Exact Categories in Functional Analysis

Transcription

1 Script Exact Categories in Functional Analysis Leonhard Frerick and Dennis Sieg June 22, 2010

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3 To Susanne Dierolf. iii

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5 Contents 1 Basic Notions Categories Morphisms and Objects Functors Additive Categories Pre-additive Categories Kernels and Cokernels Pullback and Pushout Product, Coproduct and Biproduct Additive Categories Semi-abelian Categories Exact Categories Basic Properties E-strict morphisms Maximal Exact Structure p-strict morphisms Maximal Exact Structure Quasi-abelian Categories Derived Functors Exact Functors Complexes Resolutions Derived Functors Universal δ-functors Yoneda-Ext-Functors Yoneda-Ext Yoneda-Ext n v

6 vi CONTENTS 7 Appendix 1: Projective Spectra Categories of Projective Spectra The Projective Limit

7 Preface These lecture notes have their roots in a seminar organized by the authors in the years 2008 and 2009 at the University of Trier. The present notes contain much more than the content of the original seminar, which was based on the works of Palamodov [17, 18] and the book of Wengenroth [31]. Over the course of the seminar, due to the research of the second named author for his Ph.D.-thesis, it turned out, that the language of exact categories, as introduced by Quillen [22], is a far more flexible and more useful starting point for homological algebra in functional analysis, than the setting of semi-abelian categories used in [17, 18, 31]. For example it allows one not only to treat the classical categories of functional analysis, like locally convex spaces, Banach spaches or Fréchet spaces, but also more complex non semi-abelian categories of current research, like the PLS-spaces introduced by Domanski and Vogt. A homological approach to the splitting theory of these spaces, which uses the homological methods presented in this treatise, can be found in the Ph.D.-thesis of the second named author which is available at These notes are aimed at readers who are interested in homological methods for functional analysis, who have some knowledge of functional analysis, but who lack experience in classical homological algebra. Therefore we are not assuming any knowledge of category theory or homological algebra in these notes. The classical introductory textbooks of homological algebra, like [30], always assume the categories to be abelian, which is almost never the case in the interesting categories appearing in functional analysis, hence they are only of limited use for someone who wants to learn about the latter ones. In this notes we treat non-abelian homological algebra completely on its own, without refering to the abelian case, and we prove every assumption in full detail. Furthermore, our examples are always taken from functional analysis. It is our philosophy, that the embedding theorems of category theory, like the Freyd-Mitchell full embedding theorem for abelian categories and the Gabriel-Quillen embedding theorem for exact categories, which are often used to transfer assumptions about diagrams into a category of modules and there argue by diagram chasing, are better used as an intuition than as a direct proof. Therefore, we proof everything directly from the axioms in this text by using the defining universal properties. We are convinced that this vii

8 viii PREFACE approach provides more insight into the subject than diagram chasing. This text contains part of the work of Wengenroth [31] and also uses much of the content of the survey article about exact categories of Bühler [3]. The content of the article [25] of the second named author and Wegner is also completely contained in this text. In addition this text drew much inspiration from books on classical homological algebra like Mitchell [16], Weibel [30] and Adamek, Herrlich, Strecker [1]. As a last remark, we want to remind our readers that this text is not written for publication, but only a collection of lecture notes. It is still a work in progress and we are thankful for comments and suggestions. Leonhard Frerick and Dennis Sieg.

9 Chapter 1 Basic Notions 1.1 Categories Definition 1.1. A category is a quadruple C = (Ob(C), Hom C, id, ) consisting of the following data: (1) A class Ob(C), whose members are called objects of C. (2) For each pair (X, Y ) of objects of C, a set Hom C (X, Y ), whose elements are called morphisms from X to Y. Morphisms are generally expressed by using arrows; e.g. we will often write f : X Y is a morphism instead of the statement f Hom C (X, Y ). (3) For each object X Ob(C), a morphism id X : X X, called the identity on X. (4) A composition law associating with each morphism f : X Y and each morphism g : Y Z a morphism g f : X Z, called the composite of f and g. These data have to have the following properties: (C1) The composition is associative; i.e. for morphisms f : X Y, g : Y Z, and h: Z W, the equation h (g f) = (h g) f holds. (C2) The identities act as neutral elements with respect to the composition; i.e., for a morphism f : X Y, we have id Y f = f and f id X = f. (C3) The sets Hom C (X, Y ) are pairwise disjoint. Remark 1.2. Let C = (Ob(C), Hom C, id, ) be a category. i) The class of all morphisms of C is denoted by Mor(C) and is defined to be the union of all the sets Hom C (X, Y ) in C. 1

10 2 CHAPTER 1. BASIC NOTIONS ii) If f : X Y is a morphism in C, we call X the domain of f and Y the codomain of f. The property (C3) then guarantees that each morphism has a unique domain and a unique codomain. This is given for technical convenience only, because if the other two properties are satisfied one can just replace each morphism f Hom C (X, Y ) with the triple (f, X, Y ) so that (C3) is also satisfied. Therefore when checking that an entity is a category we will only show (C1) and (C2). iii) The composition is a partial binary operation on the class Mor(C) and for a pair (f, g) of morphisms the composite g f is defined if and only if the domain of g and the codomain of f coincide. iv) For an object X of C the identity id X : X X is uniquely determined because of property (C2). Example 1.3. i) The category (Set) whose objects are sets and which has as morphisms Hom (Set) (X, Y ) the set of all mappings from X to Y. The identity morphism id X is the identity mapping on X and is the usual composition of mappings. ii) An important type of categories are those, whose objects consist of structured sets and whose morphisms are mappings between these sets that preserve the structure. These categories are called constructs. Some examples of constructs are: a) (Ab) the category of abelian groups and group morphisms. b) (T op) the category of topological spaces and continous mappings. c) (Ring 1 ) the category of commutative rings with unity together with the ring morphisms that preserve the unity. d) (F V ec) the category of vector spaces over a fixed field F and F-linear mappings. e) (T V S) the category of topological vector spaces over a fixed field F {R, C} and continous F-linear mappings. f) (LCS) the category of (not necessarily Hausdorff) locally convex vector spaces over a fixed field F and continous F-linear mappings. g) (LCS) HD the category of Hausdorff locally convex vector spaces over a fixed field F and continous F-linear mappings. iii) In the case of constructs it is often clear what the morphisms should be once the objects are defined. But this is not always the case: a) There are, at least, three different constructs having metric spaces as objects:

11 1.1. CATEGORIES 3 (M et) the category of metric spaces and contractions. (Met u ) the category of metric spaces and uniformly continous mappings. (Met c ) the category of metric spaces and continous mappings. b) The following two categories are natural constructs having as objects all Banach spaces: (Ban) the category of Banach spaces and continous linear mappings. (Ban c ) the category of Banach spaces and linear contractions. iv) Not all categories consist of structured sets and mappings preserving this structure, as the following examples show: a) (Mat) which has as objects the natural numbers N and for which Hom (Mat) (m, n) is the set of all real (m n)-matrices, id n : n n is the unit matrix, and the composition is defined by A B = BA, where BA is the usual multiplication of matrices. b) If (I, ) is a preordered set, i.e. I is a set and is a reflexive and transitive relation on I we can define a category C(I) in the following way: Ob(C(I)) = I, Hom C(I) (i, j) = { {(i, j)} if i j otherwise, as well as id i = {(i, i)} and {(j, k)} {(i, j)} = {(i, k)}. Definition 1.4. Let C = (Ob(C), Hom C, id, ) be a category. We define the dual category of C as C op = (Ob(C), Hom C op, id, op ), where the morphism sets are defined as Hom C op(x, Y ) = Hom C (Y, X) and the composition as f op g = g f. Remark 1.5. i) If C is a category, then C and C op have the same objects, only the direction of the arrows is reversed : If f : X Y is a morphism in C, then f Hom C (X, Y ) = Hom C op(y, X), hence f is a morphism from Y to X when considered as a morphism of the dual category. We write f op in place of f when we consider it as a morphism of the dual category.

12 4 CHAPTER 1. BASIC NOTIONS ii) Because of the way dual categories are defined, every statement concerning an object X in a category C can be translated into a logically equivalent statement in the dual category C op. This observation allows one to associate (in two steps) with every property P concerning objects and morphisms in categories, a dual property concerning objects and morphisms in categories, as demonstrated by the following example: Consider the following property of an object X of a category C: P C (X) For any Y Ob(C) there exists a unique λ Hom C (Y, X). Step 1: In P C (X) replace all occurrences of C by C op, thus obtaining the following property: P C op(x) For any Y Ob(C op ) there exists a unique λ Hom C op(y, X). Step 2: Translate P C op(x) into the logically equivalent statment in C: P op C (X) For any Y Ob(C) there exists a unique λ Hom C(X, Y ). C (X) is called the dual property of P C(X). Roughly speaking, C (X) is obtained from P C(X) by reversing the direction of each arrow and the order in which morphisms are composed. In general, P op C (X) is not equivalent to P C(X). For example, if C is the category (Set) then the property P C (X) holds if and only if X is a singleton P op P op set, whereas the dual property P op C (X) holds if and only if X is the empty set. iii) The dual property P op of a property P holds in a category C if and only if P holds in C op. iv) Obviously (C op ) op = C. v) From the above two observations we get the extremely useful Duality Principle for Categories, namely: Whenever a property P holds for all categories, then the property P op holds for all categories. Because of this principle, each result in category theory has two equivalent formulations. However, only one of them needs to be proved, since the other one follows by virtue of the Duality Principle. Remark 1.6. Morphisms in a category will usually be denoted by lowercase letters, while uppercase letters will be reserved for objects. The morphism

13 1.2. MORPHISMS AND OBJECTS 5 h = g f will sometimes be denoted by X f Y triangle f X Y commutes. Similarly, the statement that the square X commutes means that g f = k h. h f g Z Y h g W k Z 1.2 Morphisms and Objects g Z or by saying that the Definition 1.7. Let C be a category and φ: X Y be a morphism. i) φ is called an epimorphism, if for every two morphisms f 1, f 2 : Y W with f 1 φ = f 2 φ it follows that f 1 = f 2. ii) φ is called a monomorphism, if for every two morphisms g 1, g 2 : Z X with φ g 1 = φ g 2 it follows that g 1 = g 2. iii) φ is called a bimorphism, if φ is both a monomorphism and an epimorphism. Remark 1.8. Monomorphisms are the dual notion of epimorphisms, i.e. a morphism f : X Y in a category C is an epimorphism if and only if f op is a monomorphism in C op. The bimorphisms of C and C op are the same. Example 1.9. i) In every category the identities are bimorphisms. ii) A morphism in the category (Set) is an epimorphism if and only if it is surjective. It is a monomorphism if and only if it is injective and hence a bimorphism if and only if it is bijective. iii) In many constructs, the monomorphisms are precisely the morphisms that have injective underlying mappings; e.g. this is the case in all the constructs of example 1.3. The epimorphisms in a number of constructs are those having surjective underlying mappings; e.g. this is the case in the constructs of example 1.3.ii). However, this situation occurs less frequently than that of monomorphisms having injective

14 6 CHAPTER 1. BASIC NOTIONS underlying mappings and in quite a few familiar constructs, epimorphisms are not surjective: In the category (T op) HD of topological Hausdorff spaces and continous mappings a morphism is an epimorphism if and only if it has dense range. This is also the case in the category (LCS) HD. Proposition Let C be a category and let h = g f be a morphism in C. i) If g and f are epimorphisms, then h is an epimorphism. ii) If h is an epimorphism, then g is an epimorphism. iii) If g and f are monomorphisms, then h is a monomorphism. iv) If h is a monomorphism, then f is a monomorphism. Proof. Because of the duality principle it suffices to show i) and ii), since iii) and iv) are the dual statements. More precisely: Suppose i) and ii) have already been shown. If g and f are monomorphisms, then g op and f op are epimorphisms in C op. Since i) holds for all categories h op = f op op g op is an epimorphism and therefore a monomorphism in C. Analagously one shows iv). i) If g 1 (g f) = g 2 (g f) for two morphisms g 1 and g 2 it follows first that g 1 g = g 2 g and then g 1 = g 2, since both g and f are epimorphisms. This shows i). ii) If g 1 g = g 2 g, then g 1 (g f) = g 2 (g f), hence g 1 = g 2, since h = g f is an epimorphism. Definition Let C be a category and φ: X Y be a morphism. i) φ is called a retraction, if there exists a morphism ψ : Y X with φ ψ = id Y ; i.e. if φ has a right-inverse. ii) φ is called a coretraction, if there exists a morphism ψ : Y X with ψ φ = id X ; i.e. if φ has a left-inverse. iii) φ is called an isomorphism, if there exists a morphism ψ : Y X with φ ψ = id Y and ψ φ = id X ; i.e. if φ has an inverse. Remark and Definition The inverse of an isomorphism is uniquely determined by the defining property. Therefore, given an isomorphism φ : X Y, we will denote the inverse morphism ψ : Y X with φ ψ = id Y and ψ φ = id X by φ 1 := ψ. We will call two objects X, Y of a category C isomorphic, if there exists an isomorphism φ: X Y. This is an equivalence relation on the class Ob(C) and we will write X = Y if X and Y are isomorphic to each other. The right-inverse of a retraction and the left-inverse of a coretraction are, in general, not uniquely determined.

15 1.2. MORPHISMS AND OBJECTS 7 Proposition Let C be a category and φ: X Y be a morphism. i) If φ is retraction, then it is an epimorphism. ii) If φ is a coretraction, then it is a monomorphism. iii) If φ is an isomorphism, then it is a bimorphism. Proof. i) follows from 1.9.i) and 1.10.i), ii) is the dual statement of i) and iii) follows from i) and ii). Example i) In the category (Set) a morphism is a retraction if and only if it is surjective, it is a coretraction if and only it is injective and it is an isomorphism if and only if it is bijective. The same holds for the category (F V ec). ii) In the categories (T V S) and (LCS) a morphism is an isomorphism if and only if it is bijective and open unto its range. An isomorphism in the category (LCS) HD is characterized by being injective, open unto its range and having a dense image. The retractions and coretractions of these categories are not so easily characterized. We will come back to them in a later chapter. Proposition Let C be a category and h = g f be a morphism in C. i) If g and f are retractions, then h is a retraction. ii) If h is a retraction, then g is a retraction. iii) If g and f are coretractions, then h is a coretraction. iv) If h is a coretraction, then f is a coretraction. Proof. It suffices to show i) and ii), since iii) and iv) are the dual statements of i) and ii). i) If g r = id Z for a morphism r : Z Y and f s = id Y for a morphism s: Y X then (g f) (s r) = g (f s) r = g id Y r = g r = id Z, hence h = g f has a right-inverse. ii) Given a morphism l : Z X with (g f) l = id Z we have g (f l) = id Z, hence g is a retraction. Proposition Let C be a category and φ: X Y be a morphism in C. The following are equivalent: i) φ is a retraction and a monomorphism.

16 8 CHAPTER 1. BASIC NOTIONS ii) φ is a coretraction and an epimorphism. iii) φ is an isomorphism. iv) φ is a retraction and coretraction. Proof. i) iii) Let ψ : Y X be a morphism with φ ψ = id Y. Then we have φ ψ φ = id y φ = φ id X and since φ is a monomorphism it follows that ψ φ = id X. Hence φ is an isomorphism. The assertion ii) iii) follows by duality, iii) iv) is trivial and iv) i) as well as iv) ii) follow from proposition Definition Let C be a category. i) An object I of C is called an initial object, if for every object X Ob(C) there exists exactly one morphism i X : I X. ii) An object T of C is called a terminal object, if for every object X Ob(C) there exists exactly one morphism t X : X T. iii) An object Z of C is called a zero object, if it is both initial and terminal. Remark i) The notion of a terminal object is the dual notion of that of an initial object; i.e. if I is an initial object of a category C then I is a terminal object of C op and vice versa. ii) If I and Ĩ are initial (resp. terminal, resp. zero) objects of a category C, then there exists exactly one isomorphism λ: I Ĩ. In fact, since I and Ĩ are both initial there exists exactly one morphism i Ĩ : I Ĩ and exactly one morphism ĩ I : Ĩ I. Since id I is the only element of Hom C (I, I) and idĩ is the only element of Hom C (Ĩ, Ĩ) it follows that iĩ ĩ I = idĩ and ĩ I iĩ = id I. Hence iĩ is an isomorphism. By duality, the same holds for terminal objects. This property is characterized by saying that an initial (resp. terminal, resp. zero) object is unique up to a canonical isomorphism. iii) If I is an initial object of a category C, and f : X Y is an arbitrary morphism, the morphism f i X is a morphism from I to Y, hence f i X = i Y. Dually, if T is a terminal object of a category C we have t Y f = t X for every morphism f : X Y. Remark and Definition If a category C possesses a zero object Z, then there exists for every two objects X, Y Ob(C) a morphism 0 X,Y : X Y

17 1.3. FUNCTORS 9 called the zero morphism from X to Y, given by the composition X t X Z i X Y. 0 X,Y If f : W X and g : Y Z are morphisms in C it follows from 1.18.iii) that Example X,Y f = i Y t X f = i Y t W = 0 W,Y g 0 X,Y = g i Y t X = i Z t X = 0 X,Z. i) In the category (Set) the empty set is an initial object and every singleton is a terminal object. ii) In the category (Ring 1 ) the ring Z is an initial object and the zero ring is a terminal object. iii) In the category (Ab) the trivial group is a zero object. iv) The zero vector space is the zero object of the category (F V ec). v) The categories (T V S), (LCS) and (LCS) HD all have as a zero object the zero vector space with its unique topology. 1.3 Functors Definition Let C and D be categories. i) A covariant functor F : C D from C to D is a rule that assigns to every X Ob(C) an object F (X) Ob(D) and to every morphism f : X Y in C a morphism F (f): F (X) F (Y ) with the following properties: (F 1) F (id X ) = id F (X) (F 2) F (g f) = F (g) F (f) ii) A contravariant functor G: C D from C to D is a rule that assigns to every X Ob(C) an object G(X) Ob(D) and to every morphism f : X Y in C a morphism G(f): G(Y ) G(X) with the following properties: (F 1) G(id X ) = id G(X) (F 2) op G(g f) = G(f) G(g)

18 10 CHAPTER 1. BASIC NOTIONS Remark and Definition i) Every contravariant functor G: C D can also be seen as a covariant functor G: C op D. ii) Every functor (covariant or contravariant), preserves isomorphisms: Let ψ : X Y be an isomorphism in a category C and F : C D a covariant functor, then the functor properties yield F (ψ) F (ψ 1 ) = F (ψ ψ 1 ) = F (id Y ) = id F (Y ), F (ψ 1 ) F (ψ) = F (ψ 1 ψ) = F (id X ) = id F (X), hence F (ψ) is an isomorphism in D. iii) If C,D and E are categories, F : C D, F : D E are covariant functors and G: C D, G : D E are contravariant functors one can define the composite functor F F : C E, { F F (X) = F (F (X)) F F (f) = F (F (f)). The functor F F is covariant. In a similar way one can define the composite functors G F, F G and G G. In these cases the functors G F and F G are contravariant and the functor G G is covariant. Example i) For every category C there is the identity functor Id: C C, { Id(X) = X Id(f) = f. ii) For every category there is the duality functor op : C C op, { op (X) = X op (f) = f op. op is a contravariant functor with op op = Id C. iii) For every two categories C and D and every object Y 0 of D there is the constant functor with value Y 0 C Y0 : C D, { CY0 (X) = Y 0 Id(f) = id Y0.

19 1.3. FUNCTORS 11 iv) For every category C and A Ob(C), we have a covariant functor Hom C (A, ): C (Set), { X HomC (A, X) α Hom C (A, α), where Hom C (A, α): Hom C (A, X) Hom C (A, X ), f α f and a contravariant functor Hom C (, A): C (Set), { Z HomC (Z, A) γ Hom C (γ, A), where Hom C (γ, A): Hom C (Z, A) Hom C (Z, A), g g γ. Remark Let C be a category and let A Ob(C). Then we have Hom C (, A) = Hom C op(a, ) op. Indeed, we have Hom C (X, A) = Hom C op(a, X) for each object X of C. If f : X Y is a morphism in C, then Hom C op(a, ) op (f)(g) = f op op g. Since f op op g in C op is the morphism g op f in C, it follows that the morphisms Hom C (f, A) and Hom C op(a, f op ) coinside. Example i) There is the covariant functor F : (Ab) (Set), { F (X) = X F (f) = f. Functors of this kind, where part of the structure on the objects and morphisms are forgotten are called forgetful functors. ii) There is the contravariant duality functor L: (T V S) (T V S), { L(X) = X L(f) = f t, assigning to each topological vector space X its strong dual X and to f : X Y the transposed map f t : Y X.

20 12 CHAPTER 1. BASIC NOTIONS iii) For a locally convex space X let X be the Hausdorff completion of X, i.e. the completion of X/{0} and let j X : X X be the canonical map (which is injective if and only if X is Hausdorff and surjective if and only if X is complete). Then we have the covariant completion functor { C(X) = C : (LCS) (LCS) HD, X C(f) = f, assigning to X the space X and to a continous linear map f : X Y the canonical map f : X Ỹ. Remark and Definition We have seen above that functors act as morphisms between categories; they are closed under composition, which is associative and the identity functors act as identities with respect to the composition. Because of this, one is tempted to consider the category of all categories. However, there are two difficulties that arise when we try to form this entity. First, the category of all categories would have objects such as all sets or all vector spaces, which are not sets but proper classes. We will not concern ourselves with the set-theoretic difficulties arising in category theory. The reader who is interested in this should consult [1, 4, 5] as well as the references therein. Since proper classes cannot be elements of classes, the conglomerate of all categories would not be a class, thus violating the properties of a category. Second, given any categories C and D, it is not generally true that the conglomerate af all functors from C to D forms a set, which is another violation of the properties. However, if we restrict our attention to categories whose class of objects is actually a set, then both problems are eliminated. A category C is called small if Ob(C) is a set. Otherwise its called large. Example i) The category (Mat) of natural numbers and m n-matrices is a small category. ii) Every preordered set is small when considered as a category. iii) The category (Set) of sets and mappings is not small, since the class of all sets is not a itself a set (Russell s paradox). iv) Since for every set M there is the vector space F M, the category (F V ec) of all vector spaces is not a small category. v) Every category that contains one of the above is not small. For example, the category (T V S) of topological vector spaces is not small, since it contains all vector spaces (each vector space is a topological vector space, when considered together with the coarsest topology).

21 1.3. FUNCTORS 13 vi) The category (Ban) of Banach spaces and continous linear mappings is not small, since for every set M there is the Banach space l (M). vii) The categories (LCS) and (LCS) HD are not small, since they contain the Banach spaces. Remark and Definition The category (Cat) of small categories has as objects all small categories, as morphisms from C to D the functors from C to D, as identities the identity functors, and as composition the composition of functors. Since each small category is a set, the conglomerate of all small categories is a class, and since for each pair (C, D) of small categories the conglomerate of all functors from C to D is a set, (Cat) is indeed a category. However its not a small category. Definition A quasicategory is a quadruple C = (Ob(C), Hom C, id, ) consisting of the following data: (1) A conglomerate Ob(C), whose members are called objects of C. (2) For each pair (X, Y ) of objects of C, a conglomerate Hom C (X, Y ), whose elements are called morphisms from X to Y. (3) For each object X Ob(C), an morphism id X : X X, called the identity on X. (4) A composition law associating with each morphism f : X Y and each morphism g : Y Z a morphism g f : X Z. These data have to have the following properties: (C1) The composition is associative. (C2) The identities act as neutral elements with respect to the composition. (C3) The sets Hom C (X, Y ) are pairwise disjoint. Remark and Definition i) Every category is a quasicategory. ii) The quasicategory (CAT ) of all categories has as objects all categories, as morphisms from C to D all functors from C to D, as identities the identity functors, and as composition the composition of functors. (CAT ) is a proper quasicategory in the sense that it is not a category. iii) Virtually every categorical concept has a natural analogue or interpretation for quasicategories. The names for such quasicategorical concepts will be the same as those of their categorical analogues. Thus we have, for example, the notion of functor between quasicategories.

22 14 CHAPTER 1. BASIC NOTIONS Because the main objects of our study are categories, most notions will be specifically formulated only for categories. In fact, the only quasicategories we will introduce at all are the category (CAT ) defined above, the category which has as objects all functors between two categories C and D, the quasicategory of all classes and that of all big abelian groups, defined further below. Definition Let C and D be categories and let F : C D be a covariant functor. For X, Y Ob(C) we have the mapping φ X,Y : Hom C (X, Y ) Hom D (F (X), F (Y )), f F (f). i) The functor F is called faithful, if φ X,Y is injective for all objects X and Y. ii) The functor F is called full, if φ X,Y is surjective for all objects X and Y. iii) The functor F is called fully faithful if it is full and faithful. iv) The functor F is called essentially surjective, if for every Y Ob(D) there exists an X Ob(C) with F (X) = Y. Definition Let C be a category. A subcategory of C is a category C with the following properties: (SC1) Every object X Ob(C ) is an object of C. (SC2) For every two objects X, Y Ob(C ) there is the inclusion Hom C (X, Y ) Hom C (X, Y ). (SC3) The identity id X in C of an object X Ob(C ) is also the identity in C. (SC4) The composition of two morphisms in C is the same as the composition in C. Remark and Definition If C is a subcategory of a category C, we have a covariant faithful functor ι: C C, { ι(x) = X ι(f) = f, the so-called inclusion functor. C is called a full subcategory of C, if the functor ι is fully faithful, i.e. if Hom C (X, Y ) = Hom C (X, Y ) for all X, Y Ob(C ).

23 1.3. FUNCTORS 15 Example i) Let F be a field and let (F V ec) be the category of F-vector spaces and linear mappings. The category (F V ec) fin of finite dimensional vector spaces and linear mappings is a full subcategory of (F V ec). ii) We have the inclusions (T V S) (LCS) (LCS) HD (Ban) of categories. Since all these categories have continous linear mappings as morphisms, each of them is a full subcategory of the categories lying above them. iii) For the category (T V S) the forgetful functor F : (T V R) (F V ec), { F (X, TX ) = X F (f) = f, is full. Every vector space V can be made into a topological vector space (V, T V ) by using the coarsest topology on V, hence the functor F is essentially surjective. It is not faithful, since there are linear mappings between topological vector spaces that are not continous. iv) Let (Mat) be the category, which has as objects the natural numbers N and as morphisms Hom (Mat) (m, n) the set of all real (m n)-matrices. Then the functor with F : (Mat) (F V ec) fin, { A φ : F n F m, x Ax, is fully faithful and essentially surjective. F (n) = F n F (A F m n ) = A φ, Proposition Let C and D be categories and let F : C D be a fully faithful functor. Then the following are equivalent: i) f : X Y is an isomorphism in C. ii) F (f): F (X) F (Y ) is an isomorphism in D. Proof. i) ii) is clear from 1.22.ii). ii) i) Let u Hom D (F (Y ), F (Y ) be a morphism with u F (f) = id F (X) and F (f) u = id F (Y ). Since the mapping g F (g) is surjective, there exists an h Hom C (Y, X) with F (h) = u. Then F (h f) = F (h) F (f) = id F (X) = F (id X )

24 16 CHAPTER 1. BASIC NOTIONS and F (f h) = F (f) F (h) = id F (Y ) = F (id Y ). The mapping g F (g) is also injective, hence it follows that f h = id Y and h f = id X. Definition Let C and D be categories and let F, G: C D be covariant functors. A natural transformation τ : F G is a rule that assigns to each X Ob(C) a morphism τ(x): F (X) F (X) of D in such a way that the following condition holds: For each morphism f : X Y in C, the square F (X) τ(x) G(X) F (f) G(f) F (Y ) τ(y ) G(Y ) is commutative. We will denote natural transformations from F to G by Nat(F, G). A natural transformation between contravariant functors F, G : C D is a natural transformation between the covariant functors F, G : C op D Remark and Definition i) If τ : F G and τ : G H are natural transformations between functors F, G, H from a category C to a category D then it is clear that τ τ : F H defined by τ τ(x) := τ (X) τ(x) for each X C is again a natural transformation, since all the diagrams F (X) τ X G(X) τ (X) H(X) F (f) G(f) H(f) F (Y ) τ Y G(Y ) H(Y ) τ (Y ) are commutative. Therefore we have a composition of natural transformations, which is obviously associative. ii) Let τ : F G be a natural transformation between functors F, G: C D. If for every X Ob(C) the morphism τ(x): F (X) G(X) is an isomorphism, then τ is called a natural isomorphism.

25 1.3. FUNCTORS 17 iii) Two Functors F, G: C D are called naturally isomorphic, often denoted by F = G, if there exists a natural isomorphism φ: F G. Example i) For every functor F : C D we have the natural isomorphism id F : F F, called the identity on F, given by id F (X) := id F (X) for every object X of C. ii) Let C be a category, let A and B be objects of C and let a: A B be a morphism. Since for every morphism f : X Y the diagram Hom C (B, X) Hom C (a,x) Hom C (A, X) Hom C (B,f) Hom C (A,f) Hom C (B, Y ) Hom C (A, Y ) HomC (a,y ) is commutative, a induces a natural transformation τ a := (Hom C (a, X)) X Ob(C) : Hom C (B, ) Hom C (A, ). Definition Let C and D be categories. The functor quasicategory Fun(C, D) has as objects all functors from C to D, as morphisms from a functor F to a functor G all natural transformations from F to G, as identities the identity natural transformation, and as composition the composition of natural transformations. Remark i) If C and D are small categories, then Fun(C, D) is a category. If C is small and D is large, then Fun(C, D), though being a proper quasicategory, is isomorphic to a category in (Cat). If C and D are both large, then Fun(C, D) will generally fail to be isomorphic to a category. ii) A natural transformation between functors from C to D is a natural isomorphism if and only if it is an isomorphism in Fun(C, D). Definition A functor F : C D is called an equivalence of categories, if there exists a functor G: D C and natural isomorphisms F G = Id D and G F = Id C. The functor G is then called a quasi-inverse of F.

26 18 CHAPTER 1. BASIC NOTIONS Proposition Let C and D be categories and let F : C D be a covariant functor, then the following are equivalent: i) F is an equivalence of categories. ii) F is fully faithful and essentially surjective. Proof. i) ii) Let G: D C be a functor with F G = Id D and G F = Id C. Fix natural isomorphisms φ: F G Id D and ψ : G F Id D. If Y is an object of D, then we have the isomorphism φ(y ): F G(Y ) Y, hence the functor F is essentially surjective. Let f : X Y be a morphism in C, then the diagram G F (X) ψ(x) X G F (f) is commutative and we have G F (Y ) ψ(y ) Y f f = ψ(y ) (G F (f)) ψ(x) 1, which means that the morphism f can be recaptured from F (h), hence F is a faithful functor. Analogously one can show that G is a faithfull functor. If l : F (X) F (Y ) is a morphism in D define f := ψ(y ) G(l) ψ(x) 1, which is an element of Hom C (X, Y ). In addition we have f = ψ(y ) (G F (f)) ψ(x) 1, as was shown above. Since ψ(y ) and ψ(x) 1 are isomorphisms, it follows that G(l) = G F (f). The functor G is faithful, hence we have l = F (f), which shows that F is a full functor. ii) i) Since F is essentially surjective we can fix an object X Y Ob(C) and an isomorphism φ(y ): F (X Y ) Y for every object Y Ob(D). Define G: D C, { G(Y ) = XY G(g) = f g, where f g : X Y X X is the, because the functor F being fully faithful, uniquely determined morphism with F (f g ) = φ(z) 1 g φ(y ).

27 1.3. FUNCTORS 19 Then we have F (id XY ) = φ(y ) 1 id Y φ(y ) = id Y, hence it follows that G(id Y ) = id XY = id G(Y ). In addition it follows from F G(g g ) = φ(z) 1 g g φ(y ) = φ(z) 1 g φ(y ) φ(y ) 1 g φ(y ) = (F G(g)) (F G(g )) that G(g g ) = G(g) G(g ), hence G is functor. g : Y Z in D, the square For each morphism F G(Y ) F G(g) commutes, therefore the rule F G(Z) φ(y ) φ(z) Y Z g φ: F G Id D, φ = (φ(y )) Y Ob(D) is a natural isomorphism. Additionally the morphisms φ(f (X)): F G F (X) F (X) is an isomorphism for every object X of C. Since the functor F is fully faithful there exists a unique morphism ψ(x): G F (X) X for each X Ob(C) and by 1.35 the morphism ψ(x) is always an isomorphism. In addition we have for every morphism f : X X in C that F G F (f) = F G(φ(F (X )) 1 F (f) φ(f (X))) = F G(F (ψ(x )) 1 F (f) F (ψ(x))) = F G F (ψ(x ) 1 f ψ(x)) and since F is faithful it follows that the diagram G F (X) G F (f) G F (X ) is commutative. Therefore the rule ψ(x) ψ(x ) X f X ψ : G F Id C, ψ = (ψ(x)) X Ob(C) is a natural isomorphism, which shows that the functor F is an equivalence of categories.

28 20 CHAPTER 1. BASIC NOTIONS Definition Let C be a category. A covariant functor F : C (Set) is called representable, if there exists an A Ob(C) and a natural isomorphism F = Hom C (A, ). A contravariant functor is called representable, if there exists an B Ob(C) and a natural isomorphism F = Hom C (, B). Remark Let F : C (Set) be a representable covariant (resp. contravariant) functor with F = Hom C (A, ) (resp. F = Hom C (, A)) for an object A of C. Then the object A is uniquely determined by this property up to a unique isomorphism. In fact, if A and A are objects of C with Hom C (A, ) = F = Hom C (A, ) in Fun(C, (Set)), we have isomorphisms φ(a): Hom C (A, A) Hom C (A, A) and φ(a ): Hom C (A, A ) Hom C (A, A ) making the diagram Hom C (A, A) Hom C (A,f) Hom C (A, A) φ(a) φ(a ) Hom C (A, A) Hom C (A,f) Hom C (A, A) commutative, if we define f : A A to be the unique morphism with φ(a )(f) = id A. Then it follows that Additionally the diagram id A = φ(a )(f) = φ(a )(f) Hom C (A, f)(id A ) = Hom C (A, f) φ(a)(id A ) = f (φ(a)(id A )). Hom C (A, A ) Hom C(A,φ(A)(id X )) Hom C (A, A) φ(a ) 1 Hom C (A, A) Hom C (A,φ(A)(id X )) φ(a) 1 Hom C (A, A) commutes, hence it follows that φ(a)(id X ) f = Hom C (A, φ(a)(id A ))(f) = Hom C (A, φ(a)(id A ) φ(a )(id A ) = φ(a) 1 Hom C (A, φ(a)(id A ))(id A = φ(a) 1 φ(a)(id A ) = id A

29 1.3. FUNCTORS 21 This shows that the morphism f : A A is an isomorphism with inverse φ(a)(id A ): A A. Proposition 1.45 (Yoneda Lemma). Let C be a category and let F : C (Set) be a covariant functor. For every object A of C the map Y : Nat(Hom C (A, ), F ) F (A) defined by Y(τ) = τ(a)(id A ), is bijective. Proof. Define a mapping ψ : F (A) Nat(Hom C (A, ), F ) in the following way: For Y Ob(C) and s F (A) let ψ(s)(y ) be given by the composition Hom C (A, Y ) Hom (Set) (F (A), F (Y ) F (Y ) f F (f) F (f)(s). For a morphism g : Y Y and every f Hom C (A, Y ) we have hence the diagram (ψ(s)(y ) Hom C (A, g))(f) = ψ(s)(y )(g f) = (F (g) F (f))(s) = F (g)(f (f)(s)) = F (g)(ψ(s)(y )(f)) = (F (g) ψ(s)(y ))(f), Hom C (A, Y ) ψ(s)(y ) Hom C (A,g) Hom C (A, Y ) ψ(s)(y ) F (Y ) F (g) F (Y ) commutes. This shows that ψ(s) is indeed a natural transformation. In addition we have Y ψ(s) = Y(ψ(s)) = ψ(s)(a)(id A ) = F (id A )(s) = s for all s F (A), which shows that Y ψ = id F (A). For τ Nat(Hom C (A, ), F ) and g Hom C (A, Y ) the diagram Hom C (A, A) Hom C (A,g) Hom C (A, Y ) τ(x) τ(y ) F (A) F (g) F (A)

30 22 CHAPTER 1. BASIC NOTIONS commutes, hence we have ψ Y(τ)(g) = ψ(τ(a)(id A ))(g) = F (g)(τ(a)(id A )) = τ(y ) Hom C (A, g)(id A ) = τ(y )(g). This shows that ψ Y = id Nat(HomC (A, ),F ) and therefore the mapping Y is a bijection. Remark and Definition 1.46 (Yoneda Embedding). Let C be a category and let f : A B be a morphism in C. Define E(f)(X): Hom C (B, X) Hom C (A, X), h h f for each object X of C. If g : X Y is another morphism in C, the diagram Hom C (B, X) E(f)(X) Hom C (A, X) Hom C (B,g) Hom C (A,g) Hom C (B, Y ) E(f)(Y ) Hom C (A, Y ) is commutative, since Hom C (A, g) E(f)(X)(h) = Hom C (A, g)(h f) = g h f = E(f)(Y )(g h) = E(f)(Y ) Hom C (B, g). Therefore E(f) := (E(f)(X)) X Ob(C) is a natural transformation from the functor Hom C (B, ) to Hom C (A, ). In addition we have therefore the rule E(id A ) = id HomC (A, ) E(f f) = E(f ) E(f), E : C Fun(C, (Set)), { A HomC (A, ) f E(f) is a contravariant functor, the so-called Yoneda Embedding. Proposition For every category C the functor E : C Fun(C, (Set)) defined above is fully faithful.

31 1.3. FUNCTORS 23 Proof. By 1.45 we have a bijection Y : Nat(Hom C (B, ), Hom C (A, )) Hom C (A, B), with Y(τ) = τ(a)(id A ). Then (Y )(E(f)) = E(f)(A)(id A ) = id A f = f, which shows that the mapping f E(f) is bijective. Remark For every category C one also has the contravariant Yoneda embedding E : C Fun(C, (Set)), { A HomC (, A) f E (f) with E (f)(x)(h) = f h. In analogy to the above one can show that E is also a fully faithful functor.

32 24 CHAPTER 1. BASIC NOTIONS

33 Chapter 2 Additive Categories 2.1 Pre-additive Categories Definition 2.1. A category C is called preadditive, if it has the following properties: PA1 C possesses a zero object. PA2 Each of the sets Hom C (X, Y ) carries the structure of an abelian group, written additively, and this structure is compatible with the composition of C, in the sense that the distributive laws (f + g) h = f h + g h f (g + h) = f g + f h hold on morphisms whenever the terms are defined. Remark 2.2. If C is a preadditive category, then the dual category C op is also preadditive. Remark 2.3. Let C be a preadditive category. If e X,Y is the neutral element of the abelian group Hom C (X, Y ) and f : X Y is an arbitrary morphism it follows from e Y,Y f = (e Y,Y + e Y,Y ) f = e Y,Y f + e Y,Y f that e Y,Y f = e X,Y. Especially, if 0 X,Y is the zero morphism, we have 0 X,Y = e Y,Y 0 X,Y by 1.19, hence 0 X,Y = e Y,Y 0 X,Y = e X,Y and thus the zero morphism 0 X,Y is the neutral element of Hom C (X, Y ) for all X, Y Ob(C). From now on we will simply write 0 instead of 0 X,Y. Example

34 26 CHAPTER 2. ADDITIVE CATEGORIES i) The categories (Ab), (F V ec), (T V S), (LCS) and (LCS) HD are all preadditive categories. ii) If f, g : R S are two morphisms in the category (Ring 1 ), we have f(1 R ) + g(1 R ) = 1 S + 1 S and therefore the elementwise addition of ring morphisms that preserve the unity does not itself preserve the unity. The category (Ring 1 ) is therefore not preadditive with this addition. Definition 2.5. Let C be a preadditive category. A full preadditive subcategory C of a preadditive category C is a full subcategory that also contains the zero object. Definition 2.6. Let C and D be preadditive categories. A covariant functor F : C D is called additive if F (f + g) = F (f) + F (g) holds, for all morphisms f, g Hom C (X, Y ) and all X, Y Ob(C). A contravariant functor is called additive if it is additive as a covariant functor from C op to D. Remark 2.7. i) Let F : C D be an additive functor between preadditive categories. From F (0) = F (0 + 0) = F (0) + F (0) it follows that F (0) = 0. ii) If C is a preadditive category, then the duality functor is an additive functor. iii) If C is a preadditive category and A an object of C, the functors Hom C (A, ): C (Ab), and Hom C (, A): C (Ab), are additive functors, because of { X HomC (A, X) α Hom C (A, α) { Z HomC (Z, A) γ Hom C (γ, A),, Hom C (A, f + g)(h) = (f + g) h = f h + g h = Hom C (A, f)(h) + Hom C (A, g)(h)

35 2.2. KERNELS AND COKERNELS 27 and Hom C (f + g, A)(l) = l (f + g) = l f + l g 2.2 Kernels and Cokernels = Hom C (f, A)(l) + Hom C (g, A)(l). Definition 2.8. Let C be a preadditive category and let f : X Y be a morphism in C. i) A kernel of f is a morphism k : K X with f k = 0 that satisfies the following universal property: For every morphism t: T X with f t = 0 there exists a unique morphism λ: T K with t = k λ, i.e. making the diagram commutative. K k X f Y λ t T ii) A cokernel of f is a morphism c: Y C with c f = 0 that satisfies the following universal property: For every morphism s: Y S with s f = 0 there exists a unique morphism µ: C S with s = µ c, i.e. making the diagram commutative. X f c Y C s µ S iii) The domain of a kernel k of f is called a kernel-object of f and the codomain of a cokernel c of f is called a cokernel-object of f. iv) The category C is said to have kernels if every morphism f : X Y has a kernel. v) The category C is said to have cokernels if every morphism f : X Y has a cokernel. Remark 2.9. Let C be a preadditive category.

36 28 CHAPTER 2. ADDITIVE CATEGORIES i) A cokernel of a morphism is the dual notion of the kernel of a morphism, i.e. if k Hom C (K, X) is the kernel of a morphism f Hom C (X, Y ), then k op Hom C op(x, K) is the cokernel of the morphism f op Hom C op(y, X). ii) Kernel and cokernel of a morphism are uniquely determined up to a unique isomorphism: If k : K X and k : K X are two kernels of a morphism f : X Y, then the universal property of k gives rise to a unique morphism λ: K K with k = k λ and the universal property of k gives rise to a unique morphism λ : K K with k = k λ. In addition, the universal properties of k and k show that id K and id K are unique with k id K = k and k id K = k respectively. Since k λ λ = k λ = k k λ λ = k λ = k it follows that λ λ = id K and λ λ = id K, hence λ and λ are isomorphisms inverse to each other. The dual argument shows that the cokernel of f is uniquely determined up to a unique isomorphism. Example i) In (F V ec) the kernel of a linear mapping f : X Y is the inclusion f 1 ({0}) X, x x. The cokernel of f is the quotient map Y Y/f(X), y y + f(x). Since every morphism f has a kernel and a cokernel, the category (F V ec) has kernels as well as cokernels. The same is true in the category (Ab) of abelian groups (written additively) and group morphisms. ii) In the category (T V S) of topological vector spaces the kernel of a continous linear mapping f : X Y is also the embedding f 1 ({0}) X, where f 1 ({0}) is endowed with the topology induced by X and the cokernel of f is also the quotient map Y Y/f(X), where Y/f(X) is endowed with the quotient topology. Therefore the category (T V S) has kernels as well as cokernels. All this is the same in the category (LCS). iii) The kernel of a morphism f : X Y in the category (LCS) HD is the same as in the category (LCS), whereas the cokernel of f is the mapping Y Y/f(X), y y + f(x), where Y/f(X) is endowed with the quotient topology. Proposition Let C be a preadditive category. i) Kernels are monomorphisms. ii) Cokernels are epimorphisms.

37 2.2. KERNELS AND COKERNELS 29 Proof. It suffices to show i), since ii) is the dual statement. If f : X Y is a morphism in a category C, k : K X a kernel of f and g 1, g 2 : T K two morphisms with k g 1 = k g 2, then we have f k g 1 = f k g 2 = 0, hence there exists exactly one morphism λ: T K with k λ = k g 1 = k g 2 and therefore we have g 1 = g 2, which shows that k is a monomorphism. Definition Let C be a preadditive category and f : X Y be a morphism in C. i) An image of f is a kernel of a cokernel of f, i.e. if c: Y C is a cokernel of f, then an image of f is a morphism i: I Y with c i = 0, so that for every other morphism t: T Y with c t = 0 there exists a unique morphism λ: T I making the diagram commutative. I T λ i t Y c C ii) A coimage of f is a cokernel of a kernel of f, i.e. if k : K X is a kernel of f, then a coimage of f is a morphism q : X Q with q k = 0, so that for every other morphism s: X S with s k = 0 there exists a unique morphism µ: Q S making the diagram commutative. K k q X Q s µ S iii) The domain of an image i of f is called an image-object of f and the codomain of a coimage q of f is called a coimage-object of f. Remark i) The coimage of a morphism is the dual notion of the image of a morphism. ii) Remark 2.9.iii) shows that image and coimage of a morphism are uniquely determined uo to a unique isomorphism. Example i) In (F V ec) the image of a morphism f : X Y is the inclusion f(x) Y, y y and the coimage of f is the quotient map X X/f 1 ({0}), x x + f 1 ({0}).

38 30 CHAPTER 2. ADDITIVE CATEGORIES ii) In the category (T V S) of topological vector spaces the image of a continous linear mapping f : X Y is also the inclusion f(x) Y, where f(x) is endowed with the topology induced by Y and the coimage of f is also the quotient map X X/f 1 ({0}), where X/f 1 ({0}) is endowed with the quotient topology. All this is the same in the category (LCS). iii) The image of a morphism f : X Y in the category (LCS) HD is the inclusion f(x) Y, y y, where f(x) is endowed with the topology induced by Y. The coimage of f in (LCS) HD is the same as in the category (LCS), since f 1 ({0}) is a closed subspace of X when endowed with the induced topology. Notation Let f : X Y be a morphism in a category C. By 2.9.iii) we know that kernel and cokernel of f are unique up to a unique isomorphism. From now on k f : ker f X will denote the kernel of f, the cokernel of f, the image of f, and will denote the coimage of f. c f : Y cok f i f : im f Y ci f : X coim f Definition Let C be a preadditive category and let C be a full preadditive subcategory C of C. i) C is said to reflect kernels if whenever X f Y g Z is a sequence in C such that f is a kernel of g and Y and Z are objects of C, it follows that also X is an object of C (and therefore f is also a kernel of g in C ). ii) C is said to reflect cokernels if whenever X f Y g Z is a sequence in C such that g is a cokernel of f and X and Y are objects of C, it follows that also Z is an object of C (and therefore g is also a kernel of f in C ).

39 2.2. KERNELS AND COKERNELS 31 Example i) (LCS) is a full preadditive subcategory of (T V S) that reflects both kernels and cokernels. ii) (LCS) HD is a full preadditive subcategory of (T V S) that reflects kernels, but does not reflect cokernels. iii) Let (BOR) be the category of bornological locally convex spaces and continous linear mappings (cf. [6, 23,1.5] and [6, 11,2.3]). Since quotients of bornological spaces are again bornological (cf. [6, 23,2.93]), the category (BOR) is a full preadditive subcategory of (T V S) that reflects cokernels. On the other hand subspaces of bornological spaces need not be bornological (cf. [19, 6.3]) and therefore (BOR) does not reflect the kernels of (T V S). iv) Let (LCS) c be the category of complete locally convex spaces and continous linear mappings. It is a full preadditive subcategory of (T V S). For a morphism f : X Y in (LCS) c the subspace f 1 ({0}) is complete with regard to the topology induced by X if and only if it is closed in X, hence (LCS) c does not reflect the kernels of (T V S). In addition, the quotient Y/f(X) is in general not a complete space (cf. [20, 31, 6]) and therefore (LCS) c does not reflect the cokernels of (T V S). Remark Let C be a preadditive category and let α X X f g Y Y β be a commutative diagram in C. If f and g both possess a kernel, we have g α k f = β f k f = 0, hence the universal property of the kernel k g gives rise to a unique morphism λ: ker f ker g making the diagram k f ker f X λ α ker g kg X g f Y Y β commutative. By duality we obtain that, if f and g both possess a cokernel, there exists a