Almost All

Almost allWikipedia

Contents

1 Canonical form 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.5 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.6 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.7 Mathematical notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.8 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.9 Game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.10 Proof theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.11 Rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.12 Lambda calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.13 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.14 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.15 Dierential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.16 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Category theory 52.1 An abstraction of other mathematical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

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2.5 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Other concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6.1 Universal constructions, limits, and colimits . . . . . . . . . . . . . . . . . . . . . . . . . 92.6.2 Equivalent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.3 Further concepts and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.4 Higher-dimensional categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Cocountability 153.1 -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Coniteness 164.1 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Conite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.2 Double-pointed conite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3.1 Product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3.2 Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Complex analysis 185.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 Major results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Countable set 236.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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6.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Elementary proof 317.1 Prime number theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2 Friedmans conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Expression (mathematics) 338.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3 Syntax versus semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.3.3 Formal languages and lambda calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

8.4 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

9 Functor 369.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9.1.1 Covariance and contravariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.1.2 Opposite functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.1.3 Bifunctors and multifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.4 Relation to other categorical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.5 Computer implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10 Functor category 4110.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.3 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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11 Mathematical beauty 4411.1 Beauty in method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.2 Beauty in results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.3 Beauty in experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.4 Beauty and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.5 Beauty and mathematical information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.6 Mathematics and the arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

11.6.1 Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.6.2 Visual arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

11.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

12 Mathematical folklore 5212.1 Stories, sayings and jokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

13 Mathematical object 5413.1 Cantorian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.2 Foundational paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

14 Natural transformation 5614.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

14.2.1 Opposite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.2.2 Double dual of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.2.3 Tensor-hom adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

14.3 Unnatural isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.3.1 Example: fundamental group of torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.3.2 Example: dual of a nite-dimensional vector space . . . . . . . . . . . . . . . . . . . . . . 59

14.4 Operations with natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5914.5 Functor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.6 Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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14.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

15 Occams razor 6215.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

15.1.1 Formulations before Ockham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6315.1.2 Ockham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6415.1.3 Later formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

15.2 Justications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.2.1 Aesthetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.2.2 Empirical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.2.3 Practical considerations and pragmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.2.4 Mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.2.5 Other philosophers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

15.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.3.1 Science and the scientic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.3.2 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.3.3 Medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7015.3.4 Religion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7115.3.5 Penal ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7215.3.6 Probability theory and statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

15.4 Controversial aspects of the razor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7315.5 Anti-razors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7315.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7515.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

16 Parameter 8016.1 Mathematical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

16.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.1.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.1.3 Analytic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8116.1.4 Mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.1.5 Statistics and econometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.1.6 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

16.2 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.3 Computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8416.4 Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8416.5 Environmental science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

vi CONTENTS

16.6 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8416.7 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8416.8 Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8516.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8516.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

17 Parametrization 8617.1 Non-uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.2 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.3 Parametrization invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.4 Examples of parametrized models/objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.5 Parametrization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

18 Pathological (mathematics) 8918.1 Pathological functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.2 Prevalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.3 Pathological examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118.5 Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9218.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

19 Rigour 9319.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9319.2 Intellectual rigour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

19.2.1 Intellectual honesty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9319.2.2 Politics and law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

19.3 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.3.1 Mathematical proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.3.2 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.3.3 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

19.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

20 Statistical parameter 9620.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

21 Well-behaved 98

22 Yoneda lemma 100

CONTENTS vii

22.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10022.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

22.2.1 General version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10022.2.2 Naming conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10122.2.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10122.2.4 The Yoneda embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

22.3 Preadditive categories, rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10322.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10322.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10322.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10322.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10322.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 104

22.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10422.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10722.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Chapter 1

Canonical form

For other senses of canonical in mathematics, see Canonical (disambiguation)#Mathematics

In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standardway of presenting that object as a mathematical expression. The distinction between canonical and normal formsvaries by subeld. In most elds, a canonical form species a unique representation for every object, while a normalform simply species its form, without the requirement of uniqueness.The canonical form of a positive integer in decimal representation is a nite sequence of digits that does not beginwith zero.More generally, for a class of objects on which an equivalence relation (which can dier from standard notions ofequality, for instance by considering dierent forms of equal objects to be nonequivalent) is dened, a canonicalform consists in the choice of a specic object in each class. For example, row echelon form and Jordan normal formare canonical forms for matrices.In computer science, and more specically in computer algebra, when representing mathematical objects in a com-puter, there are usually many dierent ways to represent the same object. In this context, a canonical form is arepresentation such that every object has a unique representation. Thus, the equality of two objects can easily betested by testing the equality of their canonical forms. However canonical forms frequently depend on arbitrarychoices (like ordering the variables), and this introduces diculties for testing the equality of two objects resultingon independent computations. Therefore, in computer algebra, normal form is a weaker notion: A normal form isa representation such that zero is uniquely represented. This allows testing for equality by putting the dierence oftwo objects in normal form (see Computer algebra#Equality).Canonical form can also mean a dierential form that is dened in a natural (canonical) way; see below.In computer science, data that has more than one possible representation can often be canonicalized into a completelyunique representation called its canonical form. Putting something into canonical form is canonicalization.[1]

1.1 DenitionSuppose we have some set S of objects, with an equivalence relation. A canonical form is given by designating someobjects of S to be in canonical form, such that every object under consideration is equivalent to exactly one objectin canonical form. In other words, the canonical forms in S represent the equivalence classes, once and only once.To test whether two objects are equivalent, it then suces to test their canonical forms for equality. A canonicalform thus provides a classication theorem and more, in that it not just classies every class, but gives a distinguished(canonical) representative.In practical terms, one wants to be able to recognize the canonical forms. There is also a practical, algorithmicquestion to consider: how to pass from a given object s in S to its canonical form s*? Canonical forms are generallyused to make operating with equivalence classes more eective. For example in modular arithmetic, the canonicalform for a residue class is usually taken as the least non-negative integer in it. Operations on classes are carriedout by combining these representatives and then reducing the result to its least non-negative residue. The uniquenessrequirement is sometimes relaxed, allowing the forms to be unique up to some ner equivalence relation, like allowing

1

2 CHAPTER 1. CANONICAL FORM

reordering of terms (if there is no natural ordering on terms).A canonical form may simply be a convention, or a deep theorem.For example, polynomials are conventionally written with the terms in descending powers: it is more usual to writex2 + x + 30 than x + 30 + x2, although the two forms dene the same polynomial. By contrast, the existence of Jordancanonical form for a matrix is a deep theorem.

1.2 ExamplesNote: in this section, "up to" some equivalence relation E means that the canonical form is not unique in general, butthat if one object has two dierent canonical forms, they are E-equivalent.

1.2.1 Linear algebra

1.2.2 Classical logicMain article: Canonical form (Boolean algebra)

Negation normal form Conjunctive normal form Disjunctive normal form Algebraic normal form Prenex normal form Skolem normal form Blake canonical form, also known as the complete sum of prime implicants, the complete sum, or the disjunctive

prime form

1.2.3 Functional analysis

1.2.4 Number theory canonical representation of a positive integer canonical form of a continued fraction

1.2.5 Algebra

1.2.6 Geometry The equation of a line: Ax + By = C, with A2 + B2 = 1 and C 0

The equation of a circle: (x h)2 + (y k)2 = r2

By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written asa linear equation in point-slope and slope-intercept form.

1.2.7 Mathematical notationStandard form is used by many mathematicians and scientists to write extremely large numbers in a more concise andunderstandable way.

1.3. SEE ALSO 3

1.2.8 Set theory Cantor normal form of an ordinal number

1.2.9 Game theory Normal form game

1.2.10 Proof theory Normal form (natural deduction)

1.2.11 Rewriting systems In an abstract rewriting system a normal form is an irreducible object.

1.2.12 Lambda calculus Beta normal form if no beta reduction is possible; Lambda calculus is a particular case of an abstract rewriting

system.

1.2.13 Dynamical systems Normal form of a bifurcation

1.2.14 Graph theoryMain article: Graph canonization

1.2.15 Dierential formsCanonical dierential forms include the canonical one-form and canonical symplectic form, important in the studyof Hamiltonian mechanics and symplectic manifolds.

1.2.16 Computation Data normalization

1.3 See also Canonical class Normalization (disambiguation) Standardization

1.4 Notes[1] The term 'canonization' is sometimes incorrectly used for this.

4 CHAPTER 1. CANONICAL FORM

1.5 References Shilov, Georgi E. (1977), Silverman, Richard A., ed., Linear Algebra, Dover, ISBN 0-486-63518-X. Hansen, Vagn Lundsgaard (2006), Functional Analysis: Entering Hilbert Space, World Scientic Publishing,

ISBN 981-256-563-9.

Chapter 2

Category theory

Schematic representation of a category with objects X, Y, Z and morphisms f, g, g f. (The categorys three identity morphisms 1X,1Y and 1Z, if explicitly represented, would appear as three arrows, next to the letters X, Y, and Z, respectively, each having as itsshaft a circular arc measuring almost 360 degrees.)

Category theory[1] formalizes mathematical structure and its concepts in terms of a collection of objects and of

5

6 CHAPTER 2. CATEGORY THEORY

arrows (also called morphisms). A category has two basic properties: the ability to compose the arrows associativelyand the existence of an identity arrow for each object. Category theory can be used to formalize concepts of otherhigh-level abstractions such as sets, rings, and groups.Several terms used in category theory, including the term morphism, are used dierently from their uses in the restof mathematics. In category theory, a morphism obeys a set of conditions specic to category theory itself. Thus,care must be taken to understand the context in which statements are made.

2.1 An abstraction of other mathematical conceptsMany signicant areas of mathematics can be formalised by category theory as categories. Category theory is anabstraction of mathematics itself that allows many intricate and subtle mathematical results in these elds to be stated,and proved, in a much simpler way than without the use of categories.[2]

The most accessible example of a category is the category of sets, where the objects are sets and the arrows arefunctions from one set to another. However, the objects of a category need not be sets, and the arrows need not befunctions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour ofobjects and arrows is a valid category, and all the results of category theory will apply to it.The arrows of category theory are often said to represent a process connecting two objects, or in many cases astructure-preserving transformation connecting two objects. There are however many applications where muchmore abstract concepts are represented by objects and morphisms. The most important property of the arrows is thatthey can be composed, in other words, arranged in a sequence to form a new arrow.Categories now appear in most branches of mathematics, some areas of theoretical computer science where they cancorrespond to types, and mathematical physics where they can be used to describe vector spaces. Categories wererst introduced by Samuel Eilenberg and Saunders Mac Lane in 194245, in connection with algebraic topology.Category theory has several faces known not just to specialists, but to other mathematicians. A term dating fromthe 1940s, "general abstract nonsense", refers to its high level of abstraction, compared to more classical branchesof mathematics. Homological algebra is category theory in its aspect of organising and suggesting manipulations inabstract algebra.

2.2 Utility

2.2.1 Categories, objects, and morphisms

The study of categories is an attempt to axiomatically capture what is commonly found in various classes of relatedmathematical structures by relating them to the structure-preserving functions between them. A systematic study ofcategory theory then allows us to prove general results about any of these types of mathematical structures from theaxioms of a category.Consider the following example. The class Grp of groups consists of all objects having a group structure. Onecan proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it isimmediately proven from the axioms that the identity element of a group is unique.Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory em-phasizes the morphisms the structure-preserving mappings between these objects; by studying these morphisms,we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the grouphomomorphisms. A group homomorphism between two groups preserves the group structure in a precise sense it is a process taking one group to another, in a way that carries along information about the structure of the rstgroup into the second group. The study of group homomorphisms then provides a tool for studying general propertiesof groups and consequences of the group axioms.A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps (mor-phisms) between topological spaces in topology (the associated category is called Top), and the study of smoothfunctions (morphisms) in manifold theory.Not all categories arise as structure preserving (set) functions, however; the standard example is the category ofhomotopies between pointed topological spaces.

2.3. CATEGORIES, OBJECTS, AND MORPHISMS 7

If one axiomatizes relations instead of functions, one obtains the theory of allegories.

2.2.2 FunctorsMain article: FunctorSee also: Adjoint functors Motivation

A category is itself a type of mathematical structure, so we can look for processes which preserve this structure insome sense; such a process is called a functor.Diagram chasing is a visual method of arguing with abstract arrows joined in diagrams. Functors are representedby arrows between categories, subject to specic dening commutativity conditions. Functors can dene (construct)categorical diagrams and sequences (viz. Mitchell, 1965). A functor associates to every object of one category anobject of another category, and to every morphism in the rst category a morphism in the second.In fact, what we have done is dene a category of categories and functors the objects are categories, and the mor-phisms (between categories) are functors.By studying categories and functors, we are not just studying a class of mathematical structures and the morphismsbetween them; we are studying the relationships between various classes of mathematical structures. This is a funda-mental idea, which rst surfaced in algebraic topology. Dicult topological questions can be translated into algebraicquestions which are often easier to solve. Basic constructions, such as the fundamental group or the fundamentalgroupoid of a topological space, can be expressed as functors to the category of groupoids in this way, and theconcept is pervasive in algebra and its applications.

2.2.3 Natural transformationsMain article: Natural transformation

Abstracting yet again, some diagrammatic and/or sequential constructions are often naturally related a vaguenotion, at rst sight. This leads to the clarifying concept of natural transformation, a way to map one functor toanother. Many important constructions in mathematics can be studied in this context. Naturality is a principle, likegeneral covariance in physics, that cuts deeper than is initially apparent. An arrow between two functors is a naturaltransformation when it is subject to certain naturality or commutativity conditions.Functors and natural transformations ('naturality') are the key concepts in category theory.[3]

2.3 Categories, objects, and morphismsMain articles: Category (mathematics) and Morphism

2.3.1 CategoriesA category C consists of the following three mathematical entities:

A class ob(C), whose elements are called objects; A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a sourceobject a and target object b.The expression f : a b, would be verbally stated as "f is a morphism from a to b".The expression hom(a, b) alternatively expressed as homC(a, b), mor(a, b), or C(a, b) denotes thehom-class of all morphisms from a to b.

A binary operation , called composition of morphisms, such that for any three objects a, b, and c, we havehom(b, c) hom(a, b) hom(a, c). The composition of f : a b and g : b c is written as g f or gf,[4]governed by two axioms:


Associativity: If f : a b, g : b c and h : c d then h (g f) = (h g) f, and Identity: For every object x, there exists a morphism 1x : x x called the identity morphism for x, such

that for every morphism f : a b, we have 1b f = f = f 1a.

From the axioms, it can be proved that there is exactly one identity morphism for every object.Some authors deviate from the denition just given by identifying each object with its identitymorphism.

2.3.2 MorphismsRelations among morphisms (such as fg = h) are often depicted using commutative diagrams, with points (corners)representing objects and arrows representing morphisms.Morphisms can have any of the following properties. A morphism f : a b is a:

monomorphism (or monic) if f g1 = f g2 implies g1 = g2 for all morphisms g1, g2 : x a. epimorphism (or epic) if g1 f = g2 f implies g1 = g2 for all morphisms g1, g2 : b x. bimorphism if f is both epic and monic. isomorphism if there exists a morphism g : b a such that f g = 1b and g f = 1a.[5]

endomorphism if a = b. end(a) denotes the class of endomorphisms of a. automorphism if f is both an endomorphism and an isomorphism. aut(a) denotes the class of automorphisms

of a. retraction if a right inverse of f exists, i.e. if there exists a morphism g : b a with f g = 1b. section if a left inverse of f exists, i.e. if there exists a morphism g : b a with g f = 1a.

Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three state-ments are equivalent:

f is a monomorphism and a retraction; f is an epimorphism and a section; f is an isomorphism.

2.4 FunctorsMain article: Functor

Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category ofall (small) categories.A (covariant) functor F from a category C to a category D, written F : C D, consists of:

for each object x in C, an object F(x) in D; and for each morphism f : x y in C, a morphism F(f) : F(x) F(y),

such that the following two properties hold:

For every object x in C, F(1x) = 1Fx; For all morphisms f : x y and g : y z, F(g f) = F(g) F(f).

A contravariant functor F: C D, is like a covariant functor, except that it turns morphisms around (reverses allthe arrows). More specically, every morphism f : x y in C must be assigned to a morphism F(f) : F(y) F(x)in D. In other words, a contravariant functor acts as a covariant functor from the opposite category Cop to D.

2.5. NATURAL TRANSFORMATIONS 9

2.5 Natural transformationsMain article: Natural transformation

A natural transformation is a relation between two functors. Functors often describe natural constructions andnatural transformations then describe natural homomorphisms between two such constructions. Sometimes twoquite dierent constructions yield the same result; this is expressed by a natural isomorphism between the twofunctors.If F and G are (covariant) functors between the categories C and D, then a natural transformation from F to Gassociates to every object X in C a morphism X : F(X) G(X) in D such that for every morphism f : X Y in C,we have Y F(f) = G(f) X; this means that the following diagram is commutative:

Commutative diagram dening natural transformations

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G suchthat X is an isomorphism for every object X in C.

2.6 Other concepts

2.6.1 Universal constructions, limits, and colimits

Main articles: Universal property and Limit (category theory)

Using the language of category theory, many areas of mathematical study can be categorized. Categories includesets, groups and topologies.


Each category is distinguished by properties that all its objects have in common, such as the empty set or the product oftwo topologies, yet in the denition of a category, objects are considered to be atomic, i.e., we do not know whetheran object A is a set, a topology, or any other abstract concept. Hence, the challenge is to dene special objectswithout referring to the internal structure of those objects. To dene the empty set without referring to elements, orthe product topology without referring to open sets, one can characterize these objects in terms of their relations toother objects, as given by the morphisms of the respective categories. Thus, the task is to nd universal propertiesthat uniquely determine the objects of interest.Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The centralconcept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a colimit.

2.6.2 Equivalent categories

Main articles: Equivalence of categories and Isomorphism of categories

It is a natural question to ask: under which conditions can two categories be considered to be essentially the same, inthe sense that theorems about one category can readily be transformed into theorems about the other category? Themajor tool one employs to describe such a situation is called equivalence of categories, which is given by appropriatefunctors between two categories. Categorical equivalence has found numerous applications in mathematics.

2.6.3 Further concepts and results

The denitions of categories and functors provide only the very basics of categorical algebra; additional importanttopics are listed below. Although there are strong interrelations between all of these topics, the given order can beconsidered as a guideline for further reading.

The functor category DC has as objects the functors from C to D and as morphisms the natural transformationsof such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describesrepresentable functors in functor categories.

Duality: Every statement, theorem, or denition in category theory has a dual which is essentially obtained byreversing all the arrows. If one statement is true in a category C then its dual will be true in the dual categoryCop. This duality, which is transparent at the level of category theory, is often obscured in applications and canlead to surprising relationships.

Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction.Such a pair of adjoint functors typically arises from a construction dened by a universal property; this can beseen as a more abstract and powerful view on universal properties.

2.6.4 Higher-dimensional categories

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can besituated into the context of higher-dimensional categories. Briey, if we consider a morphism between two objects asa process taking us from one object to another, then higher-dimensional categories allow us to protably generalizethis by considering higher-dimensional processes.For example, a (strict) 2-category is a category together with morphisms between morphisms, i.e., processes whichallow us to transform one morphism into another. We can then compose these bimorphisms both horizontallyand vertically, and we require a 2-dimensional exchange law to hold, relating the two composition laws. In thiscontext, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms ofmorphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to considera 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative up toan isomorphism.This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of-category corresponding to the ordinal number .

2.7. HISTORICAL NOTES 11

Higher-dimensional categories are part of the broader mathematical eld of higher-dimensional algebra, a conceptintroduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, 'A Tale of n-categories(1996).

2.7 Historical notesIn 194245, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformationsas part of their work in topology, especially algebraic topology. Their work was an important part of the transitionfrom intuitive and geometric homology to axiomatic homology theory. Eilenberg and Mac Lane later wrote thattheir goal was to understand natural transformations; in order to do that, functors had to be dened, which requiredcategories.Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s inPoland. Eilenberg was Polish, and studied mathematics in Poland in the 1930s. Category theory is also, in somesense, a continuation of the work of Emmy Noether (one of Mac Lanes teachers) in formalizing abstract processes;Noether realized that in order to understand a type of mathematical structure, one needs to understand the processespreserving that structure. In order to achieve this understanding, Eilenberg and Mac Lane proposed an axiomaticformalization of the relation between structures and the processes preserving them.The subsequent development of category theory was powered rst by the computational needs of homological algebra,and later by the axiomatic needs of algebraic geometry, the eld most resistant to being grounded in either axiomaticset theory or the Russell-Whitehead view of united foundations. General category theory, an extension of universalalgebra having many new features allowing for semantic exibility and higher-order logic, came later; it is now appliedthroughout mathematics.Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundationof mathematics. A topos can also be considered as a specic type of category with two additional topos axioms. Thesefoundational applications of category theory have been worked out in fair detail as a basis for, and justication of,constructive mathematics. Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideassuch as pointless topology.Categorical logic is now a well-dened eld based on type theory for intuitionistic logics, with applications in functionalprogramming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambdacalculus. At the very least, category theoretic language claries what exactly these related areas have in common (insome abstract sense).Category theory has been applied in other elds as well. For example, John Baez has shown a link between Feynmandiagrams in Physics and monoidal categories.[6] Another application of category theory, more specically: topostheory, has been made in mathematical music theory, see for example the book The Topos of Music, Geometric Logicof Concepts, Theory, and Performance by Guerino Mazzola.More recent eorts to introduce undergraduates to categories as a foundation for mathematics include those ofWilliam Lawvere and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012).

2.8 See also Group theory Domain theory Enriched category theory Glossary of category theory Higher category theory Higher-dimensional algebra Important publications in category theory Outline of category theory Timeline of category theory and related mathematics


2.9 Notes[1] Awodey 2006

[2] Geroch, Robert (1985). Mathematical physics ([Repr.] ed.). Chicago: University of Chicago Press. p. 7. ISBN 0-226-28862-5. Retrieved 20 August 2012. Note that theorem 3 is actually easier for categories in general than it is for the specialcase of sets. This phenomenon is by no means rare.

[3] Mac Lane 1998, p. 18: As Eilenberg-Mac Lane rst observed, 'category' has been dened in order to be able to dene'functor' and 'functor' has been dened in order to be able to dene 'natural transformation' "

[4] Some authors compose in the opposite order, writing fg or f g for g f. Computer scientists using category theory verycommonly write f ; g for g f

[5] Note that a morphism that is both epic and monic is not necessarily an isomorphism! An elementary counterexample: inthe category consisting of two objects A and B, the identity morphisms, and a single morphism f from A to B, f is bothepic and monic but is not an isomorphism.

[6] Baez, J.C.; Stay, M. (2009). Physics, topology, logic and computation: A Rosetta stone (PDF). arXiv:0903.0340.

2.10 References Admek, Ji; Herrlich, Horst; Strecker, George E. (1990). Abstract and concrete categories. John Wiley &

Sons. ISBN 0-471-60922-6.

Awodey, Steve (2006). Category Theory. Oxford Logic Guides 49. Oxford University Press. ISBN 978-0-19-151382-4.

Barr, Michael; Wells, Charles (2012), Category Theory for Computing Science, Reprints in Theory and Appli-cations of Categories 22 (3rd ed.).

Barr, Michael; Wells, Charles (2005), Toposes, Triples and Theories, Reprints in Theory and Applications ofCategories 12 (revised ed.), MR 2178101.

Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of Mathematics and its Applications50-52. Cambridge University Press.

Bucur, Ion; Deleanu, Aristide (1968). Introduction to the theory of categories and functors. Wiley. Freyd, Peter J. (1964). Abelian Categories. New York: Harper and Row. Freyd, Peter J.; Scedrov, Andre (1990). Categories, allegories. North Holland Mathematical Library 39. North

Holland. ISBN 978-0-08-088701-2.

Goldblatt, Robert (2006) [1979]. Topoi: The Categorial Analysis of Logic. Studies in logic and the foundationsof mathematics 94 (Reprint, revised ed.). Dover Publications. ISBN 978-0-486-45026-1.

Hatcher, William S. (1982). Ch. 8. The logical foundations of mathematics. Foundations & philosophy ofscience & technology (2nd ed.). Pergamon Press.

Herrlich, Horst; Strecker, George E. (2007), Category Theory (3rd ed.), Heldermann Verlag Berlin, ISBN978-3-88538-001-6.

Kashiwara, Masaki; Schapira, Pierre (2006). Categories and Sheaves. Grundlehren der Mathematischen Wis-senschaften 332. Springer. ISBN 978-3-540-27949-5.

Lawvere, F. William; Rosebrugh, Robert (2003). Sets for Mathematics. Cambridge University Press. ISBN978-0-521-01060-3.

Lawvere, F. W.; Schanuel, Stephen Hoel (2009) [1997]. Conceptual Mathematics: A First Introduction toCategories (2nd ed.). Cambridge University Press. ISBN 978-0-521-89485-2.

Leinster, Tom (2004). Higher operads, higher categories. London Math. Society Lecture Note Series 298.Cambridge University Press. ISBN 978-0-521-53215-0.

2.11. FURTHER READING 13

Leinster, Tom (2014). Basic Category Theory. Cambridge University Press. Lurie, Jacob (2009). Higher topos theory. Annals of Mathematics Studies 170. Princeton, NJ: Princeton

University Press. arXiv:math.CT/0608040. ISBN 978-0-691-14049-0. MR 2522659.

Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5(2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. MR 1712872.

Mac Lane, Saunders; Birkho, Garrett (1999) [1967]. Algebra (2nd ed.). Chelsea. ISBN 0-8218-1646-2. Martini, A.; Ehrig, H.; Nunes, D. (1996). Elements of basic category theory. Technical Report (Technical

University Berlin) 96 (5).

May, Peter (1999). A Concise Course in Algebraic Topology. University of Chicago Press. ISBN 0-226-51183-9.

Guerino, Mazzola (2002). The Topos of Music, Geometric Logic of Concepts, Theory, and Performance.Birkhuser. ISBN 3-7643-5731-2.

Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topol-ogy, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications 97. Cambridge: CambridgeUniversity Press. ISBN 0-521-83414-7. Zbl 1034.18001.

Pierce, Benjamin C. (1991). Basic Category Theory for Computer Scientists. MIT Press. ISBN 978-0-262-66071-6.

Schalk, A.; Simmons, H. (2005). An introduction to Category Theory in four easy movements (PDF). Notesfor a course oered as part of the MSc. in Mathematical Logic, Manchester University.

Simpson, Carlos. Homotopy theory of higher categories. arXiv:1001.4071., draft of a book. Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge Studies in Advanced Mathematics 59.

Cambridge University Press. ISBN 978-0-521-63107-5.

Turi, Daniele (19962001). Category Theory Lecture Notes (PDF). Retrieved 11 December 2009. Basedon Mac Lane 1998.

2.11 Further reading Jean-Pierre Marquis (2008). From a Geometrical Point of View: A Study of the History and Philosophy ofCategory Theory. Springer Science & Business Media. ISBN 978-1-4020-9384-5.

2.12 External links Theory and Application of Categories, an electronic journal of category theory, full text, free, since 1995. nLab, a wiki project on mathematics, physics and philosophy with emphasis on the n-categorical point of view. Andr Joyal, CatLab, a wiki project dedicated to the exposition of categorical mathematics. Category Theory, a web page of links to lecture notes and freely available books on category theory. Hillman, Chris, A Categorical Primer, CiteSeerX: 10 .1 .1 .24 .3264, a formal introduction to category theory. Adamek, J.; Herrlich, H.; Stecker, G. Abstract and Concrete Categories-The Joy of Cats (PDF). Category Theory entry by Jean-Pierre Marquis in the Stanford Encyclopedia of Philosophy with an extensive

bibliography.

List of academic conferences on category theory Baez, John (1996). The Tale of n-categories. An informal introduction to higher order categories.


WildCats is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms,categories, functors, natural transformations, universal properties.

The catsterss channel on YouTube, a channel about category theory. Category Theory at PlanetMath.org. Video archive of recorded talks relevant to categories, logic and the foundations of physics. Interactive Web page which generates examples of categorical constructions in the category of nite sets. Category Theory for the Sciences, an instruction on category theory as a tool throughout the sciences.

Chapter 3

Cocountability

In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In otherwords, Y contains all but countably many elements of X. While the rational numbers are a countable subset of thereals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is nite, then onesays Y is conite.

3.1 -algebrasThe set of all subsets of X that are either countable or cocountable forms a -algebra, i.e., it is closed under theoperations of countable unions, countable intersections, and complementation. This -algebra is the countable-cocountable algebra on X. It is the smallest -algebra containing every singleton set.

3.2 TopologyThe cocountable topology (also called the countable complement topology) on any set X consists of the empty setand all cocountable subsets of X.

15

Chapter 4

Coniteness

Not to be confused with conality.

In mathematics, a conite subset of a set X is a subset A whose complement in X is a nite set. In other words, Acontains all but nitely many elements of X. If the complement is not nite, but it is countable, then one says the setis cocountable.These arise naturally when generalizing structures on nite sets to innite sets, particularly on innite products, as inthe product topology or direct sum.

4.1 Boolean algebrasThe set of all subsets ofX that are either nite or conite forms a Boolean algebra, i.e., it is closed under the operationsof union, intersection, and complementation. This Boolean algebra is the nite-conite algebra on X. A Booleanalgebra A has a unique non-principal ultralter (i.e. a maximal lter not generated by a single element of the algebra)if and only if there is an innite set X such that A is isomorphic to the nite-conite algebra on X. In this case, thenon-principal ultralter is the set of all conite sets.

4.2 Conite topologyThe conite topology (sometimes called the nite complement topology) is a topology which can be dened onevery set X. It has precisely the empty set and all conite subsets of X as open sets. As a consequence, in the conitetopology, the only closed subsets are nite sets, or the whole of X. Symbolically, one writes the topology as

T = fA X j A = ? or X nA is niteg

This topology occurs naturally in the context of the Zariski topology. Since polynomials over a eld K are zero onnite sets, or the whole of K, the Zariski topology on K (considered as ane line) is the conite topology. The sameis true for any irreducible algebraic curve; it is not true, for example, for XY = 0 in the plane.

4.2.1 Properties Subspaces: Every subspace topology of the conite topology is also a conite topology. Compactness: Since every open set contains all but nitely many points of X, the space X is compact and

sequentially compact.

Separation: The conite topology is the coarsest topology satisfying the T1 axiom; i.e. it is the smallest topologyfor which every singleton set is closed. In fact, an arbitrary topology on X satises the T1 axiom if and only if

16

4.3. OTHER EXAMPLES 17

it contains the conite topology. If X is nite then the conite topology is simply the discrete topology. If X isnot nite, then this topology is not T2, regular or normal, since no two nonempty open sets are disjoint (i.e. itis hyperconnected).

4.2.2 Double-pointed conite topologyThe double-pointed conite topology is the conite topology with every point doubled; that is, it is the topologicalproduct of the conite topology with the indiscrete topology. It is not T0 or T1, since the points of the doublet aretopologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable.An example of a countable double-pointed conite topology is the set of even and odd integers, with a topology thatgroups them together. Let X be the set of integers, and let OA be a subset of the integers whose complement is the setA. Dene a subbase of open sets Gx for any integer x to be Gx = O{x, x} if x is an even number, and Gx = O{x,x} if x is odd. Then the basis sets of X are generated by nite intersections, that is, for nite A, the open sets of thetopology are

UA :=\x2A

Gx

The resulting space is not T0 (and hence not T1), because the points x and x + 1 (for x even) are topologicallyindistinguishable. The space is, however, a compact space, since it is covered by a nite union of the UA.

4.3 Other examples

4.3.1 Product topologyThe product topology on a product of topological spacesQXi has basisQUi where Ui Xi is open, and conitelymany Ui = Xi .The analog (without requiring that conitely many are the whole space) is the box topology.

4.3.2 Direct sumThe elements of the direct sum of modulesLMi are sequences i 2Mi where conitely many i = 0 .The analog (without requiring that conitely many are zero) is the direct product.

4.4 References Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of

1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 (See example 18)

Chapter 5

Complex analysis

Complex analytic redirects here. For the class of functions often called complex analytic, see Holomorphic func-tion.Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of

mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics,including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamicsand thermodynamics and also in engineering elds such as nuclear, aerospace, mechanical and electrical engineering.Murray R. Spiegel described complex analysis as one of the most beautiful as well as useful branches of Mathemat-ics.Complex analysis is particularly concerned with analytic functions of complex variables (or, more generally, meromorphicfunctions). Because the separate real and imaginary parts of any analytic function must satisfy Laplaces equation,complex analysis is widely applicable to two-dimensional problems in physics.

5.1 HistoryComplex analysis is one of the classical branches in mathematics with roots in the 19th century and just prior. Im-portant mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, andmany more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many phys-ical applications and is also used throughout analytic number theory. In modern times, it has become very popularthrough a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions.Another important application of complex analysis is in string theory which studies conformal invariants in quantumeld theory.

5.2 Complex functionsA complex function is one in which the independent variable and the dependent variable are both complex numbers.More precisely, a complex function is a function whose domain and range are subsets of the complex plane.For any complex function, both the independent variable and the dependent variable may be separated into real andimaginary parts:

z = x+ iy andw = f(z) = u(x; y) + iv(x; y)

where x; y 2 R and u(x; y); v(x; y) are real-valued functions.

In other words, the components of the function f(z),

u = u(x; y)

18

5.3. HOLOMORPHIC FUNCTIONS 19

Plot of the function f(x) = (x2 1)(x 2 i)2 / (x2 + 2 + 2i). The hue represents the function argument, while the brightnessrepresents the magnitude.

v = v(x; y);

can be interpreted as real-valued functions of the two real variables, x and y.The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponentialfunctions, logarithmic functions, and trigonometric functions) into the complex domain.

5.3 Holomorphic functionsMain article: Holomorphic function

Holomorphic functions are complex functions dened on an open subset of the complex plane that are dierentiable.Complex dierentiability has much stronger consequences than usual (real) dierentiability. For instance, holo-morphic functions are innitely dierentiable, whereas some real dierentiable functions are not. Most elementaryfunctions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomor-phic.See also: analytic function, holomorphic sheaf and vector bundles.

20 CHAPTER 5. COMPLEX ANALYSIS

The Mandelbrot set, a fractal.

5.4 Major results

One central tool in complex analysis is the line integral. The integral around a closed path of a function that is holo-morphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem.The values of a holomorphic function inside a disk can be computed by a certain path integral on the disks bound-ary (Cauchys integral formula). Path integrals in the complex plane are often used to determine complicated realintegrals, and here the theory of residues among others is useful (see methods of contour integration). If a functionhas a pole or isolated singularity at some point, that is, at that point where its values blow up and have no nitebound, then one can compute the functions residue at that pole. These residues can be used to compute path integralsinvolving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphicfunctions near essential singularities is described by Picards Theorem. Functions that have only poles but no essentialsingularities are called meromorphic. Laurent series are similar to Taylor series but can be used to study the behaviorof functions near singularities.A bounded function that is holomorphic in the entire complex plane must be constant; this is Liouvilles theorem. Itcan be used to provide a natural and short proof for the fundamental theorem of algebra which states that the eld ofcomplex numbers is algebraically closed.If a function is holomorphic throughout a connected domain then its values are fully determined by its values onany smaller subdomain. The function on the larger domain is said to be analytically continued from its values on thesmaller domain. This allows the extension of the denition of functions, such as the Riemann zeta function, which areinitially dened in terms of innite sums that converge only on limited domains to almost the entire complex plane.Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic functionto a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on aclosely related surface known as a Riemann surface.All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more thanone complex dimension in which the analytic properties such as power series expansion carry over whereas most ofthe geometric properties of holomorphic functions in one complex dimension (such as conformality) do not carry

5.5. SEE ALSO 21

over. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane,which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.

5.5 See also Complex dynamics List of complex analysis topics Real analysis Runges theorem Several complex variables Real-valued function Function of a real variable Real multivariable function

5.6 References Ahlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979). Stephen D. Fisher, Complex Variables, 2 ed. (Dover, 1999). Carathodory, C., Theory of Functions of a Complex Variable (Chelsea, New York). [2 volumes.] Henrici, P., Applied and Computational Complex Analysis (Wiley). [Three volumes: 1974, 1977, 1986.] Kreyszig, E., Advanced Engineering Mathematics, 10 ed., Ch.13-18 (Wiley, 2011). Markushevich, A.I.,Theory of Functions of a Complex Variable (Prentice-Hall, 1965). [Three volumes.] Marsden & Homan, Basic Complex Analysis. 3 ed. (Freeman, 1999). Needham, T., Visual Complex Analysis (Oxford, 1997). Rudin, W., Real and Complex Analysis, 3 ed. (McGraw-Hill, 1986). Scheidemann, V., Introduction to complex analysis in several variables (Birkhauser, 2005) Shaw, W.T., Complex Analysis with Mathematica (Cambridge, 2006). Spiegel, Murray R. Theory and Problems of Complex Variables - with an introduction to Conformal Mappingand its applications (McGraw-Hill, 1964).

Stein & Shakarchi, Complex Analysis (Princeton, 2003).

5.7 External links Complex Analysis -- textbook by George Cain Complex analysis course web site by Douglas N. Arnold Example problems in complex analysis A collection of links to programs for visualizing complex functions (and related) Complex Analysis Project by John H. Mathews Hans Lundmarks complex analysis page (many links)

22 CHAPTER 5. COMPLEX ANALYSIS

Wolfram Researchs MathWorld Complex Analysis Page Complex function demos Application of Complex Functions in 2D Digital Image Transformation Complex Visualizer - Java applet for visualizing arbitrary complex functions Complex Map - iOS app for visualizing complex functions and iterations JavaScript complex function graphing tool Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis

Chapter 6

Countable set

Countable redirects here. For the linguistic concept, see Count noun.Not to be confused with (recursively) enumerable sets.

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the setof natural numbers. A countable set is either a nite set or a countably innite set. Whether nite or innite, theelements of a countable set can always be counted one at a time and, although the counting may never nish, everyelement of the set is associated with a natural number.Some authors use countable set to mean innitely countable alone.[1] To avoid this ambiguity, the term at mostcountable may be used when nite sets are included and countably innite, enumerable, or denumerable[2] oth-erwise.The term countable set was originated by Georg Cantor who contrasted sets which are countable with those which areuncountable (a.k.a. nonenumerable and nondenumerable[3]). Today, countable sets are researched by a branch ofmathematics called discrete mathematics.

6.1 DenitionA set S is called countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}.[4]

If such an f can be found which is also surjective (and therefore bijective), then S is called countably innite.In other words, a set is called countably innite if it has one-to-one correspondence with the natural number set, N.As noted above, this terminology is not universal: Some authors use countable to mean what is here called countablyinnite, and to not include nite sets.For alternative (equivalent) formulations of the denition in terms of a bijective function or a surjective function, seethe section Formal denition and properties below.

6.2 HistoryIn the western world, dierent innities were rst classied by Georg Cantor around 1874.[5]

6.3 IntroductionA set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements;for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}. This is only eective for smallsets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element,sometimes an ellipsis ("...) is used, if the writer believes that the reader can easily guess what is missing; for example,

23

24 CHAPTER 6. COUNTABLE SET

{1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possibleto list all the elements, because the set is nite.Some sets are innite; these sets have more than n elements for any integer n. For example, the set of natural numbers,denotable by {0, 1, 2, 3, 4, 5, ...}, has innitely many elements, and we cannot use any normal number to give itssize. Nonetheless, it turns out that innite sets do have a well-dened notion of size (or more properly, of cardinality,which is the technical term for the number of elements in a set), and not all innite sets have the same cardinality.

YX123

x

246

2x. .

. .

Bijective mapping from integer to even numbers

To understand what this means, we rst examine what it does not mean. For example, there are innitely many oddintegers, innitely many even integers, and (hence) innitely many integers overall. However, it turns out that thenumber of even integers, which is the same as the number of odd integers, is also the same as the number of integersoverall. This is because we arrange things such that for every integer, there is a distinct even integer: ... 24,12, 00, 12, 24, ...; or, more generally, n2n, see picture. What we have done here is arranged the integersand the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two setssuch that each element of each set corresponds to a single element in the other set.However, not all innite sets have the same cardinality. For example, Georg Cantor (who introduced this concept)demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.A set is countable if: (1) it is nite, or (2) it has the same cardinality (size) as the set of natural numbers. Equivalently, aset is countable if it has the same cardinality as some subset of the set of natural numbers. Otherwise, it is uncountable.

6.4 Formal denition and propertiesBy denition a set S is countable if there exists an injective function f : S N from S to the natural numbers N ={0, 1, 2, 3, ...}.

6.4. FORMAL DEFINITION AND PROPERTIES 25

It might seem natural to divide the sets into dierent classes: put all the sets containing one element together; all thesets containing two elements together; ...; nally, put together all innite sets and consider them as having the samesize. This view is not tenable, however, under the natural denition of size.To elaborate this we need the concept of a bijection. Although a bijection seems a more advanced concept than anumber, the usual development of mathematics in terms of set theory denes functions before numbers, as they arebased on much simpler sets. This is where the concept of a bijection comes in: dene the correspondence

a 1, b 2, c 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this denes abijection.We now generalize this situation and dene two sets to be of the same size if (and only if) there is a bijection betweenthem. For all nite sets this gives us the usual denition of the same size. What does it tell us about the size ofinnite sets?Consider the sets A = {1, 2, 3, ... }, the set of positive integers and B = {2, 4, 6, ... }, the set of even positive integers.We claim that, under our denition, these sets have the same size, and that therefore B is countably innite. Recallthat to prove this we need to exhibit a bijection between them. But this is easy, using n 2n, so that

1 2, 2 4, 3 6, 4 8, ....

As in the earlier example, every element of A has been paired o with precisely one element of B, and vice versa.Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets,a situation which is impossible for nite sets.Likewise, the set of all ordered pairs of natural numbers is countably innite, as can be seen by following a path likethe one in the picture:The resulting mapping is like this:

0 (0,0), 1 (1,0), 2 (0,1), 3 (2,0), 4 (1,1), 5 (0,2), 6 (3,0) ....

It is evident that this mapping will cover all such ordered pairs.Interestingly: if you treat each pair as being the numerator and denominator of a vulgar fraction, then for everypositive fraction, we can come up with a distinct number corresponding to it. This representation includes also thenatural numbers, since every natural number is also a fraction N/1. So we can conclude that there are exactly as manypositive rational numbers as there are positive integers. This is true also for all rational numbers, as can be seen below(a more complex presentation is needed to deal with negative numbers).Theorem: The Cartesian product of nitely many countable sets is countable.This form of triangular mapping recursively generalizes to vectors of nitely many natural numbers by repeatedlymapping the rst two elements to a natural number. For example, (0,2,3) maps to (5,3) which maps to 39.Sometimes more than one mapping is useful. This is where you map the set which you want to show countably innite,onto another set; and then map this other set to the natural numbers. For example, the positive rational numbers caneasily be mapped to (a subset of) the pairs of natural numbers because p/q maps to (p, q).What about innite subsets of countably innite sets? Do these have fewer elements than N?Theorem: Every subset of a countable set is countable. In particular, every innite subset of a countably innite setis countably innite.For example, the set of prime numbers is countable, by mapping the n-th prime number to n:

2 maps to 1 3 maps to 2 5 maps to 3 7 maps to 4


1

2

3

01 2 31

2

3

4

5

6

7

8

9

11

12

13 18

17

24

0

0

The Cantor pairing function assigns one natural number to each pair of natural numbers

11 maps to 5 13 maps to 6 17 maps to 7 19 maps to 8 23 maps to 9 ...

What about sets being larger than N? An obvious place to look would be Q, the set of all rational numbers, whichintuitively may seem much bigger than N. But looks can be deceiving, for we assert:Theorem: Q (the set of all rational numbers) is countable.Q can be dened as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto thesubset of ordered triples of natural numbers (a, b, c) such that a 0, b > 0, a and b are coprime, and c {0, 1} suchthat c = 0 if a/b 0 and c = 1 otherwise.

0 maps to (0,1,0)

6.4. FORMAL DEFINITION AND PROPERTIES 27

1 maps to (1,1,0) 1 maps to (1,1,1) 1/2 maps to (1,2,0) 1/2 maps to (1,2,1) 2 maps to (2,1,0) 2 maps to (2,1,1) 1/3 maps to (1,3,0) 1/3 maps to (1,3,1) 3 maps to (3,1,0) 3 maps to (3,1,1) 1/4 maps to (1,4,0) 1/4 maps to (1,4,1) 2/3 maps to (2,3,0) 2/3 maps to (2,3,1) 3/2 maps to (3,2,0) 3/2 maps to (3,2,1) 4 maps to (4,1,0) 4 maps to (4,1,1) ...

By a similar development, the set of algebraic numbers is countable, and so is the set of denable numbers.Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable.For example, given countable sets a, b, c ...Using a variant of the triangular enumeration we saw above:

a0 maps to 0 a1 maps to 1 b0 maps to 2 a2 maps to 3 b1 maps to 4 c0 maps to 5 a3 maps to 6 b2 maps to 7 c1 maps to 8 d0 maps to 9 a4 maps to 10 ...


Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore alsocountable by a previous theorem.Also note that the axiom of countable choice is needed in order to index all of the sets a, b, c,...Theorem: The set of all nite-length sequences of natural numbers is countable.This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which isa countable set (nite Cartesian product). So we are talking about a countable union of countable sets, which iscountable by the previous theorem.Theorem: The set of all nite subsets of the natural numbers is countable.If you have a nite subset, you can order the elements into a nite sequence. There are only countably many nitesequences, so also there are only countably many nite subsets.The following theorem gives equivalent formulations in terms of a bijective function or a surjective function. A proofof this result can be found in Langs text.[2]

Theorem: Let S be a set. The following statements are equivalent:

1. S is countable, i.e. there exists an injective function f : S N.

2. Either S is empty or there exists a surjective function g : N S.

3. Either S is nite or there exists a bijection h : N S.

Several standard properties follow easily from this theorem. We present them here tersely. For a gentler presentationsee the sections above. Observe that N in the theorem can be replaced with any countably innite set. In particularwe have the following Corollary.Corollary: Let S and T be sets.

1. If the function f : S T is injective and T is countable then S is countable.

2. If the function g : S T is surjective and S is countable then T is countable.

Proof: For (1) observe that if T is countable there is an injective function h : T N. Then if f : S T is injectivethe composition h o f : S N is injective, so S is countable.For (2) observe that if S is countable there is a surjective function h : N S. Then if g : S T is surjective thecomposition g o h : N T is surjective, so T is countable.Proposition: Any subset of a countable set is countable.Proof: The restriction of an injective function to a subset of its domain is still injective.Proposition: The Cartesian product of two countable sets A and B is countable.Proof: Note that N N is countable as a consequence of the denition because the function f : N N N givenby f(m, n) = 2m3n is injective. It then follows from the Basic Theorem and the Corollary that the Cartesian productof any two countable sets is countable. This follows because if A and B are countable there are surjections f : N A and g : N B. So

f g : N N A B

is a surjection from the countable set N N to the set A B and the Corollary implies A B is countable. This resultgeneralizes to the Cartesian product of any nite collection of countable sets and the proof follows by induction onthe number of sets in the collection.Proposition: The integers Z are countable and the rational numbers Q are countable.Proof: The integers Z are countable because the function f : Z N given by f(n) = 2n if n is non-negative and f(n)= 3|n| if n is negative is an injective function. The rational numbers Q are countable because the function g : Z N Q given by g(m, n) = m/(n + 1) is a surjection from the countable set Z N to the rationals Q.Proposition: If An is a countable set for each n in N then the union of all An is also countable.

6.5. MINIMAL MODEL OF SET THEORY IS COUNTABLE 29

Proof: This is a consequence of the fact that for each n there is a surjective function gn : N An and hence thefunction

G : N N![n2N

An

given by G(n, m) = gn(m) is a surjection. Since N N is countable, the Corollary implies that the union is countable.We are using the axiom of countable choice in this proof in order to pick for each n in N a surjection gn from thenon-empty collection of surjections from N to An.Cantors Theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is nosurjective function from A to P(A). A proof is given in the article Cantors Theorem. As an immediate consequenceof this and the Basic Theorem above we have:Proposition: The set P(N) is not countable; i.e. it is uncountable.For an elaboration of this result see Cantors diagonal argument.The set of real numbers is uncountable (see Cantors rst uncountability proof), and so is the set of all innitesequences of natural numbers. A topological proof for the uncountability of the real numbers is described at niteintersection property.

6.5 Minimal model of set theory is countableIf there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standardmodel (see Constructible universe). The Lwenheim-Skolem theorem can be used to show that this minimal modelis countable. The fact that the notion of uncountability makes sense even in this model, and in particular that thismodel M contains elements which are

subsets of M, hence countable,

but uncountable from the point of view of M,

was seen as paradoxical in the early days of set theory, see Skolems paradox.The minimal standard model includes all the algebraic numbers and all eectively computable transcendental num-bers, as well as many other kinds of numbers.

6.6 Total ordersCountable sets can be totally ordered in various ways, e.g.:

Well orders (see also ordinal number):

The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...) The integers in the order (0, 1, 2, 3, ...; 1, 2, 3, ...)

Other (not well orders):

The usual order of integers (..., 3, 2, 1, 0, 1, 2, 3, ...) The usual order of rational numbers (Cannot be explicitly written as a list!)

Note that in both examples of well orders here, any subset has a least element; and in both examples of non-wellorders, some subsets do not have a least element. This is the key denition that determines whether a total order isalso a well order.


6.7 See also Aleph number Counting Hilberts paradox of the Grand Hotel Uncountable set

6.8 Notes[1] For an example of this usage see (Rudin 1976, Chapter 2).

[2] See (Lang 1993, 2 of Chapter I).

[3] See (Apostol 1969, Chapter 13.19).

[4] Since there is an obvious bijection between N and N* = {1, 2, 3, ...}, it makes no dierence whether one considers 0 tobe a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic,which make 0 a natural number.

[5] Stillwell, John C. (2010), Roads to Innity: TheMathematics of Truth and Proof, CRC Press, p. 10, ISBN 9781439865507,Cantors discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before1874, innity was not even considered a legitimate mathematical subject by most people, so the need to distinguish betweencountable and uncountable innities could not have been imagined.

6.9 References Lang, Serge (1993), Real and Functional Analysis, Berlin, New York: Springer-Verlag, ISBN 0-387-94001-4 Rudin, Walter (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, ISBN 0-07-054235-X Apostol, Tom M. (June 1969), Multi-Variable Calculus and Linear Algebra with Applications, Calculus 2 (2nd

ed.), New York: John Wiley + Sons, ISBN 978-0-471-00007-5

6.10 External links Weisstein, Eric W., Countable Set, MathWorld.

Chapter 7

Elementary proof

In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specically, theterm is used in number theory to refer to proofs that make no use of complex analysis. For some time it was thoughtthat certain theorems, like the prime number theorem, could only be proved using higher mathematics. However,over time,

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