.

0 (hodnocen0 x )

BK

Second edition

Cham ; Heidelberg ; New York ; Dordrecht ; London : Springer, [2015]

x, 572 stran : ilustrace ; 24 cm

ISBN 978-3-319-11477-4 (brožováno)

Universitext, ISSN 0172-5939

Obsahuje bibliografii na stranách 531-537 a rejstřík

001643902

1 About the Course, and These Notes 1 // 1.1 Aims and prerequisites 1 // 1.2 Approach 2 // 1.3 A question a day 2 // 1.4 Homework 3 // 1.5 The name of the game 5 // 1.6 Other reading 5 // 1.7 Numeration; history; advice; web access; request for corrections 6 // 1.8 Acknowledgements 7 // Part I. Motivation and Examples 9 // 2 Making Some Things Precise 11 // 2.1 Generalities 11 // 2.2 What is a group? 11 // 2.3 Indexed sets 14 // 2.4 Arity 15 // 2.5 Group-theoretic terms 15 // 2.6 Evaluation 19 // 2.7 Terms in other families of operations 20 // 3 Free Groups 25 // 3.1 Motivation 25 // 3.2 The logician’s approach: construction from group-theoretic terms 28 // 3.3 Free groups as subgroups of big enough direct // products 32 // 3.4 The classical construction: free groups as groups of words. 37 // vii // viii Contents // 4 A Cook’s Tour of Other Universal Constructions 45 // 4.1 The subgroup and normal subgroup of G generated // by 5 C |G| 46 // 4.2 Imposing relations on a group. Quotient groups 47 // 4.3 Groups presented by generators and relations 49 // 4.4 Abelian groups, free abelian groups, and abelianizations 56 // 4.5 The Burnside problem 60 // 4.6 Products and coproducts of groups 62 // 4.7 Products and coproducts of abelian groups 69 // 4.8 Right and left universal properties 71 // 4.9 Tensor products 75 // 4.10 Monoids 79 // 4.11 Groups to monoids and back again 85 // 4.12 Associative and commutative rings 88 // 4.13 Coproducts and tensor products of rings 96 // 4.14 Boolean algebras and Boolean rings 100 // 4.15 Sets 104 // 4.16 Some algebraic structures we have not looked at 105 // 4.17 The Stone-Cech compactification of a topological space 105 // 4.18 Universal covering spaces 113 // Part II. Basic Tools and Concepts 117 // 5 Ordered Sets, Induction, and the Axiom of Choice 119 // 5.1 Partially ordered sets 119 // 5.2 Digression: preorders 128 //

5.3 Induction, recursion, and chain conditions 133 // 5.4 The axioms of set theory 141 // 5.5 Well-ordered sets and ordinals 146 // 5.6 Zorn’s Lemma 160 // 5.7 Some thoughts on set theory 168 // 6 Lattices, Closure Operators, and Galois Connections 173 // 6.1 Semilattices and lattices 173 // 6.2 0, 1, and completeness 183 // 6.3 Closure operators 194 // 6.4 Digression: a pattern of threes 201 // 6.5 Galois connections 205 // 7 Categories and Functors 213 // 7.1 What is a category? 213 // 7.2 Examples of categories 220 // 7.3 Other notations and viewpoints 228 // 7.4 Universes 232 // 7.5 Functors 238 // 7.6 Contravariant functors, and functors of several variables 247 // 7.7 Category-theoretic versions of some common mathematical notions: properties of morphisms 253 // 7.8 More categorical versions of common mathematical notions: special objects 262 // 7.9 Morphisms of functors (or “natural transformations”) 277 // 7.10 Properties of functor categories 287 // 7.11 Enriched categories (a sketch) 291 // 8 Universal Constructions in Category-Theoretic Terms 295 // 8.1 Universality in terms of initial and terminal objects 295 // 8.2 Representable functors, and Yoneda’s Lemma 296 // 8.3 Adjoint functors 305 // 8.4 Number-theoretic interlude: the p-adic numbers // and related constructions 317 // 8.5 Direct and inverse limits 323 // 8.6 Limits and colimits 334 // 8.7 What respects what? 342 // 8.8 Functors respecting limits and colimits 344 // 8.9 Interaction between limits and colimits 353 // 8.10 Some existence theorems 361 // 8.11 Morphisms involving adjunctions 368 // 8.12 Contravariant adjunctions 374 // 9 Varieties of Algebras 379 // 9.1 The category ii-Alg 379 // 9.2 Generating algebras from below 387 // 9.3 Terms and left universal constructions 391 // 9.4 Identities and varieties 399 // 9.5 Derived operations 411 //

9.6 Characterizing varieties and equational theories 415 // 9.7 Lie algebras 425 // 9.8 Some instructive trivialities 434 // 9.9 Clones and clonal theories 436 // 9.10 Structure and Semantics 449 // Part III. More on Adjunctions 455 // 10 Algebra and Coalgebra Objects in Categories, and Functors Having Adjoints 457 // 10.1 An example: SL(n) 457 // 10.2 Algebra objects in a category 461 // 10.3 Coalgebra objects in a category 466 // 10.4 Freyd’s criterion for the existence of left adjoints 470 // 10.5 Some corollaries and examples 473 // 10.6 Representable endofunctors of Monoid 480 // 10.7 Functors to and from some related categories 491 // 10.8 Representable functors among categories of abelian groups and modules 495 // 10.9 More on modules: left adjoints of representable // functors 499 // 10.10 Some general results on representable functors, mostly negative 506 // 10.11 A few ideas and techniques 510 // 10.12 Contravariant representable functors 514 // 10.13 More on commuting operations 522 // 10.14 Some further reading on representable functors, and on General Algebra 529 // References 531 // List of Exercises 539 // Symbol Index 547 // Word and Phrase Index 553