tensor categories edric wangtensor categories can be thought of as categorical generalisations of...
TRANSCRIPT
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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