Monoidal Categories and Monoidal Functors

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Ramón González Rodríguez

http://www.dma.uvigo.es/˜rgon/

Departamento de Matemática Aplicada II. Universidade de Vigo

Red Nc-Alg: Escuela de investigación avanzada en Álgebra no Conmutativa

Granada, 9-13 de noviembre de 2015

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories Outline

1 Monoidal Categories

2 Equivalences between non-strict and strict monoidal categories

3 Braided monoidal categories

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

1 Monoidal Categories

2 Equivalences between non-strict and strict monoidal categories

3 Braided monoidal categories

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. A tensor functor on C is a functor ⊗ from C × C → C. Note that this means that

For any pair (V , W ) ∈ (C × C)0 we have an object V ⊗ W ∈ C0.

For any pair (f , g) ∈ (C × C)1 we have a morphism f ⊗ g ∈ C1 such that

s(f ⊗ g) = s(f ) ⊗ s(g), t(f ⊗ g) = t(f ) ⊗ t(g).

If f 0, g 0 are morphisms in C such that s(f 0) = t(f ) and s(g 0) = t(g) we have

(f 0 ⊗ g 0) ◦ (f ⊗ g) = ((f 0 ◦ f ) ⊗ (g 0 ◦ g)).

idV ⊗W = idV ⊗ idW .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Also, if f : V → W is a morphism in C, − ⊗ f : − ⊗ V ⇒ − ⊗ W and f ⊗ − : V ⊗ − ⇒ W ⊗ − are natural transformations. For example,

(f ⊗ −)X = f ⊗ idX .

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Note that, if ⊗ is a tensor functor, for all V ∈ C0, − ⊗ V : C → C and V ⊗ − : C → C are functors. For example, − ⊗ V : C → C is deﬁned by

(− ⊗ V )(X ) = X ⊗ V

on objects and by (− ⊗ V )(f ) = f ⊗ idV on morphisms.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Note that, if ⊗ is a tensor functor, for all V ∈ C0, − ⊗ V : C → C and V ⊗ − : C → C are functors. For example, − ⊗ V : C → C is deﬁned by

(− ⊗ V )(X ) = X ⊗ V

on objects and by (− ⊗ V )(f ) = f ⊗ idV on morphisms.

Also, if f : V → W is a morphism in C, − ⊗ f : − ⊗ V ⇒ − ⊗ W and f ⊗ − : V ⊗ − ⇒ W ⊗ − are natural transformations. For example,

(f ⊗ −)X = f ⊗ idX .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Then V ⊗−

V ⊗− V ⊗− f ⊗−

g◦f ⊗− = (g⊗−)•(f ⊗−) = W ⊗− g⊗− Z⊗− Z⊗− Z⊗−

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

and in a reduced notation

V f

g◦f = W

g Z

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

On the other hand, if f : V → W , g : V 0 → W 0 are morphisms in C, for the correponding natural transformations f ⊗− : V ⊗− ⇒ W ⊗−, g⊗− : V 0⊗− ⇒ W 0⊗−,

((g ⊗ −) ∗ (f ⊗ −))X = (g ⊗ f ⊗ −)X .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

0 V ⊗− 0 V ⊗− V ⊗V ⊗−

(g⊗−)∗(f ⊗−) = g⊗− f ⊗− = g⊗f ⊗−

W 0⊗− W ⊗− W 0⊗W ⊗−

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

and in a reduced notation

V 0 V V 0⊗V

g f = g⊗f

W 0 W W 0⊗W

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. Let C be a category with a tensor functor ⊗. An associative constraint for ⊗ is a natural isomorphism a : ⊗ ◦ (⊗ × IC) ⇒ ⊗ ◦ (IC × ⊗). Then, for any triple (U, V , W ) of objects in C, there exists an isomorphism

aU,V ,W :(U ⊗ V ) ⊗ W → U ⊗ (V ⊗ W )

such that if f : U → U0, g : V → V 0, h : W → W 0 are morphisms in C,

(f ⊗ (g ⊗ h)) ◦ aU,V ,W = aU0,V 0,W 0 ◦ ((f ⊗ g) ⊗ h).

The associative constraint a satisﬁes the Pentagon Axiom if

(idU ⊗ aV ,W ,X ) ◦ aU,V ⊗W ,X ◦ (aU,V ,W ⊗ idX ) = aU,V ,W ⊗X ◦ aU⊗V ,W ,X ,

holds for all U, V , W , X ∈ C.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

We can write the previous equation in a more simple form, identifying the identities on objects wit the object. Then, in the following, for each object M in C, given objects M, N, P in C and a morphism f : M → N, we write P ⊗ f for idP ⊗ f and f ⊗ P for f ⊗ idP . As a consequence, the Pentagon Identity is

(U ⊗ aV ,W ,X ) ◦ aU,V ⊗W ,X ◦ (aU,V ,W ⊗ X ) = aU,V ,W ⊗X ◦ aU⊗V ,W ,X .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. Fixed an object K in a category C with a tensor functor ⊗, a left unit constraint with respect to K is a natural isomorphism

l : ⊗ ◦ (K × IC) ⇒ IC.

This means that for any object V ∈ C0 there exists an isomorphism

lV : K ⊗ V → V

such that, for any f : V → W in C1, the following identity holds:

lW ◦ (K ⊗ f ) = f ◦ lV .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. Fixed an object K in a category C with a tensor functor ⊗, a right unit constraint with respect to K is a natural isomorphism

r : ⊗ ◦ (IC × K) ⇒ IC.

This means that for any object V ∈ C0 there exists an isomorphism

rV : V ⊗ K → V

such that, for any f : V → W in C1, the following identity holds:

rW ◦ (f ⊗ K) = f ◦ rV .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. Let C be a category with a tensor functor ⊗. Given an associative constraint a, and left and right constraints l and r with respect to an object K, we say that they satisfy the Triangle Axiom if (V ⊗ lW ) ◦ aV ,K,W = rV ⊗ W , holds for all V , W ∈ C.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Example.

Let R be a commutative ring. Let M, N, P be objects in R M. Then, the isomorphism aM,N,P ((m ⊗ n) ⊗ p) = m ⊗ (n ⊗ p)

induces an associative constraint in R M. Also, for M the multiplication with R provides two R-bilinear maps

ml : R × M → M, ml (a, m) = am,

mr : M × R → M, mr (m, a) = ma, By the universal property of the tensor product, this maps can be transformed into a linear maps lM : R ⊗R M → M, lM (a ⊗ m) = am,

rM : M ⊗R R → M, rM (m ⊗ a) = ma, This maps are isomorphisms with inverses

−1 −1 lM : M → R ⊗R M, lM (m) = 1R ⊗ m,

−1 −1 rM : M → M ⊗R R, rM (m) = m ⊗ 1R . Then we have a left and a right constraints induced by the previous isomorphisms.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. A monoidal (or tensor) category (C, ⊗, K, a, l, r) is a category C which is equipped with a tensor product ⊗ : C × C → C, with an object K, called the unit of the monoidal category, with an associative constraint a, a left unit constraint l and a right unit constraint r with respect to K such that the Pentagon and Triangle axioms hold. The tensor category is said to be strict if the associativity and unit constraints are identities.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors If F is a ﬁeld, then (Vect(F), ⊗F, F, a, l, r) it is an example of non-strict monoidal category where the constraints are deﬁned as in the module setting.

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Examples

The most fundamental example of monoidal category is given by R M for R a commutative ring. In this case the constrains was deﬁned in the previous example. The tensor product is the tensor product of left R-modules and the unit object is R. Obviously it is not strict.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Examples

If F is a ﬁeld, then (Vect(F), ⊗F, F, a, l, r) it is an example of non-strict monoidal category where the constraints are deﬁned as in the module setting.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Let C be a category. The category of endofunctors of C is a strict monoidal category with the composition of functors, denoted by }, as the tensor product and the identity functor IC as the unit. We denote this category by End(C). The morphisms in End(C) are natural transformations between endofunctors and the composition is the vertical composition •. The tensor product of morphisms in End(C) is deﬁned by the horizontal composition of natural transformations.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Proposition. Let (C, ⊗, K, a, l, r) be a monoidal category. Then, the following identities hold:

lV ⊗ W = lV ⊗W ◦ aK,V ,W ,

rV ⊗W = (V ⊗ rW ) ◦ aV ,W ,K .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. Let (C, ⊗, K, a, l, r), (D, ⊗, I , a, l, r) be monoidal categories. A monoidal functor, or tensor functor, from C to D is a triple (F , Φ0, Φ−,−) where F : C → D is a functor, Φ0 : I → F (K) is an isomorphism and Φ−,− is a family of natural isomorphism indexed by all couples (U, V ) ∈ (C × C)0

Φ−,− = {ΦU,V : F (U) ⊗ F (V ) → F (U ⊗ W ); U, V ∈ C0}

such that the following identities hold:

ΦU,V ⊗W ◦(F (U)⊗ΦV ,W )◦aF (U),F (V ),F (W ) = F (aU,V ,W )◦ΦU⊗V ,W ◦(ΦU,V ⊗F (W )),

F (lu) ◦ ΦK,U ◦ (Φ0 ⊗ F (U)) = lF (U),

F (ru) ◦ ΦU,K ◦ (F (U) ⊗ Φ0) = rF (U),

for all objects U, V , W ∈ C0. The monoidal functor is strict if Φ0 and ΦU,V are all identities for any U, V ∈ C0.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. Let (C, ⊗, K, a, l, r), (D, ⊗, I , a, l, r) be monoidal categories and let the triples 0 0 0 (F , Φ0, Φ−,−), (F , Φ0, Φ−,−) monoidal (tensor) functors from C to D. A natural mo- 0 0 0 noidal (tensor) transformation η :(F , Φ0, Φ−,−) ⇒ (F , Φ0, Φ−,−) is a natural transformation η : F ⇒ F 0 such that the following identities hold for each couple (U, V ) of objects in C, 0 ηK ◦ Φ0 = Φ0, 0 ΦU,V ◦ (ηU ⊗ ηV ) = ηU⊗V ◦ ΦU,V .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Deﬁnition. A monoidal (tensor) equivalence between monoidal (tensor) categories is a monoidal (tensor) functor F : C → D such that there exists a monoidal (tensor) functor G : D → C and natural monoidal (tensor) isomorphisms η : ID ⇒ F ◦ G and θ : G ◦ F ⇒ IC. In the case there exists a monoidal (tensor) equivalence between C and D we say that C and D are monoidal (tensor) equivalent.

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. A natural monoidal (tensor) isomorphism is a natural monoidal (tensor) transformation that is also a natural isomorphism.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. A natural monoidal (tensor) isomorphism is a natural monoidal (tensor) transformation that is also a natural isomorphism.

Deﬁnition. A monoidal (tensor) equivalence between monoidal (tensor) categories is a monoidal (tensor) functor F : C → D such that there exists a monoidal (tensor) functor G : D → C and natural monoidal (tensor) isomorphisms η : ID ⇒ F ◦ G and θ : G ◦ F ⇒ IC. In the case there exists a monoidal (tensor) equivalence between C and D we say that C and D are monoidal (tensor) equivalent.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

1 Monoidal Categories

2 Equivalences between non-strict and strict monoidal categories

3 Braided monoidal categories

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Let S be the classes of all ﬁnite sequences S = (V1,..., Vk ) of objects of C, including the empty sequence ∅. The integer k is by deﬁnition the length of the sequence

S = (V1,..., Vk )

and the length of the empty sequence is 0 by convention. 0 If S = (V1,..., Vk ), S = (Vk+1,..., Vk+n) are nonempty sequences of S, we denote by S ∗ S0 the sequence,

0 S ∗ S = (V1,..., Vk , Vk+1,..., Vk+n)

obtaining placing S0 after S. We also agree that

S ∗ ∅ = ∅ ∗ S = S

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Let (C, ⊗, K, a, l, r) be a monoidal category. In this section we construct a strict monoidal category Cstr which is tensor equivalent to C.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Let (C, ⊗, K, a, l, r) be a monoidal category. In this section we construct a strict monoidal category Cstr which is tensor equivalent to C.

Let S be the classes of all ﬁnite sequences S = (V1,..., Vk ) of objects of C, including the empty sequence ∅. The integer k is by deﬁnition the length of the sequence

S = (V1,..., Vk )

and the length of the empty sequence is 0 by convention. 0 If S = (V1,..., Vk ), S = (Vk+1,..., Vk+n) are nonempty sequences of S, we denote by S ∗ S0 the sequence,

0 S ∗ S = (V1,..., Vk , Vk+1,..., Vk+n)

obtaining placing S0 after S. We also agree that

S ∗ ∅ = ∅ ∗ S = S

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

To any sequence S in S, we assign and object F (V ) of C deﬁned inductively by

F (∅) = K, F ((V )) = V , F (S ∗ (V )) = F (S) ⊗ V .

Then, if S = (V1,..., Vk ),

F (S) = ((... (V1 ⊗ V2) ⊗ ... ) ⊗ Vk−1) ⊗ Vk

where all opening parentheses are placed on the left hand side of V1. The category Cstr is deﬁned in the following way: str C0 = S, 0 0 HomCstr (S, S ) = HomC(F (S), F (S )), This deﬁne a category whose identities and composition is taken from C.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Proof. The map F defined above on the objects of Cstr extends to a a functor F : Cstr → C which is the identity on morphisms. Then it is fully faithful. As any object in C is clearly isomorphic to the image under F of a sequence of length one, we see that F is essentially surjective. Then, we obtain the result. Observe that G(V ) = (V ) defines a functor G : C → Cstr which is the inverse equivalence of F . We have F ◦G = IC and θ : GF ⇒ idCstr defined by the natural isomorphism

θS = idF (S) : G(F (S)) → S.

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Theorem. The categories C and Cstr are equivalent.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Theorem. The categories C and Cstr are equivalent.

Proof. The map F defined above on the objects of Cstr extends to a a functor F : Cstr → C which is the identity on morphisms. Then it is fully faithful. As any object in C is clearly isomorphic to the image under F of a sequence of length one, we see that F is essentially surjective. Then, we obtain the result. Observe that G(V ) = (V ) defines a functor G : C → Cstr which is the inverse equivalence of F . We have F ◦G = IC and θ : GF ⇒ idCstr defined by the natural isomorphism

θS = idF (S) : G(F (S)) → S.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors To deﬁne the tensor product of morphisms, we ﬁrst construct a natural isomorphism

ϕ−,− : ⊗ ◦ (F (−) × F (−)) ⇒ F (− } −) The isomorphism is deﬁned on the length of the sequence S. First we set

ϕ∅,S = lF (S), ϕS,∅ = rF (S).

Next ϕS,(V ) = idF (S)⊗V : F (S) ⊗ V → F (S } (V )) and −1 ϕS,S0∗(V ) = (ϕS,S0 ⊗ V ) ◦ aF (S),F (S0),V .

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

We now equip Cstr with the structure of strict monoidal category. We deﬁne the tensor product of objects in Cstr by 0 0 S } S = S ∗ S . This tensor product is associative on objects and if I = ∅

I } S = S = S } I . Then in this case the unit constraints are identities.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

I } S = S = S } I . Then in this case the unit constraints are identities.

To deﬁne the tensor product of morphisms, we ﬁrst construct a natural isomorphism

ϕ−,− : ⊗ ◦ (F (−) × F (−)) ⇒ F (− } −) The isomorphism is deﬁned on the length of the sequence S. First we set

ϕ∅,S = lF (S), ϕS,∅ = rF (S).

Next ϕS,(V ) = idF (S)⊗V : F (S) ⊗ V → F (S } (V )) and −1 ϕS,S0∗(V ) = (ϕS,S0 ⊗ V ) ◦ aF (S),F (S0),V .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

0 0 0 0 We can now define the tensor product f }f of two morphisms f : S → T , f : S → T of Cstr . By definition f is a morphism from F (S) to F (T ) and f 0 is another one from 0 0 0 F (S ) to F (T ) in C. We define the tensor product f } f by 0 0 −1 f } f = ϕT ,T 0 ◦ (f ⊗ f ) ◦ ϕS,S0

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Proof. str The triple (F , idK , ϕ−,−) is a monoidal functor from C to C where ϕ−,− is the natural isomorphism deﬁned above. It induces a monoidal equivalence with the functor G deﬁned previously.

Then, every non-strict monoidal category is monoidal equivalent to a strict one. Then, in general, when we work with monoidal categories, we can assume without loss of generality, that the category is strict. This lets us to treat monoidal categories as if they were strict and, as a consequence, the results proved in this course hold for every non-strict symmetric monoidal category.

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Theorem. The categories C and Cstr are monoidal equivalent.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Then, every non-strict monoidal category is monoidal equivalent to a strict one. Then, in general, when we work with monoidal categories, we can assume without loss of generality, that the category is strict. This lets us to treat monoidal categories as if they were strict and, as a consequence, the results proved in this course hold for every non-strict symmetric monoidal category.

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Theorem. The categories C and Cstr are monoidal equivalent.

Proof. str The triple (F , idK , ϕ−,−) is a monoidal functor from C to C where ϕ−,− is the natural isomorphism deﬁned above. It induces a monoidal equivalence with the functor G deﬁned previously.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Theorem. The categories C and Cstr are monoidal equivalent.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

1 Monoidal Categories

2 Equivalences between non-strict and strict monoidal categories

3 Braided monoidal categories

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Let C be a category with a tensor product ⊗ : C × C → C and an associative constraint. Denote by τ : C × C → C × C the ﬂip functor deﬁned by τ(U, V ) = (V , U) for any pair of objects in the category and by τ(f , g) = (g, f ) for any pair of morphisms in C.A commutativity constraint c is a natural isomorphism

c : ⊗ ⇒ ⊗ ◦ τ.

This means that, for any couple (V , W ) of objects in C, we have an isomorphism

cV ,W : V ⊗ W → W ⊗ V

such that (g ⊗ f ) ◦ cV ,W = cV 0,W 0 ◦ (f ⊗ g) for all morphisms f : V → V 0 and g : W → W 0. Then, we have two natural isomorphisms cV ,− : V ⊗ − ⇒ − ⊗ V , c−,W : − ⊗ W ⇒ W ⊗ −.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

We will say that the commutative constraint satisﬁes the Hexagon Axiom if

(V ⊗ cU,W ) ◦ aV ,U,W ◦ (cU,V ⊗ W ) = aV ,W ,U ◦ cU,V ⊗W ◦ aU,V ,W ,

and −1 −1 −1 aW ,U,V ◦ cU⊗V ,W ◦ aU,V ,W = (cV ,W ⊗ W ) ◦ aU,W ,V ◦ (U ⊗ cV ,W ),

holds for any objects U, V , W ∈ C0.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Deﬁnition. Let (C, ⊗, K, a, l, r) be a monoidal category. A braiding in C is a commutative constraint satisfying the Hexagon Axiom. A braided monoidal category is a monoidal category with a braiding c. We will denote it by (C, ⊗, K, a, l, r, c). Note that, if C is strict, the Hexagon Axiom are the identities

cU,V ⊗W = (V ⊗ cU,W ) ◦ (cU,V ⊗ W ),

and cU⊗V ,W = (cU,W ⊗ W ) ◦ (U ⊗ cV ,W ):

If (C, ⊗, K, a, l, r, c) is a braided monoidal category and cV ,U ◦ cU,V = idU⊗V for all U, V in C0, we will say that (C, ⊗, K, a, l, r, c) is a symmetric monoidal category.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Remember that each morphism f : V → W in C deﬁnes two natural transformations f ⊗− : V ⊗− → W ⊗− −⊗f : −⊗V → −⊗W . Using these natural transformations, in a strict setting, the naturality of cU,V ⊗ − can be expressed by the picture:

V ⊗− V ⊗−

f ⊗− f ⊗− U⊗− V ⊗− U⊗− W ⊗− W ⊗− U⊗− V ⊗− U⊗− = =

c ⊗− cU,V ⊗− cV ,W ⊗− cW ,U ⊗− V ,U V ⊗− U⊗− W ⊗− U⊗− U⊗− W ⊗− U⊗− V ⊗− f ⊗− f ⊗− W ⊗− W ⊗−

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

or by

V V

f f UV UW U UWV = =

c cU,V cV ,W cW ,U V ,U

VU WU UW UV f f

W W

−1 −1 In an analogous form we can express the naturality of −⊗cV ,U , cU,V ⊗− and −⊗cV ,U .

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With this language, the Hexagon Axiom can be represented by

U⊗− V ⊗W ⊗− U⊗− V ⊗− W ⊗− U⊗V ⊗− W ⊗− U⊗− V ⊗− W ⊗− = =

cU,V ⊗W ⊗− cU,V ⊗− cU⊗V ,W ⊗− cV ,W ⊗− V ⊗W ⊗− U⊗− V ⊗− U⊗− W ⊗− U⊗V ⊗− W ⊗− V ⊗− cU,W ⊗− cU,W ⊗− W ⊗− U⊗− W ⊗− U⊗−

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or by

U V ⊗W UVW U⊗V W UVW = =

cU,V ⊗W cU,V cU⊗V ,W cV ,W

V ⊗W U VU W U⊗V WV cU,W cU,W

WU WU

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Note that by the naturality of cU⊗V ,W ⊗ − we obtain the identity (called the Yang- Baxter identity):

U⊗V ⊗−

cU,V ⊗−

V ⊗U⊗− W ⊗− U⊗− V ⊗− W ⊗− U⊗V ⊗− W ⊗− U⊗− V ⊗− W ⊗− = = =

cV ⊗U,W ⊗− cU,V ⊗− cU⊗V ,W ⊗− cV ,W ⊗− W ⊗− V ⊗U⊗− V ⊗− U⊗− W ⊗− U⊗V ⊗− W ⊗− V ⊗− cU,W ⊗− cU,W ⊗− cU,V ⊗− W ⊗− U⊗− V ⊗U⊗− W ⊗− U⊗−

cV ,W ⊗− cU,V ⊗− W ⊗− V ⊗− V ⊗− U⊗−

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or U⊗V

cU,V

V ⊗U W UVW U⊗V W UVW = = =

cV ⊗U,W cU,V cU⊗V ,W cV ,W

W V ⊗U VU W U⊗V WV cU,W cU,W cU,V

WU V ⊗U WU cV ,W cU,V

WV VU

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Then, as a consequence, the linear form of the Yang-Baxter equation is:

(cV ,W ⊗ U) ◦ (V ⊗ cU,W ) ◦ (cU,V ⊗ W ) = (W ⊗ cU,V ) ◦ (cU,W ⊗ V ) ◦ (U ⊗ cV ,W ).

The non-strict version is

−1 aW ,V ,U ◦ (cV ,W ⊗ U) ◦ aV ,W ,U ◦ (V ⊗ cU,W ) ◦ aV ,U,W ◦ (cU,V ⊗ W )

−1 = (W ⊗ cU,V ) ◦ aW ,U,V ◦ (cU,W ⊗ V ) ◦ aU,W ,V ◦ (U ⊗ cV ,W ) ◦ aU,V ,W .

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An anologous calculus can be make for the functors − ⊗ U and then we obtain the Yang-Baxter identity for them: −⊗U −⊗V −⊗W −⊗U −⊗V −⊗W =

−⊗cV ,U −⊗cW ,V −⊗V −⊗U −⊗W −⊗V −⊗cW ,U −⊗cW ,U −⊗W −⊗U −⊗W −⊗U

−⊗cW ,V −⊗cV ,U −⊗W −⊗V −⊗V −⊗U

Recall that the Yang-Baxter identity can be obtained, in a similar way, for the natural −1 −1 −1 isomorphisms c−,U , cU,−, cU⊗V ,−, etc.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

On the other hand, by the naturality, we have the next identity U⊗− V ⊗− W ⊗− U⊗− V ⊗− W ⊗− U⊗− V ⊗− W ⊗− = =

−1 −1 c ⊗− cV ,U ⊗− cV ,U ⊗− V ,W V ⊗− U⊗− V ⊗− U⊗− W ⊗− V ⊗− cU,W ⊗− cU,W ⊗− cU,W ⊗− W ⊗− U⊗− W ⊗− U⊗− W ⊗− U⊗− c ⊗ −1 V ,W cV ,U ⊗− W ⊗− V ⊗− V ⊗− U⊗− −1 c−1 ⊗− cV ,W ⊗− V ,W V ⊗− W ⊗− V ⊗− W ⊗−

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That admits the following reduced version:

UVW UVW UVW = =

−1 −1 c cV ,U cV ,U V ,W

VU VU WV cU,W cU,W cU,W

WU WU WU c −1 V ,W cV ,U

WV VU −1 c−1 cV ,W V ,W

VWVW

and also the similar equalities for the natural transformations obtained when we change −1 c by c and c−,− ⊗ − by − ⊗ c−,−.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Deﬁnition. Let (C, ⊗, K, a, l, r, c), (D, ⊗, I , a, l, r, c) be braided monoidal categories. A monoidal functor (F , Φ0, Φ−,−) from C to D is braided if, for any pair (U, V ) of objects of C, the following equality holds:

ΦV ,U ◦ cF (U),F (V ) = F (cU,V ) ◦ ΦU,V .

Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Theorem. If (C, ⊗, K, a, l, r, c) is a braided monoidal category, (C, ⊗, K, a, l, r, c−1) also is.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

Theorem. If (C, ⊗, K, a, l, r, c) is a braided monoidal category, (C, ⊗, K, a, l, r, c−1) also is.

Deﬁnition. Let (C, ⊗, K, a, l, r, c), (D, ⊗, I , a, l, r, c) be braided monoidal categories. A monoidal functor (F , Φ0, Φ−,−) from C to D is braided if, for any pair (U, V ) of objects of C, the following equality holds:

ΦV ,U ◦ cF (U),F (V ) = F (cU,V ) ◦ ΦU,V .

Ramón González Rodríguez Monoidal Categories and Monoidal Functors Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories

References: J.N. Alonso Alvarez, J.M. Fernández Vilaboa, R. González Rodríguez, M.P. López López y E. Villanueva Novoa, A Picard-Brauer ﬁve term exact sequence for braided categories, en Rings, Hopf Algebras and Brauer Groups, eds. S. Caenepeel, A. Verschoren. Lecture Notes in Pure and Applied Mathematics (Blue Series), Marcel Dekker Inc., 197, 11-41, 1998. C. Kassel, Quantum Groups, GTM 155, Springer-Verlag, New York, 1995. M.P. López López, Álgebras de Hopf respecto a un cotriple, Álxebra 17, Universidad de Santiago de Compostela, 1976.

Ramón González Rodríguez Monoidal Categories and Monoidal Functors