sign in sign up
<<
Home , Category of modules, Closed category, Compact closed category, Endomorphism, Monoidal category, Monoidal functor

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 ...... 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 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 ∈ Ob(C), a C(X,Y ) of

3. For every X ∈ Ob(C, the set C(X,X) has at least one element 1X called the identity on X 4. For every X,Y,Z ∈ 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 ∈ C(X,Y ), g ∈ C(Y,Z), f ∈ C(Z,W )

(b) 1Y ◦ f = f and f ◦ 1X = f for f ∈ 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 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 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 ∈ C instead of the correct but longer X ∈ 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 or equivalence from the usual notions in known categories leads us to the following definition:

Definition 1.2. A morphism f ∈ C(X,Y ) is called an isomorphism if there exists g ∈ 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 between categories, which are nothing but structure preserving maps between the data that forms a category

1 Definition 1.3. A F between two categories C, D consists of an assignment F : Ob(C) → Ob(D) and for every two objects X,Y ∈ C a F : C(X,Y ) → D(F(X), F(Y )) compatible with the composition. In other words: For f ∈ C(X,Y ), g ∈ 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 ∈ 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 Π1 : C → Groups 4. The F : R − Mod → Set is the functor sending an R- 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 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 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 ∈ D there exists X ∈ 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 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 between F and G is a family of morphisms φX for every X ∈ C such that the following diagram for every f ∈ 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 . 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 |C(Z,X)| = 1 (resp |C(X,Z)| = 1 ) for every X ∈ 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 ∈ 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 ).

Its perhaps better to provide a diagram illustrating the previous definition, the following is a cocone over an object for a diagram F : D → C: 1 / 2 C / C O 1 O 2

⇒F

  0 C0 A cone over X for F: C / C > 1 6 O 2

X

(  C0

Definition 1.12. An object lim← ( resp. lim→) is called a ( resp. colimit ) for a diagram F if there is a cone over it for the diagram F such that every other cone ( resp. cocone ) factors uniquely through it. By this we mean: ( C / C C )/ C ? 1 7 O 2 = 1 6 O 2

* X limF ←

'  )(  C0 C0 A similar diagram results in the case of colimits.

3 Example 1.5. Consider a discrete category D, that is a category where the only morphisms are the identity morphisms. Then the in C ( resp. ) is the limit (resp. colimit ) of a diagram F : D → C Example 1.6. In a similar way, we can see that initial and terminal objects are a kind of limits, as well as pullbacks and pushforwards. We will leave the reader to find the appropriate diagrams for these. While not all categories have all limits and colimits, it is possible to construct a bigger category which has them formally for different kind of families of diagrams, although we will only mention the case covering colimits for finite diagrams in the following construction, there is of course more flexibility than this in the construction. Definition 1.13. Let C be a category, we will construct the category Ind(C) of ind-objects as the category whose objects are finite diagrams F : D → C and for two such diagrams F : D → C, G : E → C we write Ind(C)(F, G) := limlimC(F(d), G(e)). ← → Where the colimit is calculated over the objects e ∈ E and the limit is calculated over objects d ∈ D The following is a property that we will often desire for our categories from now own, we wont motivate or mention examples of categories satisfying these properties but we trust that this is not an issue as some of the categories mentioned so far are prototypical examples. Definition 1.14. Let k be a field. We will say that a category C is k-linear if C(X,Y ) is a vector space and is closed under countable .

2 Monoidal categories 2.1 Generalities Abstract categories cover a very wide array of situations that can be found on nature, and as such can have many interpre- tations depending where the theory is being born and where do we want to apply it. So far we have perhaps parted from the idea that categories provide a general setting in which we can study whole families of mathematical objects and relate them to each other. This is however not the only source or motivation to keep in mind while we do category theory as we can think about categories coming from a topological or homotopical perspective, or an algebraic one. A good reference on the generalities of monoidal categories is [EGNO16]. To illustrate we give the following example: Example 2.1. A category C is a if every morphism is an isomorphism It is not hard to see that a group is then nothing but a groupoid with one object. In fact let G be an object, then we can construct a category BG whose only object we denote by ∗ and BG(∗, ∗) = G, we declare then composition to be the group operation and it is routine to check that this indeed forms a category. So if we can think of as generalized groups ( and categories perhaps as generalized ), it is perhaps possible to come up with a categorized version of a ring.

Definition 2.1. A is the following data: 1. A category C 2. A ⊗ : C × C → C

3. An object 1C called the unit

4. For every X,Y,Z ∈ C a isomorphism αX,Y,Z : X ⊗ (Y ⊗ Z) → (X × Y ) ⊗ Z

5. Isomorphisms λ : 1C ⊗ X → X and ρ : X ⊗ 1C → X Making the following diagrams commute

(X ⊗ (Y ⊗ Z)) ⊗ W / X ⊗ ((Y ⊗ Z) ⊗ W ) 4

((X ⊗ Y ) ⊗ Z) ⊗ W

*  (X ⊗ Y ) ⊗ (Z ⊗ W ) / (X ⊗ (Y ⊗ (Z ⊗ W )))

4 And

(X ⊗ 1C) ⊗ Y / X ⊗ (1C ⊗ Y )

' w X ⊗ Y Remark. What we mean by ⊗ : C × C → C being a functor is that it is a functor in the C × C which is, by definition, the disjoint union of the categories.

It is now possible to realize a ring as a special kind of monoidal category, more specificly we have : Example 2.2. Let R be a , the category MR is then a monoidal category whose underlying category is the groupoid BR as an and whose tensor ⊗ is the multiplication of R on morphisms. There are of course plenty of other naturally occurring monoidal categories:

Example 2.3. 1. The category Ens of sets with the as a tensor. 2. The category R-Mod of modules over a commutative ring with the usual . 3. The category of chain complexes over a commutative ring. Now that we have this extra structure on our categories it would be good, just as with regular categories, to be able to compare them through functors and compare the functors themselves. We arrive then to the apropriate definitions: Definition 2.2. A F : C → D between monoidal categories is a functor with natural isomorphisms ∼ ∼ F(X ⊗ Y ) = F(X) ⊗ F(Y ), F(1C = 1F making the following diagrams commutative:

F(X ⊗ Y ) ⊗ F(Z) / F((X ⊗ Y ) ⊗ Z)) / F(X ⊗ (Y ⊗ Z)) O O

(F(X) ⊗ F(Y )) ⊗ F(Z) / F(X) ⊗ (F(Y ) ⊗ F(Y )) / F(X) ⊗ F(Y ⊗ Z))

And 1D ⊗ F(X) / F(1C) ⊗ F(X) F(X) ⊗ 1D / F(X) ⊗ 1C

λ ρ     F(X) / F(1C ⊗ X) F(X) / F(X ⊗ 1C)

Remark. Often the previous definition is refered as a strong monoidal functor and monoidal functors just require natural transformations F(X ⊗ Y ) → F(X) ⊗ F(Y ) and F(1F → 1D Definition 2.3. Let F, G : C → D be two monoidal functors between monoidal categories, a monoidal natural transfor- mation is a family of morphisms φX for each X ∈ C such that it is a natural transformation and the following diagrams commute: φX ⊗φY F(X) ⊗ F(Y ) / G(X) ⊗ G(Y ) 1D

 φ $  φX⊗Y  1C F(X ⊗ Y ) / G(X ⊗ Y ) F(1C) / G(1C)

Example 2.4. 1. Let R, S be commutative rings, then a monoidal functor between categories MR → MS is a morphism of rings.

2. Let K/k be a field extension, then the functor from V ectk to V ectK is a monoidal functor Example 2.5. The double dual construction of finite dimensional vector spaces is a monoidal natural transformation One last thing to say about monoidal categories in this generalty is that they give a context in which it is possible to do algebra in a very general way. By this we mean that it is possible to define and monoids over them just as we do in classical mathematics.

5 Definition 2.4. A in a monoidal category C is an object X ∈ C and a couple of morphisms µ : X ⊗ X → X, u:1C → X such that the following diagrams commute:

id⊗µ u⊗id (X ⊗ X) ⊗ X / X ⊗ (X ⊗ X) / X ⊗ X , 1C ⊗ X / X ⊗ X o X ⊗ 1C id⊗u λ µ µ ρ  µ  %  y X ⊗ X / X X Definition 2.5. Let C be a monoidal category and R,S ∈ C two monoids over C then a morphism of monoids is a morphism f : R → S such that the following diagrams commute

f⊗f R ⊗ R / S ⊗ S 1C / R

s  f   R / S S Example 2.6. 1. In the monoidal category Ens, of sets, a monoid coincides with the usual notion. A morphism of monoids coincides with the usual definition too.

2. In the monoidal category of Z-Modules, a monoid is a ring. A morphism of monoids is a morphism of rings. Just as in the traditional , it is possible to study the over a monoid. Definition 2.6. Let C be a monoidal category and R ∈ C a monoid over C, a left module over R is an object M ∈ C and a morphism ∆ : R ⊗ M → M making the following diagrams commutative: R ⊗ R ⊗ M / R ⊗ M

  R ⊗ M / M Definition 2.7. A morphism of monoids f : M → N over a monoid R ∈ C in a monoidal category is a morphism satisfying the following diagram R ⊗ M / R ⊗ N

 f  M / N Example 2.7. 1. Let X be a scheme, then the category of abelian sheaves over X is a monoidal category and the structure sheaf OX is a monoid over it and the category of OX − Modules is the category of modules over it. 2. Let C be the category of chain complexes over a ring R. Then a monoid A over C is a dg-algebra and a module over A is a dg-module.

2.2 Symmetry and linear algebra It is important to note that we are not giving any condition on the ’commutativty’ of the tensor product and in principle there is no reason to expect any kind of symmetry when interchanging the inputs. It is however common in nature that some kind of relation arises. We want to consider the following condition on the symmetry of the inputs Definition 2.8. A monoidal category C is said to be a symmetric monoidal category if there is a natural isomorphism ∼ βX,Y : X ⊗ Y = Y ⊗ X satisfying βX,Y ◦ βY,X = IdY ⊗X and making the following diagrams commute:

β 1C ⊗ X / X ⊗ 1C

λ ρ  y X And (X ⊗ Y ) ⊗ Z / (Y ⊗ X) ⊗ Z / Y × (X ⊗ Z)

  X ⊗ (Y ⊗ Z) / (Y ⊗ Z) ⊗ X / Y ⊗ (Z ⊗ X)

6 The following two examples of symmetric monoidal categories are essential in what we will be interested in the rest of these notes. Example 2.8. 1. The category of finite dimensional vector spaces over a field k is a symmetric monoidal category 2. The category Rep(G) of finite dimensional representations over a field k of an algebraic group G is a symmetric monoidal category It is maybe appropriate to recall the basic structure of the category of representations of a group. We will do this quickly here: Let G be an algebraic group, a representation of dimension n over a field k is a morphism of groups ψ : G → GLn(V ) where V is a vector space over k of dimension n. Morphism of representations are given by equivariant maps between the groups of matrices, that is, maps GLn(V ) → GLn(W ). The monoidal structure on Rep(G) is given by the following: If ψ : G → GLn(V ) and φ : G → GLm(W ) are two representations, the tensor product is the morphism of groups ψ ⊗ φ : G → GLnm(V ⊗ W ) defined by the formula (ψ ⊗ φ)(g)(Σvi ⊗ wj) := Σψ(g)vi ⊗ φ(g)wj. It An important result about the category of representations of a group is that this category is equivalent to the module category kG − Mod. What we will be looking for in the rest of these notes is to present good properties from the category of finite dimensional vector spaces in the general context of symmetric monoidal categories. To be more specific we are looking for a notion of dimension and duality.

Definition 2.9. Let C be a symmetric monoidal category, an object X ∈ C is said to have a dual X∗ ∈ C if there are ∗ ∗ morphisms coev : 1C → X ⊗ X and ev : X ⊗ X → 1C rendering the following diagrams commutative:

∗ ∗ ∗ ∗ X ⊗ (X ⊗ X ) o X ⊗ 1C (X ⊗ X ) ⊗ X o 1C ⊗ X

    ∗ ∗ ∗ ∗ (X ⊗ X) ⊗ X / 1C ⊗ X X ⊗ (X ⊗ X) / 1C ⊗ X

Example 2.9. Let C be the category of finite dimensional vector spaces F inV ectk, then every object has a dual defined in ∗ the usual way with ev being the evaluation map and k → V ⊗ V being the coevaluation given by choosing a basis {vi} ⊂ V ∗ for V and sending 1 ∈ k to Σvi ⊗ vi Example 2.10. Consider X to be a scheme or a smooth manifold, then in the symmetric monoidal category of line bundles there exists duals for every object. Definition 2.10. We will say that a symmetric monoidal category is called a compacty if every object has a dual. Remark. In some literature an object is said to be finite if it has a dual. Definition 2.11. Let C be a and let f : X → Y be a morphism. The dual morphism f ∗ : Y ∗ → X∗ ∗ ∗ ∗ ∗ f⊗id⊗id ∗ ∗ ∗ ∗ ∗ ∗ is the morphism Y → 1C ⊗ Y → X ⊗ X ⊗ Y −→ Y ⊗ X ⊗ Y → X ⊗ Y ⊗ Y → X ⊗ 1C → X Now that we have the concept of duals in a symmetric monoidal categy, we are able to talk about the trace of an endomorphism: Definition 2.12. Let C be a compact closed category and let f : X → X be a morphism. The trace tr(f) of f is the ∗ f⊗id ∗ endomorphism 1C → X ⊗ X −→ X ⊗ X → 1C Definition 2.13. Let C be a compact closed category and let X ∈ C, then the dimension of X dim(X) is defined as the trace of the identity endomorphism 1X .

Example 2.11. Let k be a field of characteristic 0 and let C be the category F inV ectk, we know from linear algebra that this is a compact closed category and that the identity endomorphism corresponds to the identity . Moreso, as the trace of a matrix is the sum of the values along the diagonal, we see that the dimension of a space V in the traditional sense coincides with the definition given for the compact closed structure. An immediate remark about the previous definition is that even in nice cases the usual definitions of dimension and the dimension given by the compact closed structure may diverge. We can see this by simply considering the category of finite dimensional vector spaces over a finite field, in this case there are only a finite number of endomorphisms between the ground field and so there is a bound on the dimensions that the spaces can take, which is far from true for the dimension of a space in the usual sense.

7 It is also true that there is no restriction given on the of the unit and then the dimension can be fractional, negative or simply an abstract endomorphism. We arrive at last to the final pieces of abstract linear algebra before moving on to the main results of these notes.

L ⊗n Definition 2.14. Let C be a k-linear monoidal category, the of an object X ∈ C is the object n∈ X ⊗0 ⊗1 N where X = 1C and X = X. Remark. Suppose the tensor product commutes with countable direct sums, then the tensor algebra is indeed a monoid in ⊗i ⊗j C, we have multiplication T (X) ⊗ T (X) → T (X) given by Σi,ji ⊗ j where i ⊗ j : X ⊗ X → T (X) is the injection to ⊗i+j X . The unit morphism 1C → T (X) is given by 0 in the previous notation.

Given a σ ∈ Sn and a symmetric monoidal category C we can consider the permutation morphism associated to σ, βσ : X ⊗ · · · ⊗ X → X ⊗ · · · ⊗ X constructed in the following way: First for a transposition τ = (i(i + 1)) we associate the | {z } | {z } n n ⊗n ⊗n ⊗n morphism βτ := id ⊗ · · · ⊗ ⊗βX,X id ⊗ · · · ⊗ id : X → x transposing the ith term with the (i+1)th-term in X , we note | {z } | {z } i−1 n−i−1 that any transposition τ = (ij) ( with i < j ) can be written as a composition ((i+1)i) ... ((j −2)(j −1))((j −1)j) ... (i(i+1)) ⊗n ⊗n and so we can associate it a morphism βτ : X → X . To a permutation σ ∈ Sn written as σ = τ1 . . . τk we then associate ⊗n a morphism βσ := βτ1 ◦ · · · ◦ βτk which will permute the terms of X according to the permutation σ Definition 2.15. Let C be a k-linear symmetric monoidal category for k a field of characteristic 0, X ∈ C and n a natural Vn ⊗n ⊗ number we construct the nth exterior product X as the of the morphism Σσ∈Sn sgn(σ)βσ : X → X Vn Remark. In the case that (1C) = k we can show that dim( ) = C(ndim(X))

This is simply an abstract way of quoting by the relation x⊗y−y⊗x. For example let n=2 then the morphism Σσ∈S2 sgn(σ)βσ Vn is nothing but Id − βX,X which intuitively sends an element x ⊗ y to x ⊗ y − y ⊗ x, then X is the quotient of X ⊗ X by the of this morphism.

3 Tannakian reconstruction and a theorem of Deligne

Tannakian phenomena appears in many aspects of mathematics, in a way it is a formalism in which we can express the idea that symmetries of objects describe the objects in which we take interests. The Tannakian reconstruction presented here is already a classical result in algebraic geometry and since Saavedra’s doctoral thesis ([SR72]) and Deligne and Milne’s famous paper ([DM82]). Here we will reproduce the results presented in [Ros00] which in turn are notes taken during a course lectured by Kazhdan. Definition 3.1. Let C and C be two k-linear symmetric monoidal categories, we say that a monoidal functor F : C → D is a fiber functor if it is exact and faithful.

Definition 3.2. If a k-linear symmetric monoidal category C with k a field of characteristic zero and End(1C) = k has a fiber functor to the category of finite dimensional vector spaces, we say that it is a neutral Tannakian category.

Example 3.1. Let G be a finite group and k a field of characteristic zero then the forgetful functor F : Rep(G) → F inV ectk is a fiber functor and thus Rep(G) is a Tannakian category. In the case of the previous example we get a morphism of groups G → Aut⊗(F ) between G and the monoidal natural isomorphisms of F. This morphism is given by the assignment g → φ(V,φ) where φ(V,φ) is the linear transformation v → φ(g)v. By the equivariant nature of morphisms of representations these linear transformations form a natural . Theorem 3.1. The morphism G → Aut⊗(F ) is an isomorphism. In other words, knowing the category of representations of a finite group together with a fiber functor towards the monoidal category of finite dimensional vector spaces is enough to determine the group. We make this remark because it is not true that the category of representations alone determines the group, as there are known counterexamples provided by nonisomorphic finite groups with equivalent categories of representations as monoidal categories. Let us be more precise, in [EG01] the concept of Isocategorical groups is introduced. Since the category of representations of a finite group always has a forgetful functor to the category of finite dimensional vector spaces, however if for two nonisomorphic finite groups G,H one has a (not necessarely) monoidal equivalence of categories Rep(G) =∼ Rep(H) then the composition of this equivalence with the forgetful functor will not necessarely be monoidal and so it wont be possible to reconstruct the group, as expected. We finish these notes with the following theorem due to Deligne which gives conditions for a category to be a neutral Tannakian category.

8 Theorem 3.2. Let C be an abelian symmetric monoidal category over a field of characteristic zero and such that End(1C) = k. Then the following are equivalent: 1. C is a neutral Tannakian category 2. The dimension of every object in C is nonnegative

3. The dimension of every nonzero object in C is positive 4. There exists an n such that Vn X = 0 for every object X ∈ C Proof. We will only sketch in very rough terms how the nontrivial equivalences are deduced as we are lacking a good number of technical results and remarks.

1. 1) ⇒ 2) Follows from observing that fiber functors preserve the dimension. 2. 2) ⇒ 3) This follows again by a dimension argument 3. 2) ⇒ 4) This follows from a dimension argument by calculating the dimension of the exterior product

4. 4) ⇒ 1) Is the nontrivial step: First we realize C as a of Ind(C) and we note that we get an induced symmetric monoidal structure on Ind(C). Then a series of technical results allows us to find a monoid R ∈ Ind(C) such that the functor X 7→ Ind(C)(1,R ⊗ X) is a fiber functor.

As a last comment we should say that this is in many ways not the strongest form of the theorem. In fact in [Del02] Deligne gives conditions for a category as in the hypothesis of the theorem to be equivalent to the category of representations of a super algebraic group. There exists also partial results ([Wed04]) on weakening the notion of Tannakian category by considering fiber functors with values on categories of R-Modules over a k-algebra and there are also results on working over positive characteristic ([Cou18] ). The Tannakian reconstruction theorem can also be stated in very general terms of enriched categories [Day96].

References

[Cou18] Kevin Coulembier. Tannakian categories in positive characteristic. arXiv preprint arXiv:1812.02452, 2018. [Day96] Brian J. Day. Enriched tannaka reconstruction. Journal of Pure and Applied Algebra, 108(1):17–22, April 1996.

[Del02] Pierre Deligne. Cat´egoriestensorielles. Moscow Mathematical Journal, 2(2):227–248, 2002. [DM82] Pierre Deligne and James S Milne. Tannakian categories. In Hodge cycles, motives, and Shimura varieties, pages 101–228. Springer, 1982. [EG01] Pavel Etingof and Shlomo Gelaki. Isocategorical groups. International Mathematics Research Notices, 2001(2):59– 76, 2001. [EGNO16] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor categories, volume 205. American Mathematical Soc., 2016. [ML13] . Categories for the working mathematician, volume 5. Springer Science & Business Media, 2013. [Rie17] Emily Riehl. Category theory in context. Courier Dover Publications, 2017. [Ros00] Alexander L Rosenberg. The existence of fiber functors. In The Gelfand Mathematical Seminars, 1996–1999, pages 145–154. Springer, 2000.

[Rot08] Joseph J Rotman. An introduction to homological algebra. Springer Science & Business Media, 2008. [SR72] Neantro Saavedra Rivano. Cat´egories tannakiennes. Bulletin de la Soci´et´eMath´ematiquede France, 100:417–430, 1972. [Wed04] Torsten Wedhorn. On tannakian duality over valuation rings. Journal of Algebra, 282(2):575–609, 2004.

9