Intro to Category Theory 1 / 17 0 0 f - 0 Introduction to A B 0 a h b Abstract Nonsense - - f g0 A - B Christoph Rauch g ? ? k0 FAU Erlangen-Nürnberg C0 - D0 h Theorieseminar SS2012 c d - - July 11, 2012 ? k ? C - D . Origins. foundations laid by Eilenberg, Mac Lane Grothendieck, Lawvere . Abstract. Language convenient notation huge amount of definitions only a handful of results concerning the theory itself . Categories. in Computer Science Haskell automata theory, program semantics, logics . Intro to Category Theory | Motivation 2 / 17 Category Theory as a Mindset . Abstract. Language convenient notation huge amount of definitions only a handful of results concerning the theory itself . Categories. in Computer Science Haskell automata theory, program semantics, logics . Intro to Category Theory | Motivation 2 / 17 Category Theory as a Mindset . Origins. foundations laid by Eilenberg, Mac Lane Grothendieck, Lawvere . Categories. in Computer Science Haskell automata theory, program semantics, logics . Intro to Category Theory | Motivation 2 / 17 Category Theory as a Mindset . Origins. foundations laid by Eilenberg, Mac Lane Grothendieck, Lawvere . Abstract. Language convenient notation huge amount of definitions only a handful of results concerning the theory itself . Intro to Category Theory | Motivation 2 / 17 Category Theory as a Mindset . Origins. foundations laid by Eilenberg, Mac Lane Grothendieck, Lawvere . Abstract. Language convenient notation huge amount of definitions only a handful of results concerning the theory itself . Categories. in Computer Science Haskell automata theory, program semantics, logics . where f : A ! B and g : B ! C ) 9 g ◦ f : A ! C (composition) for each object A there is an arrow idA : A ! A (identity) and h ◦ (g ◦ f) = (h ◦ g) ◦ f for compatible morphisms (associativity) f ◦ idA = f = idB ◦f for all f : A ! B (unit) Intro to Category Theory | Definitions and Concepts 3 / 17 Categories . A. Category C consists of a class Ob(C) of objects a class Hom(C) of morphisms, where each morphism has a domain and codomain in Ob(C), and we write f : A ! B for a morphism f with domain A and codomain B . and h ◦ (g ◦ f) = (h ◦ g) ◦ f for compatible morphisms (associativity) f ◦ idA = f = idB ◦f for all f : A ! B (unit) Intro to Category Theory | Definitions and Concepts 3 / 17 Categories . A. Category C consists of a class Ob(C) of objects a class Hom(C) of morphisms, where each morphism has a domain and codomain in Ob(C), and we write f : A ! B for a morphism f with domain A and codomain B where f : A ! B and g : B ! C ) 9 g ◦ f : A ! C (composition) for each object A there is an arrow idA : A ! A (identity) . Intro to Category Theory | Definitions and Concepts 3 / 17 Categories . A. Category C consists of a class Ob(C) of objects a class Hom(C) of morphisms, where each morphism has a domain and codomain in Ob(C), and we write f : A ! B for a morphism f with domain A and codomain B where f : A ! B and g : B ! C ) 9 g ◦ f : A ! C (composition) for each object A there is an arrow idA : A ! A (identity) and h ◦ (g ◦ f) = (h ◦ g) ◦ f for compatible morphisms (associativity) f ◦ id = f = id ◦f for all f : A ! B (unit) . A B . Haskell:. Haskell types and Haskell functions in theory form a category Hask, but note that the actual implementations .have minor issues (right identity, limits) . Posets. as Categories Each poset (S; ≤) defines a category C(S; ≤) with the elements of S as .objects. There is a morphism between p; q 2 S iff p ≤ q. Games. (Conway) Mike Shulman at the n-Category Café stated that combinatorial games can .be seen as the objects in a category with winning strategies as morphisms. Intro to Category Theory | Definitions and Concepts 4 / 17 Example Categories . Sets. The category Set consists of sets as objects, total functions as morphisms .and function composition. Haskell:. Haskell types and Haskell functions in theory form a category Hask, but note that the actual implementations .have minor issues (right identity, limits) . Games. (Conway) Mike Shulman at the n-Category Café stated that combinatorial games can .be seen as the objects in a category with winning strategies as morphisms. Intro to Category Theory | Definitions and Concepts 4 / 17 Example Categories . Sets. The category Set consists of sets as objects, total functions as morphisms .and function composition. Posets. as Categories Each poset (S; ≤) defines a category C(S; ≤) with the elements of S as .objects. There is a morphism between p; q 2 S iff p ≤ q. Haskell:. Haskell types and Haskell functions in theory form a category Hask, but note that the actual implementations .have minor issues (right identity, limits) . Intro to Category Theory | Definitions and Concepts 4 / 17 Example Categories . Sets. The category Set consists of sets as objects, total functions as morphisms .and function composition. Posets. as Categories Each poset (S; ≤) defines a category C(S; ≤) with the elements of S as .objects. There is a morphism between p; q 2 S iff p ≤ q. Games. (Conway) Mike Shulman at the n-Category Café stated that combinatorial games can .be seen as the objects in a category with winning strategies as morphisms. Intro to Category Theory | Definitions and Concepts 4 / 17 Example Categories . Sets. The category Set consists of sets as objects, total functions as morphisms .and function composition. Haskell:. Posets. as Categories Haskell types and Haskell functions in theory form a Each posetcategory(S; ≤)Haskdefines, but a category note thatC the(S; actual≤) with implementations the elements of S as 2 ≤ .objects..have There minor is a morphism issues (right between identity,p; qlimits)S iff p q. Games. (Conway) Mike Shulman at the n-Category Café stated that combinatorial games can .be seen as the objects in a category with winning strategies as morphisms. Mac. Lane: .Show that every diagram commutes. Usage. replacement for equations in category theory “pasting together” of diagrams . Intro to Category Theory | Definitions and Concepts 5 / 17 Commutative Diagrams . Principle. h - A B This diagram is said to commute if the equation f = g ◦ h f g - ? holds. C . Mac. Lane: .Show that every diagram commutes. Intro to Category Theory | Definitions and Concepts 5 / 17 Commutative Diagrams . Principle. h - A B This diagram is said to commute if the equation f = g ◦ h f g - ? holds. C . Usage. replacement for equations in category theory “pasting together” of diagrams . Intro to Category Theory | Definitions and Concepts 5 / 17 Commutative Diagrams . Principle. h - A B This diagram is said to commute if the equation f = g ◦ h f g . Mac. Lane:- ? holds. .Show thatC every diagram commutes. Usage. replacement for equations in category theory “pasting together” of diagrams . Haskell:. class Functor f where fmap :: (a ! b) ! (f a ! f b) .Type constructors are endofunctors! . Category. of Categories .The category Cat has small categories as objects and functors as arrows. Intro to Category Theory | Definitions and Concepts 6 / 17 Functors . A. Functor F between categories C and D is a a pair of maps (F0 : Ob(C) ! Ob(D);F1 : Hom(C) ! Hom(D)) where if f : A ! B in C, then F (f): F (A) ! F (B) in D if f : B ! C and g : A ! B in C, then F (f) ◦ F (g) = F (f ◦ g) For all A in C, idF (A) = F (idA) a C . We let F (A) = F0(A) for objects A and F (f) = F1(f) for arrows f of . Haskell:. class Functor f where fmap :: (a ! b) ! (f a ! f b) .Type constructors are endofunctors! . Intro to Category Theory | Definitions and Concepts 6 / 17 Functors . A. Functor F between categories C and D is a a pair of maps (F0 : Ob(C) ! Ob(D);F1 : Hom(C) ! Hom(D)) where if f : A ! B in C, then F (f): F (A) ! F (B) in D if f : B ! C and g : A ! B in C, then F (f) ◦ F (g) = F (f ◦ g) For all A in C, idF (A) = F (idA) a C . We let F (A) = F0(A) for objects A and F (f) = F1(f) for arrows f of . Category. of Categories .The category Cat has small categories as objects and functors as arrows. Intro to Category Theory | Definitions and Concepts 6 / 17 Functors . A. Functor F between categories C and D is a a pair of maps (F0 : Ob(C) ! Ob(D);F1 : Hom(C) ! Hom(D)) where if f. Haskell:. : A ! B in C, then F (f): F (A) ! F (B) in D ! ! C ◦ ◦ if f :classB C Functorand g : Af whereB in , then F (f) F (g) = F (f g) For all Afmapin C, ::idF ( (aA)! = Fb)(idA!) (f a ! f b) a Type constructors are endofunctors! C . We let. F (A) = F0(A) for objects A and F (f) = F1(f) for arrows f of . Category. of Categories .The category Cat has small categories as objects and functors as arrows. such that. Haskell:. maybeToList map even $ maybeToListF (f) $ Just 5 maybeToList $ fmapF (A) even- F$ (BJust) 5 . ηA ηB ? ? G(f) G(A) - G(B) commutes for any A; B 2 Ob(C) and morphisms f : A ! B. Intro to Category Theory | Definitions and Concepts 7 / 17 Natural Transformations . For. categories C, D and functors F; G : C!D, a natural transformation η : F ! G is defined camponentwise on the objects of C: ηA : F (A) ! G(A) 8A 2 Ob(C) . Haskell:. maybeToList map even $ maybeToList $ Just 5 maybeToList $ fmap even $ Just 5 . Intro to Category Theory | Definitions and Concepts 7 / 17 Natural Transformations . For.