to gwW,i. On the left, we have the map

∆ w1⊗w2 αW,i Fun G Fun G ⊗ Fun G W1 ⊗ W2 W

δg δg ⊗ δg gw1 ⊗ gw2 αW,i(g(w1 ⊗ w2))

Since αW,i is a map of representations, and recalling that wW,i = αW,i(w1 ⊗ w2), we see that these maps are indeed equal, and hence T is functorial.

To construct the inverse functor S : Cfcomod-A → C ⊗RepG Vec, note that from Lemma 4.13, we have ! M A = F W · W ∗ W ∈Irrep G M = F (W ) · W ∗. W ∈Irrep G It follows that ! M C(xA → y) = C xF (W ) · W ∗ → y W ∈Irrep G M = W ∗∗ ⊗ C(xF (W ) → y) W ∈Irrep G M = W ⊗ C(xF (W ) → y) W ∈Irrep G and therefore any f ∈ C(xA → y) is given by X f = fW , W ∈Irrep G where fW ∈ W ⊗ C(xF (W ) → y). Each fW will be a ﬁnite sum of pure tensors X fW = wW,i ⊗ fW,i i for wW,i ∈ W and fW,i ∈ C(xF (W ) → y). We deﬁne the inverse linear functor

Cfcomod-A → C ⊗RepG Vec on objects by sending x to (x, k), and on morphisms 52 CHAPTER 4. DE-EQUIVARIANTISATION f ∈ C(xA → y) by

y k

P fW,i W f 7→ W,i wW,i

x k

where fW,i and wW,i come from the decomposition of f as described above. That S is functorial follows from the same calculation as for T . It is immediate that T and S are mutually inverse on objects, and to see that they are mutually inverse on morphisms it suﬃces to recognise that the morphism

F (w)

x A

L in C(xA → y) decomposes as w ⊗ f in W ∈Irrep G W ⊗ C(xF (W ) → y). Hence T is an equivalence. To see that T is monoidal, we note that it suﬃces to deﬁne the tensorator on the full subcategory of objects (x, k), as it then extends to all direct sums of these objects, and the equations the tensorator must satisfy will decompose into equations on the objects (x, k). We note that on the full subcategory of objects (x, k) we have that

T (x, k) ⊗ T (y, k) = xy = T ((x, k) ⊗ (y, k)), and so we can use the identity as the tensorator. So to see that T is monoidal it suﬃces to show that this is natural, which is to say that for any (f1,W1, w1): (x1, k) → (y1, k) and (f2,W2, w2):(x2, k) → (y2, k), we have

T ((f1,W1, w1) ⊗ (f2,W2, w2)) = T (f1,W1, w1) ⊗ T (f2,W2, w2).

The calculation is analogous to the case for composition. We see that (f1,W1, w1)⊗ 4.3. LADDERS AND DE-EQUIVARIANTISATION 53

(f2,W2, w2) is given by

y1 k y2 k y1 y2 k k y1 y2 k W f1 W1 f2 W2 W2 2 ⊗ = = w2 w1 w2 W1 W1 w1

x1 k x2 k x1 x2 k k x1 x2 k y1 y2 k

∗ αW,i P P = W . W ∈Irrep G i αW,i w2 w1

x1 x2 k

Hence

y1 y2

fW,i P P F (W ) T ((f1,W1, w1) ⊗ (f2,W2, w2)) = , W ∈Irrep G i F (wW,i)

x1 x2 A where

y1 y2 W k

f1 f2 ∗ α w2 fW,i = F (αW,i) and wW,i = W,i . w1

x1 x2 F (W ) k

The tensor product T (f1,W1, w1) ⊗ T (f2,W2, w2) is given by

y1 y2 y1 y2 f1 f2 f1 f2 F (W1) ⊗ F (W2) = F (w1) F (w2) F (w1) F (w2) x1 A x2 A F (∆)

x1 x2 A 54 CHAPTER 4. DE-EQUIVARIANTISATION

y1 y2 y1 y2 fW,i f1 f2 P P ∗ = F (αW,i) = F (αW,i) . W ∈Irrep G i F (αW,i) F (w1) F (w2) F (w1) F (w2) F (∆) F (∆) x1 x2 A x1 x2 A Hence the equality

T ((f1,W1, w1) ⊗ (f2,W2, w2)) = T (f1,W1, w1) ⊗ T (f2,W2, w2) follows from the same calculation as for composition, and so T is monoidal.

4.4 The Enrichment De-equivariantisation

We can give a third deﬁnition of de-equivariantisation, which we will call the enrichment de-equivariantisation. Using the central functor F : RepG → C, we can construct a category with the same objects as C, but in which the hom-spaces are representations. Such a category (satisfying certain conditions) is said to be enriched in RepG. By taking the idempotent completion of this enriched version of C, we obtain the de-equivariantisation. First, however, we will introduce the notion of enriched categories.

4.4.1 Enriched Categories

In the deﬁnition of an additive category, we require that the hom-sets have the structure of an abelian group, and that composition is a bi-homomorphism. Sim- ilarly, for linear categories we require that the hom-sets have the structure of a vector space, and that composition is bilinear. These are both examples of enriched categories, where the hom-sets of a category C are objects in another category V. Ordinary categories are enriched in Set, additive categories are enriched in Ab, and linear categories are enriched in Vec. There is no reason, however, to restrict ourselves to enriching in categories in which the objects have underlying sets. We can in fact enrich in any monoidal category.

Deﬁnition 4.15. Let V be a monoidal category. A V-enriched category C consists of 4.4. THE ENRICHMENT DE-EQUIVARIANTISATION 55

• a collection of objects ob C, • a hom-object C(x → y) ∈ V for each pair x, y ∈ C,

• an identity morphism jx ∈ V(1V → C(x → x)) for each x ∈ C, and

• a composition law −◦C − ∈ V (C(x → y)C(y → z) → C(x → z)) for each triple x, y, z ∈ C, such that the following two axioms hold: • (Identity) For all x, y ∈ C:

C(x → y) C(x → y) C(x → y)

− ◦C − − ◦C − = = jy jx

C(x → y) C(x → y) C(x → y)

• (Associativity) For all w, x, y, z ∈ C:

C(w → z) C(w → z)

− ◦C − − ◦C − = − ◦C − − ◦C −

C(w → x) C(x → y) C(y → z) C(w → x) C(x → y) C(y → z)

More information on enriched categories can be found in [Kel05]. It is worth noting that we write composition in enriched categories in the “di- agrammatic” style; given f : x → y and g : y → z, we write f ◦ g : x → z for their composition, rather than g ◦ f as is the usual convention. This is done to align with the graphical methods we use when working with enriched categories, as composition should be read left to right as it is in diagrams

f g x y z.

f◦g

In most cases we will be treating composition as a morphism − ◦C − in V, rather than writing f ◦C g directly, and so this should not cause confusion. In the case of additive categories, we deﬁned the enrichment simply by say- ing that the hom-sets are actually abelian groups, and that the composition map is a bi-homomorphism. The existence of an identity morphism, as well as the identity and associativity axioms, followed from the corresponding axioms in ordinary categories. However, if C is enriched in a category where the objects do not have underlying sets, then C is not actually a category, as there are no underlying hom-sets. Hence we must impose the identity and associativity axioms directly. 56 CHAPTER 4. DE-EQUIVARIANTISATION

4.4.2 Monoidally Enriched Categories

Suppose we wish to give a V-enriched category C a monoidal structure. In ordinary monoidal categories, the tensor product ⊗ must be functorial, and so we require that

(f1 ⊗ g1) ◦ (f2 ⊗ g2) = (f1 ◦ f2) ⊗ (g1 ◦ g2).

Observe that this involves exchanging the positions of f2 and g1, and so to create an analogous axiom we must be able to exchange the positions of hom-objects. Therefore V must be braided, which leads us to the following deﬁnition. Let (V, ⊗, β) be a braided monoidal category.

Deﬁnition 4.16. A (strict)V-monoidal category C is a V-enriched category C with

• a unit object 1C ∈ C, • an object xy ∈ C for every pair x, y ∈ C,

• a tensor product law − ⊗C − ∈ V(C(a → b)C(x → y) → C(ax → by)) for all a, b, x, y ∈ C, satisfying the following axioms:

• (Strict unitor for objects) For all x ∈ C, 1Cx = x = x1C. • (Strict associator for objects) For all x, y, z ∈ C,(xy)z = x(yz). • (Unitality) For all x, y ∈ C:

C(x → y) C(x → y) C(x → y)

− ⊗C − − ⊗C − = =

j1C j1C

C(x → y) C(x → y) C(x → y)

C(xy → xy) C(xy → xy)

− ⊗C − and = jx jy jxy

• (Associativity of − ⊗C −) For all a, b, c, x, y, z ∈ C:

C(abc → xyz) C(abc → xyz)

− ⊗C − − ⊗C − = − ⊗C − − ⊗C −

C(a → x) C(b → y) C(c → z) C(a → x) C(b → y) C(c → z) 4.4. THE ENRICHMENT DE-EQUIVARIANTISATION 57

• (Braided interchange) For all a, b, c, x, y, z ∈ C:

C(ax → cz) C(ax → cz)

− ◦C − − ⊗C −

= − ⊗C − − ⊗C − − ◦C − − ◦C −

C(a → b) C(x → y) C(b → c) C(y → z) C(a → b) C(x → y) C(b → c) C(y → z)

This deﬁnition, together with some fundamental results on V-monoidal categories, is given in [MP19]. Once again, in the case of additive (or linear) categories which already have a strict monoidal structure as an ordinary category, it suﬃces to require that the tensor product be a bi-homomorphism (respectively bilinear), as the rest of this structure is given by the underlying monoidal structure.

4.4.3 Transporting Enrichment

Let C be a V-monoidal category, and let ρ : V → W be a braided lax monoidal functor with tensorator µ : ρ(−)ρ(−) → ρ(−).

We can form a W-monoidal category ρ∗C which has the same objects as C, with hom-objects ρ∗C(x → y) = ρ(C(x → y)), and with composition and tensor product ρ∗C given by ρ(− ◦C −) ◦ µ and ρ(− ⊗C −) ◦ µ respectively. Identity morphisms jx are C given by composing ρ(jx ) with the canonical morphism 1W → ρ(1V ). The identity and associativity axioms (for both composition and tensor product) follow immediately because ρ is lax monoidal. The braided interchange axiom is almost as immediate, and follows because ρ is braided. We will use β to denote the braiding in V. We have

ρ(C(ax → cz))

ρ∗C(ax → cz) ρ(− ◦C −) µ − ◦ρ∗C −

= ρ(− ⊗C −) ρ(− ⊗C −)

− ⊗ρ∗C − − ⊗ρ∗C − µ µ

ρ∗C(a → b) ρ∗C(x → y) ρ∗C(b → c) ρ∗C(y → z) ρ(C(a → b)) ρ(C(x → y)) ρ(C(b → c)) ρ(C(y → z)) 58 CHAPTER 4. DE-EQUIVARIANTISATION

ρ(− ◦C −) ρ((− ⊗C −)) ρ((− ⊗C −))

ρ((− ⊗C −)(− ⊗C −)) ρ((− ◦C −)(− ◦C −)) ρ((− ◦C −)(− ◦C −)) = = = ρ(id β id) µ µ µ

ρ∗C(ax → cz) ρ(− ⊗C −) − ⊗ − µ ρ∗C

= ρ(− ◦C −) ρ(− ◦C −) = − ◦ρ∗C − − ◦ρ∗C − .

µ µ

ρ∗C(a → b) ρ∗C(x → y) ρ∗C(b → c) ρ∗C(y → z)

4.4.4 The Enrichment De-equivariantisation

The following deﬁnition is ﬁrst described in [MP19].

We let C be a fusion category and let F : RepG → C be a k-linear central functor. Then since RepG and C are ﬁnitely semisimple, F has a right adjoint. For x ∈ C, let Lx : RepG → C be the functor given by Lx(W ) = xF (W ). We can construct a right adjoint Rx. To do so, let R1C : C → RepG be the right adjoint of F : RepG → C, and deﬁne Rx(y) = R1C (x∗y). Then for all x, y ∈ C and W ∈ RepG, we have

C(Lx(W ) → y) = C(xF (W ) → y) ∼= C(F (W ) → x∗y) ∼= RepG(W → R1C (x∗y)) = RepG(W → Rx(y)).

Deﬁnition 4.17. Let CgG be the category with objects the same as C, and hom- x objects CgG(x → y) := R (y). Then CgG(x → y) satisﬁes the adjunction

RepG(W → CgG(x → y)) := C(xF (W ) → y) (4.1)

for all W ∈ RepG. This gives a RepG-monoidal category, with the following deﬁni- tions of identity, composition, and tensor product. 4.4. THE ENRICHMENT DE-EQUIVARIANTISATION 59

We will follow [MP19] and use the following notation for clarity.

[x → y] := F (CgG(x → y))

[a → b; x → y; ... ] := F (CgG(a → b)CgG(x → y) ... ).

x The identity element j ∈ RepG(1RepG → CgG(x → x)) is the transpose of the identity idx ∈ C(x → x) under the adjunction (4.1).

The evaluation morphism εx→y : x[x → y] → y is the transpose of the identity

RepG(CgG(x → y) → CgG(x → y)) under the adjunction (4.1). By Proposition x x x A.3, this is precisely the counit y at y of the adjunction L a R . The composition law

− ◦ − : C G(x → y)C G(y → z) → C G(x → z) CgG g g g is given by the transpose of the following map in C(x[x → y; y → z] → z) under the adjunction (4.1): z

εy→z

εx→y

µ CgG(x→y),CgG(y→z)

x [x → y; y → z] . The tensor product law

− ⊗ − : C G(a → b)C G(x → y) → C G(ax → by) CgG g g g is the transpose of the following map in C(ax[a → b; x → y] → by) under the adjunction (4.1): b y

εa→b εx→y

µ CgG(a→b),CgG(x→y)

a x [a → b; x → y] .

Z Note that since CgG(a → b) ∈ RepG, the central functor F : RepG → Z(C) includes a half-braiding of [a → b] past x. The proof that this satisﬁes the associativity and braided interchange axioms required to make CgG into a RepG-monoidal category can be found in Section 6 of

[MP19]. 60 CHAPTER 4. DE-EQUIVARIANTISATION

We then forget the RepG-monoidal structure on CgG and take the idempotent completion. More precisely, let ρ : RepG → Vec be the forgetful functor. The enrichment de-equivariantisation C G is given by Kar(ρ∗(CgG)).

In [MP19] it is asserted without proof that this is equivalent to the standard deﬁnition of de-equivariantisation. The purpose of the remainder of this chapter is to provide this proof.

4.4.5 The Underlying Monoidal Categories

As described above, when the objects of V do not have underlying sets (that is, V is not a concrete category), a V-monoidal category is not a category in the ordinary sense. However, for any object v ∈ V, the collection

V(v → C(x → y)) of v-morphisms x → y is indeed a set. If v is a coalgebra, we can obtain an ordinary monoidal category by taking the hom-sets to be the collections of v-morphisms.

Deﬁnition 4.18. Let C be a V-monoidal category, and let A ∈ V be a coalgebra in C with counit and comultiplication ∆. The underlying monoidal category CA has the same objects as C, and hom-sets

CA(x → y) := V(A → C(x → y)).

A The identity morphism idx ∈ C (x → x) is given by jx ◦ , where jx ∈ V(1V → C(x → x)) is the identity morphism in C. Composition of morphisms f ∈ CA(x → y) and g ∈ CA(y → z) is given by

fg A ∆ AA C(x → y)C(y → z) −◦C− C(x → z).

The tensor product of the morphisms f ∈ CA(a → b) and g ∈ CA(x → y) is given by fg A ∆ AA C(a → b)C(x → y) −⊗C− C(ax → by).

One can then check that these deﬁnitions deﬁne a monoidal category, though we will not provide this calculation here. When V is linear, the underlying monoidal category CA is also linear. Let V be a braided monoidal linear category, and let C be V-monoidal.

Proposition 4.19. Transporting enrichment along the linear braided lax monoidal functor V → Vec given by V(A → −) gives the underlying monoidal category CA. 4.4. THE ENRICHMENT DE-EQUIVARIANTISATION 61

A Proof. Let ρ : V → Vec be given by V(A → −). Then ρ∗C and C both have the same objects as C, and morphisms x → y given by V(A → C(x → y)). The tensorator of ρ is given by

µv,w : V(A → v)V(A → w) → V(A → vw)

∆ f⊗V g f ⊗Vec g 7→ A −→ AA −−−→ vw .

Hence the composition law in ρ∗C, given by ρ(− ◦C −) ◦ µ, is precisely the compo- A sition law in C . Similarly the tensor product in ρ∗C, given by ρ(− ⊗C −) ◦ µ, is precisely the tensor product in CA.

Let ρ : V → W be a linear braided lax monoidal functor with left adjoint λ, and let C be a V-monoidal category. Then λ is an oplax monoidal functor with oplax tensorator νx,y : λ(xy) → λ(x)λ(y) deﬁned to be the transpose of

η η µ xy −−→x y ρλ(x)ρλ(y) −→ ρ(λ(x)λ(y)), where η is the unit of the adjunction and µ is the lax tensorator of ρ. The identity comparison ι : F (1C) → 1D of λ is the transpose of the identity comparison of ρ. Further, suppose A ∈ V is a coalgebra with multiplication ∆ and counit e. Then

λ(A) is also a coalgebra, with comultiplication νA,A ◦ λ(∆) and counit λ(e) ◦ ι. There is a certain sense in which taking the underlying monoidal category “commutes” with transporting enrichment along ρ, which we describe in the following proposition.

Lemma 4.20. Let ρ : V → W be a linear braided lax monoidal functor with left adjoint λ, let C be a V-monoidal category, and let A ∈ W be a coalgebra. Then there is a monoidal equivalence

A λ(A) (ρ∗C) 'C .

A Proof. Unfolding deﬁnitions, we see that (ρ∗C) has the same objects as C and morphisms A (ρ∗C) (x → y) = W(A → ρ(C(x → y))), while Cλ(A) has the same objects as C and morphisms

Cλ(A)(x → y) = V(λ(A) → C(x → y)).

These vector spaces are naturally isomorphic by the adjunction λ a ρ, and so we A λ(A) deﬁne a functor T :(ρ∗C) → C which is the identity on objects, and takes a morphism f ∈ W(A → ρ(C(x → y))) to its transpose f ∈ V(λ(A) → C(x → y)). 62 CHAPTER 4. DE-EQUIVARIANTISATION

To see that this is functorial, we need it to respect identities and composition.

Each object x ∈ C has an identity morphism jx ∈ V(1V → C(x → x)), and we have

A ρ(j ) (ρ∗C) x idx = A −→ ρ(1V ) −−−→ ρ(V(x → x)) Cλ(A) jx idx = λ(A) −→ 1V −→C(x → x), where is the unitor of ρ. Indeed, the transpose of the ﬁrst is precisely the second, by the naturality of the adjunction.

A For composition, we need that for f : x → y and g : y → z we have f ◦(ρ∗C) g = f ◦Cλ(A) g. However, since f = f, it is equivalent to show that

A f ◦(ρ∗C) g = f ◦Cλ(A) g. Unfolding deﬁnitions, we ﬁnd that given

A f ∈ (ρ∗C) (x → y) = W(A → ρ(C(x → y))) A g ∈ (ρ∗C) (y → z) = W(A → ρ(C(y → z))),

A their composition f ◦ g in (ρ∗C) is given by

µ◦fg ρ(−◦ −) A ∆ AA ρ(C(x → y)C(y → z)) C ρ(C(x → z)) where ∆ is the comultiplication on A and µ is the lax tensorator for ρ. The composition f ◦ g in Cλ(A) is correspondingly given by

λ(∆) fg◦ν λ(A) λ(AA) C(x → y)C(y → z) −◦C− C(x → z).

Therefore by the naturality of the adjunction, to show that f ◦ g = f ◦ g it suﬃces to show that fg µ AA ρ(C(x → y))ρ(C(y → z)) ρ(C(x → y)C(y → z))

fg transpose λ(AA) ν λ(A)λ(A) C(x → y)C(y → z)

By Proposition A.3, we can unfold fg ◦ ν as

η ρλ(ηη) AA ρλ(AA) ρλ(ρλ(A)ρλ(A))

ρλ(µ) ρ() ρλρ(λ(A)λ(A)) ρ(λ(A)λ(A))

ρ(λ(f)λ(g)) ρ() ρ(λρ(C(x → y))λρ(C(y → z))) ρ(C(x → y)C(y → z)). 4.4. THE ENRICHMENT DE-EQUIVARIANTISATION 63

That this is precisely µ ◦ fg follows from the commutativity of the following diagram:

ρλ(ηη) ρλ(µ) ρλ(AA) ρλ(ρλ(A)ρλ(A)) ρλρ(λ(A)λ(A)) ρ() η η η ηη µ AA ρλ(A)ρλ(A) ρ(λ(A)λ(A)) ρ(λ(A)λ(A))

fg ρλ(f)ρλ(g) ρ(λ(f)λ(g)) ηη µ ρ(C(x → y))ρ(C(y → z)) ρλρ(C(x → y))ρλρ(C(y → z)) ρ(λρ(C(x → y))λρ(C(y → z)))

ρ()ρ() ρ() µ ρC(x → y)ρC(y → z) ρ(C(x → y)C(y → z))

Every square in this diagram commutes by the naturality of either η, µ, or ηη. The triangles are precisely components of the triangle identities. Hence this diagram commutes, and so we have

gf ◦ ν = µ ◦ gf.

A λ(A) Therefore the map T :(ρ∗C) → C which is the identity on objects and takes morphisms to their transpose is indeed a functor. The definition of the tensor product in CA is almost identical to the definition of composition, just replacing − ◦C − with − ⊗C −. The same is true of the definition of ρ∗(C). Therefore the above proof also shows that

A T (f ⊗(ρ∗C) g) = T (f) ⊗Cλ(A) T (g), and so T is a monoidal functor. λ(A) A There is an obvious inverse functor S : C → (ρ∗C) which is the identity on objects and takes morphisms to their transpose. The same proof as above shows that this is a monoidal functor, and since f = f this is immediately an inverse to T . Hence T is a monoidal equivalence.

4.4.6 Equivalence to De-equivariantisation

We can now apply the previous result to the enrichment de-equivariantisation. This approach (using Lemma 4.20) was suggested by David Penneys.

Theorem 4.21. There is a monoidal equivalence between the standard deﬁnition of de-equivariantisation Cmod-F (Fun G) and the enrichment de-equivariantisation

C G.

64 CHAPTER 4. DE-EQUIVARIANTISATION

Proof. Recall that the enrichment de-equivariantisation is the idempotent completion Kar(ρ∗(CgG)), where CgG has the same objects as C with hom-spaces determined by

RepG(W → CgG(x → y)) := C(xF (W ) → y), and ρ : RepG → Vec is the forgetful functor.

Furthermore, we observe that since 1Vec = k is the only simple object in Vec, we have ∼ Vec(k → V ) = V.

Therefore for any Vec-monoidal category D, we have that Dk = D, where we are taking the underlying monoidal category with respect to the algebra k. The forgetful functor ρ : RepG → Vec has a left adjoint λ by 2.49, and

M λ(k) = W · Vec(W → k) W ∈Irrep G M = W · W ∗ W ∈Irrep G = Fun G by Lemma 4.13. We can therefore apply Lemma 4.20 to obtain

k C G = (ρ∗(CgG))

Fun G ' (CgG) .

Fun G We see that CgG has the same objects as C, and hom-spaces

Fun G CgG (x → y) = RepG(Fun G → CgG(x → y)) = C(xF (Fun G) → y).

Given f ∈ C(xF (Fun G) → y) and g ∈ C(yF (Fun G) → z), their composition is determined by the adjunctions

RepG(Fun G → CgG(x → y)) = C(xF (Fun G) → y).

In particular, we have a collection of adjunctions for every x ∈ C, given by

Lx : RepG → C Rx : C → RepG

A 7→ xF (A) y 7→ CgG(x → y).

4.4. THE ENRICHMENT DE-EQUIVARIANTISATION 65

In both f and g the representation is Fun G, so let A := Fun G. The composition g ◦ f is therefore given by g ◦ f, where the transposes are

f : A → CgG(x → y)

g : A → CgG(y → z).

One subtlety here is that the transposes are actually in diﬀerent adjunctions; f and g ◦ f are along the Lx a Rx adjunction, whereas g is along the Ly a Ry adjunction. A The composition f ◦ g is the composition in (CgG) , which is

∆ fg −◦− A AA CgG(x → y)CgG(y → z) CgG(x → z).

where this last map is composition in CgG. By the deﬁnition of composition in

CgG, and unfolding the deﬁnition of the transpose according to Proposition A.3, we get that composition in CgG is given by

x x η x R (idx µ) x CgG(x → y)CgG(y → z) R (x[x → y; y → z]) R (x[x → y][y → z])

Rx(x id ) x y y [y→z] x R (z ) x R (y[y → z]) R (z) = CgG(x → z).

Therefore the composition f ◦ g : xF (A) = Lx(A) → z is given by the upper composition in the diagram below.

x x x L (∆) x L (fg) x L (A) L (AA) L (CgG(x → y)CgG(y → z))

Lx(ηx)

x x idx µ x[x → y; y → z] L R (x[x → y; y → z]) x x x idx µ L R (idx µ) id F (f)F (g) xF (A)F (A) x x[x → y][y → z] LxRx(x[x → y][y → z])

x x x x y id[y→z] L R (y id[y→z]) y[y → z] LxRx(y[y → z])

y LxRx(y) z z x x z x L R (z) z The square in the bottom right commutes by naturality of x, the square in the top left commutes by naturality of idx µ, and the triangle in the top right is one of the triangle identities for Lx a Rx. 66 CHAPTER 4. DE-EQUIVARIANTISATION

We can simplify this even further. By Proposition A.3, idx F (f)F (g) is given by

x y x y idx F (η )F (η ) id FR (f)FR (g) xF (A)F (A) A A xF RxLx(A)FRyLy(A) x x[x → y][y → z].

We have

LxRx(f)FRy(g) xF RxLx(A)FRyLy(A) x[x → y][y → z] id F (ηx )F (ηy ) x A A x x Lx(A) id y id y id F (η ) f id LyRy(g) xF (A)F (A) A Lx(A)FRyLy(A) yF RyLy(A) y[y → z] y idy F (ηA) y y y L (A) z f idF (A) g yF (A) Ly(A) z.

The upper right square commutes by the naturality of x id, the bottom right square commutes by the naturality of y, the triangles commute by the triangle identities, and the bottom left square is simply that the tensor product of morphisms can be exchanged. Hence the composition f ◦ g is given by

(idx µ )◦(idx F (∆)) f idF (A) g xF (A) A,A xF (A)F (A) yF (A) z.

Since µA,A◦F (∆) is the comultiplication on F (A), this is precisely the statement that z y z g f ◦ g = f , ∆ x F (A) y F (A) F (A) x F (A)

Fun G which is precisely the composition in Cfcomod-F (Fun G). Hence (CgG) is precisely Cfcomod-F (Fun G) as ordinary categories. We now consider the monoidal structure. Given f ∈ C(aF (Fun G) → b) and g ∈ C(xF (Fun G) → y), their tensor product is determined by adjunctions in precisely the same way as with composition; that is, f ⊗ g = f ⊗ g. A The tensor product f ◦ g is the tensor product in (CgG) , which is

∆ fg −⊗− A AA CgG(a → b)CgG(x → y) CgG(ax → bz).

4.4. THE ENRICHMENT DE-EQUIVARIANTISATION 67 where this last map is the tensor product in CgG. By the deﬁnition of the tensor product in CgG, and unfolding the deﬁnition of the transpose according to

Proposition A.3, we get that the tensor product in CgG is given by

ax ax η ax R (idax µ) ax CgG(a → b)CgG(x → y) R (ax[a → b; x → y]) R (ax[a → b][x → y])

Rax(id β id) Rax(ax) x,[a→b] ax b y ax R (ax[a → b][x → y]) R (by) = CgG(ax → bz).

In particular, we note that the tensor product f ⊗ g is given by almost exactly the same map as composition. The diﬀerence is that we use the Lax a Rax adjunction rather than the Lx a Rx adjunction, and we replace the section of the composition map given by

x x y x x x L R (z) ◦ L R (y id) (which represents composition) with

x x a x ax ax L R (b y ) ◦ L R (id βx,[a→b] id) (which represents the tensor product). We can therefore do the same calculation as we did for composition to ﬁnd that the tensor product f ⊗ g is given by

(idax µ )◦(idax F (∆)) id βx,F (A) id fg axF (A) A,A axF (A)F (A) aF (A)xF (A) by.

b y y b f g f ⊗ g = , ∆ a F (A) x F (A) F (A) a x F (A)

Fun G which is the tensor product in Cfcomod-F (Fun G). Hence (CgG) is precisely

Cfcomod-F (Fun G) as a monoidal category. Therefore we have a chain of monoidal equivalences

k C G = Kar((ρ∗(CgG)) )

Fun G ' Kar((CgG) )

= Kar(Cfcomod-F (Fun G))

' Ccomod-F (Fun G) which is the standard deﬁnition of de-equivariantisation.

5 Tensor Categories and the Deligne Product

In this chapter we give an overview of the basic deﬁnitions of tensor categories and the Deligne tensor product. Tensor categories are a special type of linear rigid monoidal category, of which fusion categories are a particularly well-behaved example. Given two tensor categories we can construct the Deligne tensor product, which is also a tensor category. We ﬁnish this chapter by showing that the Deligne tensor product agrees with the monoidal ladder category construction for fusion categories, though it does not agree for all tensor categories.

5.1 Tensor Categories

We have already deﬁned an additive category to be a category with direct sums such that the hom-sets are abelian groups, satisfying certain nice properties. The motivating example is of course Ab, the category of abelian groups. We can go further in generalising the properties of Ab; a category is said to be abelian if all kernels and cokernels exist, and satisfy a nice property which is analogous to the existence of images of group homomorphisms in Ab.

Deﬁnition 5.1. Let f : x → y be a morphism in an additive category C. The kernel ker(f) of f is an object ker(f) ∈ C with a morphism k : ker(f) → x such that f ◦ k = 0, and if given any morphism k0 : z → x such that f ◦ k0 = 0 there is a unique morphism u : z → ker(f) such that k ◦ u = k0. x f k

0 k ker(f) 0 y

u 0 z Dually, the cokernel of f is an object coker(f) ∈ C together with a morphism c : y → coker(f) such that c ◦ f = 0, and if given any morphism c0 : y → z there exists a unique morphism u : coker(f) → c0 such that u ◦ c = c0.

69 70 CHAPTER 5. TENSOR CATEGORIES AND THE DELIGNE PRODUCT

Kernels and cokernels do not necessarily exist, but if they do then they are unique up to unique isomorphism.

Deﬁnition 5.2. An additive category is abelian if all kernels and cokernels exist and for every morphism f : x → y there exists a decomposition

f j ker(f) k x i I y c coker(f) where k and c are the morphisms associated to ker(f) and coker(f) respectively, and I = coker(k) = ker(c) with associated morphisms i and j. Such a decomposition is called a canonical decomposition of f. The object I is called the image of f and is denoted im(f).

Of course, not all abelian categories are as well-behaved as Ab; however, we do get additional nice properties if we restrict ourselves to abelian categories with certain ﬁniteness conditions.

Deﬁnition 5.3. Let C be an abelian category. A morphism f : x → y is said to be a monomorphism if ker(f) = 0. It is said to be an epimorphism if coker(f) = 0.

In the case of Ab, a monomorphism is precisely an injective homomorphism, and similarly an epimorphism is a surjective homomorphism.

Deﬁnition 5.4. Let C be an abelian category. A subobject of an object y ∈ C is an object x ∈ C with a monomorphism i : x → y.A quotient object of y is an object z with an epimorphism p : y → z. For a subobject x ⊂ y we deﬁne the quotient object x/y to be the cokernel of the monomorphism i : x → y.

Proposition 5.5. Let C be an abelian category. We will say a nonzero object x ∈ C is simple if 0 and x are its only subobjects, and that an object in C is semisimple if it is a direct sum of simple objects. Then C is semisimple (in the sense of Deﬁnition 2.25) if and only if every object of C is semisimple.

When working with abelian categories, it is more common to make the definition that a category is semisimple if all of its objects are semisimple. The proof that these two definitions are equivalent can be found in §2.1 of [Müg03].In particular, we note that in a semisimple abelian category, the nonzero objects x which have only 0 and x as subojects are precisely the simple objects in the sense of Definition 2.25, and so there is no confusion in using the word “simple” in both cases. 5.1. TENSOR CATEGORIES 71

Deﬁnition 5.6. We say that an object x in an abelian category C has ﬁnite length if there exists a sequence of objects

0 = x0 ⊂ x1 ⊂ · · · ⊂ xn = x such that xi/xi−1 is simple.

Definition 5.7. A linear abelian category C is said to be locally finite if all hom- spaces C(x → y) are finite dimensional, and every object in C has finite length.

Definition 5.8. An abelian category C is said to be finite if it is equivalent to the category A-mod of finite-dimensional modules over a finite-dimensional algebra A.

It is worth noting that every finite category is locally finite – see Definition 1.8.6 of [EGNO15]. Let k be an algebraically closed field.

Deﬁnition 5.9. A multitensor category C is a locally ﬁnite k-linear abelian rigid ∼ category. If additionally EndC(1C) = k then we call C a tensor category.

5.1.1 Fusion Categories as Tensor Categories

In Definition 2.29, we introduced fusion categories as finitely semisimple rigid k- linear monoidal categories in which 1C is a simple object. In fact, we could have made an alternate definition as a tensor category.

Proposition 5.10. A rigid k-linear monoidal category is a fusion category if and only if it is a ﬁnite semisimple tensor category.

Proof. For one direction, we note that any fusion category is equivalent (though not necessarily monoidally equivalent) to Vec⊕n, by Proposition 2.28. Hence it is abelian, as Vec (and therefore Vec⊕n) is clearly abelian. Furthermore, we see that ⊕n n Vec ' k-mod, and so Vec ' k -mod, and so fusion categories are finite. Since 1C ∼ is a simple object, we have that End(1C) = k by definition. Hence fusion categories are finite semisimple tensor categories. Now suppose C is a finite semisimple tensor category. From Remark 2.26, we know that every object c ∈ C is isomorphic to a unique direct sum of simple objects L ⊕ni i xi , and that C(c → c) is isomorphic to

M M M 2 ∼ ⊕ni ⊕ni ∼ ni C(c → xi)C(xi → c) = C(xi → xi)C(xi → xi ) = k . i i i 72 CHAPTER 5. TENSOR CATEGORIES AND THE DELIGNE PRODUCT

∼ So End(1C) = k if and only if 1C is a simple object (up to isomorphism). Now since C is finite, we have that C' A-mod for some finite-dimensional algebra A. Consider A as an A-module and let xi be a simple A-module. Then we can define a nonzero map A → xi by choosing s ∈ xi and sending a ∈ A to a · s. Hence dim(A-mod(A → xi)) is nonzero, and so [A : xi] > 0 for all simple xi. ∼ L [A:xi] However, we know that A = i xi , and this must be a finite direct sum, so there must be finitely many simple objects in C.

5.2 The Deligne Product

The Deligne product is another way to take the tensor product of categories, which works for locally ﬁnite abelian categories.

Deﬁnition 5.11. Let C and D be abelian categories. An additive functor F : C → D is called right (respectively, left) exact if for any short exact sequence

0 → x → y → z → 0 in C, the sequence

F (x) → F (y) → F (z) → 0 (respectively, 0 → F (x) → F (y) → F (z)) is exact in D. A functor is said to be exact if it is both right and left exact.

Definition 5.12. Let C and D be locally finite abelian categories. The Deligne tensor product is a locally finite abelian category C D together with a bifunctor

: C × D → C D which is right exact in both variables, and is such that for any bifunctor F : C × D → A which is right exact in both variables, there exists a unique right exact ¯ ¯ functor F : C D → A satisfying F ◦ = F .

To prove the existence of such a category we will make use of the following theorem, which is Theorem 5.1 of [Tak77].

Theorem 5.13. Any (essentially small1) locally ﬁnite abelian category C is equivalent to the category C-comod for some coalgebra C. 1A category is essentially small if it is equivalent to a category where the collection of all morphisms is a set. 5.2. THE DELIGNE PRODUCT 73

Proposition 5.14. i) The Deligne tensor product exists. If C' C-comod and D' D-comod then

C D = (C ⊗ D)-comod.

ii) If C and D are both ﬁnite, so C' A-mod and D' B-mod, then

C D = (A ⊗ B)-mod.

iii) The Deligne tensor product is unique up to unique equivalence.

This construction is originally given in [Del90]. For proofs see [EGNO15] and [EGNO09]. In particular, if we let C' C-comod and D' D-comod, then the functor : C × D → C D is given by x y = x ⊗ y, where x is a C-comodule and y is a D-comodule (and likewise in the ﬁnite case). The Deligne product can also be applied to functors, using the universal property. If F : C → C0 and G : D → D0 are right exact functors between locally ﬁnite abelian categories, then

0 0 F × G : C × D → C D (c, d) 7→ F (c) F (d)

0 0 is a right exact bifunctor, and so there exists a unique F G : C D → C D .

5.2.1 Monoidal Structure on the Deligne Product

Proposition 5.15. If C and D are multitensor categories, then C D is a multitensor category.

Proof (adapted from [EGNO15]). The functor ⊗ : C ×C → C is exact (see Proposi- tion 4.2.1 of [EGNO15]). Hence by the universal property of the Deligne product it extends to a functor TC : C C → C, and the associativity isomorphism extends to ∼ an isomorphism of functors TC ◦(idC TC) = TC ◦(TC idC), satisfying the pentagon identity. If C = C-comod and D = D-comod, then there is a canonical isomorphism

(23) : C ⊗ D ⊗ C ⊗ D → C ⊗ C ⊗ D ⊗ D, given by swapping the second and third components. This gives a canonical exact functor (23) : C D C D → C C D D. 74 CHAPTER 5. TENSOR CATEGORIES AND THE DELIGNE PRODUCT

For X,Y ∈ C D, we deﬁne a functor

⊗ :(C D) × (C D) → C D X ⊗ Y := ((TC TD) ◦ (23))(X Y ). The associativity isomorphism is given simply by the Deligne product of the associativity isomorphisms for TC and TD, and it satisfies the pentagon identity precisely because TC and TD satisfy the pentagon identity. The unit object is given by 1C 1D, and it satisfies the left and right unitors because 1C and 1D satisfy the left and right unitors. Since C and D are rigid, we can define left and right duality functors on C D by taking the Deligne product of the duality functors on C and D, and so C D is also rigid. Hence C D is a multitensor category, as the Deligne product of locally finite abelian categories is always a locally finite abelian category.

Corollary 5.16. The Deligne tensor product C D is a tensor category if C and D are tensor categories, and is a fusion category if C and D are fusion categories.

For more details, see Section 4.6 of [EGNO15].

5.3 Ladder Categories and the Deligne Product

Recall that we deﬁned C ⊗ D to be the category with objects given by pairs (c, d) for c ∈ C and d ∈ D, and hom-spaces given by the tensor product of hom-spaces. Unfortunately, while this deﬁnition (which we will henceforth refer to as the naive tensor product) agrees with the Deligne product in some cases, they do not always agree. We will use the following standard lemma, which appears as Theorem 3.10.2 in [EGHLSVY11].

Lemma 5.17. Let x be a simple A-module and let y be a simple B-module, where A and B are algebras over an algebraically closed ﬁeld. Then x ⊗ y is a simple (A ⊗ B)-module. Furthermore, every simple (A ⊗ B)-module has the form x ⊗ y for unique simple x ∈ A-mod and simple y ∈ B-mod.

The following theorem is well known to those working with fusion categories.

Theorem 5.18. Let C and D be fusion categories over an algebraically closed ﬁeld k. Then there is a canonical monoidal equivalence

C ⊗ D ' C D. 5.3. LADDER CATEGORIES AND THE DELIGNE PRODUCT 75

Proof. Let C and D have tensor product ⊗C and ⊗D respectively. Recall from Proposition 5.15 that for X,Y ∈ C D, we deﬁne X ⊗ Y to be

X ⊗ Y := ((TC TD) ◦ (23))(X Y ), 0 C 0 where TC : C C → C is an exact functor with the property that TC(cc ) = c⊗ c for all c, c0 ∈ C. There is a canonical monoidal functor

P : C ⊗ D → C D (c, d) 7→ c d. To see that this is monoidal, we note that F ((c, d) ⊗ (c0, d0)) = F (cc0, dd0) 0 0 = cc dd , 0 0 0 0 F ((c, d)) ⊗ F ((c , d )) = (c d) ⊗ (c d ) 0 0 = TC(c c ) TD(d d ) 0 0 = cc dd . It therefore suffices to show that P is an equivalence of categories. The Deligne product of fusion categories is fusion, and so in particular is finitely semisimple. To show that P is essentially surjective it therefore suffices to show that every simple object in C D is isomorphic to an object in the image of P , as P is additive. We can consider C = A-mod and D = B-mod, and recall that c d is given by the (A ⊗ B)-module c ⊗ d. Then by Lemma 5.17, we know that every simple object in C D = (A ⊗ B)-mod has the form c ⊗ d for simple objects c ∈ A-mod and d ∈ b-mod. Hence every simple object in C D has the form P (c, d) for simple objects c and d. To show that P is fully faithful, we need that P : A-mod(c → c0) ⊗ B-mod(d → d0) → (A ⊗ B)-mod(c ⊗ c → d ⊗ d0) is an isomorphism for all c, c0, d, d0. For f ∈ A-mod(c → c0) and g ∈ B-mod(d → d0), we have that P (f, g) = f ⊗ g : c ⊗ d → c0 ⊗ d0. Hence P is faithful, as the tensor product of vector spaces is a faithful bifunctor. In particular, this means that the map P on hom-spaces is injective, so to show that it is full it suffices to show that dim (A-mod(c → c0) ⊗ B-mod(d → d0)) = dim ((A ⊗ B)-mod(c ⊗ d → c0 ⊗ d0)) . (5.1) 76 CHAPTER 5. TENSOR CATEGORIES AND THE DELIGNE PRODUCT

This is immediate if c, c0, d, d0 are all simple, as c ⊗ d and c0 ⊗ d0 are simple modules which are isomorphic if and only if c ∼= c0 and d ∼= d0. Hence both sides are 1-dimensional if c ∼= c0 and d ∼= d0, and 0-dimensional otherwise. L ⊕ni L ⊕mj If c = i ci and d = j dj as sums of simples, then by the additivity of P we know that

M ⊕nimj c ⊗ d = (ci ⊗ dj) . i,j In particular, for we can decompose the hom-spaces on both sides of (5.1) to sums of hom-spaces of simple objects, and by the equation both sides will have the same dimension. Hence P is full, and so is an equivalence of categories.

However, if C and D are arbitrary tensor categories, we cannot expect C ⊗ D and C D to be equivalent in this way, as the following counterexample shows.

Example 5.19. Let A = C[x]/(x2) and B = C[y]/(y2), so A⊗B = C[x, y]/(x2, y2). 2 Then C[z]/(z ) is an (A ⊗ B)-module (and so is an object in A-mod B-mod), with the action given by m · x = mz and m · y = mz for all m ∈ C[z]/(z2). It is clear from the action that C[z]/(z2) does not arise as a direct sum of 1-dimensional A ⊗ B-modules, as this would require that x and y act by scalar multiplication. Hence if C[z]/(z2) is in the image of the functor A-mod⊗B-mod → A-mod B-mod which sends (V,W ) to V ⊗ W , it must arise as V ⊗ W for an A-module V and a B-module W . However, this cannot be the case, since C[z]/(z2) is 2-dimensional, this would require that either V or W be 1-dimensional. Assuming without loss of generality that V is 1-dimensional, this means that x acts by scalar multiplication on V , and so this must then be true for the action of A ⊗ B on V ⊗ W as well. However, this is not the case for C[z]/(z2), and so it does not arise as V ⊗ W . 6 An Explicit Example: − + Ad E8 ⊗Fib Ad E8

In this chapter, we further describe the theory of fusion categories, speciﬁcally the unitary ADE fusion categories. We then give an explicit construction of the − + monoidal ladder category Ad E8 ⊗Fib Ad E8 .

6.1 Classiﬁcation of “Small” Fusion Categories

6.1.1 Frobenius-Perron Dimension

n In this chapter, we let C be a fusion category with simple objects {xi}i=1. Recall from Remark 2.26 that for any object c ∈ C, we have a unique decomposition ∼ M ⊕[c:xi] c = xi , i where [c : xi] = dim(C(c → xi)) = dim(C(xi → c)). We can therefore think of

[c : xi] as the number of copies of xi in c. We can construct a ring which reﬂects this information.

Deﬁnition 6.1. The Grothendieck group Gr(C) of C is the free abelian group generated by the simple objects in C. We can associate every object c ∈ C with an element [c] ∈ Gr(C) by X [c] := [c : xi]xi. i We will sometimes drop the brackets and write c ∈ Gr(C). The tensor product on C gives a multiplication on Gr(C) deﬁned by

X xixj := [xixj] = [xixj : xk]xk. (6.1) k ∼ This is clearly associative, since if c = d then [c : xi] = [d : xi]. We therefore call

Gr(C) the Grothendieck ring of C. It has unit 1C.

77 − + 78 CHAPTER 6. AN EXPLICIT EXAMPLE: Ad E8 ⊗Fib Ad E8

The multiplication (6.1) on Gr(C) is called the fusion rule (or fusion rules) of C. It is important to note that the fusion rules of a fusion category do not determine it up to monoidal equivalence. We will soon meet an example of this + − in the categories Ad E8 and Ad E8 , which have the same fusion rules but are not monoidally equivalent. If C has n simple objects, then for any object c ∈ C we can construct the left multiplication matrix Mc by deﬁning

(Mc)ij = [cxj : xi], and so we have ⊕(M ) ∼ M c ij cxj = xi . i

Note that the entries in Mc are non-negative integers.

Deﬁnition 6.2. The Frobenius-Perron dimension of c ∈ C is the maximal non- negative eigenvalue of Mc.

A proof that such an eigenvalue always exists follows from the Perron-Frobenius theorem [Per07; Fro12]. A more detailed discussion, together with a proof that the Frobenius-Perron dimension of an object is an algebraic integer greater than or equal to 1, can be found in chapter 3 of [EGNO15].

6.1.2 Principal Graphs

Deﬁnition 6.3. Let C be a fusion category and x ∈ C a simple object. We say

C is ⊗-generated by x if for all simple objects xi ∈ C, there exists some positive n ⊗n such that [x : xi] 6= 0.

Deﬁnition 6.4. An object x in a pivotal category C is said to be self-dual if it comes equipped with an isomorphism ψ : x → x∗. It is said to be symmetically ∗ ∗∗ self-dual if ψ ◦φx = ψ, where φx : x → x is the isomorphism given by the pivotal structure, and ψ∗ : x∗∗ → x∗ is

m∗

evm∗

ψ .

coevm m∗∗ 6.1. CLASSIFICATION OF “SMALL” FUSION CATEGORIES 79

If an object c ∈ C is self-dual, then Mc will be a symmetric matrix, as for all simple objects xi, xj we have

[cxi : xj] = dim (C(cxi → xj)) = dim (C(xi → cxj)) = [cxj : xi].

Deﬁnition 6.5. Let C be a fusion category, and let x1 ∈ C be self-dual. The principal graph for the pair (C, x1) has vertices given by the simple objects of C, and the number of edges between xi and xj given by [x1xi : xj].

For example, if the principal graph of (C, x1) is

2 , 1C x1 x2 x3 then in Gr(C) we have

x1x2 = x1 + 2x2 + x3. Hence in C we have, ∼ x1 ⊗ x2 = x1 ⊕ x2 ⊕ x2 ⊕ x3.

If C is ⊗-generated by x1, we can often use the principal graph for (C, x1) to determine the fusion rules for C entirely. For example, to calculate x2x2 ∈ Gr(C) with the principal graph above, we note that x2 = x1x1 − 1C − x1, and then can easily calculate x2x2 = (x1x1 − 1C − x1)x2.

6.1.3 The ADE Fusion Categories

One important class of fusion categories are the ADE fusion categories. These are fusion categories ⊗-generated by a symmetrically self-dual simple object (which in each case we will label x1) such that the principal graph of the pair (C, x1) is a simply-laced Dynkin diagram, which are:

AN : 1 x1 x2 xN−2 xN−1

xN−2 DN : 1 x1 x2 xN−3 xN−1

E6: 1 x1 x2 x3 x4

E8: . 1 x1 x2 x3 x4 x5 x6 − + 80 CHAPTER 6. AN EXPLICIT EXAMPLE: Ad E8 ⊗Fib Ad E8

As we mentioned above, fusion rules do not determine the category up to monoidal equivalence, and indeed the ADE fusion categories are not unique.

Deﬁnition 6.6. Let C be a fusion category. The adjoint subcategory Ad C of C is the full subcategory generated by simple objects xi such that

∗ [xx : xi] 6= 0 for some x ∈ C.

Here the full subcategory generated by a collection of simple objects is the full subcategory of objects which are isomorphic to a sum of simples in the collection.

For example, the principal graphs for Ad A2N and Ad E8 are

Ad A2N : 1C x2 x4 x2N−2 2

Ad E8: . 1C x2 x4 x6

6.1.4 Planar Algebras and a Classiﬁcation

We have deliberately avoided giving a construction of the ADE fusion categories, merely claiming that they exist. The standard construction of these categories is as the idempotent completion of the ADE planar algebras. Planar algebras were ﬁrst introduced in [Jon99] for use in the ﬁeld of subfactors, and for brevity we will not describe them here. Skein-theoretic constructions of the ADE planar algebras are given in [MPS10; Big10]. We have the following theorem about ADE planar algebras.

Theorem 6.7. The number of planar algebras realising each of the ADE Dynkin diagrams is given by the following table:

Principal graph An D2n+1 D2n E6 E7 E8 Realizations 1 0 1 2 0 2

This theorem is stated in this form in the review paper [JMS14]. In the language of subfactors, an outline of this proof is given in [Ocn88], and more details given in [GHJ89; Bio91; Izu91; Izu94; Kaw95].

Deﬁnition 6.8. A C∗-category D is a C-linear abelian category with an involutive antilinear contravariant endofunctor ∗ which is the identity on objects, such that 6.1. CLASSIFICATION OF “SMALL” FUSION CATEGORIES 81 the hom-spaces D(x → y) are Hilbert spaces and the norms on these Hilbert spaces satisfy kg ◦ fk ≤ kfkkgk and kf ◦ f ∗k = kfk2.

∗ ∗ A morphism f : x → y in a C -category is called unitary if f ◦ f = idx and ∗ f ◦ f = idy. A C∗ fusion category is called unitary if all of the structure isomorphisms (such as associators) are unitary and (f ⊗ g)∗ = f ∗ ⊗ g∗ for all morphisms f and g.

Note that a full subcategory of a unitary category is automatically unitary, as the ∗ endofunctor is the identity on objects and so restricts to the full subcategory. In particular, the adjoint subcategory of a unitary category is unitary. For each ADE planar algebra, their idempotent completion is a unitary fusion category with the corresponding principal graph, as seen in the construction given in section 2 of [Edi18]. We will follow the standard notation and refer to the two + − distinct unitary E8 fusion categories as E8 and E8 . From the Dynkin diagrams for the ADE fusion categories we can easily read oﬀ

Mx1 , and note that in each case the Frobenius-Perron dimension of x1 is less than 2. In fact the unitary ADE fusion categories, together with some of the unitary

Ad A2N categories, are the only unitary categories where this is the case.

Theorem 6.9. Let C be a unitary fusion category generated by a symmetrically self-dual object x1 of dimension less than 2. Then C is the idempotent completion of one of the AN , Ad A2N , D2N , E6, or E8 planar algebras.

This theorem is well known to those working on subfactors, but translating from the language of subfactors to the language of fusion categories can be diﬃcult. A proof in the language of fusion categories is given as Theorem 2.0.3 in [Edi18].

6.1.5 Fib and Ad E8

Deﬁnition 6.10. A Fibonacci fusion category has two simple objects, 1 and τ, and satisﬁes the fusion rule ττ ∼= 1 ⊕ τ.

A classiﬁcation of Fibonacci categories can be found in [BD12], which shows that there are two possible associators for this fusion rule. Both associators have a unique pivotal structure, and we can distinguish these two categories by the cate- − + 82 CHAPTER 6. AN EXPLICIT EXAMPLE: Ad E8 ⊗Fib Ad E8

√ √ 1 1+ 5 1− 5 gorical dimension of τ, which is either 2 or 2 . Both of these categories are braided. We will write Fib for the Fibonacci category in which τ has categorical dimen- √ 1+ 5 sion 2 . This category is unitary, and so by Theorem 6.9 must be the unitary Ad(A4) fusion category. The Fibonacci category in which τ has categorical di- √ 1− 5 mension 2 is not unitary (indeed, if it were then by Theorem 6.9 it must be equivalent to Fib).

+ − The ﬁrst constructions of the unitary fusion categories E8 and E8 are given in [Izu94], which was one of the results needed for the subfactor literature predecessor of Theorem 6.9. Another construction, this time as a skein theory, is given in [Big10].

The fusion rules for Ad E8 fusion categories are given by principal graph

1 X Y τ

From this, we see that in Gr(Ad E8),

τ 2 = (XY − 2Y − X)τ = (X3 − 3X2 + 2)τ = 1 + τ.

Hence by the classiﬁcation of fusion categories with Fibonacci fusion rules, we see that the full subcategory of a unitary Ad E8 generated by 1 and τ is precisely Fib (since it will be unitary). + − In fact, this Fib subcategory lifts to the centre in both Ad E8 and Ad E8 by the following proposition, which is Lemma 3.1.3 in [Edi18].

Proposition 6.11. There are braided monoidal equivalences

+ rev Z(Ad E8 ) ' Ad D16 ⊗ Fib − rev Z(Ad E8 ) ' Ad D16 ⊗ Fib .

1The categorical dimension of an object x in a pivotal category C with pivotality isomorphism ψ is the scalar

evx∗ ◦(ψx ⊗ idx∗ ) ◦ coevx : k → k.

For unitary categories it agrees with the Frobenius-Perron dimension. − + 6.2. FUSION RULES FOR Ad E8 ⊗Fib Ad E8 83

Hence there are central functors

rev + Fib → Ad E8 − Fib → Ad E8 , and so we can construct − + Ad E8 ⊗Fib Ad E8 .

− + 6.2 Fusion Rules for Ad E8 ⊗Fib Ad E8

− + Remark 6.12. In this chapter, we make the assumption that Ad E8 ⊗Fib Ad E8 is a fusion category. We conjecture that if C, D, and V are fusion categories then

C ⊗V D will also be a fusion category, but we are not able to prove this at this − + time. The statement that Ad E8 ⊗Fib Ad E8 speciﬁcally is fusion could of course be proved without this; in §6.2.1 we describe precisely what is required to show this directly.

− + In this section, we calculate the fusion rules for Ad E8 ⊗Fib Ad E8 . When calculating the fusion rules, we can ignore the associators, and so will refer to both + − − + Ad E8 and Ad E8 as Ad E8. We will use C to denote Ad E8 ⊗Fib Ad E8 .

− + 6.2.1 Simple Objects of Ad E8 ⊗Fib Ad E8 We will begin by determining the simple objects in C. All objects in C are direct sums of pairs (x1, x2) where x1, x2 are simple objects in Ad E8, and so to ﬁnd the simple objects it suﬃces to consider these objects.

− + Lemma 6.13. There are eight simple objects in Ad E8 ⊗Fib Ad E8 , which are

1 := (1, 1) XL := (X, 1) XR := (1,X) X := (X,X)

τ := (1, τ) YL := (Y, 1) YR := (1,Y ) Y := (X,Y ).

Proof. Since Fib is semisimple with simple objects 1 and τ, we have by the argument in §3.1.2 that

C((x1, x2) → (x3, x4)) = Ad E8(x1 → x3) Ad E8(x2 → x4)

⊕ Ad E8(x1τ → x3) Ad E8(x2 → τx4). − + 84 CHAPTER 6. AN EXPLICIT EXAMPLE: Ad E8 ⊗Fib Ad E8

Where the xi are simple, we have that  C if x1 = x3 and x2 = x4 Ad E8(x1 → x3) Ad E8(x2 → x4) = (6.2) 0 otherwise. Similarly,

dim(Ad E8(x1τ → x3) Ad E8(x2 → τx4)) = [x1τ : x3][τx4 : x2]

= [τx1 : x3][τx4 : x2]. ∼ This last equality holds because there exists a half-braiding on τ, and so x1τ = τx1.

From the principal graph for Ad E8, we can determine that 0 1 0 0 1 1 0 0   Mτ =   0 0 0 1 0 0 1 1 where we order the simple objects in Ad E8 as 1, τ, X, Y . For clarity, we will also express this as a graph, which is

. 1 τ X Y

There are two things we learn from this immediately. Firstly, since [τxi : xj] is either 1 or 0 for all simple xi, xj, we must have that

Ad E8(x1τ → x3) Ad E8(x2 → τx4) = C or 0. (6.3) Secondly, we have that  C if x1, x2 ∈ {τ, Y } C((x1, x2) → (x1, x2)) = 0 otherwise.

Combining this with (6.2), we see that C((x1, x2) → (x3, x4)) is 2-dimensional if and only if

x1 = x3 = τ or Y

x2 = x4 = τ or Y, and is 0 or 1-dimensional otherwise. Hence we can form the table on the following page, which shows the dimension of the space of maps from the row-label to the column-label. From (6.2) we obtain a copy of C in every point along the main diagonal; those oﬀ the diagonal and the second copy of C in the 2-dimensional entries come from (6.3). Omitted entries represent a 0-dimensional hom-space. (1, 1) (1, τ)(τ, 1) (τ, τ) (X, 1) (Y, 1) (τ, X)(τ, Y ) (1,X) (1,Y )(X, τ)(Y, τ) (X,X)(X,Y )(Y,X)(Y,Y ) (1, 1) 1 1 (1, τ) 1 1 1 (τ, 1) 1 1 1 (τ, τ) 1 1 1 2 (X, 1) 1 1 (Y, 1) 1 1 1 (X, τ) 1 1 1 (Y, τ) 1 1 1 2 (1,X) 1 1 (1,Y ) 1 1 1 (τ, X) 1 1 1 (τ, Y ) 1 1 1 2 (X,X) 1 1 (X,Y ) 1 1 1 (Y,X) 1 1 1 (Y,Y ) 1 1 1 2 − + 86 CHAPTER 6. AN EXPLICIT EXAMPLE: Ad E8 ⊗Fib Ad E8

What can we say about a collection of objects A, B, C, D which have hom-spaces with the following dimensions? ABCD A 1 1 B 1 1 1 C 1 1 1 D 1 1 1 2 Firstly, an object in a semisimple category is simple if and only if its space of ∼ L ⊕ni endomorphisms is 1-dimensional, since if X = i xi as a direct sum of simples, then M M 2 ⊕ni ⊕ni ni C(X → X) = C(xi → xi ) = C . i i Hence A, B, and C must all be simple. Since B and C are simple, and C(B → ∼ C) = C, it must be the case that B = C. We then have [D : A] = 1 and [D : B] = [D : C] = 1, so it must be the case that

D ∼= A ⊕ B ∼= A ⊕ C.

Indeed, the calculation of the endomorphism space above shows that there can be no other simples in D, since the dimension of the endomorphism space of D is 2-dimensional. Hence in C we have precisely eight simple objects, which we will label

1 := (1, 1) XL := (X, 1) XR := (1,X) X := (X,X)

τ := (1, τ) YL := (Y, 1) YR := (1,Y ) Y := (X,Y ).

We also have the following decompositions into simples:

∼ ∼ ∼ ∼ (τ, 1) = τ (X, τ) = YL (τ, X) = YR (Y,X) = Y ∼ 1 ∼ ∼ ∼ (τ, τ) = ⊕ τ (Y, τ) = XL ⊕ YL (τ, Y ) = XR ⊕ YR (Y,Y ) = X ⊕ Y, which determine the decompositions into simples of all objects of C.

− + Remark 6.14. In this section we have assumed that Ad E8 ⊗Fib Ad E8 is semisimple. We could, however, show this directly simply by showing that all four of the 2-dimensional spaces of endomorphisms in the table above are isomorphic to C2 as an algebra (where the multiplication is given by composition), and then checking some simple compositions are nonzero. − + 6.2. FUSION RULES FOR Ad E8 ⊗Fib Ad E8 87

For example, since End(Y,Y ) is 2-dimensional we can choose a basis consist- ∗ ∗ ing of (Y, 1Fib,Y ) and (e , τ, e) for some particular e, e . Calculating the algebra structure then requires only the explicit calculation of the decomposition

Y Y Y Y Y Y e∗ τ e∗ τ e∗ . e = α +β e τ e Y Y Y Y Y Y In particular, we see that End(Y,Y ) is isomorphic to

2 C[z]/(z − βz − α), where z corresponds to (e∗, τ, e). We can then factorise

2 z − βz − α = (z − λ1)(z − λ2), and it is then straightforward to show (for example, by the Chinese remainder theorem for rings) that  C ⊕ C if λ1 6= λ2 End(Y,Y ) ∼= 2 C[z]/(z ) if λ1 = λ2. − + For Ad E8 ⊗Fib Ad E8 to be semisimple, we claim that we will get C ⊕ C. It is then a matter of further calculation to construct isomorphisms between our supposed simple objects. To see that (X,Y ) ∼= (Y,X), we would simply need to show that there exist f :(X,Y ) → (Y,X) and g :(Y,X) → (X,Y ) such that the compositions f ◦ g and g ◦ f are non-zero. Then since the hom-spaces between (X,Y ) and (Y,X) are 1-dimensional, there exists some scaling of f and g such that they are isomorphisms. To show that (Y,Y ) ∼= (X,X)⊕(X,Y ), consider maps f :(X,X) → (Y,Y ) and ∼ g :(Y,Y ) → (X,X). Since End(Y,Y ) = C ⊕ C as an algebra under composition, the composition f ◦ g must land wholly inside one of the C components. We can then make the same argument as in the previous paragraph. Doing the same calculation for all four 2-dimensional spaces of endomorphisms − + would show that Ad E8 ⊗Fib Ad E8 is semisimple.

− + 6.2.2 Calculating Fusion Rules of Ad E8 ⊗Fib Ad E8 We now know the simple objects in C, and so determining the fusion rules is a simple calculation. We will do the example of calculating MX, and then give the full result. − + 88 CHAPTER 6. AN EXPLICIT EXAMPLE: Ad E8 ⊗Fib Ad E8

From the principal graph of Ad E8, we can determine that the left-multiplication graphs for X, Y , and τ in Ad E8 are

2 X : 1 X Y τ

3 2 X Y : 1 Y τ

τ : . 1 τ X Y

We can therefore calculate that in Gr(C)

X1 = (X,X) Xτ = (Xτ, X) = X = (Y,X) = Y

2 XXL = (X ,X) XYL = (XY,X) = (1 + X + Y,X) = (X + 2Y + τ, X)

= XR + X + Y = X + 2Y + YR

2 XXR = (X,X ) XYR = (X,XY ) = (X, 1 + X + Y ) = (X,X + 2Y + τ)

= XL + X + Y = X + 2Y + YL

XX = (XX,XX) = (1 + X + Y, 1 + X + Y )

= 1 + XL + YL + XR + YR + 2X + 3Y

XY = (XX,XY ) = (1 + X + Y,X + 2Y + τ)

= τ + XL + 2YL + XR + 2YR + 3X + 5Y. − + 6.2. FUSION RULES FOR Ad E8 ⊗Fib Ad E8 89 and so the left-multiplication graph of X is given by the graph

2 XR 5 : 3 (6.4) X 1 X 2 Y τ

YR 2

YL Rather than provide the rest of the calculations, we will simply give the fusion rules.

− + Lemma 6.15. The fusion rules for Ad E8 ⊗Fib Ad E8 are given by the following left-multiplication graphs. We will omit the graph for 1, as it is the tensor unit.

τ : 1 τ XL YL XR YR X Y 2 2 X L 1 τ XL YL XR X Y YR

3 3 2 XL 2 X Y : L 1 YL τ XR Y YR 2 2 X R 1 τ XR YR XL X Y YL

3 3 2 XR 2 X Y : R 1 YR τ XL Y YL

2 XR 5 3 X : 1 X 2 Y τ

YR 2

YL − + 90 CHAPTER 6. AN EXPLICIT EXAMPLE: Ad E8 ⊗Fib Ad E8

8 2 3 3 YL 2 : Y 1 Y 3 YR 2 X 2

τ 5 One can then verify that these fusion rules are indeed associative.

− + 6.3 Ad E8 ⊗Fib Ad E8 as a Category of Modules

We now have suﬃcient theory that it is fairly easy to give an alternate deﬁnition of − + Ad E8 ⊗FibAd E8 . Recall from Proposition 3.11 that we have a dominant monoidal functor − + − + T : Ad E8 ⊗ Ad E8 → Ad E8 ⊗Fib Ad E8 . By our assumption in Remark 6.12 these are both fusion categories, and so by − Lemma 2.49 T has a right adjoint S. We see that S is exact because Ad E8 ⊗Fib + Ad E8 is semisimple, and is faithful because T is dominant. We then have the following proposition, which appears as Proposition 5.1 in [BN11]. Recall that a central algebra A in C is an algebra (A, σ) in Z(C).

Proposition 6.16. Let L : C → D be a linear monoidal functor between tensor categories over a ﬁeld k, admitting a right adjoint R. Then A = R(1D) has a natural structure of a commutative central algebra in C. Moreover, if R is faithful exact, then Cmod-A is a tensor category and we have a monoidal equivalence K :

D → Cmod-A such that the following triangle of monoidal functors commutes up to monoidal isomorphism: C L D

K FA

Cmod-A where FA is the free module functor for A.

This means that

− + Ad E8 ⊗Fib Ad E8 ' − +mod-S(1, 1), Ad E8 ⊗Ad E8 − + 6.3. Ad E8 ⊗Fib Ad E8 AS A CATEGORY OF MODULES 91 and T is the free module functor. We have

M − + S(1, 1) = (xi, xj) · Ad E8 ⊗Fib Ad E8 (F (xi, xj) → (1, 1)) i,j

− + where the sum is taken over the simple objects in Ad E8 ⊗ Ad E8 , which are + − precisely the pairs (xi, xj) where xi is simple in Ad E8 and xj is simple in Ad E8 . Noting that F is the identity on objects, we see from our previous calculation of the hom-spaces between such objects that  − + C if (xi, xj) = (1, 1) or (τ, τ) Ad E8 ⊗Fib Ad E8 ((xi, xj) → (1, 1)) = 0 otherwise.

Hence S(1, 1) = (1, 1) ⊕ (τ, τ).

Proposition 6.17. There is a monoidal equivalence

− + Ad E8 ⊗Fib Ad E8 ' − +mod-((1, 1) ⊕ (τ, τ)), Ad E8 ⊗Ad E8 and the standard dominant monoidal functor

− + − + T : Ad E8 ⊗ Ad E8 → Ad E8 ⊗Fib Ad E8 is the free module functor for (1, 1) ⊕ (τ, τ).

We will not explicitly calculate the commutative central algebra structure on (1, 1) ⊕ (τ, τ) here. This appears to be a previously unstudied fusion category (pending a veriﬁca- tion that it is indeed semisimple). The immediate problem for future study is of course this veriﬁcation; subsequent to that, we would like to calculate the centre − + of Ad E8 ⊗Fib Ad E8 and see whether we obtain another new fusion category.

A Adjunctions

We give a brief summary of adjunctions. For more details see Chapter 4 of [Mac98]. Definition A.1. An adjunction is a pair of functors L : C → D and R : D → C such that D(L(c) → d) ∼= C(c → R(d)) naturally in c ∈ C and d ∈ D. We say L is the left adjoint of R and R is the right adjoint of L, and write L a R. For f ∈ D(L(c) → d), we call the corresponding map in C(c → R(d)) the transpose of f and write f, and similarly for g ∈ C(c → R(d)). The naturality of the isomorphism amounts to the assertion that for any f : L(c) → d and k : d → d0, we have that k ◦ f = R(k) ◦ f, and for any g : c → R(d) and h : c0 → c, we have that g ◦ h = g ◦ L(h). We may make an alternative definition of an adjunction which is equivalent to the first. Definition A.2. An adjunction is a pair of functors L : C → D and R : D → C, together with natural transformations η : idC → RL and : LR → idD that satisfy the triangle identities Lη ηR L LRL R RLR

L and R . idL idR L R With this deﬁnition, we have the following result, which is useful when calculating transposes.

Proposition A.3. Let η : idC → RL be the unit of the adjunction, and : LR → idD the counit. Then • The transpose of f : X → R(Y ) is the composite L(f) L(X) −−→ LR(Y ) −→Y Y. • The transpose of g : L(X) → Y is the composite η R(g) X −→X RL(X) −−→ R(Y ).

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