A brief Introduction to Category Theory

Transcription

1 A brief Introduction to Category Theory Dirk Hofmann CIDMA, Department of Mathematics, University of Aveiro, Aveiro, Portugal Office: , October 9, 2017

2 Motivation I John Hughes (1989). Why functional programming matters. In: The Computer Journal 32.(2), pp Modular design is the key to successful programming... The ways in which one can divide up the original problem depend directly on the ways in which one can glue solutions together. Therefore, to increase ones ability to modularise a problem conceptually, one must provide new kinds of glue in the programming language.... Now let us return to functional programming. We shall argue in the remainder of this paper that functional languages provide two new, very important kinds of glue.

3 Motivation I Whitehead, Alfred North Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specifications of definite particular things.

4 Motivation I Whitehead, Alfred North Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specifications of definite particular things. Saunders MacLane The basic insight is that a mathematical structure is a scientific structure but one which has many different empirical realizations. Mathematics provides common overreaching forms, each of which can and does serve to describe different aspects of the external world. Thus mathematics is that part of science which applies in more than one empirical context. Saunders MacLane (1997). Despite physicist, proof is essential in mathematics. In: Synthese 111.(2), pp

5 Motivation II A seemingly paradoxical observation... an equation is only interesting or useful to the extent that the two sides are different! John Baez and James Dolan (2001). From finite sets to Feynman diagrams. In: Mathematics Unlimited and Beyond. Springer, Berlin, pp

6 Motivation II A seemingly paradoxical observation... an equation is only interesting or useful to the extent that the two sides are different! John Baez and James Dolan (2001). From finite sets to Feynman diagrams. In: Mathematics Unlimited and Beyond. Springer, Berlin, pp Just compare: Numbers: 3 = 3 vs. e iω = cos(ω) + i sin(ω).

7 Motivation II A seemingly paradoxical observation... an equation is only interesting or useful to the extent that the two sides are different! John Baez and James Dolan (2001). From finite sets to Feynman diagrams. In: Mathematics Unlimited and Beyond. Springer, Berlin, pp Just compare: Numbers: 3 = 3 vs. e iω = cos(ω) + i sin(ω). Spaces: V V vs. V R n (for dim V = n).

8 Motivation II A seemingly paradoxical observation... an equation is only interesting or useful to the extent that the two sides are different! John Baez and James Dolan (2001). From finite sets to Feynman diagrams. In: Mathematics Unlimited and Beyond. Springer, Berlin, pp Just compare: Numbers: 3 = 3 vs. e iω = cos(ω) + i sin(ω). Spaces: V V vs. V R n (for dim V = n). More general: linear maps = matrices.

9 A comercial break Vladimir Voevodsky (Fields Medal in 2002) At the heart of 20th century mathematics lies one particular notion, and that is the notion of a category.

10 A comercial break Vladimir Voevodsky (Fields Medal in 2002) At the heart of 20th century mathematics lies one particular notion, and that is the notion of a category. Jean Bénabou Analogies are useful in mathematics for generalising a class of well-known examples to a wider class of equally or even more useful structures. Category Theory is particularly well suited for this purpose which is no wonder as it has been developed for precisely this purpose

11 A comercial break Vladimir Voevodsky (Fields Medal in 2002) At the heart of 20th century mathematics lies one particular notion, and that is the notion of a category. Jean Bénabou Analogies are useful in mathematics for generalising a class of well-known examples to a wider class of equally or even more useful structures. Category Theory is particularly well suited for this purpose which is no wonder as it has been developed for precisely this purpose Bill Lawvere The kinds of structures which actually arise in the practice of geometry and analysis are far from being arbitrary..., as concentrated in the thesis that fundamental structures are themselves categories.

12 Category Theory Sammy Eilenberg ( ) and Saunders MacLane ( ) Started in the 1940 s in their work about algebraic topology.

13 Category Theory Sammy Eilenberg ( ) and Saunders MacLane ( ) Started in the 1940 s in their work about algebraic topology. Is by now present in (almost) all areas of mathematics and also extensively used in physics and in computer science.

14 Bibliography Saunders MacLane (1998). Categories for the working mathematician. 2nd ed. New York: Springer-Verlag, pp. ix Graduate Texts in Mathematics, Vol. 5. Tom Leinster (2014a). Basic Category Theory. Cambridge University Press. 190 pp. Tom Leinster (2014b). Rethinking set theory. In: American Mathematical Monthly 121.(5), pp F. William Lawvere and Robert Rosebrugh (2003). Sets for mathematics. English. Cambridge: Cambridge University Press, pp. xi F. William Lawvere and Stephen H. Schanuel (2009). Conceptual mathematics. A first introduction to categories. English. 2nd ed. Cambridge University Press, pp. xii Andrea Asperti and Giuseppe Longo (1991). Categories, types, and structures. Foundations of Computing Series. Cambridge, MA: MIT Press, pp. xii+306. An introduction to category theory for the working computer scientist.

15 So what is it about?

16 So what is it about? Definition A category X consists of

17 So what is it about? Definition A category X consists of a collection of objects X, Y,..., Think of vector spaces, ordered sets, topological spaces,...

18 So what is it about? Definition A category X consists of a collection of objects X, Y,..., arrows (morphisms) f : X Y between objects, Think of vector spaces, ordered sets, topological spaces,... linear maps, monotone maps, continuous maps,...

19 So what is it about? Definition A category X consists of a collection of objects X, Y,..., arrows (morphisms) f : X Y between objects, arrows can be composed (associativity) X f g f Y g Z Think of vector spaces, ordered sets, topological spaces,... linear maps, monotone maps, continuous maps,... the composite of linear maps is linear,...

20 So what is it about? Definition A category X consists of a collection of objects X, Y,..., arrows (morphisms) f : X Y between objects, arrows can be composed (associativity) X f g f Y g Z for every object there is an identity arrow 1 X : X X. Think of vector spaces, ordered sets, topological spaces,... linear maps, monotone maps, continuous maps,... the composite of linear maps is linear,... The identity map is linear,....

21 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin,

22 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin, Ban,

23 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin, Ban, Met,

24 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin, Ban, Met, Met,...

25 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin, Ban, Met, Met,..., Rel, Mat...

26 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin, Ban, Met, Met,..., Rel, Mat... An abstract category...

27 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin, Ban, Met, Met,..., Rel, Mat... An abstract category... Definition For every category X, there is the dual category X op with the same objects but all arrows point in the opposite direction.

28 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin, Ban, Met, Met,..., Rel, Mat... An abstract category... Definition For every category X, there is the dual category X op with the same objects but all arrows point in the opposite direction. Examples Top op, Grp op, Vec op fin,..., Relop, Mat op...

29 So what is it about? Every field of mathematics defines (at least) one category Top, Grp, Vec fin, Ban, Met, Met,..., Rel, Mat... An abstract category and its dual Definition For every category X, there is the dual category X op with the same objects but all arrows point in the opposite direction. Examples Top op, Grp op, Vec op fin,..., Relop, Mat op...

30 Some typical categorical notions

31 Some typical categorical notions Isomorphism An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y

32 Some typical categorical notions Isomorphism An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y Product in X X π 1 Z pairing X Y Y π 2

33 Some typical categorical notions Isomorphism An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y Product in X Sum in X X X π 1 i 1 Z pairing X Y X + Y if then else Z π 2 i 2 Y Y

34 Some typical categorical notions Isomorphism An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y Product in X Sum in X = product in X op X X π 1 i 1 Z pairing X Y X + Y if then else Z π 2 i 2 Y Y

35 When are two categories equal? Equivalence X Y of categories

36 When are two categories equal? Equivalence X Y of categories A functor F : X Y: (X 1 f X2 ) (FX 1 Ff FX 2 ) so that F(g f ) = Fg Ff and F1 X = 1 FX.

37 When are two categories equal? Equivalence X Y of categories A functor F : X Y: (X 1 f X2 ) (FX 1 Ff FX 2 ) so that F(g f ) = Fg Ff and F1 X = 1 FX. A functor G : Y X.

38 When are two categories equal? Equivalence X Y of categories A functor F : X Y: (X 1 f X2 ) (FX 1 Ff FX 2 ) so that F(g f ) = Fg Ff and F1 X = 1 FX. A functor G : Y X. Natural isomorphisms η X : X GFX and ε Y : FGY Y...

39 When are two categories equal? Equivalence X Y of categories A functor F : X Y: (X 1 f X2 ) (FX 1 Ff FX 2 ) so that F(g f ) = Fg Ff and F1 X = 1 FX. A functor G : Y X. Natural isomorphisms η X : X GFX and ε Y : FGY Y... Adjunction As above but the arrows η X : X GFX and ε Y : FGY Y need not be isomorphisms...

40 When are two categories equal? Equivalence X Y of categories A functor F : X Y: (X 1 f X2 ) (FX 1 Ff FX 2 ) so that F(g f ) = Fg Ff and F1 X = 1 FX. A functor G : Y X. Natural isomorphisms η X : X GFX and ε Y : FGY Y... Adjunction As above but the arrows η X : X GFX and ε Y : FGY Y need not be isomorphisms... Definition Let F, G : X Y be functors. An natural transformation α is a family (α X : FX GX) X which commutes with arrows in X.

41 An elementary example Linear algebra via matrices Let X = Vec fin be the category of finite dimensional vector spaces (over R) and linear maps.

42 An elementary example Linear algebra via matrices Let X = Vec fin be the category of finite dimensional vector spaces (over R) and linear maps. Let Y = Mat be the category of natural numbers and matrices. (Here A: n m means A is a matrix of type m n.)

43 An elementary example Linear algebra via matrices Let X = Vec fin be the category of finite dimensional vector spaces (over R) and linear maps. Let Y = Mat be the category of natural numbers and matrices. (Here A: n m means A is a matrix of type m n.) Theorem. Vec fin Mat.

44 An elementary example Linear algebra via matrices Let X = Vec fin be the category of finite dimensional vector spaces (over R) and linear maps. Let Y = Mat be the category of natural numbers and matrices. (Here A: n m means A is a matrix of type m n.) Theorem. Vec fin Mat. F : Mat Vec fin, (A: n m) (f A :R n R m ).

45 An elementary example Linear algebra via matrices Let X = Vec fin be the category of finite dimensional vector spaces (over R) and linear maps. Let Y = Mat be the category of natural numbers and matrices. (Here A: n m means A is a matrix of type m n.) Theorem. Vec fin Mat. F : Mat Vec fin, (A: n m) (f A :R n R m ). G : Vec fin Mat, (f : V W ) (matrix of f: dim V dim W ). Requires choosing a base for every space.

46 An elementary example Linear algebra via matrices Let X = Vec fin be the category of finite dimensional vector spaces (over R) and linear maps. Let Y = Mat be the category of natural numbers and matrices. (Here A: n m means A is a matrix of type m n.) Theorem. Vec fin Mat. F : Mat Vec fin, (A: n m) (f A :R n R m ). G : Vec fin Mat, (f : V W ) (matrix of f: dim V dim W ). Requires choosing a base for every space. n = dimr n and V R n

47 An elementary example Linear algebra via matrices Let X = Vec fin be the category of finite dimensional vector spaces (over R) and linear maps. Let Y = Mat be the category of natural numbers and matrices. (Here A: n m means A is a matrix of type m n.) Theorem. Vec fin Mat. F : Mat Vec fin, (A: n m) (f A :R n R m ). G : Vec fin Mat, (f : V W ) (matrix of f: dim V dim W ). Requires choosing a base for every space. n = dimr n and V R n Theorem. Mat Mat op. Here: (A: n m) (A T : m n), (B A) T = A T B T ; I T = I.

48 An elementary example Linear algebra via matrices Let X = Vec fin be the category of finite dimensional vector spaces (over R) and linear maps. Let Y = Mat be the category of natural numbers and matrices. (Here A: n m means A is a matrix of type m n.) Theorem. Vec fin Mat. F : Mat Vec fin, (A: n m) (f A :R n R m ). G : Vec fin Mat, (f : V W ) (matrix of f: dim V dim W ). Requires choosing a base for every space. n = dimr n and V R n Theorem. Mat Mat op. Here: (A: n m) (A T : m n), (B A) T = A T B T ; I T = I. Corollary. Vec fin Vec op fin.

49 A brief Introduction to Category Theory Aula 2 Dirk Hofmann CIDMA, Department of Mathematics, University of Aveiro, Aveiro, Portugal Office: , dirk@ua.pt, October 16, 2017

50 Recall from last week 1. A category X consists of a collection of objects X, Y,... ; for each pair of objects, a set of arrows (morphisms) f : X Y (denoted as X(X, Y ) or hom(x, Y )); arrows can be composed (associativity) and for every object there is an identity arrow 1 X : X Y.

51 Recall from last week 1. A category X consists of a collection of objects X, Y,... ; for each pair of objects, a set of arrows (morphisms) f : X Y (denoted as X(X, Y ) or hom(x, Y )); arrows can be composed (associativity) and for every object there is an identity arrow 1 X : X Y. 2. For every category X, there is the dual category X op with the same objects but all arrows point in the opposite direction.

52 Recall from last week 1. A category X consists of a collection of objects X, Y,... ; for each pair of objects, a set of arrows (morphisms) f : X Y (denoted as X(X, Y ) or hom(x, Y )); arrows can be composed (associativity) and for every object there is an identity arrow 1 X : X Y. 2. For every category X, there is the dual category X op with the same objects but all arrows point in the opposite direction. 3. Functor F : X Y: (X 1 f X2 ) (FX 1 Ff FX 2 ) so that F(g f ) = Fg Ff and F1 X = 1 FX.

53 Recall from last week 1. A category X consists of a collection of objects X, Y,... ; for each pair of objects, a set of arrows (morphisms) f : X Y (denoted as X(X, Y ) or hom(x, Y )); arrows can be composed (associativity) and for every object there is an identity arrow 1 X : X Y. 2. For every category X, there is the dual category X op with the same objects but all arrows point in the opposite direction. 3. Functor F : X Y: (X 1 f X2 ) (FX 1 Ff FX 2 ) so that F(g f ) = Fg Ff and F1 X = 1 FX. 4. Let F, G : X Y be functors. An natural transformation α is a family (α X : FX GX) X of Y-arrows which commutes with arrows in X.

54 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X:

55 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X.

56 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y.

57 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition.

58 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition. Reflexivity identities.

59 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition. Reflexivity identities. Then: Functor means monotone map.

60 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition. Reflexivity identities. Then: Functor means monotone map. Monoids Every monoid (M,, e) can be viewed as a category X:

61 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition. Reflexivity identities. Then: Functor means monotone map. Monoids Every monoid (M,, e) can be viewed as a category X: X has one object (denoted here as ).

62 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition. Reflexivity identities. Then: Functor means monotone map. Monoids Every monoid (M,, e) can be viewed as a category X: X has one object (denoted here as ). X(, ) = M.

63 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition. Reflexivity identities. Then: Functor means monotone map. Monoids Every monoid (M,, e) can be viewed as a category X: X has one object (denoted here as ). X(, ) = M. Composition of X = multiplication of the monoid.

64 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition. Reflexivity identities. Then: Functor means monotone map. Monoids Every monoid (M,, e) can be viewed as a category X: X has one object (denoted here as ). X(, ) = M. Composition of X = multiplication of the monoid. identity e :.

65 Two elementary examples Ordered sets Every ordered set (X, ) can be viewed as a category X: Objects of X = elements of X. There is an arrow x y whenever x y. Transitivity composition. Reflexivity identities. Then: Functor means monotone map. Monoids Every monoid (M,, e) can be viewed as a category X: X has one object (denoted here as ). X(, ) = M. Composition of X = multiplication of the monoid. identity e :. Then: Functor means monoid homomorphism.

66 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y.

67 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y

68 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y f monomorphism: f h = f k = h = k.

69 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y f monomorphism: f h = f k = h = k. f split monomorphism: exists g : Y X with g f = 1 X.

70 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y f monomorphism: f h = f k = h = k. f split monomorphism: exists g : Y X with g f = 1 X. f epimorphism: h f = k f = h = k.

71 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y f monomorphism: f h = f k = h = k. f split monomorphism: exists g : Y X with g f = 1 X. f epimorphism: h f = k f = h = k. f split epimorphism: exists g : Y X with f g = 1 Y.

72 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y f monomorphism: f h = f k = h = k. f split monomorphism: exists g : Y X with g f = 1 X. f epimorphism: h f = k f = h = k. f split epimorphism: exists g : Y X with f g = 1 Y. Some properties

73 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y f monomorphism: f h = f k = h = k. f split monomorphism: exists g : Y X with g f = 1 X. f epimorphism: h f = k f = h = k. f split epimorphism: exists g : Y X with f g = 1 Y. Some properties split mono = mono (and split epi = epi).

74 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y f monomorphism: f h = f k = h = k. f split monomorphism: exists g : Y X with g f = 1 X. f epimorphism: h f = k f = h = k. f split epimorphism: exists g : Y X with f g = 1 Y. Some properties split mono = mono (and split epi = epi). (split mono & epi) = iso = (split epi & mono).

75 Special morphisms Definition An arrow f : X Y in a category X is called an isomorphism whenever there is some arrow g : Y X with g f = 1 X and f g = 1 Y. More arrows f : X Y f monomorphism: f h = f k = h = k. f split monomorphism: exists g : Y X with g f = 1 X. f epimorphism: h f = k f = h = k. f split epimorphism: exists g : Y X with f g = 1 Y. Some properties split mono = mono (and split epi = epi). (split mono & epi) = iso = (split epi & mono). Every functor preserves split monos/split epis/isos.

76 Limits and colimits Some constructions in categories terminal object

77 Limits and colimits Some constructions in categories terminal object product

78 Limits and colimits Some constructions in categories terminal object product initial object sum

79 Limits and colimits Some constructions in categories terminal object product pullback initial object sum

80 Limits and colimits Some constructions in categories terminal object product pullback initial object sum pushout

81 Limits and colimits Some constructions in categories terminal object product pullback equaliser initial object sum pushout

82 Limits and colimits Some constructions in categories terminal object product pullback equaliser initial object sum pushout coequaliser

83 Limits and colimits Some constructions in categories terminal object product pullback equaliser limit initial object sum pushout coequaliser

84 Limits and colimits Some constructions in categories terminal object product pullback equaliser limit initial object sum pushout coequaliser colimit

85 Limits and colimits Some constructions in categories terminal object product pullback equaliser limit initial object sum pushout coequaliser colimit Definition A category X is called (co)complete whenever X has all (co)limits.

86 Limits and colimits Some constructions in categories terminal object product pullback equaliser limit initial object sum pushout coequaliser colimit Definition A category X is called (co)complete whenever X has all (co)limits. Some facts Limit = terminal object in some category (uniqueness!!).

87 Limits and colimits Some constructions in categories terminal object product pullback equaliser limit initial object sum pushout coequaliser colimit Definition A category X is called (co)complete whenever X has all (co)limits. Some facts Limit = terminal object in some category (uniqueness!!). X is complete X has products and equalisers.

88 Limits and colimits Some constructions in categories terminal object product pullback equaliser limit initial object sum pushout coequaliser colimit Definition A category X is called (co)complete whenever X has all (co)limits. Some facts Limit = terminal object in some category (uniqueness!!). X is complete X has products and equalisers. X is finitely complete X has a terminal object and pullbacks.

89 Limits and colimits Some constructions in categories terminal object product pullback equaliser limit initial object sum pushout coequaliser colimit Definition A category X is called (co)complete whenever X has all (co)limits. Some facts Limit = terminal object in some category (uniqueness!!). X is complete X has products and equalisers. X is finitely complete X has a terminal object and pullbacks. mono vs. pullback.

90 Representable functors Example For every category X, X(, ): X op X Set is a functor.

91 Representable functors Example For every category X, X(, ): X op X Set is a functor. Remark X(X, ) X(Y, ) = X Y.

92 Representable functors Example For every category X, X(, ): X op X Set is a functor. Remark X(X, ) X(Y, ) = X Y. Definition A functor F : X Set is called representable whenever F X(X, ).

93 Representable functors Example For every category X, X(, ): X op X Set is a functor. Remark X(X, ) X(Y, ) = X Y. Definition A functor F : X Set is called representable whenever F X(X, ). Examples For F : Ord Set: F Vec(1, ).

94 Representable functors Example For every category X, X(, ): X op X Set is a functor. Remark X(X, ) X(Y, ) = X Y. Definition A functor F : X Set is called representable whenever F X(X, ). Examples For F : Ord Set: F Vec(1, ). For F : Vec Set: F Vec(R, ).

95 Representable functors Example For every category X, X(, ): X op X Set is a functor. Remark X(X, ) X(Y, ) = X Y. Definition A functor F : X Set is called representable whenever F X(X, ). Examples For F : Ord Set: F Vec(1, ). For F : Vec Set: F Vec(R, ). Some facts

96 Representable functors Example For every category X, X(, ): X op X Set is a functor. Remark X(X, ) X(Y, ) = X Y. Definition A functor F : X Set is called representable whenever F X(X, ). Examples For F : Ord Set: F Vec(1, ). For F : Vec Set: F Vec(R, ). Some facts Representable functors preserve limits.

97 Representable functors Example For every category X, X(, ): X op X Set is a functor. Remark X(X, ) X(Y, ) = X Y. Definition A functor F : X Set is called representable whenever F X(X, ). Examples For F : Ord Set: F Vec(1, ). For F : Vec Set: F Vec(R, ). Some facts Representable functors preserve limits. Representable functors preserve monos.

98 Adjunction The slogan is Adjoint functors arise everywhere. a a Saunders MacLane (1998). Categories for the working mathematician. 2nd ed. New York: Springer-Verlag, pp. ix

99 Adjunction The slogan is Adjoint functors arise everywhere. a a Saunders MacLane (1998). Categories for the working mathematician. 2nd ed. New York: Springer-Verlag, pp. ix Definition Let G : B A and F : B A be functors. Then F is left adjoint to G, or G is right adjoint to F, written as F G, whenever B(FA, B) A(A, GB), (f f ) naturally in A and B. An adjunction between F and G is a choice of such a natural isomorphism.

100 Adjunction The slogan is Adjoint functors arise everywhere. a a Saunders MacLane (1998). Categories for the working mathematician. 2nd ed. New York: Springer-Verlag, pp. ix Definition Let G : B A and F : B A be functors. Then F is left adjoint to G, or G is right adjoint to F, written as F G, whenever B(FA, B) A(A, GB), (f f ) naturally in A and B. An adjunction between F and G is a choice of such a natural isomorphism. Remark The naturality of f f can be expressed as:

101 Adjunction The slogan is Adjoint functors arise everywhere. a a Saunders MacLane (1998). Categories for the working mathematician. 2nd ed. New York: Springer-Verlag, pp. ix Definition Let G : B A and F : B A be functors. Then F is left adjoint to G, or G is right adjoint to F, written as F G, whenever B(FA, B) A(A, GB), (f f ) naturally in A and B. An adjunction between F and G is a choice of such a natural isomorphism. Remark The naturality of f f can be expressed as: 1. For every h: A A: f Fh = f h.

102 Adjunction The slogan is Adjoint functors arise everywhere. a a Saunders MacLane (1998). Categories for the working mathematician. 2nd ed. New York: Springer-Verlag, pp. ix Definition Let G : B A and F : B A be functors. Then F is left adjoint to G, or G is right adjoint to F, written as F G, whenever B(FA, B) A(A, GB), (f f ) naturally in A and B. An adjunction between F and G is a choice of such a natural isomorphism. Remark The naturality of f f can be expressed as: 1. For every h: A A: f Fh = f h. 2. For every k : B B : k f = Gk f.

103 Adjunction via (co)units Theorem Let G : B A and F : B A be functors. There is a bijective correspondence between 1. Adjunctions F G. 2. Natural transformations η : 1 GF and ε: FG 1 so that F Fη FGF and G ε G GFG 1 F ε F 1 Gε G

104 Adjunction via (co)units Theorem Let G : B A and F : B A be functors. There is a bijective correspondence between 1. Adjunctions F G. 2. Natural transformations η : 1 GF and ε: FG 1 so that F Fη FGF and G ε G GFG 1 F ε F 1 Gε G Corollary F G and F G implies F F.

105 Adjunction via (co)units Theorem Let G : B A and F : B A be functors. There is a bijective correspondence between 1. Adjunctions F G. 2. Natural transformations η : 1 GF and ε: FG 1 so that F Fη FGF and G ε G GFG 1 F ε F 1 Gε G Corollary F G and F G implies F F. Remark A category C has limits of type I if and only if : C C I has a right adjoint.

106 Adjunctions via initial objects Theorem A functor G : B A is right adjoint if and only if the category (A G) has an initial object. Theorem (General Adjoint Functor Theorem) Let G : B A be a functor so that (A G) has a weak initial set. Then G is right adjoint if and only if G preserves limits.

107 Adjunctions via initial objects Theorem A functor G : B A is right adjoint if and only if the category (A G) has an initial object. Theorem (General Adjoint Functor Theorem) Let G : B A be a functor so that (A G) has a weak initial set. Then G is right adjoint if and only if G preserves limits. The proof is based on the following lemmas.

108 Adjunctions via initial objects Theorem A functor G : B A is right adjoint if and only if the category (A G) has an initial object. Theorem (General Adjoint Functor Theorem) Let G : B A be a functor so that (A G) has a weak initial set. Then G is right adjoint if and only if G preserves limits. The proof is based on the following lemmas. Lemma Let C be a complete category with a weak initial set S. Then C has an initial object.

109 Adjunctions via initial objects Theorem A functor G : B A is right adjoint if and only if the category (A G) has an initial object. Theorem (General Adjoint Functor Theorem) Let G : B A be a functor so that (A G) has a weak initial set. Then G is right adjoint if and only if G preserves limits. The proof is based on the following lemmas. Lemma Let C be a complete category with a weak initial set S. Then C has an initial object. Lemma Let G : B A be a limit-preserving functor and assume that B is complete. Then (A G) is complete.