Tensor Categories

Edric WangSupervised by Prof. Scott Morrison

Australian National University

Vacation Research Scholarships are funded jointly by the Department of Education and Training

and the Australian Mathematical Sciences Institute.

Abstract

In this report we lay out the basics of tensor categories and all of the concepts needed to define them.

Beginning with the definition of a category, we visit abelian categories, monoidal categories, tensor categories,

module categories and algebra objects. We end with a theorem of Ostrik which provides a powerful tool for

the classification of algebra objects.

Introduction

Category theory was first introduced in the context of algebraic topology to understand relations between

different mathematical structures. Category theory provides a framework which allows us to generalise the key

properties of mathematical objects such as sets, groups, topological spaces and so on. Tensor categories can

be thought of as categorical generalisations of vector spaces. They have a wide range of applications including

group representation theory, operator algebras, algebraic topology and algebraic geometry. In the same way

that a monoidal category can be seen as a category endowed with the structure of a monoid, a tensor category

can be seen as a category endowed with the structure of a ring.

Definition 1. A category C is:

• A collection Obj(C) of objects

• A collection HomC(A,B) of morphisms for each A,B ∈ Obj(C)

• A composition rule: if f ∈ HomC(A,B) and g ∈ HomC(B,C) then gf ∈ HomC(A,C)

such that composition is associative, and for all A ∈ Obj(C) we have a (two-sided) identity element 1A (with

respect to composition).

Definition 2. Let C and D be categories. A functor F : C → D is:

• A function F : Obj(C)→ Obj(D) such that F (1A) = 1F (A) for all A ∈ Obj(C)

• A function F : HomC(A,B) → HomD(F (A), F (B)) for all A,B ∈ C such that F (gf) = F (g)F (f) for

morphisms f and g whenever gf is defined

Definition 3. Let C and D be categories and let F : C → D and G : C → D be functors. A natural transformation

a : F → G is a morphism aA ∈ HomD(F (A), G(A)) for each A ∈ Obj(C) such that the following square

F (A) F (B)

G(A) G(B)

aA

F (f)

aB

G(f)

commutes for all f ∈ HomC(A,B).

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Remark 4. Let C and D be categories. There exists a category [C,D] whose objects are functors from C to D

and whose morphisms are the natural transformations between those functors.

Definition 5. A morphism f ∈ HomC(A,B) is an isomorphism if there exists a morphism g ∈ HomC(B,A)

with f ◦ g = 1B and g ◦ f = 1A.

Definition 6. A natural transformation a : F → G is a natural isomorphism if a is an isomorphism in [C,D].

Definition 7. Let C and D be categories and let F : C → D and G : D → C be functors. Then C and D are

equivalent if there exist natural isomorphisms η : 1C → G ◦ F and ε : F ◦ G → 1D. Such a functor (or pair of

functors) is called an equivalence between C and D.

Definition 8. An object A in a category C is initial if there is a unique morphism in HomC(A,B) for each

B ∈ Obj(C). An object A in a category C is initial if there is a unique morphism in HomC(B,A) for each

B ∈ Obj(C). An object A is a zero object if it is both initial and terminal.

Definition 9. Let A,B ∈ Obj(C). The product AuB is an object in C plus projection morphisms p and q such

that for all X ∈ Obj(C) there exists a unique morphism θ making the following diagram commute:

X

A A uB B

f gθ

p q

The coproduct A t B is an object in C plus injection morphisms α and β such that for all X ∈ Obj(C) there

exists a unique morphism θ making the following diagram commute:

X

A A tB B

f

α

θg

β

Remark 10. As solutions to universal mapping problems, the product and coproduct are unique up to unique

isomorphism.

Proof of the above remark for the product. Suppose X and X ′ are two solutions to the universal mapping prob-

lem. Then we have the following commutative diagrams:

X

A X ′ B

X

p q∃!θ

p′ q′

∃!ψp q

=⇒

X

A X B

p q∃!ψθ

p q

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but in the right hand diagram, the identity morphism 1X also makes the diagram commute. By the universal

mapping property of the product, this morphism should be unique. Hence ψθ = 1X . A similar argument shows

that θψ = 1X′ , hence θ : X → X ′ and ψ : X ′ → X are isomorphisms. But θ and ψ are unique by the universal

mapping property of the product.

Definition 11. A category C is additive if:

• HomC(A,B) is an abelian group under pointwise addition for each A,B ∈ Obj(C)

• Composition is distributive: for morphisms f , g and h we have h(f+g) = hf+hg and (f+g)h = fh+gh

whenever the compositions are defined

• C has a zero object

• A uB and A tB exist for all A,B ∈ Obj(C)

Definition 12. Let C be an additive category. Consider the following universal mapping problem: there exists

a unique θ such that the following diagram commutes:

X

K A B

θ r 0

i f

Let (K, i) a solution to the above problem. Then (K, i) is the kernel of f .

Consider the following universal mapping problem: there exists a unique ψ such that the following diagram

commutes:Y

B C Q

0

g

s

π

ψ

Let (Q, π) a solution to the above problem. Then (Q, π) is the kernel of g.

Remark 13. As solutions to universal mapping problems, the kernel and cokernel are unique up to unique

isomorphism.

Proof. This is proved analagously to the product and coproduct.

Definition 14. Let C be a category. A morphism f ∈ HomC(A,B) is monic (a monomorphism) if for all

C ∈ Obj(C) and for all g, h ∈ HomC(C,A), we have that fg = fh implies g = h. A morphism f ∈ HomC(A,B)

is epic (an epimorphism) if for all C ∈ Obj(C) and for all g, h ∈ HomC(B,C), we have that gf = hf implies

g = h.

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Definition 15. A category C is abelian if for all morphisms f ∈ HomC(A,B) there exists a sequence of mor-

phisms (a canonical decomposition)

Kk−→ A

i−→ Ij−→ Y

c−→ C

such that ji = f , (K, k) = ker f , (C, c) = coker f , (I, i) = coker k and (I, j) = ker c.

Definition 16. A monoidal category is:

• A category C

• A bifunctor (functor of two variables) ⊗ : C × C → C called the tensor product

• A natural isomorphism

aX,Y,Z : (X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z)

for all X,Y, Z ∈ Obj(C) called the associativity constraint

• An object 1 ∈ Obj(C) and an isomorphism ι : 1⊗ 1→ 1, together called a unit object

such that

• The diagram

((W ⊗X)⊗ Y )⊗ Z

(W ⊗ (X ⊗ Y ))⊗ Z (W ⊗X)⊗ (Y ⊗ Z)

W ⊗ ((X ⊗ Y )⊗ Z) W ⊗ (X ⊗ (Y ⊗ Z))

aW,X,Y ⊗idZ aW⊗X,Y,Z

aW,X⊗Y,Z aW,X,Y⊗Z

idQ⊗aX,Y,Z

commutes for all W,X, Y, Z ∈ Obj(C) (the pentagon axiom)

• The functors L1 : X → 1⊗X and R1 : X → X ⊗ 1 are autoequivalences of C.

Definition 17. Let C be a monoidal category. The left and right unit constraints lX : 1 ⊗ X → X and

rX : X ⊗ 1→ X are natural isomorphisms such that

1⊗ (1⊗X)a−11,1,X−−−−→ (1⊗ 1)⊗X ι⊗idX−−−−→ 1⊗X = L1(l1)

and

(X ⊗ 1)⊗ 1aX,1,1−−−−→ X ⊗ (1⊗ 1)

idX ⊗ι−−−−→ X ⊗ 1 = R1(r1)

Definition 18. Let k be a field. An additive category C is k-linear if HomC(A,B) is a k-vector space for all

A,B ∈ Obj(C) such that composition of morphisms is k-linear.

Definition 19. A k-linear abelian category C is locally finite if:

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• HomC(A,B) is finite dimensional for all A,B ∈ C, and

• The Jordan-Holder series of every object has finite length

Definition 20. Let C be a monoidal category. An object X∗ in C is a left dual of X if there exist evaluation

and coevaluation morphisms evX : X∗ ⊗X → 1 and coevX : 1→ X∗ ⊗X such that

XcoevX ⊗idX−−−−−−−−→ (X ⊗X∗)⊗X

aX,X∗,X−−−−−→ X ⊗ (X∗ ⊗X)idX ⊗ evX−−−−−−→ X = 1X

and

X∗idX∗ ⊗ coevX−−−−−−−−−→ X∗ ⊗ (X ⊗X∗)

a−1X∗,X,X∗−−−−−−→ (X∗ ⊗X)⊗X∗ evX ⊗ idX∗−−−−−−−→ X∗ = 1X∗

An object ∗X in C is a right dual of X if there exist evaluation and coevaluation morphisms ev′X : X ⊗ ∗X → 1

and coev′X : 1→ X ⊗ ∗X such that

XidX ⊗ coev′X−−−−−−−−→ X ⊗ (∗X ⊗X)

a−1X,∗X,X−−−−−→ (X ⊗ ∗X)⊗X ev′X ⊗ idX−−−−−−→ X = 1X

and

∗Xcoev′X ⊗ id∗X−−−−−−−−−→ (∗X ⊗X)⊗ ∗X

a∗X,X,∗X−−−−−−→ ∗X ⊗ (X ⊗ ∗X)id∗X ⊗ ev′X−−−−−−−→ ∗X = 1∗X

Definition 21. A rigid object in a monoidal category is one with both left and right duals. A rigid monoidal

category is one in which all objects are rigid.

Definition 22. Let k be an algebraically closed field. A multitensor category over k is a locally finite k-linear

abelian rigid monoidal category such that the bifunctor ⊗ : C × C → C is bilinear on morphisms. A multitensor

category is called a tensor category if EndC(1) ∼= k. A multifusion category is a finite semisimple multitensor

category. A fusion category is a finite semisimple tensor category.

Definition 23. Let C be a monoidal category. A left module category over C is:

• A category M

• A module product bifunctor ⊗ : C ×M→M

• A natural isomorphism (the module associativity constraint)

mX,Y,M : (X ⊗ Y )⊗M → X ⊗ (Y ⊗M)

for all X,Y ∈ C and for all M ∈M

such that the functor M 7→ 1⊗M is an autoequivalence of M, and the diagram

((X ⊗ Y )⊗ Z)⊗M

(X ⊗ (Y ⊗ Z))⊗M (X ⊗ Y )⊗ (Z ⊗M)

X ⊗ ((Y ⊗ Z)⊗M) X ⊗ (Y ⊗ (Z ⊗M))

mX,Y,Z⊗idM mX⊗Y,Z,M

mX,Y⊗Z,M mX,Y,Z⊗M

idZ ⊗mY,Z,M

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commutes for all X,Y, Z ∈ Obj(C) and for all M ∈ Obj(M). Right module categories are defined analogously.

Definition 24. Let C be a multitensor category. A module category over C is a locally finite abelian categoryM

over k which is a module category over C considered as a monoidal category and such that the module product

bifunctor ⊗ : C ×M→M is bilinear on morphisms and exact in the first variable.

Definition 25. Let C be a multitensor category. An algebra in C is:

• An object A ∈ Obj(C)

• A multiplication morphism m : A⊗A→ A

• A unit morphism u : 1→ A

such that the diagrams

(A⊗A)⊗A

A⊗ (A⊗A) A⊗A

A⊗A A

aA,A,A m⊗idA

idA⊗m m

m

1⊗A A

A⊗A A

u⊗idA

lA

idA

m

A⊗ 1 A

A⊗A A

idA⊗u

rA

idA

m

commute.

Definition 26. A right module over an algebra A is an object M ∈ Obj(C) and a morphism p : M ⊗ A → M

such that the diagrams

(M ⊗A)⊗A

M ⊗A M ⊗ (A⊗A)

M M ⊗A

p⊗idA aM,A,A

p idM ⊗m

p

M ⊗ 1 M

M ⊗A M

idM ⊗u

rM

idM

p

commute. Left modules are defined analogously.

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Definition 27. Let M be a module category over a multitensor category C. Let M1,M2 ∈ Obj(M). Then the

internal Hom Hom(M1,M2) is the object in C representing the functor X 7→ HomM(X ⊗M1,M2). That is,

HomM(X ⊗M1,M2) ∼= HomC(X,Hom(M1,M2))

is a natural isomorphism.

Theorem 28. There is a multiplication morphism

Hom(M2,M3)⊗Hom(M1,M2)→ Hom(M1,M3)

for all M1,M2,M3 ∈ Obj(M) and a canonical unit morphism uM : 1 → Hom(M,M) for all M ∈ Obj(M).

This makes Hom(M,M) an algebra object.

Theorem 29 (Theorem 1, [3]). Let M be a simple left (right) module category over a fusion category C and let

X be a simple object inM. ThenM is equivalent as a module category to the category of right (left) Hom(X,X)

modules in C.

The above theorem allows us to classify simple algebra objects and indecomposable module categories over

them in a given category. A subfactor is a unital inclusion of von Neumann algebras with trivial centres.

There is a correspondence between subfactors and algebra objects in tensor categories which allows to use these

algebraic tools to study subfactors. A more detailed picture is given in [2].

Conclusion

In this report we have outlined the basics of tensor categories and all of the relevant background, including basic

category theory, abelian categories and monoidal categories. We have also provided a view toward applications

of the theory of tensor categories.

References

[1] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor categories, volume 205. American

Mathematical Soc., 2016.

[2] Pinhas Grossman and Noah Snyder. Quantum subgroups of the haagerup fusion categories. Communications

in Mathematical Physics, 311(3):617–643, 2012.

[3] Victor Ostrik. Module categories, weak hopf algebras and modular invariants. Transformation Groups,

8(2):177–206, 2003.

[4] Joseph J Rotman. Advanced modern algebra, volume 114. American Mathematical Soc., 2010.

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