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Conceptual Mathematics: A First Introduction to Categories 2nd Edition by F. William Lawvere, ISBN-13: 978-0521719162

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Conceptual Mathematics: A First Introduction to Categories 2nd Edition by F. William Lawvere, ISBN-13: 978-0521719162

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  • Publisher: ‎ Cambridge University Press; 2nd edition (August 31, 2009)
  • Language: ‎ English
  • 404 pages
  • ISBN-10: ‎ 1107654165
  • ISBN-13: ‎ 978-0521719162

Conceptual Mathematics introduces the concept of category to beginning students, general readers, and practicing mathematical scientists based on a leisurely introduction to the important categories of directed graphs and discrete dynamical systems. The expanded second edition approaches more advanced topics via historical sketches and a concise introduction to adjoint functors.

In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics, Second Edition, introduces the concept of ‘category’ for the learning, development, and use of mathematics, to both beginning students and general readers, and to practicing mathematical scientists. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories.

Table of Contents:

Preface

xiii

Organisation of the book

xv

Acknowledgements

xvii

Preview

Session 1 Galileo and multiplication of objects

3

1 Introduction

3

2 Galileo and the flight of a bird

3

3 Other examples of multiplication of objects

7

Part I The category of sets

Article I Sets, maps, composition

13

1 Guide

20

Summary: Definition of category

21

Session 2 Sets, maps, and composition

22

1 Review of Article I

22

2 An example of different rules for a map

27

3 External diagrams

28

4 Problems on the number of maps from one set to another

29

Session 3 Composing maps and counting maps

31

Part II The algebra of composition

Article II Isomorphisms

39

1 Isomorphisms

39

2 General division problems: Determination and choice

45

3 Retractions, sections, and idempotents

49

4 Isomorphisms and automorphisms

54

5 Guide

58

Summary: Special properties a map may have

59

Session 4 Division of maps: Isomorphisms

60

1 Division of maps versus division of numbers

60

2 Inverses versus reciprocals

61

3 Isomorphisms as ‘divisors’

63

4 A small zoo of isomorphisms in other categories

64

Session 5 Division of maps: Sections and retractions

68

1 Determination problems

68

2 A special case: Constant maps

70

3 Choice problems

71

4 Two special cases of division: Sections and retractions

72

5 Stacking or sorting

74

6 Stacking in a Chinese restaurant

76

Session 6 Two general aspects or uses of maps

81

1 Sorting of the domain by a property

81

2 Naming or sampling of the codomain

82

3 Philosophical explanation of the two aspects

84

Session 7 Isomorphisms and coordinates

86

1 One use of isomorphisms: Coordinate systems

86

2 Two abuses of isomorphisms

89

Session 8 Pictures of a map making its features evident

91

Session 9 Retracts and idempotents

99

1 Retracts and comparisons

99

2 Idempotents as records of retracts

100

3 A puzzle

102

4 Three kinds of retract problems

103

5 Comparing infinite sets

106

Quiz

108

How to solve the quiz problems

109

Composition of opposed maps

114

Summary/quiz on pairs of ‘opposed’ maps

116

Summary: On the equation p = 1A

117

Review of 1-words’

118

Test 1

119

Session 10 Brouwer’s theorems

120

1 Balls, spheres, fixed points, and retractions

120

2 Digression on the contrapositive rule

124

3 Brouwer’s proof

124

4 Relation between fixed point and retraction theorems

126

5 How to understand a proof: The objectification and ‘mapification’ of concepts

127

6 The eye of the storm

130

7 Using maps to formulate guesses

131

Part III Categories of structured sets

Article III Examples of categories

135

1 The category Sp of endomaps of sets

136

2 Typical applications of S°

137

3 Two subcategories of S°

138

4 Categories of endomaps

138

5 Irreflexive graphs

141

6 Endomaps as special graphs

143

7 The simpler category SI: Objects are just maps of sets

144

8 Reflexive graphs

145

9 Summary of the examples and their general significance

146

10 Retractions and injectivity

146

11 Types of structure

149

12 Guide

151

Session 11 Ascending to categories of richer structures

152

1 A category of richer structures: Endomaps of sets

152

2 Two subcategories: Idempotents and automorphisms

155

3 The category of graphs

156

Session 12 Categories of diagrams

161

1 Dynamical systems or automata

161

2 Family trees

162

3 Dynamical systems revisited

163

Session 13 Monoids

166

Session 14 Maps preserve positive properties

170

1 Positive properties versus negative properties

173

Session 15 Objectification of properties in dynamical systems

175

1 Structure-preserving maps from a cycle to another endomap

175

2 Naming elements that have a given period by maps

176

3 Naming arbitrary elements

177

4 The philosophical role of N

180

5 Presentations of dynamical systems

182

Session 16 Idempotents, involutions, and graphs

187

1 Solving exercises on idempotents and involutions

187

2 Solving exercises on maps of graphs

189

Session 17 Some uses of graphs

196

1 Paths

196

2 Graphs as diagram shapes

200

3 Commuting diagrams

201

4 Is a diagram a map?

203

Test 2

204

Session 18 Review of Test 2

205

Part IV Elementary universal mapping properties

Article IV Universal mapping properties

213

1 Terminal objects

213

2 Separating

215

3 Initial object

215

4 Products

216

5 Commutative, associative, and identity laws for multiplication of objects

220

6 Sums

222

7 Distributive laws

222

8 Guide

223

Session 19 Terminal objects

225

Session 20 Points of an object

230

Session 21 Products in categories

236

Session 22 Universal mapping properties and incidence relations

245

1 A special property of the category of sets

245

2 A similar property in the category of endomaps of sets

246

3 Incidence relations

249

4 Basic figure-types, singular figures, and incidence, in the category of graphs

250

Session 23 More on universal mapping properties

254

1 A category of pairs of maps

255

2 How to calculate products

256

session 24 Uniqueness of products and definition of sum

261

1 The terminal object as an identity for multiplication

261

2 The uniqueness theorem for products

263

3 Sum of two objects in a category

265

Session 25 Labelings and products of graphs

269

1 Detecting the structure of a graph by means of labelings

270

2 Calculating the graphs A x Y

273

3 The distributive law

275

Session 26 Distributive categories and linear categories

276

1 The standard map Ax B1 + A X B2 -> A x (B1+ B2)

276

2 Matrix multiplication in linear categories

279

3 Sum of maps in a linear category

279

4 The associative law for sums and products

281

Session 27 Examples of universal constructions

284

1 Universal constructions

284

2 Can objects have negatives?

287

3 Idempotent objects

289

4 Solving equations and picturing maps

292

Session 28 The category of pointed sets

295

1 An example of a non-distributive category

295

Test 3

299

Test 4

300

Test 5

301

Session 29 Binary operations and diagonal arguments

302

1 Binary operations and actions

302

2 Cantor’s diagonal argument

303

Part V Higher universal mapping properties

Article V Map objects

313

1 Definition of map object

313

2 Distributivity

315

3 Map objects and the Diagonal Argument

316

4 Universal properties and `observables’

316

5 Guide

319

Session 30 Exponentiation

320

1 Map objects, or function spaces

320

2 A fundamental example of the transformation of map objects

323

3 Laws of exponents

324

4 The distributive law in cartesian closed categories

327

Session 31 Map object versus product

328

1 Definition of map object versus definition of product

329

2 Calculating map objects

331

Article VI The contravariant parts functor

335

1 Parts and stable conditions

335

2 Inverse Images and Truth

336

Session 32 Subobject, logic, and truth

339

1 Subobjects

339

2 Truth

342

3 The truth value object

344

Session 33 Parts of an object: Toposes

348

1 Parts and inclusions

348

2 Toposes and logic

352

Article VII The Connected Components Functor

358

1 Connectedness versus discreteness

358

2 The points functor parallel to the components functor

359

3 The topos of right actions of a monoid

360

Session 34 Group theory and the number of types of connected objects

362

Session 35 Constants, codiscrete objects, and many connected objects

366

1 Constants and codiscrete objects

366

2 Monoids with at least two constants

367

Appendices

368

Appendix I Geometery of figures and algebra of functions

369

1 Functors

369

2 Geometry of figures and algebra of functions as categories themselves

370

Appendix II Adjoint functors with examples from graphs and dynamical systems

372

Appendix III The emergence of category theory within mathematics

378

Appendix IV Annotated Bibliography

381

Index

385

F. William Lawvere is a Professor Emeritus of Mathematics at the State University of New York. He has previously held positions at Reed College, the University of Chicago and the City University of New York, as well as visiting Professorships at other institutions worldwide. At the 1970 International Congress of Mathematicians in Nice, Prof. Lawvere delivered an invited lecture in which he introduced an algebraic version of topos theory which united several previously ‘unrelated’ areas in geometry and in set theory; over a dozen books, several dozen international meetings, and hundreds of research papers have since appeared, continuing to develop the consequences of that unification.

Stephen H. Schanuel is a Professor of Mathematics at the State University of New York at Buffalo. He has previously held positions at Johns Hopkins University, Institute for Advanced Study and Cornell University, as well as lecturing at institutions in Denmark, Switzerland, Germany, Italy, Colombia, Canada, Ireland, and Australia. Best known for Schanuel’s Lemma in homological algebra (and related work with Bass on the beginning of algebraic K–theory), and for Schanuel’s Conjecture on algebraic independence and the exponential function, his research thus wanders from algebra to number theory to analysis to geometry and topology.

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