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Exact Categories in Functional Analysis Script Exact Categories in Functional Analysis Leonhard Frerick and Dennis Sieg June 22, 2010 ii To Susanne Dierolf. iii iv Contents 1 Basic Notions 1 1.1 Categories . 1 1.2 Morphisms and Objects . 5 1.3 Functors . 9 2 Additive Categories 25 2.1 Pre-additive Categories . 25 2.2 Kernels and Cokernels . 27 2.3 Pullback and Pushout . 36 2.4 Product, Coproduct and Biproduct . 42 2.5 Additive Categories . 50 2.6 Semi-abelian Categories . 58 3 Exact Categories 65 3.1 Basic Properties . 65 3.2 E-strict morphisms . 88 4 Maximal Exact Structure 103 4.1 p-strict morphisms . 103 4.2 Maximal Exact Structure . 109 4.3 Quasi-abelian Categories . 115 5 Derived Functors 121 5.1 Exact Functors . 121 5.2 Complexes . 126 5.3 Resolutions . 142 5.4 Derived Functors . 157 5.5 Universal δ-Functors . 171 6 Yoneda-Ext-Functors 181 6.1 Yoneda-Ext1 . 181 6.2 Yoneda-Extn . 195 v vi CONTENTS 7 Appendix 1: Projective Spectra 225 7.1 Categories of Projective Spectra . 225 7.2 The Projective Limit . 231 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 con- tain 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 start- ing 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 con- vex spaces, Banach spaches or Fr´echet 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 avail- able at http://ubt.opus.hbz-nrw.de/volltexte/2010/572/pdf/SiegDiss.pdf. 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 analy- sis. 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 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 B¨uhler[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 inspira- tion 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. Chapter 1 Basic Notions 1.1 Categories Definition 1.1. A category is a quadruple C = (Ob(C); HomC; id; ◦) con- sisting 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 HomC(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 2 HomC(X; Y )". (3) For each object X 2 Ob(C), a morphism idX : 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 idY ◦f = f and f ◦ idX = f. (C3) The sets HomC(X; Y ) are pairwise disjoint. Remark 1.2. Let C = (Ob(C); HomC; 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 HomC(X; Y ) in C. 1 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 2 HomC(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 idX : 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 mor- phism idX 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) (Ring1) 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) (TVS) the category of topological vector spaces over a fixed field F 2 fR; Cg 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: 1.1. CATEGORIES 3 • (Met) the category of metric spaces and contractions. • (Metu) the category of metric spaces and uniformly conti- nous mappings. • (Metc) the category of metric spaces and continous map- pings. b) The following two categories are natural constructs having as objects all Banach spaces: • (Ban) the category of Banach spaces and continous linear mappings. • (Banc) the category of Banach spaces and linear contrac- tions. 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, idn : 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: f(i; j)g if i ≤ j Ob(C(I)) = I; Hom (i; j) = ; C(I) ; otherwise as well as idi = f(i; i)g and f(j; k)g ◦ f(i; j)g = f(i; k)g.
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