- NOTE TO USERS This reproduction is the best 'copy available Acyclic Models May 1997 Department of Mathematics ancl S tatistics McGill University, Montréal Canada A thesis submitted to the Faculty of Graduate S tudies and Research in partial fuEllment of the requirements for the degree of Master of Science National Library Bibliothique nationale I*m of Canada du Canada Acquisitions and Acquisitions et Bibliographie Seivices services bibliographiques 395 Wdtimgton Street 395, nie Wellington ûttawa ON K1A ON4 ûttawaON K1AON4 Canada i;ariada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant a la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sel1 reproduire, prêter, distribuer ou copies of this thesis in rnicroform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othefwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation, Dedicated to my husband Robi and daughter Jita, with love and affection. Acknowledgement s I wish to express my deepest gratitude to my superviser Prof. M. Barr for his guidance, invaluable suggestions and encouragement throughout the preparation of this thesis. He also greatly helped in making the diagram. 1 wish to express my sincere gratitude to Prof. S. Majumder. who introcluced me to homology group and to Prof. A. H. Beg, Prof. S. Bhattacharjee and Prof. A. S. A. Noor, mho encouraged me to corne here. 1 would like to thank my student friends Lassina Dembele and Gülhan Alpargu for their help and numerous advices on cornputer materials. 1 specidly acknowledge their friendly attitude and the nice and pleasent atmosphere they made during the years we spent togather in this department. 1am also grateful to Lassinn, who reads my thesis very carefuly and made some grammatical corrections and translated the abstract into French. Very special thanks go to Hasiba Islam for her warmest friendship Ùuring my studying and to my parents and my brothers for their mental support and for t&ng care of my daughter during my absence. I thank rny husband for his understanding and also 'apologize' to him for dl the hard time he endured when 1 was away. My studies at McGill University are supported by Canadian Commonwealth Scholarship and Felloship Program. Abstract In this thesis we describe a new version of acyclic models, which was first given by Barr, that gives the Theorem of Barr and Beck, and of André as special cases. We begin the thesis with Beck's definition of module and we describe how he used this definition, in conjunction with the theory of triples, to define homology t heories. The t heory described here is based on the notion of acyclic classe^. An acyclic class is a class of objects in a category of chah complexes and corresponds to a class of arrows, whose mapping cones they are. We also give an answer to the question. where do acyclic classes corne fkom. We conclude the thesis by showing that Cartan- Eilenberg cohomology of groups, of associative algebras. and of Lie dgebras are the sarne as cotnple ones, using proofs based on those presented by Barr. Résumé Dans ce mémoire, nous décrivons une nouvelle version de modèles acycliques ini- tialement introduite par Barr. Cela conduit aux Théorèmes de Barr et Beck, et de André comme cas particuliers. La thèse commence par la définition des modules. par Beck, qui en conjonction avec la théorie des triples conduit aux théories de l'homologie. La théorie décrite ici est basée sur la notion de classes acycliques. Une classe acyclique est une classe d'objets dans une catégorie de chaines complexes correspondant à une classe de flêches dont elles sont les 'mapping cones'. Nous répondons également à la question à savoir d'où viennent les classes acycliques. Nous concluons le mémoire en montrant que la cohomologie de Cartan-Eilenberg pour les groupes. les algèbres associatives. et les algèbres de Lie sont les mêmes que celles de leurs cotriples. La preuve est basée sur des résultats de Bm. Contents Acknowledgements iii Abstract Résumé Introduction 1 Triple Cohomology 1.1 Simplicial objects 2 Acyclic Models 2.1 Acyclic classes 2.2 Properties of acyclic classes 2.3 Examples of acyclic classes 3 Cohomology of Groups and Algebras 4 Calculuses of fractions 4.1 Some more theorems on double complexes 5 Application to homology on manifolds 6 Cartan-Eilenberg cohomology and cotriples 6.1 Regular epimorphisms and regular categories 6.2 The Cart an-Eilenberg setting 6.3 Themaintheorem 6.4 Applications of the main theorem Bibliograp hy Introduction The main purpose of this thesis is to fill in details in papers [3] and [4],we also add some new material. Chapter 1 presents triples and cotriples, and their relation with a pair of adjoint furictors [19]. It also describes Beck module and the way a cotriple G gives rise to a cohomology theory in a tripleable category dT [SI. In Chapter 2, we define e-presentable and G-contractible fùnctors, where G is a endofunctor and E is a counit. We also define Acyclic classes and Mapping cone sequence, and discuss some of their properties; some examples of Acyclic classes are given. Almost all results are fkom [41. The question of where do acyclic classes corne fkom is discussed, and the way they match with functors as weU. The Barr-Beck Theorern and its proof by Barr (41 using acyclic classes is also presented. In Chapter 3, we discuss the naturality of the isomorphism between the cotriple cohomology of groups and Eilenberg-Mac Lane cohomology group [13],and between the Hochschild cohomology group [18] and cotnple cohomology of associative alge- bras. These are results from [5]. Chapter 4 discusses the properties of C, the class of arrows whose mapping cone is a member of an acyclic class. Most of the results are from Barr [4]. Here we prove C-"& E C-'(%/.-), where - is defined by (f,o) .- (f',~') (each pair has same codomain) if a f a'f and aa a'of E C and Y'%?is a category of fractions [Ir]. We also present some properties of double complexes 131. In Chapter 5, we state André's Theorem and then find the homology of manifolds in detail. We follow the proofs given by Barr in [4]. In Chapter 6. we present Cartan-Eilenberg cohomology of groups. associative algebras and Lie algebras [8] are the same as (with sorne shifted dimensions) the cotriple cohomology of groups, associative algebras and Lie dgebras respectively. The discussion follows [3]. Chapter 1 Triple Cohomology Let J3/ be a category. T = (T, q, p) is a triple in d if T:s;/ -t .d is a functor. 7: ld -t T and p: TT -+ T are natural transformations such that p Q qi? = p 0 TV= 1~:T -, T,p 0 pT = p O Tp: TTT + S. The natural transformation rl is called the unit and p is called the multiplication of T. 1.0.1 Relation between adjoint functors and triples Let F be a left adjoint of U. wbere F: d + 9, then there is a natural isomorphism Homd(.l, [TB) 9 Homa(FA,B)for A E Ob(&), B E Ob@'). Putting B = FA, we get a natural transformation q: ld -t UF and for -4 = UB we have s: FU -, la: and they satisfy EFoF~= IF, UEO~U= lu. P. Huber [19] observed that T=UF:d-td T= -A' and G=FU:&?+&? t:G+ lS 6=FlrU:G-GG T is a triple in d and G is a cotriple in 9. 1.0.2 Definition Let V be a category and S E Ob(%). The cotego~y(%.-.Y) qwho~eobjects are aiiowJ C X and anows are commutative triangles -y + C -, C' -t X; C,C' E Ob(%) is called a dice category over -y. Here is Beck's definition of a module. His definition is appropriate for the kind of module that is a coefficient module for cohomology. For groups, commutative 3 algebras and Lie dgebras, these are left modules; for associative algebras, the ap- propriate notion is that of two-sided module. 1.0.3 Definition Let a! be a category and let A E Ob(&). The category of Mod(A) is the category of abelian group objects of (d,-4) Assuming that a category d has finite products, by the abelian group objects of a! we mean that for each A E Ob(&) there are riaps rn: A x -4 -t A, i: .1 -4 and u: 1 -t A called multiplication, inverse and zero maps respectively. They are assumed to make the following dia.grarns are commutative. -4 x -4 *4 x -4 .4 x -4 -lxa -2x1 A~1~~4x.A-1x4uxl Here A: A + A x A denotes the arrow whose components are the ideutity on each 11 coordinate and O: A + A denotes the composite -4-, 1 -4 A. So the objects of Ab(&) are (A, m, i, u) which satisfy the above conditions. An arrow f: (A,n, u) -t (B, m, u) is f: 4 + B, such that the following diagrams conunute. Now, we show how a category of Mod(R) is the category of abelian group objects of (9,R), where 9 is a category of commutative rings and R is a commutative ring with unity.