RigourWikipedia

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 Complex analysis 153.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Major results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Elementary proof 204.1 Prime number theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Friedmans conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Expression (mathematics) 225.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Syntax versus semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.3 Formal languages and lambda calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.4 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Functor 256.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.1.1 Covariance and contravariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.1.2 Opposite functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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6.1.3 Bifunctors and multifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.4 Relation to other categorical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.5 Computer implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Functor category 307.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Mathematical beauty 338.1 Beauty in method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Beauty in results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.3 Beauty in experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.4 Beauty and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.5 Beauty and mathematical information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.6 Mathematics and the arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.6.1 Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.6.2 Visual arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9 Mathematical folklore 419.1 Stories, sayings and jokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

10 Mathematical object 4310.1 Cantorian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.2 Foundational paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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10.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

11 Natural transformation 4511.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

11.2.1 Opposite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.2.2 Double dual of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.2.3 Tensor-hom adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

11.3 Unnatural isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.3.1 Example: fundamental group of torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.3.2 Example: dual of a nite-dimensional vector space . . . . . . . . . . . . . . . . . . . . . . 48

11.4 Operations with natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.5 Functor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.6 Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

12 Occams razor 5112.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

12.1.1 Formulations before Ockham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.1.2 Ockham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.1.3 Later formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

12.2 Justications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5412.2.1 Aesthetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5412.2.2 Empirical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5412.2.3 Practical considerations and pragmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . 5412.2.4 Mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.2.5 Other philosophers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

12.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.3.1 Science and the scientic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.3.2 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.3.3 Medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.3.4 Religion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.3.5 Penal ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.3.6 Probability theory and statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12.4 Controversial aspects of the razor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.5 Anti-razors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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12.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6412.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

13 Parameter 6913.1 Mathematical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

13.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.1.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.1.3 Analytic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.1.4 Mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.1.5 Statistics and econometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.1.6 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13.2 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.3 Computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.4 Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.5 Environmental science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.6 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.7 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.8 Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

14 Parametrization 7514.1 Non-uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.2 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.3 Parametrization invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.4 Examples of parametrized models/objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.5 Parametrization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

15 Pathological (mathematics) 7815.1 Pathological functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7815.2 Prevalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.3 Pathological examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.5 Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8115.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

16 Rigour 8216.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.2 Intellectual rigour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

16.2.1 Intellectual honesty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

vi CONTENTS

16.2.2 Politics and law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.3 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

16.3.1 Mathematical proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.3.2 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.3.3 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

16.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8416.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17 Statistical parameter 8517.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

18 Yoneda lemma 8718.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

18.2.1 General version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8718.2.2 Naming conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.2.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.2.4 The Yoneda embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

18.3 Preadditive categories, rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 91

18.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9418.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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:

8 CHAPTER 2. CATEGORY THEORY

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.

10 CHAPTER 2. CATEGORY THEORY

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

12 CHAPTER 2. CATEGORY THEORY

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.

14 CHAPTER 2. CATEGORY THEORY

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

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.

3.1 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.

3.2 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)

15

16 CHAPTER 3. COMPLEX ANALYSIS

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.

3.3 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.

3.4. MAJOR RESULTS 17

The Mandelbrot set, a fractal.

3.4 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

18 CHAPTER 3. COMPLEX ANALYSIS

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.

3.5 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

3.6 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).

3.7 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)

3.7. EXTERNAL LINKS 19

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 4

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, many of these results have been reproved using only elementary techniques.While the meaning has not always been dened precisely, the term is commonly used in mathematical jargon. Anelementary proof is not necessarily simple, in the sense of being easy to understand: some elementary proofs can bequite complicated.[1]

4.1 Prime number theoremThe distinction between elementary and non-elementary proofs has been considered especially important in regardto the prime number theorem. This theorem was rst proved in 1896 by Jacques Hadamard and Charles Jean de laValle-Poussin using complex analysis. Many mathematicians then attempted to construct elementary proofs of thetheorem, without success. G. H. Hardy expressed strong reservations; he considered that the essential "depth" of theresult ruled out elementary proofs:

No elementary proof of the prime number theorem is known, and one may ask whether it is reason-able to expect one. Now we know that the theorem is roughly equivalent to a theorem about an analyticfunction, the theorem that Riemanns zeta function has no roots on a certain line. A proof of such atheorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely.It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seemsquite clear. We have certain views about the logic of the theory; we think that some theorems, as wesay lie deep and others nearer to the surface. If anyone produces an elementary proof of the primenumber theorem, he will show that these views are wrong, that the subject does not hang together in theway we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten.G. H. Hardy (1921). Lecture to Mathematical Society of Copenhagen. Quoted in Goldfeld (2003),p. 3

However, in 1948, Atle Selberg produced new methods which led him and Paul Erds to nd elementary proofs ofthe prime number theorem.[2]

A possible formalization of the notion of elementary in connection to a proof of a number-theoretical result is therestriction that the proof can be carried out in Peano arithmetic. Also in that sense, these proofs are elementary.

4.2 Friedmans conjectureMain article: Grand conjecture

20

4.3. REFERENCES 21

Harvey Friedman conjectured, Every theorem published in the Annals of Mathematics whose statement involvesonly nitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in elementaryarithmetic.[3] The form of elementary arithmetic referred to in this conjecture can be formalized by a small setof axioms concerning integer arithmetic and mathematical induction. For instance, according to this conjecture,Fermats Last Theorem should have an elementary proof; Wiles proof of Fermats Last Theorem is not elementary.However, there are other simple statements about arithmetic such as the existence of iterated exponential functionsthat cannot be proven in this theory.

4.3 References[1] Diamond, Harold G. (1982), Elementary methods in the study of the distribution of prime numbers, Bulletin of the

American Mathematical Society 7 (3): 55389, doi:10.1090/S0273-0979-1982-15057-1, MR 670132.

[2] Goldfeld, Dorian M. (2003), The Elementary Proof of the Prime Number Theorem: An Historical Perspective (PDF), p. 3,retrieved October 31, 2009

[3] Avigad, Jeremy (2003), Number theory and elementary arithmetic (PDF), Philosophia Mathematica 11 (3): 257, at 258,doi:10.1093/philmat/11.3.257.

Chapter 5

Expression (mathematics)

In mathematics, an expression (or mathematical expression) is a nite combination of symbols that is well-formedaccording to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables,operations, functions, punctuation, grouping, and other aspects of logical syntax.

5.1 ExamplesThe use of expressions ranges from the simple:

0 + 0

8x 5 (linear polynomial)

7x2 + 4x 10 (quadratic polynomial)

x1x2+12 (rational expression)

to the complex:

f(a) +nX

k=1

1

k!

dk

dtk

t=0

f(u(t)) +

Z 10

(1 t)nn!

dn+1

dtn+1f(u(t)) dt:

5.2 FormsMathematical expressions include arithmetic expressions, polynomials, algebraic expressions, closed-form expres-sions, and analytical expressions. The table below highlights some similarities and dierences between these dierenttypes.

5.3 Syntax versus semantics

5.3.1 SyntaxMain article: Syntax

22

5.4. VARIABLES 23

Being an expression is a syntactic concept.An expression must be well-formed; i.e., the operators must have the correct number of inputs, in the correct places.Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions.For example, in the usual notation of arithmetic, the expression 2 + 3 is well formed, but the expression * 2 + is not.Similarly,

4)x+; /ywould not be considered a mathematical expression but only a meaningless jumble.

5.3.2 SemanticsMain articles: Semantics and Formal semantics (logic)

Semantics is the study of meaning. Formal semantics is about attaching meaning to expressions.In algebra, an expression may be used to designate a value, which might depend on values assigned to variablesoccurring in the expression. The determination of this value depends on the semantics attached to the symbols of theexpression. These semantic rules may declare that certain expressions do not designate any value (for instance whenthey involve division by 0); such expressions are said to have an undened value, but they are well-formed expressionsnonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expressionmight designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right thatcan be manipulated according to certain rules. Certain expressions that designate a value simultaneously express acondition that is assumed to hold, for instance those involving the operator to designate an internal direct sum.

5.3.3 Formal languages and lambda calculusMain articles: Formal language, Formal system and Lambda calculus

Formal languages are concerned by how expressions are constructed. They form a key element of formal systems.In the 1930s, Alonzo Church and Stephen Kleene have formalized expressions and their evaluation by introducinglambda calculus.The equivalence of two expressions in the lambda calculus is undecidable. This is also the case for the expressionsrepresenting real numbers, which are built from the integers by using the arithmetical operations, the logarithm andthe exponential.

5.4 VariablesMany mathematical expressions include variables. Any variable can be classied as being either a free variable or abound variable.For a given combination of values for the free variables, an expression may be evaluated, although for some combi-nations of values of the free variables, the value of the expression may be undened. Thus an expression represents afunction whose inputs are the value assigned the free variables and whose output is the resulting value of the expres-sion.For example, the expression

x/y

evaluated for x = 10, y = 5, will give 2; but it is undened for y = 0.The evaluation of an expression is dependent on the denition of the mathematical operators and on the system ofvalues that is its context.

24 CHAPTER 5. EXPRESSION (MATHEMATICS)

Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the sameoutput, i.e., they represent the same function. Example:The expression

3Xn=1

(2nx)

has free variable x, bound variable n, constants 1, 2, and 3, two occurrences of an implicit multiplication operator,and a summation operator. The expression is equivalent to the simpler expression 12x. The value for x = 3 is 36.

5.5 See also Algebraic closure Algebraic expression Analytical expression Computer algebra#Expressions Closed-form expression Combinator Dened and undened Equation Expression (programming) Formula Formal grammar Functional programming Logical expression Term (mathematics)

5.6 Notes

5.7 References Redden, John. Elementary Algebra. Flat World Knowledge, 2011.

Chapter 6

Functor

This article is about the mathematical concept. For other uses, see Functor (disambiguation).

In mathematics, a functor is a type of mapping between categories, which is applied in category theory. Functors canbe thought of as homomorphisms between categories. In the category of small categories, functors can be thought ofmore generally as morphisms.Functors were rst considered in algebraic topology, where algebraic objects (like the fundamental group) are asso-ciated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functorsare used throughout modern mathematics to relate various categories. Thus, functors are generally applicable in areaswithin mathematics that category theory can make an abstraction of.The word functor was borrowed by mathematicians from the philosopher Rudolf Carnap,[1] who used the term in alinguistic context:[2] see function word.

6.1 DenitionLet C and D be categories. A functor F from C to D is a mapping that[3]

associates to each object X 2 C an object F (X) 2 D ,

associates to each morphism f : X ! Y 2 C a morphism F (f) : F (X) ! F (Y ) 2 D such that thefollowing two conditions hold:

F (idX) = idF (X) for every object X 2 C F (g f) = F (g) F (f) for all morphisms f : X ! Y and g : Y ! Z:

That is, functors must preserve identity morphisms and composition of morphisms.

6.1.1 Covariance and contravariance

There are many constructions in mathematics that would be functors but for the fact that they turn morphisms aroundand reverse composition. We then dene a contravariant functor F from C to D as a mapping that

associates to each object X 2 C an object F (X) 2 D;

associates to each morphism f : X ! Y 2 C a morphism F (f) : F (Y )! F (X) 2 D such that

F (idX) = idF (X) for every object X 2 C , F (g f) = F (f) F (g) for all morphisms f : X ! Y and g : Y ! Z:

25

26 CHAPTER 6. FUNCTOR

Note that contravariant functors reverse the direction of composition.Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Notethat one can also dene a contravariant functor as a covariant functor on the opposite category Cop .[4] Some authorsprefer to write all expressions covariantly. That is, instead of saying F : C ! D is a contravariant functor, theysimply write F : Cop ! D (or sometimes F : C ! Dop ) and call it a functor.Contravariant functors are also occasionally called cofunctors.

6.1.2 Opposite functorEvery functor F : C ! D induces the opposite functor F op : Cop ! Dop , where Cop and Dop are the oppositecategories to C and D .[5] By denition, F op maps objects and morphisms identically to F . Since Cop does notcoincide with C as a category, and similarly for D , F op is distinguished from F . For example, when composingF : C0 ! C1 with G : Cop1 ! C2 , one should use either G F op or Gop F . Note that, following the property ofopposite category, (F op)op = F .

6.1.3 Bifunctors and multifunctorsA bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, theHom functor is of the type Cop C Set. It can be seen as a functor in two arguments. The Hom functor is a naturalexample; it is contravariant in one argument, covariant in the other.Amultifunctor is a generalization of the functor concept to n variables. So, for example, a bifunctor is a multifunctorwith n = 2.

6.2 ExamplesDiagram: For categories C and J, a diagram of type J in C is a covariant functor D : J ! C .(Category theoretical) presheaf: For categories C and J, a J-presheaf on C is a contravariant functor D : C ! J .Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open(X) under inclusion.Like every partially ordered set, Open(X) forms a small category by adding a single arrow U V if and only ifU V . Contravariant functors on Open(X) are called presheaves on X. For instance, by assigning to every open setU the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.Constant functor: The functor C D which maps every object of C to a xed object X in D and every morphismin C to the identity morphism on X. Such a functor is called a constant or selection functor.Endofunctor: A functor that maps a category to itself.Identity functor in category C, written 1C or idC, maps an object to itself and a morphism to itself. Identity functoris an endofunctor.Diagonal functor: The diagonal functor is dened as the functor from D to the functor category DC which sendseach object in D to the constant functor at that object.Limit functor: For a xed index category J, if every functor JC has a limit (for instance if C is complete), thenthe limit functor CJC assigns to each functor its limit. The existence of this functor can be proved by realizing thatit is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem. This requires a suitableversion of the axiom of choice. Similar remarks apply to the colimit functor (which is covariant).Power sets: The power set functor P : Set Set maps each set to its power set and each function f : X ! Y to themap which sends U X to its image f(U) Y . One can also consider the contravariant power set functor whichsends f : X ! Y to the map which sends V Y to its inverse image f1(V ) X:Dual vector space: The map which assigns to every vector space its dual space and to every linear map its dual ortranspose is a contravariant functor from the category of all vector spaces over a xed eld to itself.Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguishedpoints. The objects are pairs (X, x0), where X is a topological space and x0 is a point in X. A morphism from (X, x0)to (Y, y0) is given by a continuous map f : X Y with f(x0) = y0.

6.3. PROPERTIES 27

To every topological space X with distinguished point x0, one can dene the fundamental group based at x0, denoted1(X, x0). This is the group of homotopy classes of loops based at x0. If f : X Y morphism of pointed spaces, thenevery loop in X with base point x0 can be composed with f to yield a loop in Y with base point y0. This operation iscompatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphismfrom (X, x0) to (Y, y0). We thus obtain a functor from the category of pointed topological spaces to the categoryof groups.In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves,but they cannot be composed unless they share an endpoint. Thus one has the fundamental groupoid instead of thefundamental group, and this construction is functorial.Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuousmaps as morphisms) to the category of real associative algebras is given by assigning to every topological space Xthe algebra C(X) of all real-valued continuous functions on that space. Every continuous map f : X Y induces analgebra homomorphism C(f) : C(Y) C(X) by the rule C(f)() = o f for every in C(Y).Tangent and cotangent bundles: The map which sends every dierentiable manifold to its tangent bundle and everysmooth map to its derivative is a covariant functor from the category of dierentiable manifolds to the category ofvector bundles.Doing this constructions pointwise gives the tangent space, a covariant functor from the category of pointed dieren-tiable manifolds to the category of real vector spaces. Likewise, cotangent space is a contravariant functor, essentiallythe composition of the tangent space with the dual space above.Group actions/representations: Every group G can be considered as a category with a single object whose mor-phisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set,i.e. a G-set. Likewise, a functor from G to the category of vector spaces, VectK, is a linear representation of G. Ingeneral, a functor G C can be considered as an action of G on an object in the category C. If C is a group, thenthis action is a group homomorphism.Lie algebras: Assigning to every real (complex) Lie group its real (complex) Lie algebra denes a functor.Tensor products: If C denotes the category of vector spaces over a xed eld, with linear maps as morphisms, thenthe tensor product V W denes a functor C C C which is covariant in both arguments.[6]Forgetful functors: The functorU :Grp Setwhich maps a group to its underlying set and a group homomorphismto its underlying function of sets is a functor.[7] Functors like these, which forget some structure, are termed forgetfulfunctors. Another example is the functor Rng Ab which maps a ring to its underlying additive abelian group.Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms).Free functors: Going in the opposite direction of forgetful functors are free functors. The free functor F : Set Grp sends every set X to the free group generated by X. Functions get mapped to group homomorphisms betweenfree groups. Free constructions exist for many categories based on structured sets. See free object.Homomorphism groups: To every pairA, B of abelian groups one can assign the abelian group Hom(A,B) consistingof all group homomorphisms from A to B. This is a functor which is contravariant in the rst and covariant in thesecond argument, i.e. it is a functor Abop Ab Ab (where Ab denotes the category of abelian groups with grouphomomorphisms). If f : A1 A2 and g : B1 B2 are morphisms in Ab, then the group homomorphism Hom(f,g): Hom(A2,B1) Hom(A1,B2) is given by g f. See Hom functor.Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objectsin C one can assign the set Hom(X,Y) of morphisms from X to Y. This denes a functor to Set which is contravariantin the rst argument and covariant in the second, i.e. it is a functor Cop C Set. If f : X1 X2 and g : Y1 Y2are morphisms in C, then the group homomorphism Hom(f,g) : Hom(X2,Y1) Hom(X1,Y2) is given by g f.Functors like these are called representable functors. An important goal in many settings is to determine whether agiven functor is representable.

6.3 PropertiesTwo important consequences of the functor axioms are:

F transforms each commutative diagram in C into a commutative diagram in D;

28 CHAPTER 6. FUNCTOR

if f is an isomorphism in C, then F(f) is an isomorphism in D.

One can compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form thecomposite functor GF from A to C. Composition of functors is associative where dened. Identity of compositionof functors is identity functor. This shows that functors can be considered as morphisms in categories of categories,for example in the category of small categories.A small category with a single object is the same thing as a monoid: the morphisms of a one-object category canbe thought of as elements of the monoid, and composition in the category is thought of as the monoid operation.Functors between one-object categories correspond to monoid homomorphisms. So in a sense, functors betweenarbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.

6.4 Relation to other categorical conceptsLet C and D be categories. The collection of all functors C D form the objects of a category: the functor category.Morphisms in this category are natural transformations between functors.Functors are often dened by universal properties; examples are the tensor product, the direct sum and direct productof groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limitand colimit generalize several of the above.Universal constructions often give rise to pairs of adjoint functors.

6.5 Computer implementationsFunctors sometimes appear in functional programming. For instance, the programming language Haskell has a classFunctor where fmap is a polytypic function used to map functions (morphisms on Hask, the category of Haskell types)between existing types to functions between some new types.

6.6 See also Functor category

Kan extension

Pseudofunctor

6.7 Notes[1] Mac Lane, Saunders (1971), Categories for the Working Mathematician, Springer-Verlag: New York, p. 30, ISBN 978-3-

540-90035-1

[2] Carnap, The Logical Syntax of Language, p. 1314, 1937, Routledge & Kegan Paul

[3] Jacobson (2009), p. 19, def. 1.2.

[4] Jacobson (2009), p. 1920.

[5] Mac Lane, Saunders; Moerdijk, Ieke (1992), Sheaves in geometry and logic: a rst introduction to topos theory, Springer,ISBN 978-0-387-97710-2

[6] Hazewinkel, Michiel; Gubareni, Nadezhda Mikhalovna; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004), Algebras,rings and modules, Springer, ISBN 978-1-4020-2690-4

[7] Jacobson (2009), p. 20, ex. 2.

6.8. REFERENCES 29

6.8 References Jacobson, Nathan (2009), Basic algebra 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7.

6.9 External links Hazewinkel, Michiel, ed. (2001), Functor, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-

4 see functor in nLab and the variations discussed and linked to there. Andr Joyal, CatLab, a wiki project dedicated to the exposition of categorical mathematics Hillman, Chris. A Categorical Primer. CiteSeerX: 10 .1 .1 .24 .3264: formal introduction to category theory. J. Adamek, H. Herrlich, G. Stecker, Abstract and Concrete Categories-The Joy of Cats Stanford Encyclopedia of Philosophy: "Category Theory" by Jean-Pierre Marquis. 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 catsters, a YouTube 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.

Chapter 7

Functor category

In category theory, a branch of mathematics, the functors between two given categories form a category, where theobjects are the functors and the morphisms are natural transformations between the functors. Functor categories areof interest for two main reasons:

many commonly occurring categories are (disguised) functor categories, so any statement proved for generalfunctor categories is widely applicable;

every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicerproperties than the original category, allowing certain operations that were not available in the original setting.

7.1 DenitionSuppose C is a small category (i.e. the objects and morphisms form a set rather than a proper class) and D is anarbitrary category. The category of functors from C to D, written as Fun(C, D), Funct(C,D) or DC , has as objectsthe covariant functors from C to D, and as morphisms the natural transformations between such functors. Note thatnatural transformations can be composed: if (X) : F(X) G(X) is a natural transformation from the functor F : C D to the functor G : C D, and (X) : G(X) H(X) is a natural transformation from the functor G to the functorH, then the collection (X)(X) : F(X) H(X) denes a natural transformation from F to H. With this compositionof natural transformations (known as vertical composition, see natural transformation), DC satises the axioms of acategory.In a completely analogous way, one can also consider the category of all contravariant functors from C to D; we writethis as Funct(Cop,D).If C and D are both preadditive categories (i.e. their morphism sets are abelian groups and the composition ofmorphisms is bilinear), then we can consider the category of all additive functors from C to D, denoted by Add(C,D).

7.2 Examples If I is a small discrete category (i.e. its only morphisms are the identity morphisms), then a functor from I toC essentially consists of a family of objects of C, indexed by I; the functor category CI can be identied withthe corresponding product category: its elements are families of objects in C and its