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 denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Automorphism 132.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Automorphism group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Inner and outer automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Category theory 163.1 An abstraction of other mathematical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Other concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6.1 Universal constructions, limits, and colimits . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.2 Equivalent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.3 Further concepts and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.4 Higher-dimensional categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Dual (category theory) 234.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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4.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Endomorphism 255.1 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Endomorphism ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Operator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Endofunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Epimorphism 276.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Equivalence of categories 307.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Functor category 338.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9 Identity function 359.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.2 Algebraic property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10 Isomorphism 37

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10.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.1.1 Logarithm and exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.1.2 Integers modulo 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.1.3 Relation-preserving isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10.2 Isomorphism vs. bijective morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.4 Relation with equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11 Isomorphism of categories 4211.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12 Mathematical structure 4312.1 Example: the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

13 Monomorphism 4513.1 Relation to invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.5 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

14 Section (category theory) 4714.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

15 Yoneda lemma 4815.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.2 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

15.2.1 General version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4815.2.2 Naming conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4915.2.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4915.2.4 The Yoneda embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

15.3 Preadditive categories, rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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15.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 51

15.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Chapter 1

Adjoint functors

For the construction in eld theory, see Adjunction (eldtheory). For the construction in topology, see Adjunctionspace.

In mathematics, specically category theory, adjunctionis a possible relationship between two functors.Adjunction is ubiquitous in mathematics, as it speciesintuitive notions of optimization and eciency.In the most concise symmetric denition, 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 a G:This denition and others are made precise below.

1.1 IntroductionThe 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 denitions 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 dierent 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 ecient 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 ecient 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 ecient if it satises a universal property,and is formulaic if it denes 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 nd an initial object of E. This

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2 CHAPTER 1. ADJOINT FUNCTORS

has an advantage that the optimization the sense that weare nding the most ecient 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 denition 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 ecient 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 ecient solution to our problem.The two facts that this method of turning rngs into ringsis most ecient and formulaic can be expressed simulta-neously by saying that it denes 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 ecient solution?The notion that F is the most ecient solution to the prob-lem posed by G is, in a certain rigorous sense, equivalentto the notion that G poses the most dicult 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 denitions. Theequivalent symmetric denitions 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 denitionsThere are various denitions for adjoint functors. Theirequivalence is elementary but not at all trivial and in facthighly useful. This article provides several such deni-tions:

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

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

The denition 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 denition 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 denitions 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 ConventionsThe 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 morphismsA 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.

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 denitions given below. Thedenitions 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 nding a universalmorphism is like solving an optimization problem.

1.3.3 Counit-unit adjunctionA 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 ! GFrespectively called the counit and the unit of the adjunc-tion (terminology from universal algebra), such that thecompositions

FF!FGF "F!F

GG!GFG G"!G

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 a G , or simply F a G .In equation form, the above conditions on (,) are thecounit-unit equations

1F = "F F1G = G" Gwhich mean that for each X in C and each Y in D,

1FY = "FY F (Y )1GX = G("X) GXNote 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 rst write down the nonsensical equa-tion 1 = " and then ll in either F or G in one of thetwo simple ways which make the compositions dened.

Note: The use of the prex co in counit here is notconsistent with the terminology of limits and colimits,because a colimit satises 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 species 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 a G , or simply F a G .This denition is a logical compromise in that it is some-what more dicult to satisfy than the universal morphismdenitions, and has fewer immediate implications thanthe counit-unit denition. It is useful because of its obvi-ous symmetry, and as a stepping-stone between the otherdenitions.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.

4 CHAPTER 1. ADJOINT FUNCTORS

1.4 Adjunctions in fullThere are hence numerous functors and natural trans-formations associated with every adjunction, and only asmall portion is sucient 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) 2 homC(F (Y ); X)g = Y;X(f) = G(f) Y 2 homD(Y;G(X))1GX;X(1GX) = "X 2 homC(FG(X); X)Y;FY (1FY ) = Y 2 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 dene ,, and in terms of any one of the three. However, theadjoint functors F and G alone are in general not su-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 aninitial 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 factorizationimplies the commuting diagram of naturaltransformations, so : 1D G o F is a naturaltransformation.

Uniqueness of that factorization and that G isa 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 dened inclusion or forgetfulfunctor.)

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 a G, we can construct a hom-set adjunction by nding the natural transformation :homC(F-,-) homD(-,G-) in the following steps:

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

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 a G ,

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

Let "X = 1GX;X(1GX) 2 homC(FGX;X) foreach X in C, where 1GX 2 homD(GX;GX) is theidentity morphism.

Let Y = Y;FY (1FY ) 2 homD(Y;GFY ) foreach Y in D, where 1FY 2 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 rst counit-unitequation

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

1.5 History

1.5.1 UbiquityThe 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 withA), andG 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, rstly, 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 hTx; yi = hx;Uyi , 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 quitedierent from F (see below for an example). The analogy

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 nally 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 rening a correspondence to a duality (i.e. antitoneorder isomorphism). A treatment of Galois theory alongthese lines by Kaplansky was inuential in the recognitionof the general structure here.The partial order case collapses the adjunction denitionsquite 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 groupX, 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 through"X : 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.

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 verication thatthey form a counit-unit adjunction ("; ) : F a G is asfollows:The rst counit-unit equation 1F = "F F says thatfor each set Y the composition

FYF (Y )!FGFY "FY!FY

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

GXGX!GFGX 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 limitsProducts, bred 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 den-ing maps from the limit object (i.e. from the diagonalfunctor on the limit, in the functor category). Below aresome specic 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 thedening pair of projection maps from X1X2 to X1and X2 which dene 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 den-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 functorsCoproducts, bred 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 dening maps into the colimitobject. Below are some specic examples.

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 thedening 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 examplesAlgebra

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 dening 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, dened 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 eld 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 elds has aleft adjoint - it assigns to every integral domain itseld of fractions.

Polynomial rings. Let Ring* be the category ofpointed commutative rings with unity (pairs (A,a)where A is a ring, a 2 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 coecients 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 nitary 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 fromX to the loop space Yof Y. This is an important fact in homotopy theory.

1.6. EXAMPLES 9

Stone-ech compactication. Let KHaus bethe category of compact Hausdor 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 compactica-tion. The unit of this adjoint pair yields a continuousmap from every topological space X into its Stone-ech compactication. This map is an embedding(i.e. injective, continuous and open) if and only if Xis a Tychono 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 dier from that for sheaves (ofsets).

Soberication. 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 soberication. 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 nally 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

Quantication. If Y is a unary predicate express-ing some property, then a suciently strong set the-ory may prove the existence of the set Y = fy jY (y)g of terms that fulll 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 quantiers 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 quantier to close X , wecan form the set

fy 2 Y j 9x: f (x; y) ^ S(x)g

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 subeldof topos theory, quantiers are identied 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 denition 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 9f and 8f ,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/quantier closes X inX Y T and returns the thereby specied 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 gure out the left adjoinet,which is dened via

Hom(9fS; T ) = Hom(S; fT );

which here just means

9fS T $ S f1[T ]

10 CHAPTER 1. ADJOINT FUNCTORS

Consider f [S] T . We see S f1[f [S]] f1[T ] . Conversely, If for anx 2 S we also have x 2 f1[T ] , then clearlyf(x) 2 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 9f is the full set of y 's, suchthat f1[fyg] \ S is non-empty. This worksbecause it neglects exactly those y 2 Y whichare in the complement of f [S] . So

9fS = fy 2 Y j 9(x 2 f1[fyg]): x 2 S g = f [S]:Put this in analogy to our motivation fy 2 Y j9x: f (x; y) ^ S(x)g .The right adjoint to the inverse image functoris given (without doing the computation here)by8fS = fy 2 Y j 8(x 2 f1[fyg]): x 2 S g:The subset 8fS of Y is characterized as the fullset of y 's with the property that the inverse im-age of fyg with respect to f is fully containedwithin S . Note how the predicate determiningthe set is the same as above, except that 9 isreplaced by 8 .

See also powerset.

1.7 Properties

1.7.1 ExistenceNot 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 satised:for every object Y ofD 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 UniquenessIf 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

0 = ( ) "0 = " (1 1):Here denotes vertical composition of natural transfor-mations, and denotes horizontal composition.

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

F 0 F : C E

is left adjoint to

G G0 : C ! E :

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

1E!GF G

0F!GG0F 0FF 0FGG0 F

0"G0!F 0G0 "0!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 preservationThe most important property of adjoints is their continu-ity: every functor that has a left adjoint (and therefore is

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 AdditivityIf 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 nite biproducts), then anypair of adjoint functors between them are automaticallyadditive.

1.8 Relationships

1.8.1 Universal constructionsAs 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 categoriesIf 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. Dene C1 as the full subcategoryof C consisting of those objects X of C for which X isan isomorphism, and dene 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 MonadsEvery adjunction F, G, , gives rise to an associatedmonad T, , in the category D. The functor

T : D ! Dis given by T = GF. The unit of the monad

: 1D ! Tis just the unit of the adjunction and the multiplicationtransformation

: T 2 ! Tis given by = GF. Dually, the triple FG, , FGdenes 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 dierent 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.

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.

Chapter 2

Automorphism

In mathematics, an automorphism is an isomorphismfrom a mathematical object to itself. It is, in some sense,a symmetry of the object, and a way of mapping the ob-ject to itself while preserving all of its structure. The setof all automorphisms of an object forms a group, calledthe automorphism group. It is, loosely speaking, thesymmetry group of the object.

2.1 Denition

The exact denition of an automorphism depends on thetype of mathematical object in question and what, pre-cisely, constitutes an isomorphism of that object. Themost general setting in which these words have mean-ing is an abstract branch of mathematics called categorytheory. Category theory deals with abstract objects andmorphisms between those objects.In category theory, an automorphism is an endomorphism(i.e. a morphism from an object to itself) which is also anisomorphism (in the categorical sense of the word).This is a very abstract denition since, in category theory,morphisms aren't necessarily functions and objects aren'tnecessarily sets. In most concrete settings, however, theobjects will be sets with some additional structure and themorphisms will be functions preserving that structure.In the context of abstract algebra, for example, a mathe-matical object is an algebraic structure such as a group,ring, or vector space. An isomorphism is simply abijective homomorphism. (The denition of a homomor-phism depends on the type of algebraic structure; see, forexample: group homomorphism, ring homomorphism,and linear operator).The identity morphism (identity mapping) is called thetrivial automorphism in some contexts. Respectively,other (non-identity) automorphisms are called nontrivialautomorphisms.

2.2 Automorphism groupIf the automorphisms of an objectX form a set (instead ofa proper class), then they form a group under compositionof morphisms. This group is called the automorphismgroup of X. That this is indeed a group is simple to see:

Closure: composition of two endomorphisms is an-other endomorphism.

Associativity: composition of morphisms is alwaysassociative.

Identity: the identity is the identity morphism froman object to itself, which exists by denition.

Inverses: by denition every isomorphism has an in-verse which is also an isomorphism, and since theinverse is also an endomorphism of the same objectit is an automorphism.

The automorphism group of an object X in a category Cis denoted AutC(X), or simply Aut(X) if the category isclear from context.

2.3 Examples In set theory, an arbitrary permutation of the ele-

ments of a set X is an automorphism. The automor-phism group of X is also called the symmetric groupon X.

In elementary arithmetic, the set of integers, Z, con-sidered as a group under addition, has a unique non-trivial automorphism: negation. Considered as aring, however, it has only the trivial automorphism.Generally speaking, negation is an automorphism ofany abelian group, but not of a ring or eld.

A group automorphism is a group isomorphismfrom a group to itself. Informally, it is a permu-tation of the group elements such that the structureremains unchanged. For every group G there is anatural group homomorphism G Aut(G) whoseimage is the group Inn(G) of inner automorphisms

13

14 CHAPTER 2. AUTOMORPHISM

and whose kernel is the center of G. Thus, if G hastrivial center it can be embedded into its own auto-morphism group.[1]

In linear algebra, an endomorphism of a vector spaceV is a linear operator V V. An automorphism isan invertible linear operator on V. When the vectorspace is nite-dimensional, the automorphism groupof V is the same as the general linear group, GL(V).

A eld automorphism is a bijective ring homomor-phism from a eld to itself. In the cases of therational numbers (Q) and the real numbers (R)there are no nontrivial eld automorphisms. Somesubelds of R have nontrivial eld automorphisms,which however do not extend to all of R (becausethey cannot preserve the property of a number hav-ing a square root in R). In the case of the complexnumbers, C, there is a unique nontrivial automor-phism that sendsR intoR: complex conjugation, butthere are innitely (uncountably) many wild au-tomorphisms (assuming the axiom of choice).[2][3]Field automorphisms are important to the theory ofeld extensions, in particular Galois extensions. Inthe case of a Galois extension L/K the subgroup ofall automorphisms of L xing K pointwise is calledthe Galois group of the extension.

The eld Qp of p-adic numbers has no nontrivialautomorphisms.

In graph theory an automorphism of a graph is a per-mutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by anedge, so are their images under the permutation.

For relations, see relation-preserving automorphism. In order theory, see order automorphism.

In geometry, an automorphism may be called amotion of the space. Specialized terminology is alsoused:

In metric geometry an automorphism is a self-isometry. The automorphism group is alsocalled the isometry group.

In the category of Riemann surfaces, an au-tomorphism is a bijective biholomorphic map(also called a conformal map), from a surfaceto itself. For example, the automorphisms ofthe Riemann sphere are Mbius transforma-tions.

An automorphism of a dierentiable manifoldM is a dieomorphism from M to itself. Theautomorphism group is sometimes denotedDi(M).

In topology, morphisms between topologi-cal spaces are called continuous maps, andan automorphism of a topological space is

a homeomorphism of the space to itself, orself-homeomorphism (see homeomorphismgroup). In this example it is not sucient fora morphism to be bijective to be an isomor-phism.

2.4 HistoryOne of the earliest group automorphisms (automorphismof a group, not simply a group of automorphisms ofpoints) was given by the Irish mathematician WilliamRowan Hamilton in 1856, in his icosian calculus, wherehe discovered an order two automorphism,[4] writing:

so that is a new fth root of unity, con-nected with the former fth root by relationsof perfect reciprocity.

2.5 Inner and outer automor-phisms

In some categoriesnotably groups, rings, and Lie alge-brasit is possible to separate automorphisms into twotypes, called inner and outer automorphisms.In the case of groups, the inner automorphisms are theconjugations by the elements of the group itself. For eachelement a of a group G, conjugation by a is the operationa : G G given by a(g) = aga1 (or a1ga; usagevaries). One can easily check that conjugation by a is agroup automorphism. The inner automorphisms form anormal subgroup of Aut(G), denoted by Inn(G); this iscalled Goursats lemma.The other automorphisms are called outer automor-phisms. The quotient group Aut(G) / Inn(G) is usuallydenoted by Out(G); the non-trivial elements are the cosetsthat contain the outer automorphisms.The same denition holds in any unital ring or algebrawhere a is any invertible element. For Lie algebras thedenition is slightly dierent.

2.6 See also Endomorphism ring

Antiautomorphism

Frobenius automorphism

Morphism

Characteristic subgroup

2.8. EXTERNAL LINKS 15

2.7 References[1] PJ Pahl, R Damrath (2001). "7.5.5 Automorphisms.

Mathematical foundations of computational engineering(Felix Pahl translation ed.). Springer. p. 376. ISBN 3-540-67995-2.

[2] Yale, Paul B. (May 1966). Automorphisms of the Com-plex Numbers (PDF). Mathematics Magazine 39 (3):135141. doi:10.2307/2689301. JSTOR 2689301.

[3] Lounesto, Pertti (2001), Cliord Algebras and Spinors(2nd ed.), Cambridge University Press, pp. 2223, ISBN0-521-00551-5

[4] Sir William Rowan Hamilton (1856). Memorandumrespecting a new System of Roots of Unity (PDF).Philosophical Magazine 12: 446.

2.8 External links Automorphism at Encyclopaedia of Mathematics Weisstein, Eric W., Automorphism, MathWorld.

Chapter 3

Category theory

Schematic representation of a category with objects X, Y, Z andmorphisms f, g, g f. (The categorys three identity morphisms1X, 1Y and 1Z, if explicitly represented, would appear as threearrows, next to the letters X, Y, and Z, respectively, each havingas its shaft a circular arc measuring almost 360 degrees.)

Category theory[1] formalizes mathematical structureand its concepts in terms of a collection of objects andof arrows (also called morphisms). A category hastwo basic properties: the ability to compose the arrowsassociatively and the existence of an identity arrow foreach object. Category theory can be used to formal-ize concepts of other high-level abstractions such as sets,rings, and groups.Several terms used in category theory, including the termmorphism, are used dierently from their uses in therest of mathematics. In category theory, a morphismobeys a set of conditions specic to category theory it-self. Thus, care must be taken to understand the contextin which statements are made.

3.1 An abstraction of other mathe-matical concepts

Many signicant areas of mathematics can be formalisedby category theory as categories. Category theory is anabstraction of mathematics itself that allows many intri-cate and subtle mathematical results in these elds to bestated, and proved, in a much simpler way than withoutthe use of categories.[2]

The most accessible example of a category is the categoryof sets, where the objects are sets and the arrows are func-tions from one set to another. However, the objects ofa category need not be sets, and the arrows need not befunctions; any way of formalising a mathematical conceptsuch that it meets the basic conditions on the behaviour ofobjects and arrows is a valid category, and all the resultsof category theory will apply to it.The arrows of category theory are often said to repre-sent a process connecting two objects, or in many cases astructure-preserving transformation connecting two ob-jects. There are however many applications where muchmore abstract concepts are represented by objects andmorphisms. The most important property of the arrowsis that they can be composed, in other words, arrangedin a sequence to form a new arrow.Categories now appear in most branches of mathematics,some areas of theoretical computer science where theycan correspond to types, and mathematical physics wherethey can be used to describe vector spaces. Categorieswere rst introduced by Samuel Eilenberg and SaundersMac Lane in 194245, in connection with algebraictopology.Category theory has several faces known not just to spe-cialists, but to other mathematicians. A term dating fromthe 1940s, "general abstract nonsense", refers to its highlevel of abstraction, compared to more classical branchesof mathematics. Homological algebra is category theoryin its aspect of organising and suggesting manipulationsin abstract algebra.

16

3.3. CATEGORIES, OBJECTS, AND MORPHISMS 17

3.2 Utility

3.2.1 Categories, objects, and morphisms

The study of categories is an attempt to axiomatically cap-ture what is commonly found in various classes of relatedmathematical structures by relating them to the structure-preserving functions between them. A systematic studyof category theory then allows us to prove general resultsabout any of these types of mathematical structures fromthe axioms of a category.Consider the following example. The classGrp of groupsconsists of all objects having a group structure. Onecan proceed to prove theorems about groups by makinglogical deductions from the set of axioms. For example,it is immediately proven from the axioms that the identityelement 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 map-pings between these objects; by studying these mor-phisms, we are able to learn more about the structure ofthe objects. In the case of groups, the morphisms arethe group homomorphisms. A group homomorphism be-tween two groups preserves the group structure in a pre-cise sense it is a process taking one group to another,in a way that carries along information about the struc-ture of the rst group into the second group. The studyof group homomorphisms then provides a tool for study-ing general properties of groups and consequences of thegroup axioms.A similar type of investigation occurs in many mathemat-ical theories, such as the study of continuous maps (mor-phisms) between topological spaces in topology (the as-sociated 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 categoryof homotopies between pointed topological spaces.If one axiomatizes relations instead of functions, one ob-tains the theory of allegories.

3.2.2 Functors

Main article: FunctorSee also: Adjoint functors Motivation

A category is itself a type of mathematical structure, sowe can look for processes which preserve this structurein some sense; such a process is called a functor.Diagram chasing is a visual method of arguing with ab-stract arrows joined in diagrams. Functors are rep-resented by arrows between categories, subject to spe-cic dening commutativity conditions. Functors can de-

ne (construct) categorical diagrams and sequences (viz.Mitchell, 1965). A functor associates to every object ofone category an object of another category, and to everymorphism in the rst category a morphism in the second.In fact, what we have done is dene a category of cate-gories and functors the objects are categories, and themorphisms (between categories) are functors.By studying categories and functors, we are not juststudying a class of mathematical structures and the mor-phisms between them; we are studying the relationshipsbetween various classes of mathematical structures. Thisis a fundamental idea, which rst surfaced in algebraictopology. Dicult topological questions can be trans-lated into algebraic questions which are often easier tosolve. Basic constructions, such as the fundamental groupor the fundamental groupoid of a topological space, canbe expressed as functors to the category of groupoids inthis way, and the concept is pervasive in algebra and itsapplications.

3.2.3 Natural transformationsMain article: Natural transformation

Abstracting yet again, some diagrammatic and/or sequen-tial constructions are often naturally related a vaguenotion, at rst sight. This leads to the clarifying conceptof natural transformation, a way to map one functor toanother. Many important constructions in mathematicscan be studied in this context. Naturality is a princi-ple, like general covariance in physics, that cuts deeperthan is initially apparent. An arrow between two functorsis a natural transformation when it is subject to certainnaturality or commutativity conditions.Functors and natural transformations ('naturality') are thekey concepts in category theory.[3]

3.3 Categories, objects, and mor-phisms

Main articles: Category (mathematics) and Morphism

3.3.1 CategoriesA category C consists of the following three mathematicalentities:

A class ob(C), whose elements are called objects; A class hom(C), whose elements are called

morphisms or maps or arrows. Each morphism fhas a source object a and target object b.

18 CHAPTER 3. CATEGORY THEORY

The expression f : a b, would be verbally statedas "f is a morphism from a to b".The expression hom(a, b) alternatively ex-pressed as homC(a, b), mor(a, b), or C(a, b) denotes the hom-class of all morphisms from a to b.

A binary operation , called composition of mor-phisms, such that for any three objects a, b, and c,we have hom(b, c) hom(a, b) hom(a, c). Thecomposition of f : a b and g : b c is written asg f or gf,[4] governed by two axioms:

Associativity: If f : a b, g : b c and h : c d then h (g f) = (h g) f, and

Identity: For every object x, there exists a mor-phism 1x : x x called the identity morphismfor x, such that for every morphism f : a b,we have 1b f = f = f 1a.

From the axioms, it can be provedthat there is exactly one identitymorphism for every object. Someauthors deviate from the denitionjust given by identifying each ob-ject with its identity morphism.

3.3.2 Morphisms

Relations among morphisms (such as fg = h) are of-ten depicted using commutative diagrams, with points(corners) representing objects and arrows representingmorphisms.Morphisms can have any of the following properties. Amorphism f : a b is a:

monomorphism (or monic) if f g1 = f g2 impliesg1 = 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 ofendomorphisms of a.

automorphism if f is both an endomorphism and anisomorphism. aut(a) denotes the class of automor-phisms of a.

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

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

Every retraction is an epimorphism, and every section is amonomorphism. 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.

3.4 FunctorsMain article: Functor

Functors are structure-preserving maps between cate-gories. They can be thought of as morphisms in the cate-gory of all (small) categories.A (covariant) functor F from a category C to a categoryD, 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 covariantfunctor, except that it turns morphisms around (re-verses all the arrows). More specically, every mor-phism f : x y in C must be assigned to a morphismF(f) : F(y) F(x) in D. In other words, a contravari-ant functor acts as a covariant functor from the oppositecategory Cop to D.

3.5 Natural transformationsMain article: Natural transformation

A natural transformation is a relation between two func-tors. Functors often describe natural constructions andnatural transformations then describe natural homomor-phisms between two such constructions. Sometimes twoquite dierent constructions yield the same result; thisis expressed by a natural isomorphism between the twofunctors.If F andG are (covariant) functors between the categoriesC and D, then a natural transformation from F to Gassociates to every object X in C a morphism X : F(X)

3.6. OTHER CONCEPTS 19

G(X) in D such that for every morphism f : X Y inC, we have Y F(f) = G(f) X; this means that thefollowing diagram is commutative:

Commutative diagram dening natural transformations

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

3.6 Other concepts

3.6.1 Universal constructions, limits, andcolimits

Main articles: Universal property and Limit (categorytheory)

Using the language of category theory, many areas ofmathematical study can be categorized. Categories in-clude sets, groups and topologies.Each category is distinguished by properties that all itsobjects have in common, such as the empty set or theproduct of two topologies, yet in the denition of a cat-egory, objects are considered to be atomic, i.e., we donot know whether an object A is a set, a topology, or anyother abstract concept. Hence, the challenge is to denespecial objects without referring to the internal structureof those objects. To dene the empty set without refer-ring to elements, or the product topology without refer-ring to open sets, one can characterize these objects interms of their relations to other objects, as given by themorphisms of the respective categories. Thus, the taskis to nd universal properties that uniquely determine theobjects of interest.Indeed, it turns out that numerous important construc-tions can be described in a purely categorical way. Thecentral concept which is needed for this purpose is calledcategorical limit, and can be dualized to yield the notionof a colimit.

3.6.2 Equivalent categoriesMain articles: Equivalence of categories andIsomorphism of categories

It is a natural question to ask: under which conditions cantwo categories be considered to be essentially the same,in the sense that theorems about one category can readilybe transformed into theorems about the other category?The major tool one employs to describe such a situationis called equivalence of categories, which is given by ap-propriate functors between two categories. Categoricalequivalence has found numerous applications in mathe-matics.

3.6.3 Further concepts and resultsThe denitions of categories and functors provide onlythe very basics of categorical algebra; additional impor-tant topics are listed below. Although there are stronginterrelations between all of these topics, the given ordercan be considered as a guideline for further reading.

The functor category DC has as objects the functorsfrom C to D and as morphisms the natural transfor-mations of such functors. The Yoneda lemma is oneof the most famous basic results of category theory;it describes representable functors in functor cate-gories.

Duality: Every statement, theorem, or denition incategory theory has a dual which is essentially ob-tained by reversing all the arrows. If one statementis true in a category C then its dual will be true in thedual category Cop. This duality, which is transparentat the level of category theory, is often obscured inapplications and can lead to surprising relationships.

Adjoint functors: A functor can be left (or right)adjoint to another functor that maps in the oppo-site direction. Such a pair of adjoint functors typi-cally arises from a construction dened by a univer-sal property; this can be seen as a more abstract andpowerful view on universal properties.

3.6.4 Higher-dimensional categoriesMany of the above concepts, especially equivalence ofcategories, adjoint functor pairs, and functor categories,can be situated into the context of higher-dimensionalcategories. Briey, if we consider a morphism betweentwo objects as a process taking us from one objectto another, then higher-dimensional categories allowus to protably generalize this by considering higher-dimensional processes.For example, a (strict) 2-category is a category togetherwith morphisms between morphisms, i.e., processes

20 CHAPTER 3. CATEGORY THEORY

which allow us to transform one morphism into another.We can then compose these bimorphisms both hor-izontally and vertically, and we require a 2-dimensionalexchange law to hold, relating the two compositionlaws. In this context, the standard example is Cat, the2-category of all (small) categories, and in this example,bimorphisms of morphisms are simply natural transfor-mations of morphisms in the usual sense. Another basicexample is to consider a 2-category with a single object;these are essentially monoidal categories. Bicategoriesare a weaker notion of 2-dimensional categories in whichthe composition of morphisms is not strictly associative,but only associative up to an isomorphism.This process can be extended for all natural numbers n,and these are called n-categories. There is even a notionof -category corresponding to the ordinal number .Higher-dimensional categories are part of the broadermathematical eld of higher-dimensional algebra, a con-cept introduced by Ronald Brown. For a conversationalintroduction to these ideas, see John Baez, 'A Tale of n-categories (1996).

3.7 Historical notesIn 194245, Samuel Eilenberg and Saunders Mac Laneintroduced categories, functors, and natural transfor-mations as part of their work in topology, especiallyalgebraic topology. Their work was an important partof the transition from intuitive and geometric homologyto axiomatic homology theory. Eilenberg and Mac Lanelater wrote that their goal was to understand natural trans-formations; in order to do that, functors had to be dened,which required categories.Stanislaw Ulam, and some writing on his behalf, haveclaimed that related ideas were current in the late 1930sin Poland. Eilenberg was Polish, and studied mathematicsin Poland in the 1930s. Category theory is also, in somesense, a continuation of the work of Emmy Noether (oneof Mac Lanes teachers) in formalizing abstract processes;Noether realized that in order to understand a type ofmathematical structure, one needs to understand the pro-cesses preserving that structure. In order to achieve thisunderstanding, Eilenberg and Mac Lane proposed an ax-iomatic formalization of the relation between structuresand the processes preserving them.The subsequent development of category theory waspowered rst by the computational needs of homologicalalgebra, and later by the axiomatic needs of algebraic ge-ometry, the eld most resistant to being grounded in ei-ther axiomatic set theory or the Russell-Whitehead viewof united foundations. General category theory, an exten-sion of universal algebra having many new features allow-ing for semantic exibility and higher-order logic, camelater; it is now applied throughout mathematics.Certain categories called topoi (singular topos) can even

serve as an alternative to axiomatic set theory as a foun-dation of mathematics. A topos can also be consideredas a specic type of category with two additional toposaxioms. These foundational applications of category the-ory have been worked out in fair detail as a basis for, andjustication of, constructive mathematics. Topos theoryis a form of abstract sheaf theory, with geometric origins,and leads to ideas such as pointless topology.Categorical logic is now a well-dened eld based ontype theory for intuitionistic logics, with applications infunctional programming and domain theory, where acartesian closed category is taken as a non-syntactic de-scription of a lambda calculus. At the very least, categorytheoretic language claries what exactly these related ar-eas have in common (in some abstract sense).Category theory has been applied in other elds aswell. For example, John Baez has shown a link betweenFeynman diagrams in Physics and monoidal categories.[6]Another application of category theory, more speci-cally: topos theory, has been made in mathematical mu-sic theory, see for example the book The Topos of Music,Geometric Logic of Concepts, Theory, and Performanceby Guerino Mazzola.More recent eorts to introduce undergraduates to cat-egories as a foundation for mathematics include thoseof William Lawvere and Rosebrugh (2003) and Law-vere and Stephen Schanuel (1997) and Mirroslav Yotov(2012).

3.8 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 mathemat-

ics

3.9 Notes[1] Awodey 2006

[2] Geroch, Robert (1985). Mathematical physics ([Repr.]ed.). Chicago: University of Chicago Press. p. 7. ISBN0-226-28862-5. Retrieved 20 August 2012. Note thattheorem 3 is actually easier for categories in general than

3.10. REFERENCES 21

it is for the special case of sets. This phenomenon is byno means rare.

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

[4] Some authors compose in the opposite order, writing fg orf g for g f. Computer scientists using category theoryvery commonly write f ; g for g f

[5] Note that a morphism that is both epic and monic is notnecessarily an isomorphism! An elementary counterex-ample: in the category consisting of two objects A and B,the identity morphisms, and a single morphism f from Ato B, f is both epic and monic but is not an isomorphism.

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

3.10 References Admek, Ji; Herrlich, Horst; Strecker, George E.

(1990). Abstract and concrete categories. John Wi-ley & Sons. ISBN 0-471-60922-6.

Awodey, Steve (2006). Category Theory. OxfordLogic Guides 49. Oxford University Press. ISBN978-0-19-151382-4.

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

Barr, Michael; Wells, Charles (2005), Toposes,Triples and Theories, Reprints in Theory and Ap-plications of Categories 12 (revised ed.), MR2178101.

Borceux, Francis (1994). Handbook of categoricalalgebra. Encyclopedia of Mathematics and its Ap-plications 50-52. Cambridge University Press.

Bucur, Ion; Deleanu, Aristide (1968). Introductionto the theory of categories and functors. Wiley.

Freyd, Peter J. (1964). Abe