Monoidal Categories Equivalences between non-strict and strict monoidal categories Braided monoidal categories Monoidal Categories and Monoidal Functors 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 ) 2 (C × C)0 we have an object V ⊗ W 2 C0. For any pair (f ; g) 2 (C × C)1 we have a morphism f ⊗ g 2 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 2 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 2 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 V f g◦f = W g Z 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 2 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 2 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 2 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 2 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 isomorp- hism 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 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. 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.