Category theory abFrom Wikipedia, the free encyclopedia

Contents

1 Adjoint functors 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Spelling (or morphology) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Solutions to optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Symmetry of optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Universal morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Counit-unit adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 Hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Adjunctions in full . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Universal morphisms induce hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . . 41.4.2 Counit-unit adjunction induces hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . 51.4.3 Hom-set adjunction induces all of the above . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5.1 Ubiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5.2 Problems formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.3 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6.1 Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6.2 Free constructions and forgetful functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6.3 Diagonal functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6.4 Colimits and diagonal functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7.4 Limit preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7.5 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.8 Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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1.8.1 Universal constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8.2 Equivalences of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8.3 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Algebraic topology 132.1 Main branches of algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.4 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.5 Knot theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.6 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Method of algebraic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Setting in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Applications of algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Notable algebraic topologists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Important theorems in algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Automorphism 183.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Automorphism group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Inner and outer automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Axiom 214.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.1 Early Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2.2 Modern development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.3 Other sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.1 Logical axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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4.3.2 Non-logical axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.3 Role in mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.4 Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Category theory 275.1 An abstraction of other mathematical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2.1 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.6 Other concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.6.1 Universal constructions, limits, and colimits . . . . . . . . . . . . . . . . . . . . . . . . . 305.6.2 Equivalent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.6.3 Further concepts and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.6.4 Higher-dimensional categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Dual (category theory) 346.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Endomorphism 367.1 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.2 Endomorphism ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.3 Operator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.4 Endofunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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7.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8 Epimorphism 388.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.3 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9 Equivalence of categories 419.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.2 Equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

10 Functor category 4410.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.3 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

11 Homology (mathematics) 4611.1 Informal examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.3 Construction of homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.4 Types of homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

11.4.1 Simplicial homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.4.2 Singular homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.4.3 Group homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.4.4 Other homology theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

11.5 Homology functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

11.7.1 Application in science and engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.8 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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11.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

12 Identity function 5212.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.2 Algebraic property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

13 Isomorphism 5413.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

13.1.1 Logarithm and exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.1.2 Integers modulo 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.1.3 Relation-preserving isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

13.2 Isomorphism vs. bijective morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5513.4 Relation with equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5613.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5813.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

14 Isomorphism of categories 5914.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5914.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

15 Mathematical structure 6015.1 Example: the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

16 Monomorphism 6216.1 Relation to invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.5 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

17 Section (category theory) 6417.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

vi CONTENTS

17.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

18 Yoneda lemma 6518.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

18.2.1 General version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.2.2 Naming conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.2.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.2.4 The Yoneda embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

18.3 Preadditive categories, rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 68

18.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Chapter 1

Adjoint functors

For the construction in field theory, see Adjunction (fieldtheory). For the construction in topology, see Adjunctionspace.

In mathematics, specifically category theory, adjunctionis a possible relationship between two functors.Adjunction is ubiquitous in mathematics, as it specifiesintuitive notions of optimization and efficiency.In the most concise symmetric definition, an adjunctionbetween categories C and D is a pair of functors,

F : D C and G : C D

and a family of bijections

homC(FY,X) = homD(Y,GX)

which is natural in the variables X and Y. The functor Fis called a left adjoint functor, while G is called a rightadjoint functor. The relationship F is left adjoint to G(or equivalently, G is right adjoint to F) is sometimeswritten

F G.

This definition and others are made precise below.

1.1 Introduction

The slogan is Adjoint functors arise everywhere.(Saunders Mac Lane, Categories for the working math-ematician)The long list of examples in this article is only a partialindication of how often an interesting mathematical con-struction is an adjoint functor. As a result, general theo-rems about left/right adjoint functors, such as the equiv-alence of their various definitions or the fact that they re-spectively preserve colimits/limits (which are also foundin every area of mathematics), can encode the details ofmany useful and otherwise non-trivial results.

1.1.1 Spelling (or morphology)

One can observe (e.g. in this article), two different rootsare used: adjunct and adjoint. From Oxford shorterEnglish dictionary, adjunct is from Latin, adjoint isfrom French.In Mac Lane, Categories for the working mathematician,chap. 4, Adjoints, one can verify the following usage. : homC(FY,X) = homD(Y,GX)The hom-set bijection is an adjunction.If f an arrow in homC(FY,X) , f is the right adjunctof f (p. 81).The functor F is left adjoint for G .

1.2 Motivation

1.2.1 Solutions to optimization problems

It can be said that an adjoint functor is a way of givingthe most efficient solution to some problem via a methodwhich is formulaic. For example, an elementary problemin ring theory is how to turn a rng (which is like a ring thatmight not have a multiplicative identity) into a ring. Themost efficient way is to adjoin an element '1' to the rng,adjoin all (and only) the elements which are necessary forsatisfying the ring axioms (e.g. r+1 for each r in the ring),and impose no relations in the newly formed ring that arenot forced by axioms. Moreover, this construction is for-mulaic in the sense that it works in essentially the sameway for any rng.This is rather vague, though suggestive, and can be madeprecise in the language of category theory: a construc-tion is most efficient if it satisfies a universal property,and is formulaic if it defines a functor. Universal prop-erties come in two types: initial properties and terminalproperties. Since these are dual (opposite) notions, it isonly necessary to discuss one of them.The idea of using an initial property is to set up the prob-lem in terms of some auxiliary category E, and then iden-tify that what we want is to find an initial object of E. This

1

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2 CHAPTER 1. ADJOINT FUNCTORS

has an advantage that the optimization the sense that weare finding the most efficient solution means somethingrigorous and is recognisable, rather like the attainment ofa supremum. The category E is also formulaic in thisconstruction, since it is always the category of elementsof the functor to which one is constructing an adjoint. Infact, this latter category is precisely the comma categoryover the functor in question.As an example, take the given rng R, and make a categoryE whose objects are rng homomorphisms R S, with S aring having a multiplicative identity. The morphisms in Ebetween R S1 and R S2 are commutative triangles ofthe form (R S1,R S2, S1 S2) where S1 S2 is aring map (which preserves the identity). Note that this isprecisely the definition of the comma category of R overthe inclusion of unitary rings into rng. The existence ofa morphism between R S1 and R S2 implies that S1is at least as efficient a solution as S2 to our problem: S2can have more adjoined elements and/or more relationsnot imposed by axioms than S1. Therefore, the assertionthat an object R R* is initial in E, that is, that there is amorphism from it to any other element of E, means thatthe ring R* is a most efficient solution to our problem.The two facts that this method of turning rngs into ringsis most efficient and formulaic can be expressed simulta-neously by saying that it defines an adjoint functor.

1.2.2 Symmetry of optimization problems

Continuing this discussion, suppose we started with thefunctor F, and posed the following (vague) question: isthere a problem to which F is the most efficient solution?The notion that F is the most efficient solution to the prob-lem posed by G is, in a certain rigorous sense, equivalentto the notion that G poses the most difficult problem thatF solves.This has the intuitive meaning that adjoint functorsshould occur in pairs, and in fact they do, but this isnot trivial from the universal morphism definitions. Theequivalent symmetric definitions involving adjunctionsand the symmetric language of adjoint functors (we cansay either F is left adjoint to G or G is right adjoint to F)have the advantage of making this fact explicit.

1.3 Formal definitions

There are various definitions for adjoint functors. Theirequivalence is elementary but not at all trivial and in facthighly useful. This article provides several such defini-tions:

The definitions via universal morphisms are easy tostate, and require minimal verifications when con-structing an adjoint functor or proving two functors

are adjoint. They are also the most analogous to ourintuition involving optimizations.

The definition via counit-unit adjunction is conve-nient for proofs about functors which are known tobe adjoint, because they provide formulas that canbe directly manipulated.

The definition via hom-sets makes symmetry themost apparent, and is the reason for using the wordadjoint.

Adjoint functors arise everywhere, in all areas of mathe-matics. Their full usefulness lies in that the structure inany of these definitions gives rise to the structures in theothers via a long but trivial series of deductions. Thus,switching between them makes implicit use of a greatdeal of tedious details that would otherwise have to berepeated separately in every subject area. For example,naturality and terminality of the counit can be used toprove that any right adjoint functor preserves limits.

1.3.1 Conventions

The theory of adjoints has the terms left and right at itsfoundation, and there are many components which live inone of two categories C and D which are under consid-eration. It can therefore be extremely helpful to chooseletters in alphabetical order according to whether they livein the lefthand category C or the righthand categoryD, and also to write them down in this order wheneverpossible.In this article for example, the letters X, F, f, will con-sistently denote things which live in the category C, theletters Y, G, g, will consistently denote things whichlive in the category D, and whenever possible such thingswill be referred to in order from left to right (a functorF:CD can be thought of as living where its outputsare, in C).

1.3.2 Universal morphisms

A functor F : C D is a left adjoint functor if for eachobject X in C, there exists a terminal morphism from Fto X. If, for each object X in C, we choose an object G0Xof D for which there is a terminal morphism X : F(G0X) X from F to X, then there is a unique functor G : C D such that GX = G0X and X FG(f) = f X for f :X X a morphism in C; F is then called a left adjointto G.A functor G : C D is a right adjoint functor if foreach object Y in D, there exists an initial morphism fromY to G. If, for each object Y in D, we choose an objectF0Y of C and an initial morphism Y : Y G(F0Y) fromY to G, then there is a unique functor F : C D such thatFY = F0Y and GF(g) Y = Y g for g : Y Y amorphism in D; G is then called a right adjoint to F.

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1.3. FORMAL DEFINITIONS 3

Remarks:

It is true, as the terminology implies, that F is left ad-joint to G if and only if G is right adjoint to F. This is ap-parent from the symmetric definitions given below. Thedefinitions via universal morphisms are often useful forestablishing that a given functor is left or right adjoint, be-cause they are minimalistic in their requirements. Theyare also intuitively meaningful in that finding a universalmorphism is like solving an optimization problem.

1.3.3 Counit-unit adjunction

A counit-unit adjunction between two categories C andD consists of two functors F : C D and G : C D andtwo natural transformations

: FG 1C : 1D GF

respectively called the counit and the unit of the adjunc-tion (terminology from universal algebra), such that thecompositions

FFFGF FF

GGGFG GG

are the identity transformations 1F and 1G on F and Grespectively.In this situation we say that F is left adjoint toG and Gis right adjoint to F , and may indicate this relationshipby writing (, ) : F G , or simply F G .In equation form, the above conditions on (,) are thecounit-unit equations

1F = F F1G = G G

which mean that for each X in C and each Y in D,

1FY = FY F (Y )1GX = G(X) GX

Note that here 1 denotes identity functors, while abovethe same symbol was used for identity natural transfor-mations.These equations are useful in reducing proofs about ad-joint functors to algebraic manipulations. They are some-times called the zig-zag equations because of the appear-ance of the corresponding string diagrams. A way to re-member them is to first write down the nonsensical equa-tion 1 = and then fill in either F or G in one of thetwo simple ways which make the compositions defined.

Note: The use of the prefix co in counit here is notconsistent with the terminology of limits and colimits,because a colimit satisfies an initial property whereas thecounit morphisms will satisfy terminal properties, and du-ally. The term unit here is borrowed from the theory ofmonads where it looks like the insertion of the identity 1into a monoid.

1.3.4 Hom-set adjunction

A hom-set adjunction between two categories C and Dconsists of two functors F : C D and G : C D and anatural isomorphism

: homC(F,) homD(, G)

This specifies a family of bijections

Y,X : homC(FY,X) homD(Y,GX)

for all objects X in C and Y in D.In this situation we say that F is left adjoint toG and Gis right adjoint to F , and may indicate this relationshipby writing : F G , or simply F G .This definition is a logical compromise in that it is some-what more difficult to satisfy than the universal morphismdefinitions, and has fewer immediate implications thanthe counit-unit definition. It is useful because of its obvi-ous symmetry, and as a stepping-stone between the otherdefinitions.In order to interpret as a natural isomorphism, one mustrecognize homC(F, ) and homD(, G) as functors. Infact, they are both bifunctors from Dop C to Set (thecategory of sets). For details, see the article on hom func-tors. Explicitly, the naturality of means that for allmorphisms f : X X in C and all morphisms g : Y Y in D the following diagram commutes:

Naturality of

The vertical arrows in this diagram are those inducedby composition with f and g. Formally, Hom(Fg, f) :HomC(FY, X) HomC(FY, X ) is given by h f o h oFg for each h in HomC(FY, X). Hom(g, Gf) is similar.

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4 CHAPTER 1. ADJOINT FUNCTORS

1.4 Adjunctions in full

There are hence numerous functors and natural trans-formations associated with every adjunction, and only asmall portion is sufficient to determine the rest.An adjunction between categories C and D consists of

A functor F : C D called the left adjoint

A functor G : C D called the right adjoint

A natural isomorphism : homC(F,) homD(,G)

A natural transformation : FG 1C called thecounit

A natural transformation : 1D GF called theunit

An equivalent formulation, where X denotes any objectof C and Y denotes any object of D:For every C-morphism f : FY X, there is a unique D-morphismY, X(f) = g : Y GX such that the diagramsbelow commute, and for every D-morphism g : Y GX,there is a unique C-morphism 1Y, X(g) = f : FY Xin C such that the diagrams below commute:

From this assertion, one can recover that:

The transformations , , and are related by theequations

f = 1Y,X(g) = X F (g) homC(F (Y ), X)g = Y,X(f) = G(f) Y homD(Y,G(X))1GX,X(1GX) = X homC(FG(X), X)Y,FY (1FY ) = Y homD(Y,GF (Y ))

The transformations , satisfy the counit-unitequations

1FY = FY F (Y )1GX = G(X) GX

Each pair (GX, X) is a terminal morphism from Fto X in C

Each pair (FY, Y) is an initial morphism from Y toG in D

In particular, the equations above allow one to define ,, and in terms of any one of the three. However, theadjoint functors F and G alone are in general not suffi-cient to determine the adjunction. We will demonstratethe equivalence of these situations below.

1.4.1 Universalmorphisms induce hom-setadjunction

Given a right adjoint functor G : C D; in the sense ofinitial morphisms, one may construct the induced hom-set adjunction by doing the following steps.

Construct a functor F : C D and a natural trans-formation .

For each object Y in D, choose an initial mor-phism (F(Y), Y) from Y to G, so we have Y: Y G(F(Y)). We have the map of F on ob-jects and the family of morphisms . For each f : Y0 Y1, as (F(Y0), Y0) is an

initial morphism, then factorize Y1 o f withY0 and get F(f) : F(Y0) F(Y1). This isthe map of F on morphisms. The commuting diagram of that factorization

implies the commuting diagram of naturaltransformations, so : 1D G o F is a naturaltransformation. Uniqueness of that factorization and that G is

a functor implies that the map of F on mor-phisms preserves compositions and identities.

Construct a natural isomorphism : homC(F-,-) homD(-,G-).

For each object X in C, each object Y in D, as(F(Y), Y) is an initial morphism, then Y, Xis a bijection, where Y, X(f : F(Y) X) =G(f) o Y. is a natural transformation, G is a functor,

then for any objects X0, X1 in C, any objectsY0, Y1 in D, any x : X0 X1, any y : Y1 Y0, we have Y1, X1(x o f o F(y)) = G(x) oG(f) o G(F(y)) o Y1 = G(x) o G(f) o Y0 o y= G(x) o Y0, X0(f) o y, and then is naturalin both arguments.

A similar argument allows one to construct a hom-set ad-junction from the terminal morphisms to a left adjointfunctor. (The construction that starts with a right adjointis slightly more common, since the right adjoint in manyadjoint pairs is a trivially defined inclusion or forgetfulfunctor.)

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1.5. HISTORY 5

1.4.2 Counit-unit adjunction induces hom-set adjunction

Given functors F : C D, G : C D, and a counit-unit adjunction (, ) : F G, we can construct a hom-set adjunction by finding the natural transformation :homC(F-,-) homD(-,G-) in the following steps:

For each f : FY X and each g : Y GX, define

Y,X(f) = G(f) YY,X(g) = X F (g)The transformations and are natural be-cause and are natural.

Using, in order, that F is a functor, that is natural,and the counit-unit equation 1FY = FY o F(Y),we obtain

f = X FG(f) F (Y )= f FY F (Y )= f 1FY = f

hence is the identity transformation.

Dually, using that G is a functor, that is natural,and the counit-unit equation 1GX = G(X) o GX,we obtain

g = G(X) GF (g) Y= G(X) GX g= 1GX g = g

hence is the identity transformation. Thus is a natural isomorphism with inverse 1 =.

1.4.3 Hom-set adjunction induces all of theabove

Given functors F : C D, G : C D, and a hom-setadjunction : homC(F-,-) homD(-,G-), we can con-struct a counit-unit adjunction

(, ) : F G ,

which defines families of initial and terminal morphisms,in the following steps:

Let X = 1GX,X(1GX) homC(FGX,X) foreach X in C, where 1GX homD(GX,GX) is theidentity morphism.

Let Y = Y,FY (1FY ) homD(Y,GFY ) foreach Y in D, where 1FY homC(FY, FY ) is theidentity morphism.

The bijectivity and naturality of imply that each(GX, X) is a terminal morphism from F to X in C,and each (FY, Y) is an initial morphism from Y toG in D.

The naturality of implies the naturality of and, and the two formulas

Y,X(f) = G(f) Y1Y,X(g) = X F (g)

for each f: FY X and g: Y GX (whichcompletely determine ).

Substituting FY for X and Y = Y, FY(1FY) forg in the second formula gives the first counit-unitequation

1FY = FY F (Y ) ,and substituting GX for Y and X = 1GX,X(1GX) for f in the first formula gives the sec-ond counit-unit equation1GX = G(X) GX .

1.5 History

1.5.1 Ubiquity

The idea of an adjoint functor was formulated by DanielKan in 1958. Like many of the concepts in categorytheory, it was suggested by the needs of homological al-gebra, which was at the time devoted to computations.Those faced with giving tidy, systematic presentations ofthe subject would have noticed relations such as

hom(F(X), Y) = hom(X, G(Y))

in the category of abelian groups, where F was the functorA (i.e. take the tensor product with A), and G was thefunctor hom(A,). The use of the equals sign is an abuseof notation; those two groups are not really identical butthere is a way of identifying them that is natural. It can beseen to be natural on the basis, firstly, that these are twoalternative descriptions of the bilinear mappings from X A to Y. That is, however, something particular to thecase of tensor product. In category theory the 'naturality'of the bijection is subsumed in the concept of a naturalisomorphism.The terminology comes from the Hilbert space idea ofadjoint operators T, U with Tx, y = x,Uy , which isformally similar to the above relation between hom-sets.We say that F is left adjoint to G, and G is right adjoint toF. Note that G may have itself a right adjoint that is quitedifferent from F (see below for an example). The analogy

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6 CHAPTER 1. ADJOINT FUNCTORS

to adjoint maps of Hilbert spaces can be made precise incertain contexts.[1]

If one starts looking for these adjoint pairs of functors,they turn out to be very common in abstract algebra, andelsewhere as well. The example section below providesevidence of this; furthermore, universal constructions,which may be more familiar to some, give rise to numer-ous adjoint pairs of functors.In accordance with the thinking of Saunders Mac Lane,any idea such as adjoint functors that occurs widelyenough in mathematics should be studied for its own sake.

1.5.2 Problems formulations

Mathematicians do not generally need the full adjointfunctor concept. Concepts can be judged according totheir use in solving problems, as well as for their use inbuilding theories. The tension between these two moti-vations was especially great during the 1950s when cat-egory theory was initially developed. Enter AlexanderGrothendieck, who used category theory to take com-pass bearings in other work in functional analysis,homological algebra and finally algebraic geometry.It is probably wrong to say that he promoted the adjointfunctor concept in isolation: but recognition of the roleof adjunction was inherent in Grothendiecks approach.For example, one of his major achievements was the for-mulation of Serre duality in relative form loosely, ina continuous family of algebraic varieties. The entireproof turned on the existence of a right adjoint to a cer-tain functor. This is something undeniably abstract, andnon-constructive, but also powerful in its own way.

1.5.3 Posets

Every partially ordered set can be viewed as a category(with a single morphism between x and y if and only ifx y). A pair of adjoint functors between two partiallyordered sets is called a Galois connection (or, if it is con-travariant, an antitone Galois connection). See that arti-cle for a number of examples: the case of Galois theoryof course is a leading one. Any Galois connection givesrise to closure operators and to inverse order-preservingbijections between the corresponding closed elements.As is the case for Galois groups, the real interest lies oftenin refining a correspondence to a duality (i.e. antitoneorder isomorphism). A treatment of Galois theory alongthese lines by Kaplansky was influential in the recognitionof the general structure here.The partial order case collapses the adjunction definitionsquite noticeably, but can provide several themes:

adjunctions may not be dualities or isomorphisms,but are candidates for upgrading to that status

closure operators may indicate the presence ofadjunctions, as corresponding monads (cf. theKuratowski closure axioms)

a very general comment of William Lawvere[2] isthat syntax and semantics are adjoint: take C to bethe set of all logical theories (axiomatizations), andD the power set of the set of all mathematical struc-tures. For a theory T in C, let F(T) be the set ofall structures that satisfy the axioms T ; for a set ofmathematical structures S, let G(S) be the minimalaxiomatization of S. We can then say that F(T) is asubset of S if and only if T logically implies G(S):the semantics functor F is left adjoint to the syn-tax functor G.

division is (in general) the attempt to invert multi-plication, but many examples, such as the introduc-tion of implication in propositional logic, or the idealquotient for division by ring ideals, can be recog-nised as the attempt to provide an adjoint.

Together these observations provide explanatory value allover mathematics.

1.6 Examples

1.6.1 Free groups

The construction of free groups is a common and illumi-nating example.Suppose that F : Grp Set is the functor assigning toeach set Y the free group generated by the elements of Y,and that G : Grp Set is the forgetful functor, whichassigns to each group X its underlying set. Then F is leftadjoint to G:Terminalmorphisms. For each group X, the group FGXis the free group generated freely by GX, the elementsof X. Let X : FGX X be the group homomor-phism which sends the generators of FGX to the elementsof X they correspond to, which exists by the universalproperty of free groups. Then each (GX, X) is a ter-minal morphism from F to X, because any group homo-morphism from a free group FZ to X will factor throughX : FGX X via a unique set map from Z to GX.This means that (F,G) is an adjoint pair.Initial morphisms. For each set Y, the set GFY is justthe underlying set of the free group FY generated by Y.Let Y : Y GFY be the set map given by in-clusion of generators. Then each (FY, Y ) is an ini-tial morphism from Y to G, because any set map from Yto the underlying set GW of a group will factor throughY : Y GFY via a unique group homomorphismfrom FY to W. This also means that (F,G) is an adjointpair.

https://en.wikipedia.org/wiki/Abstract_algebrahttps://en.wikipedia.org/wiki/Universal_constructionhttps://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/Alexander_Grothendieckhttps://en.wikipedia.org/wiki/Alexander_Grothendieckhttps://en.wikipedia.org/wiki/Functional_analysishttps://en.wikipedia.org/wiki/Homological_algebrahttps://en.wikipedia.org/wiki/Algebraic_geometryhttps://en.wikipedia.org/wiki/Serre_dualityhttps://en.wikipedia.org/wiki/Partially_ordered_sethttps://en.wikipedia.org/wiki/Galois_connectionhttps://en.wikipedia.org/wiki/Galois_theoryhttps://en.wikipedia.org/wiki/Closure_operatorhttps://en.wikipedia.org/wiki/Duality_(mathematics)https://en.wikipedia.org/wiki/Irving_Kaplanskyhttps://en.wikipedia.org/wiki/Monad_(category_theory)https://en.wikipedia.org/wiki/Kuratowski_closure_axiomshttps://en.wikipedia.org/wiki/William_Lawverehttps://en.wikipedia.org/wiki/Division_(mathematics)https://en.wikipedia.org/wiki/Material_conditionalhttps://en.wikipedia.org/wiki/Propositional_calculushttps://en.wikipedia.org/wiki/Ideal_quotienthttps://en.wikipedia.org/wiki/Ideal_quotienthttps://en.wikipedia.org/wiki/Ring_idealhttps://en.wikipedia.org/wiki/Free_grouphttps://en.wikipedia.org/wiki/Category_of_groupshttps://en.wikipedia.org/wiki/Category_of_setshttps://en.wikipedia.org/wiki/Free_grouphttps://en.wikipedia.org/wiki/Forgetful_functor

1.6. EXAMPLES 7

Hom-set adjunction. Maps from the free group FY to agroup X correspond precisely to maps from the set Y tothe set GX: each homomorphism from FY to X is fullydetermined by its action on generators. One can verifydirectly that this correspondence is a natural transforma-tion, which means it is a hom-set adjunction for the pair(F,G).Counit-unit adjunction. One can also verify directlythat and are natural. Then, a direct verification thatthey form a counit-unit adjunction (, ) : F G is asfollows:The first counit-unit equation 1F = F F says thatfor each set Y the composition

FYF (Y )FGFY FYFY

should be the identity. The intermediate group FGFY isthe free group generated freely by the words of the freegroup FY. (Think of these words as placed in parenthe-ses to indicate that they are independent generators.) Thearrow F (Y ) is the group homomorphism from FY intoFGFY sending each generator y of FY to the correspond-ing word of length one (y) as a generator of FGFY. Thearrow FY is the group homomorphism from FGFY toFY sending each generator to the word of FY it corre-sponds to (so this map is dropping parentheses). Thecomposition of these maps is indeed the identity on FY.The second counit-unit equation 1G = G G saysthat for each group X the composition

GXGXGFGX G(X)GX

should be the identity. The intermediate set GFGX is justthe underlying set of FGX. The arrow GX is the inclu-sion of generators set map from the set GX to the setGFGX. The arrow G(X) is the set map from GFGX toGX which underlies the group homomorphism sendingeach generator of FGX to the element of X it correspondsto (dropping parentheses). The composition of thesemaps is indeed the identity on GX.

1.6.2 Free constructions and forgetfulfunctors

Free objects are all examples of a left adjoint to a forgetfulfunctor which assigns to an algebraic object its underlyingset. These algebraic free functors have generally the samedescription as in the detailed description of the free groupsituation above.

1.6.3 Diagonal functors and limits

Products, fibred products, equalizers, and kernels are allexamples of the categorical notion of a limit. Any limit

functor is right adjoint to a corresponding diagonal func-tor (provided the category has the type of limits in ques-tion), and the counit of the adjunction provides the defin-ing maps from the limit object (i.e. from the diagonalfunctor on the limit, in the functor category). Below aresome specific examples.

Products Let : Grp2 Grp the functor whichassigns to each pair (X1, X2) the product groupX1X2, and let : Grp2 Grp be the diagonalfunctor which assigns to every group X the pair (X,X) in the product category Grp2. The universalproperty of the product group shows that is right-adjoint to . The counit of this adjunction is thedefining pair of projection maps from X1X2 to X1and X2 which define the limit, and the unit is the di-agonal inclusion of a group X into X1X2 (mappingx to (x,x)).

The cartesian product of sets, the product ofrings, the product of topological spaces etc.follow the same pattern; it can also be extendedin a straightforward manner to more than justtwo factors. More generally, any type of limitis right adjoint to a diagonal functor.

Kernels. Consider the category D of homomor-phisms of abelian groups. If f1 : A1 B1 and f2: A2 B2 are two objects of D, then a morphismfrom f1 to f2 is a pair (gA, gB) of morphisms suchthat gBf1 = f2gA. Let G : D Ab be the functorwhich assigns to each homomorphism its kernel andlet F : D Ab be the functor which maps the groupA to the homomorphism A 0. Then G is right ad-joint to F, which expresses the universal property ofkernels. The counit of this adjunction is the defin-ing embedding of a homomorphisms kernel into thehomomorphisms domain, and the unit is the mor-phism identifying a group A with the kernel of thehomomorphism A 0.

A suitable variation of this example also showsthat the kernel functors for vector spaces andfor modules are right adjoints. Analogously,one can show that the cokernel functors forabelian groups, vector spaces and modules areleft adjoints.

1.6.4 Colimits and diagonal functors

Coproducts, fibred coproducts, coequalizers, andcokernels are all examples of the categorical notionof a colimit. Any colimit functor is left adjoint to acorresponding diagonal functor (provided the categoryhas the type of colimits in question), and the unit of theadjunction provides the defining maps into the colimitobject. Below are some specific examples.

https://en.wikipedia.org/wiki/Free_objecthttps://en.wikipedia.org/wiki/Forgetful_functorhttps://en.wikipedia.org/wiki/Forgetful_functorhttps://en.wikipedia.org/wiki/Free_functorhttps://en.wikipedia.org/wiki/Product_(category_theory)https://en.wikipedia.org/wiki/Pullback_(category_theory)https://en.wikipedia.org/wiki/Equalizer_(mathematics)https://en.wikipedia.org/wiki/Kernel_(algebra)https://en.wikipedia.org/wiki/Limit_(category_theory)https://en.wikipedia.org/wiki/Diagonal_functorhttps://en.wikipedia.org/wiki/Diagonal_functorhttps://en.wikipedia.org/wiki/Cartesian_producthttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Product_topologyhttps://en.wikipedia.org/wiki/Kernel_(algebra)https://en.wikipedia.org/wiki/Coproducthttps://en.wikipedia.org/wiki/Pushout_(category_theory)https://en.wikipedia.org/wiki/Coequalizerhttps://en.wikipedia.org/wiki/Cokernelhttps://en.wikipedia.org/wiki/Limit_(category_theory)

8 CHAPTER 1. ADJOINT FUNCTORS

Coproducts. If F : Ab Ab2 assigns to every pair(X1, X2) of abelian groups their direct sum, and ifG : Ab Ab2 is the functor which assigns to everyabelian group Y the pair (Y, Y), then F is left adjointto G, again a consequence of the universal propertyof direct sums. The unit of this adjoint pair is thedefining pair of inclusion maps from X1 and X2 intothe direct sum, and the counit is the additive mapfrom the direct sum of (X,X) to back to X (sendingan element (a,b) of the direct sum to the elementa+b of X).

Analogous examples are given by the directsum of vector spaces and modules, by the freeproduct of groups and by the disjoint union ofsets.

1.6.5 Further examples

Algebra

Adjoining an identity to a rng. This example wasdiscussed in the motivation section above. Givena rng R, a multiplicative identity element can beadded by taking RxZ and defining a Z-bilinear prod-uct with (r,0)(0,1) = (0,1)(r,0) = (r,0), (r,0)(s,0) =(rs,0), (0,1)(0,1) = (0,1). This constructs a left ad-joint to the functor taking a ring to the underlyingrng.

Ring extensions. Suppose R and S are rings, and : R S is a ring homomorphism. Then S canbe seen as a (left) R-module, and the tensor productwith S yields a functor F : R-Mod S-Mod. ThenF is left adjoint to the forgetful functor G : S-Mod R-Mod.

Tensor products. If R is a ring and M is a rightR module, then the tensor product with M yields afunctor F : R-Mod Ab. The functor G : Ab R-Mod, defined by G(A) = homZ(M,A) for everyabelian group A, is a right adjoint to F.

From monoids and groups to rings The integralmonoid ring construction gives a functor frommonoids to rings. This functor is left adjoint to thefunctor that associates to a given ring its underly-ing multiplicative monoid. Similarly, the integralgroup ring construction yields a functor from groupsto rings, left adjoint to the functor that assigns to agiven ring its group of units. One can also start witha field K and consider the category of K-algebras in-stead of the category of rings, to get the monoid andgroup rings over K.

Field of fractions. Consider the category Domof integral domains with injective morphisms. The

forgetful functor Field Dom from fields has aleft adjoint - it assigns to every integral domain itsfield of fractions.

Polynomial rings. Let Ring* be the category ofpointed commutative rings with unity (pairs (A,a)where A is a ring, a A and morphisms preservethe distinguished elements). The forgetful functorG:Ring* Ring has a left adjoint - it assigns toevery ring R the pair (R[x],x) where R[x] is thepolynomial ring with coefficients from R.

Abelianization. Consider the inclusion functor G: Ab Grp from the category of abelian groupsto category of groups. It has a left adjoint calledabelianization which assigns to every group G thequotient group Gab=G/[G,G].

The Grothendieck group. In K-theory, the pointof departure is to observe that the category ofvector bundles on a topological space has a com-mutative monoid structure under direct sum. Onemay make an abelian group out of this monoid, theGrothendieck group, by formally adding an additiveinverse for each bundle (or equivalence class). Al-ternatively one can observe that the functor that foreach group takes the underlying monoid (ignoringinverses) has a left adjoint. This is a once-for-allconstruction, in line with the third section discussionabove. That is, one can imitate the construction ofnegative numbers; but there is the other option of anexistence theorem. For the case of finitary algebraicstructures, the existence by itself can be referred touniversal algebra, or model theory; naturally there isalso a proof adapted to category theory, too.

Frobenius reciprocity in the representation theoryof groups: see induced representation. This exam-ple foreshadowed the general theory by about half acentury.

Topology

A functor with a left and a right adjoint. Let Gbe the functor from topological spaces to sets thatassociates to every topological space its underlyingset (forgetting the topology, that is). G has a leftadjoint F, creating the discrete space on a set Y, anda right adjoint H creating the trivial topology on Y.

Suspensions and loop spaces Given topologicalspaces X and Y, the space [SX, Y] of homotopyclasses of maps from the suspension SX of X to Yis naturally isomorphic to the space [X, Y] of ho-motopy classes of maps from X to the loop space Yof Y. This is an important fact in homotopy theory.

https://en.wikipedia.org/wiki/Category_of_abelian_groupshttps://en.wikipedia.org/wiki/Direct_sum_of_groupshttps://en.wikipedia.org/wiki/Direct_sum_of_moduleshttps://en.wikipedia.org/wiki/Direct_sum_of_moduleshttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Module_(mathematics)https://en.wikipedia.org/wiki/Free_producthttps://en.wikipedia.org/wiki/Free_producthttps://en.wikipedia.org/wiki/Rng_(algebra)https://en.wikipedia.org/wiki/Ring_homomorphismhttps://en.wikipedia.org/wiki/Tensor_producthttps://en.wikipedia.org/wiki/Tensor-hom_adjunctionhttps://en.wikipedia.org/wiki/Integral_monoid_ringhttps://en.wikipedia.org/wiki/Integral_monoid_ringhttps://en.wikipedia.org/wiki/Monoidhttps://en.wikipedia.org/wiki/Integral_group_ringhttps://en.wikipedia.org/wiki/Integral_group_ringhttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Group_of_unitshttps://en.wikipedia.org/wiki/Field_(mathematics)https://en.wikipedia.org/wiki/Associative_algebrahttps://en.wikipedia.org/wiki/Field_of_fractionshttps://en.wikipedia.org/wiki/Polynomial_ringhttps://en.wikipedia.org/wiki/Category_of_abelian_groupshttps://en.wikipedia.org/wiki/Category_of_groupshttps://en.wikipedia.org/wiki/Abelianizationhttps://en.wikipedia.org/wiki/K-theoryhttps://en.wikipedia.org/wiki/Vector_bundlehttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Direct_sum_of_moduleshttps://en.wikipedia.org/wiki/Abelian_grouphttps://en.wikipedia.org/wiki/Grothendieck_grouphttps://en.wikipedia.org/wiki/Negative_numberhttps://en.wikipedia.org/wiki/Existence_theoremhttps://en.wikipedia.org/wiki/Universal_algebrahttps://en.wikipedia.org/wiki/Model_theoryhttps://en.wikipedia.org/wiki/Group_representationhttps://en.wikipedia.org/wiki/Group_representationhttps://en.wikipedia.org/wiki/Induced_representationhttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Discrete_spacehttps://en.wikipedia.org/wiki/Trivial_topologyhttps://en.wikipedia.org/wiki/Topological_spaceshttps://en.wikipedia.org/wiki/Topological_spaceshttps://en.wikipedia.org/wiki/Homotopy_classeshttps://en.wikipedia.org/wiki/Homotopy_classeshttps://en.wikipedia.org/wiki/Suspension_(topology)https://en.wikipedia.org/wiki/Loop_spacehttps://en.wikipedia.org/wiki/Homotopy_theory

1.6. EXAMPLES 9

Stone-ech compactification. Let KHaus bethe category of compact Hausdorff spaces and G :KHaus Top be the inclusion functor to the cate-gory of topological spaces. Then G has a left adjointF : Top KHaus, the Stoneech compactifica-tion. The unit of this adjoint pair yields a continuousmap from every topological space X into its Stone-ech compactification. This map is an embedding(i.e. injective, continuous and open) if and only if Xis a Tychonoff space.

Direct and inverse images of sheaves Everycontinuous map f : X Y between topologicalspaces induces a functor f from the category ofsheaves (of sets, or abelian groups, or rings...) on Xto the corresponding category of sheaves on Y, thedirect image functor. It also induces a functor f 1from the category of sheaves of abelian groups on Yto the category of sheaves of abelian groups on X,the inverse image functor. f 1 is left adjoint to f.Here a more subtle point is that the left adjoint forcoherent sheaves will differ from that for sheaves (ofsets).

Soberification. The article on Stone duality de-scribes an adjunction between the category of topo-logical spaces and the category of sober spaces thatis known as soberification. Notably, the article alsocontains a detailed description of another adjunc-tion that prepares the way for the famous dualityof sober spaces and spatial locales, exploited inpointless topology.

Category theory

A series of adjunctions. The functor 0 which as-signs to a category its set of connected componentsis left-adjoint to the functor D which assigns to a setthe discrete category on that set. Moreover, D isleft-adjoint to the object functor U which assigns toeach category its set of objects, and finally U is left-adjoint to A which assigns to each set the indiscretecategory on that set.

Exponential object. In a cartesian closed categorythe endofunctor C C given by A has a rightadjoint A.

Categorical logic

Quantification. If Y is a unary predicate express-ing some property, then a sufficiently strong set the-ory may prove the existence of the set Y = {y |Y (y)} of terms that fulfill the property. A propersubset T Y and the associated injection of Tinto Y is characterized by a predicate T (y) =Y (y) (y) expressing a strictly more restrictiveproperty.

The role of quantifiers in predicate logics is informing propositions and also in expressing so-phisticated predicates by closing formulas withpossibly more variables. For example, considera predicate f with two open variables of sortX and Y . Using a quantifier to close X , wecan form the set

{y Y | x. f (x, y) S(x)}

of all elements y of Y for which there is an xto which it is f -related, and which itself ischaracterized by the property S . Set theo-retic operations like the intersection of twosets directly corresponds to the conjunction of predicates. In categorical logic, a subfieldof topos theory, quantifiers are identified withadjoints to the pullback functor. Such a real-ization can be seen in analogy to the discussionof propositional logic using set theory but, in-terestingly, the general definition make for aricher range of logics.

So consider an object Y in a category with pull-backs. Any morphism f : X Y induces afunctor

f : Sub(Y ) Sub(X)

on the category that is the preorder of subob-jects. It maps subobjects T of Y (technically:monomorphism classes of T Y ) to thepullback X Y T . If this functor has a left-or right adjoint, they are called f and f ,respectively.[3] They both map from Sub(X)back to Sub(Y ) . Very roughly, given a do-main S X to quantify a relation expressedvia f over, the functor/quantifier closes X inX Y T and returns the thereby specified sub-set of Y .

Example: In Set , the category of sets andfunctions, the canonical subobjects are the sub-set (or rather their canonical injections). Thepullback fT = X Y T of an injection ofa subset T into Y along f is characterized asthe largest set which knows all about f andthe injection of T into Y . It therefore turnsout to be (in bijection with) the inverse imagef1[T ] X .For S X , let us figure out the left adjoinet,which is defined via

Hom(fS, T ) = Hom(S, fT ),

which here just means

fS T S f1[T ]

https://en.wikipedia.org/wiki/Compact_spacehttps://en.wikipedia.org/wiki/Hausdorff_spacehttps://en.wikipedia.org/wiki/Topological_spaceshttps://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactificationhttps://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactificationhttps://en.wikipedia.org/wiki/Continuous_function_(topology)https://en.wikipedia.org/wiki/Embeddinghttps://en.wikipedia.org/wiki/Injectivehttps://en.wikipedia.org/wiki/Tychonoff_spacehttps://en.wikipedia.org/wiki/Continuous_maphttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Sheaf_(mathematics)https://en.wikipedia.org/wiki/Direct_image_functorhttps://en.wikipedia.org/wiki/Inverse_image_functorhttps://en.wikipedia.org/wiki/Coherent_sheafhttps://en.wikipedia.org/wiki/Stone_dualityhttps://en.wikipedia.org/wiki/Sober_spacehttps://en.wikipedia.org/wiki/Duality_(category_theory)https://en.wikipedia.org/wiki/Pointless_topologyhttp://ncatlab.org/nlab/show/indiscrete+categoryhttp://ncatlab.org/nlab/show/indiscrete+categoryhttps://en.wikipedia.org/wiki/Cartesian_closed_categoryhttps://en.wikipedia.org/wiki/Quantifier_(logic)https://en.wikipedia.org/wiki/Categorical_logichttps://en.wikipedia.org/wiki/Topos_theory

10 CHAPTER 1. ADJOINT FUNCTORS

Consider f [S] T . We see S f1[f [S]] f1[T ] . Conversely, If for anx S we also have x f1[T ] , then clearlyf(x) T . So S f1[T ] implies f [S] T. We concude that left adjoint to the inverseimage functor f is given by the direct image.Here is a characterization of this result, whichmatches more the logical interpretation: Theimage of S under f is the full set of y 's, suchthat f1[{y}] S is non-empty. This worksbecause it neglects exactly those y Y whichare in the complement of f [S] . SofS = {y Y | (x f1[{y}]). x S } = f [S].

Put this in analogy to our motivation {y Y |x. f (x, y) S(x)} .The right adjoint to the inverse image functoris given (without doing the computation here)byfS = {y Y | (x f1[{y}]). x S }.

The subset fS of Y is characterized as the fullset of y 's with the property that the inverse im-age of {y} with respect to f is fully containedwithin S . Note how the predicate determiningthe set is the same as above, except that isreplaced by .

See also powerset.

1.7 Properties

1.7.1 Existence

Not every functor G : C D admits a left adjoint. If C isa complete category, then the functors with left adjointscan be characterized by the adjoint functor theorem ofPeter J. Freyd: G has a left adjoint if and only if it iscontinuous and a certain smallness condition is satisfied:for every object Y of D there exists a family of morphisms

fi : Y G(Xi)

where the indices i come from a set I, not a proper class,such that every morphism

h : Y G(X)

can be written as

h = G(t) o fi

for some i in I and some morphism

t : Xi X in C.

An analogous statement characterizes those functors witha right adjoint.

1.7.2 Uniqueness

If the functor F : C D has two right adjoints G and G,then G and G are naturally isomorphic. The same is truefor left adjoints.Conversely, if F is left adjoint to G, and G is naturallyisomorphic to G then F is also left adjoint to G. Moregenerally, if F, G, , is an adjunction (with counit-unit (,)) and

: F F : G G

are natural isomorphisms then F, G, , is an ad-junction where

= ( ) = (1 1).

Here denotes vertical composition of natural transfor-mations, and denotes horizontal composition.

1.7.3 Composition

Adjunctions can be composed in a natural fashion.Specifically, if F, G, , is an adjunction between Cand D and F, G, , is an adjunction between Dand E then the functor

F F : C E

is left adjoint to

G G : C E .

More precisely, there is an adjunction between F F andG G with unit and counit given by the compositions:

1EGF G

FGGF F

F FGGF GF G

1C .

This new adjunction is called the composition of the twogiven adjunctions.One can then form a category whose objects are all smallcategories and whose morphisms are adjunctions.

1.7.4 Limit preservation

The most important property of adjoints is their continu-ity: every functor that has a left adjoint (and therefore is

https://en.wikipedia.org/wiki/Powersethttps://en.wikipedia.org/wiki/Complete_categoryhttps://en.wikipedia.org/wiki/Peter_J._Freydhttps://en.wikipedia.org/wiki/Limit_(category_theory)#Preservation_of_limitshttps://en.wikipedia.org/wiki/Class_(set_theory)https://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Small_categoryhttps://en.wikipedia.org/wiki/Small_category

1.9. REFERENCES 11

a right adjoint) is continuous (i.e. commutes with limitsin the category theoretical sense); every functor that hasa right adjoint (and therefore is a left adjoint) is cocontin-uous (i.e. commutes with colimits).Since many common constructions in mathematics arelimits or colimits, this provides a wealth of information.For example:

applying a right adjoint functor to a product of ob-jects yields the product of the images;

applying a left adjoint functor to a coproduct of ob-jects yields the coproduct of the images;

every right adjoint functor is left exact;

every left adjoint functor is right exact.

1.7.5 Additivity

If C and D are preadditive categories and F : C D is anadditive functor with a right adjoint G : C D, then G isalso an additive functor and the hom-set bijections

Y,X : homC(FY,X) = homD(Y,GX)

are, in fact, isomorphisms of abelian groups. Dually, if Gis additive with a left adjoint F, then F is also additive.Moreover, if both C and D are additive categories (i.e.preadditive categories with all finite biproducts), then anypair of adjoint functors between them are automaticallyadditive.

1.8 Relationships

1.8.1 Universal constructions

As stated earlier, an adjunction between categories C andD gives rise to a family of universal morphisms, one foreach object in C and one for each object in D. Conversely,if there exists a universal morphism to a functor G : C D from every object of D, then G has a left adjoint.However, universal constructions are more general thanadjoint functors: a universal construction is like an opti-mization problem; it gives rise to an adjoint pair if andonly if this problem has a solution for every object of D(equivalently, every object of C).

1.8.2 Equivalences of categories

If a functor F: CD is one half of an equivalence of cat-egories then it is the left adjoint in an adjoint equivalenceof categories, i.e. an adjunction whose unit and counitare isomorphisms.

Every adjunction F, G, , extends an equivalence ofcertain subcategories. Define C1 as the full subcategoryof C consisting of those objects X of C for which X isan isomorphism, and define D1 as the full subcategory ofD consisting of those objects Y of D for which Y is anisomorphism. Then F and G can be restricted to D1 andC1 and yield inverse equivalences of these subcategories.In a sense, then, adjoints are generalized inverses. Notehowever that a right inverse of F (i.e. a functor G suchthat FG is naturally isomorphic to 1D) need not be a right(or left) adjoint of F. Adjoints generalize two-sided in-verses.

1.8.3 Monads

Every adjunction F, G, , gives rise to an associatedmonad T, , in the category D. The functor

T : D D

is given by T = GF. The unit of the monad

: 1D T

is just the unit of the adjunction and the multiplicationtransformation

: T 2 T

is given by = GF. Dually, the triple FG, , FGdefines a comonad in C.Every monad arises from some adjunctionin fact, typi-cally from many adjunctionsin the above fashion. Twoconstructions, called the category of EilenbergMoore al-gebras and the Kleisli category are two extremal solutionsto the problem of constructing an adjunction that givesrise to a given monad.

1.9 References[1] arXiv.org: John C. Baez Higher-Dimensional Algebra II:

2-Hilbert Spaces.

[2] William Lawvere, Adjointness in foundations, Dialectica,1969, available here. The notation is different nowa-days; an easier introduction by Peter Smith in these lecturenotes, which also attribute the concept to the article cited.

[3] Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Ge-ometry and Logic Springer-Verlag. ISBN 0-387-97710-4See page 58

Admek, Ji; Herrlich, Horst; Strecker, George E.(1990). Abstract and Concrete Categories. The joyof cats (PDF). John Wiley & Sons. ISBN 0-471-60922-6. Zbl 0695.18001.

https://en.wikipedia.org/wiki/Limit_(category_theory)https://en.wikipedia.org/wiki/Limit_(category_theory)https://en.wikipedia.org/wiki/Product_(category_theory)https://en.wikipedia.org/wiki/Coproducthttps://en.wikipedia.org/wiki/Left_exact_functorhttps://en.wikipedia.org/wiki/Right_exact_functorhttps://en.wikipedia.org/wiki/Preadditive_categorieshttps://en.wikipedia.org/wiki/Additive_functorhttps://en.wikipedia.org/wiki/Additive_categorieshttps://en.wikipedia.org/wiki/Biproducthttps://en.wikipedia.org/wiki/Universal_morphismhttps://en.wikipedia.org/wiki/Equivalence_of_categorieshttps://en.wikipedia.org/wiki/Equivalence_of_categorieshttps://en.wikipedia.org/wiki/Full_subcategoryhttps://en.wikipedia.org/wiki/Monad_(category_theory)https://en.wikipedia.org/wiki/Comonadhttps://en.wikipedia.org/wiki/Eilenberg%E2%80%93Moore_algebrahttps://en.wikipedia.org/wiki/Eilenberg%E2%80%93Moore_algebrahttps://en.wikipedia.org/wiki/Kleisli_categoryhttp://www.arxiv.org/abs/q-alg/9609018http://www.arxiv.org/abs/q-alg/9609018https://en.wikipedia.org/wiki/William_Lawverehttp://www.tac.mta.ca/tac/reprints/articles/16/tr16abs.htmlhttp://www.logicmatters.net/resources/pdfs/Galois.pdfhttp://www.logicmatters.net/resources/pdfs/Galois.pdfhttps://en.wikipedia.org/wiki/Special:BookSources/0387977104http://katmat.math.uni-bremen.de/acc/acc.pdfhttp://katmat.math.uni-bremen.de/acc/acc.pdfhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-471-60922-6https://en.wikipedia.org/wiki/Special:BookSources/0-471-60922-6https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0695.18001

12 CHAPTER 1. ADJOINT FUNCTORS

Mac Lane, Saunders (1998). Categories for theWorking Mathematician. Graduate Texts in Math-ematics 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.

1.10 External links Adjunctions Seven short lectures on adjunctions.

https://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematicianhttps://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematicianhttps://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematicshttps://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematicshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-98403-8https://en.wikipedia.org/wiki/Special:BookSources/0-387-98403-8https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0906.18001http://www.youtube.com/view_play_list?p=54B49729E5102248

Chapter 2

Algebraic topology

For the topology of pointwise convergence, see Algebraictopology (object).Algebraic topology is a branch of mathematics that uses

A torus, one of the most frequently studied objects in algebraictopology

tools from abstract algebra to study topological spaces.The basic goal is to find algebraic invariants that classifytopological spaces up to homeomorphism, though usuallymost classify up to homotopy equivalence.Although algebraic topology primarily uses algebra tostudy topological problems, using topology to solve al-gebraic problems is sometimes also possible. Algebraictopology, for example, allows for a convenient proof thatany subgroup of a free group is again a free group.

2.1 Main branches of algebraictopology

Below are some of the main areas studied in algebraictopology:

2.1.1 Homotopy groups

Main article: Homotopy group

In mathematics, homotopy groups are used in algebraictopology to classify topological spaces. The first and sim-plest homotopy group is the fundamental group, which

records information about loops in a space. Intuitively,homotopy groups record information about the basicshape, or holes, of a topological space.

2.1.2 Homology

Main article: Homology

In algebraic topology and abstract algebra, homology (inpart from Greek homos identical) is a certain gen-eral procedure to associate a sequence of abelian groupsor modules with a given mathematical object such as atopological space or a group.[1]

2.1.3 Cohomology

Main article: Cohomology

In homology theory and algebraic topology, cohomologyis a general term for a sequence of abelian groups de-fined from a co-chain complex. That is, cohomology isdefined as the abstract study of cochains, cocycles, andcoboundaries. Cohomology can be viewed as a methodof assigning algebraic invariants to a topological spacethat has a more refined algebraic structure than doeshomology. Cohomology arises from the algebraic dual-ization of the construction of homology. In less abstractlanguage, cochains in the fundamental sense should as-sign 'quantities to the chains of homology theory.

2.1.4 Manifolds

Main article: Manifold

A manifold is a topological space that near each pointresembles Euclidean space. More precisely, each pointof an n-dimensional manifold has a neighbourhood thatis diffeomorphic to the Euclidean space of dimensionn. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds arealso called surfaces. Examples include the plane, the

13

https://en.wikipedia.org/wiki/Algebraic_topology_(object)https://en.wikipedia.org/wiki/Algebraic_topology_(object)https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Torushttps://en.wikipedia.org/wiki/Abstract_algebrahttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Invariant_(mathematics)https://en.wikipedia.org/wiki/Classification_theoremhttps://en.wikipedia.org/wiki/Up_tohttps://en.wikipedia.org/wiki/Homeomorphismhttps://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence_and_null-homotopyhttps://en.wikipedia.org/wiki/Free_grouphttps://en.wikipedia.org/wiki/Homotopy_grouphttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Fundamental_grouphttps://en.wikipedia.org/wiki/Homology_(mathematics)https://en.wikipedia.org/wiki/Abstract_algebrahttps://en.wikipedia.org/wiki/Greek_languagehttps://en.wikipedia.org/wiki/Sequencehttps://en.wikipedia.org/wiki/Abelian_grouphttps://en.wikipedia.org/wiki/Module_(mathematics)https://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Cohomologyhttps://en.wikipedia.org/wiki/Homology_theoryhttps://en.wikipedia.org/wiki/Sequencehttps://en.wikipedia.org/wiki/Abelian_grouphttps://en.wikipedia.org/wiki/Chain_complexhttps://en.wikipedia.org/wiki/Chain_complexhttps://en.wikipedia.org/wiki/Coboundaryhttps://en.wikipedia.org/wiki/Algebraic_invarianthttps://en.wikipedia.org/wiki/Algebraic_structurehttps://en.wikipedia.org/wiki/Homology_(mathematics)https://en.wikipedia.org/wiki/Chain_(algebraic_topology)https://en.wikipedia.org/wiki/Manifoldhttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Neighbourhood_(mathematics)https://en.wikipedia.org/wiki/Diffeomorphichttps://en.wikipedia.org/wiki/Line_(geometry)https://en.wikipedia.org/wiki/Circlehttps://en.wikipedia.org/wiki/Lemniscatehttps://en.wikipedia.org/wiki/Surfacehttps://en.wikipedia.org/wiki/Plane_(geometry)

14 CHAPTER 2. ALGEBRAIC TOPOLOGY

sphere, and the torus, which can all be realized in threedimensions, but also the Klein bottle and real projectiveplane which cannot be realized in three dimensions, butcan be realized in four dimensions.

2.1.5 Knot theory

Main article: Knot theory

Knot theory is the study of mathematical knots. Whileinspired by knots that appear in daily life in shoelaces andrope, a mathematicians knot differs in that the ends arejoined together so that it cannot be undone. In precisemathematical language, a knot is an embedding of a circlein 3-dimensional Euclidean space, R3. Two mathemati-cal knots are equivalent if one can be transformed intothe other via a deformation of R3 upon itself (known asan ambient isotopy); these transformations correspond tomanipulations of a knotted string that do not involve cut-ting the string or passing the string through itself.

2.1.6 Complexes

Main articles: Simplicial complex and CW complex

A simplicial complex is a topological space of a cer-tain kind, constructed by gluing together points, linesegments, triangles, and their n-dimensional counterparts(see illustration). Simplicial complexes should not beconfused with the more abstract notion of a simplicial setappearing in modern simplicial homotopy theory. Thepurely combinatorial counterpart to a simplicial complexis an abstract simplicial complex.A CW complex is a type of topological space introducedby J. H. C. Whitehead to meet the needs of homotopytheory. This class of spaces is broader and has some bet-ter categorical properties than simplicial complexes, butstill retains a combinatorial nature that allows for compu-tation (often with a much smaller complex).

2.2 Method of algebraic invariants

An older name for the subject was combinatorial topol-ogy, implying an emphasis on how a space X was con-structed from simpler ones[2] (the modern standard toolfor such construction is the CW-complex). In the 1920sand 1930s, there was growing emphasis on investigatingtopological spaces by finding correspondences from themto algebraic groups, which led to the change of name toalgebraic topology.[3] The combinatorial topology nameis still sometimes used to emphasize an algorithmic ap-proach based on decomposition of spaces.[4]

In the algebraic approach, one finds a correspondencebetween spaces and groups that respects the relation of

homeomorphism (or more general homotopy) of spaces.This allows one to recast statements about topologicalspaces into statements about groups, which have a greatdeal of manageable structure, often making these state-ment easier to prove. Two major ways in which thiscan be done are through fundamental groups, or moregenerally homotopy theory, and through homology andcohomology groups. The fundamental groups give us ba-sic information about the structure of a topological space,but they are often nonabelian and can be difficult to workwith. The fundamental group of a (finite) simplicial com-plex does have a finite presentation.Homology and cohomology groups, on the other hand,are abelian and in many important cases finitely gener-ated. Finitely generated abelian groups are completelyclassified and are particularly easy to work with.

2.3 Setting in category theory

In general, all constructions of algebraic topology arefunctorial; the notions of category, functor and naturaltransformation originated here. Fundamental groups andhomology and cohomology groups are not only invari-ants of the underlying topological space, in the sense thattwo topological spaces which are homeomorphic have thesame associated groups, but their associated morphismsalso correspond a continuous mapping of spaces in-duces a group homomorphism on the associated groups,and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.One of the first mathematicians to work with differenttypes of cohomology was Georges de Rham. One canuse the differential structure of smooth manifolds viade Rham cohomology, or ech or sheaf cohomologyto investigate the solvability of differential equations de-fined on the manifold in question. De Rham showed thatall of these approaches were interrelated and that, fora closed, oriented manifold, the Betti numbers derivedthrough simplicial homology were the same Betti num-bers as those derived through de Rham cohomology. Thiswas extended in the 1950s, when Eilenberg and Steenrodgeneralized this approach. They defined homology andcohomology as functors equipped with natural transfor-mations subject to certain axioms (e.g., a weak equiva-lence of spaces passes to an isomorphism of homologygroups), verified that all existing (co)homology theoriessatisfied these axioms, and then proved that such an ax-iomatization uniquely characterized the theory.

2.4 Applications of algebraic topol-ogy

Classic applications of algebraic topology include:

https://en.wikipedia.org/wiki/Spherehttps://en.wikipedia.org/wiki/Torushttps://en.wikipedia.org/wiki/Klein_bottlehttps://en.wikipedia.org/wiki/Real_projective_planehttps://en.wikipedia.org/wiki/Real_projective_planehttps://en.wikipedia.org/wiki/Knot_theoryhttps://en.wikipedia.org/wiki/Knot_(mathematics)https://en.wikipedia.org/wiki/Embeddinghttps://en.wikipedia.org/wiki/Circlehttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Ambient_isotopyhttps://en.wikipedia.org/wiki/Simplicial_complexhttps://en.wikipedia.org/wiki/CW_complexhttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Point_(geometry)https://en.wikipedia.org/wiki/Line_segmenthttps://en.wikipedia.org/wiki/Line_segmenthttps://en.wikipedia.org/wiki/Trianglehttps://en.wikipedia.org/wiki/Simplexhttps://en.wikipedia.org/wiki/Simplicial_sethttps://en.wikipedia.org/wiki/Abstract_simplicial_complexhttps://en.wikipedia.org/wiki/J._H._C._Whiteheadhttps://en.wikipedia.org/wiki/Homotopy_theoryhttps://en.wikipedia.org/wiki/Homotopy_theoryhttps://en.wikipedia.org/wiki/Category_theoryhttps://en.wikipedia.org/wiki/Simplicial_complexhttps://en.wikipedia.org/wiki/Combinatorial_topologyhttps://en.wikipedia.org/wiki/Combinatorial_topologyhttps://en.wikipedia.org/wiki/CW_complexhttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Homeomorphismhttps://en.wikipedia.org/wiki/Homotopyhttps://en.wikipedia.org/wiki/Fundamental_grouphttps://en.wikipedia.org/wiki/Homotopy_theoryhttps://en.wikipedia.org/wiki/Homology_(mathematics)https://en.wikipedia.org/wiki/Cohomologyhttps://en.wikipedia.org/wiki/Abelian_grouphttps://en.wikipedia.org/wiki/Simplicial_complexhttps://en.wikipedia.org/wiki/Simplicial_complexhttps://en.wikipedia.org/wiki/Presentation_of_a_grouphttps://en.wikipedia.org/wiki/Finitely_generated_abelian_grouphttps://en.wikipedia.org/wiki/Category_theoryhttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Functorhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Homeomorphichttps://en.wikipedia.org/wiki/Group_homomorphismhttps://en.wikipedia.org/wiki/Georges_de_Rhamhttps://en.wikipedia.org/wiki/Differentiable_manifoldhttps://en.wikipedia.org/wiki/De_Rham_cohomologyhttps://en.wikipedia.org/wiki/%C4%8Cech_cohomologyhttps://en.wikipedia.org/wiki/Sheaf_cohomologyhttps://en.wikipedia.org/wiki/Differential_equationhttps://en.wikipedia.org/wiki/Functorhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)

2.5. NOTABLE ALGEBRAIC TOPOLOGISTS 15

The Brouwer fixed point theorem: every continuousmap from the unit n-disk to itself has a fixed point.

The free rank of the nth homology group of asimplicial complex is the n-th Betti number, whichallows one to calculate the Euler-Poincar charac-teristic.

One can use the differential structure of smoothmanifolds via de Rham cohomology, or ech orsheaf cohomology to investigate the solvability ofdifferential equations defined on the manifold inquestion.

A manifold is orientable when the top-dimensionalintegral homology group is the integers, and is non-orientable when it is 0.

The n-sphere admits a nowhere-vanishing continu-ous unit vector f