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A Few Comments About Monoidal Categories

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A Few Comments About Monoidal Categories A few comments about monoidal categories Angel Toledo April 30, 2019 Contents 1 Categories 1 2 Monoidal categories 4 2.1 Generalities . 4 2.2 Symmetry and linear algebra . 6 3 Tannakian reconstruction and a theorem of Deligne 8 1 Categories We will recall a few basic definitions and remarks about general (locally small) categories just sake of consistency and to fix the notation for the following sections. General references on category theory include [ML13] [Rie17] and for [Rot08] for abelian categories. Definition 1.1. A category C consists of the following data: 1. A collection Ob(C) of objects 2. For each two such objects X; Y 2 Ob(C), a set C(X; Y ) of morphisms 3. For every X 2 Ob(C, the set C(X; X) has at least one element 1X called the identity morphism on X 4. For every X; Y; Z 2 Ob(C) there is an operator ◦ : C(X; Y )×C(Y; Z) !C(X; Z) called composition which is associative and for which the identity morphisms act as identity elements, more explicitly we have: (a) ((f ◦ g) ◦ h) = (f ◦ (g ◦ h)) for h 2 C(X; Y ); g 2 C(Y; Z); f 2 C(Z; W ) (b) 1Y ◦ f = f and f ◦ 1X = f for f 2 C(X; Y ) Example 1.1. The following are well known examples of categories 1. Let C be a category, then the category Cop consisting of the same objects of C and Cop(X; Y ) := C(Y; X) is a category called the opposite category of C 2. The category Ens of sets consisting of sets as objects and functions of sets as morphisms 3. The category Top of topological spaces consisting of topological spaces as objects and continuous functions as morphisms 4. The category R-Mod of R-modules over a ring 5. A partial order (P; ≤) with P(x; y) = ∗ iff x ≤ y and P(x; y) = ; otherwise. Note. From now on we will write, for an object X, X 2 C instead of the correct but longer X 2 Ob(C). Being able to treat two objects as essentially the same will be of great importance, and so a way to abstract the notion of isomorphism or equivalence from the usual notions in known categories leads us to the following definition: Definition 1.2. A morphism f 2 C(X; Y ) is called an isomorphism if there exists g 2 D(Y; X) such that f ◦ g = IdY and g ◦ f = IdX While categories are very natural and flexible structures, its important to note that an essential tool in the theory and in its applications is the ability to compare categories in a certain way. The way in which we do this is by using functors between categories, which are nothing but structure preserving maps between the data that forms a category 1 Definition 1.3. A functor F between two categories C; D consists of an assignment F : Ob(C) ! Ob(D) and for every two objects X; Y 2 C a function F : C(X; Y ) !D(F(X); F(Y )) compatible with the composition. In other words: For f 2 C(X; Y ); g 2 C(Y; Z) we have (F(f ◦ g) = F(f) ◦ F(g) and F(1X ) = 1F(X) As with categories, functors occur very naturally all over mathematics in many ways. We have the following examples: Example 1.2. 1. Let C be a category, the identity functor 1C is the functor sending objects and morphisms to themselves. 2. Let C be a category, then for every object X 2 C there are functors C(X; ) and C(;X) acting on morphisms via adequate composition. 3. Let C = T op∗ the category of pointed topological spaces, then we can consider the fundamental group Π1 : C! Groups 4. The forgetful functor F : R − Mod ! Set is the functor sending an R-Module to its underlying set and a morphism of modules to its underlying function. Remark. The forgetful functor can be defined for every category C which has as objects sets with certain structure and structure preserving morphisms as morphisms. Definition 1.4. A functor F : Cop !D is called a contravariant functor from C to D Now that we have a way to compare categories we need to specify what we will require for two categories to be 'the same' in a categorical sense. There is a naive although valid definition: Definition 1.5. Let C and D be two categories, we will say that C and D are isomorphic if there exist functors F : C!D and G : D!C such that F ◦ G = IdD and G ◦ F = 1C Even if this definition makes sense it is a very rigid one and doesn't really capture what we want precisely to identify two categories as having the same categorical properties. To illustrate this we look into the following easily generalizable example: Example 1.3. Let k be a field and C = F inV ectk the category of finite dimensional vector spaces and linear transformations over k. Consider now the category Sk(C) with objects the vector spaces kn and linear transformations between them. It would make sense to think of this category as having the same categorical properties than the category C since every finite dimensional vector space is isomorphic to a vector space of the form kn, however it is not true that these two categories are isomorphic. To see this, suppose there are a pair of functors F : C! Sk(C) and G : Sk(C) !C) as in the definition and consider a n-dimensional vector space V , then F(V ) = km for some m and every endomorphism f : V ! V corresponds to an endomorphism km ! km. From linear algebra we deduce now that km = kn and so n=m, but then if we chose a different n-dimensional vector space V 0 we have F(V ) = F(V 0) and so G can't be an inverse. We thus need a weaker notion which still captures the categorical behaviour of categories but which is much less restrictive. Definition 1.6. Let F : C!D be a functor, we will say that it is faithful if the map C(X; Y ) !D(F(X); F(Y )) is injective. Similarly we will say that it is full if the map is surjective. We will say that F is essentially surjective on objects if for every A 2 D there exists X 2 C such that F(X) =∼ A Definition 1.7. We will say two categories C; D are equivalent if there exists a functor F : C!D which is full, faithful and essentially surjective. We will call this functor an equivalence. It is however not functors on their own that provide the full power of category theory but we also need to introduce a way to compare functors with each other. Definition 1.8. Let F; G : C!D be two functors, a natural transformation between F and G is a family of morphisms φX for every X 2 C such that the following diagram for every f 2 C(X; Y ) F(f) F(X) / F(Y ) φX φY G(f) G(X) / G(Y ) Natural transformations are also very common in nature, as we can see from the following example: 2 Example 1.4. Let C = F inV ectk the category of finite dimensional vector spaces over a field k, then there is a natural transformation between functor ∗∗ : C!C sending a vector space to its double dual and a linear transformation to its associated double dual transformation, and the identity functor. In fact this natural equivalence is a natural isomorphism meaning that all the morphisms φX are isomorphisms. This previous fact comes from unwrapping the definitions, as all we have to check is that for every linear transformation f : V ! W the linear transformation f ∗∗ : V ∗∗ ! W ∗∗ commutes with the isomorphisms V ! V ∗∗ given by sending a vector to its evaluation functional evv(f) := f(v). ∗∗ This is clear since f (evv) is the map sending a functional g : W ! k to evv(f ◦ g) = g(f(v)) = evf(v)(g) which is what we wanted to show. To illustrate fully the details of the definition, it is important to note that although a finite dimensional vector space is isomorphic to its dual vector space Hom(V; k), it is not true that this assignment is a natural transformation. Finally we introduce a general construction of inductive limits within categories which are a broad concept covering a lot of phenomena occurring within different categories as we will show through some examples. Definition 1.9. An initial ( resp. terminal ) object Z in a category C is an object such that jC(Z; X)j = 1 (resp jC(X; Z)j = 1 ) for every X 2 C If an object is both initial and terminal, we will say that it is a zero. As usual, these universal properties imply that these objects are unique up to isomorphism. Definition 1.10. A diagram in a category C is a functor F : D!C, if we want to make explicit the role of the category D we say that F is a D-shaped diagram. Definition 1.11. A cone (resp. cocone ) over an object X 2 C for a diagram F is a natural transformation between the constant functor CX : D!C sending every object to X and every morphism to the identity on X, and F ( resp. a natural transformation between F and the constant functor CX ).
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