Abstract Nonsense Introduction to 1 / 17 Intro to Category Theory - PowerPoint PPT Presentation

. . July 11, 2012 Theorieseminar SS2012 FAU Erlangen-Nürnberg Christoph Rauch Abstract Nonsense Introduction to 1 / 17 Intro to Category Theory . . . . f ′ ✲ B ′ A ′ h ′ a b ✲ ✲ f ✲ g ′ A B g ❄ ❄ k ′ ✲ D ′ C ′ h d c ✲ ✲ ❄ ❄ k ✲ C D

. Grothendieck , Lawvere Haskell . Categories in Computer Science .. only a handful of results concerning the theory itself huge amount of definitions convenient notation . Abstract Language .. foundations laid by Eilenberg , Mac Lane . . Origins .. Category Theory as a Mindset 2 / 17 Intro to Category Theory | Motivation . . . . automata theory, program semantics, logics

. Grothendieck , Lawvere Haskell . Categories in Computer Science .. only a handful of results concerning the theory itself huge amount of definitions convenient notation . Abstract Language .. foundations laid by Eilenberg , Mac Lane . . Origins .. Category Theory as a Mindset 2 / 17 Intro to Category Theory | Motivation . . . . automata theory, program semantics, logics

. Grothendieck , Lawvere Haskell . Categories in Computer Science .. only a handful of results concerning the theory itself huge amount of definitions convenient notation . Abstract Language .. foundations laid by Eilenberg , Mac Lane . . Origins .. Category Theory as a Mindset 2 / 17 Intro to Category Theory | Motivation . . . . automata theory, program semantics, logics

. Grothendieck , Lawvere Haskell . Categories in Computer Science .. only a handful of results concerning the theory itself huge amount of definitions convenient notation . Abstract Language .. foundations laid by Eilenberg , Mac Lane . . Origins .. Category Theory as a Mindset 2 / 17 Intro to Category Theory | Motivation . . . . automata theory, program semantics, logics

. . for all id id for compatible morphisms (associativity) and (identity) there is an arrow id for each object (composition) and where (unit) . .. Categories 3 / 17 Intro to Category Theory | Definitions and Concepts . . . . 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

. . for all id id for compatible morphisms (associativity) and where . (unit) .. Categories 3 / 17 Intro to Category Theory | Definitions and Concepts . . . . 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 f : A → B and g : B → C ⇒ ∃ g ◦ f : A → C (composition) for each object A there is an arrow id A : A → A (identity)

. Categories and where . .. . 3 / 17 Intro to Category Theory | Definitions and Concepts . . . . 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 f : A → B and g : B → C ⇒ ∃ g ◦ f : A → C (composition) for each object A there is an arrow id A : A → A (identity) h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f for compatible morphisms (associativity) f ◦ id A = f = id B ◦ f for all f : A → B (unit)

. Mike Shulman at the n-Category Café stated that combinatorial games can objects. There is a morphism between iff . .. Games (Conway) . be seen as the objects in a category with winning strategies as morphisms. with the elements of . .. Haskell: . Haskell types and Haskell functions in theory form a category Hask , but note that the actual implementations as defines a category . Example Categories . . . . Intro to Category Theory | Definitions and Concepts 4 / 17 .. Each poset Sets . The category Set consists of sets as objects, total functions as morphisms and function composition. .. Posets as Categories . have minor issues (right identity, limits)

. . category Hask , but note that the actual implementations Haskell types and Haskell functions in theory form a . Haskell: .. . be seen as the objects in a category with winning strategies as morphisms. Mike Shulman at the n-Category Café stated that combinatorial games can . Games (Conway) .. . Posets as Categories .. and function composition. The category Set consists of sets as objects, total functions as morphisms . Sets .. Example Categories 4 / 17 Intro to Category Theory | Definitions and Concepts . . . . have minor issues (right identity, limits) Each poset ( S, ≤ ) defines a category C ( S, ≤ ) with the elements of S as objects. There is a morphism between p, q ∈ S iff p ≤ q .

. . category Hask , but note that the actual implementations Haskell types and Haskell functions in theory form a . Haskell: .. . . be seen as the objects in a category with winning strategies as morphisms. Mike Shulman at the n-Category Café stated that combinatorial games can . Games (Conway) .. . Posets as Categories Example Categories . . . . Intro to Category Theory | Definitions and Concepts 4 / 17 .. .. Sets . The category Set consists of sets as objects, total functions as morphisms and function composition. have minor issues (right identity, limits) Each poset ( S, ≤ ) defines a category C ( S, ≤ ) with the elements of S as objects. There is a morphism between p, q ∈ S iff p ≤ q .

. . category Hask , but note that the actual implementations Haskell types and Haskell functions in theory form a . Haskell: .. . . be seen as the objects in a category with winning strategies as morphisms. Mike Shulman at the n-Category Café stated that combinatorial games can . Games (Conway) .. . Posets as Categories Example Categories . . . . Intro to Category Theory | Definitions and Concepts 4 / 17 .. .. Sets . The category Set consists of sets as objects, total functions as morphisms and function composition. have minor issues (right identity, limits) Each poset ( S, ≤ ) defines a category C ( S, ≤ ) with the elements of S as objects. There is a morphism between p, q ∈ S iff p ≤ q .

. . . Mac Lane: .. . “pasting together” of diagrams replacement for equations in category theory . Usage .. holds. This diagram is said to commute if the equation Show that every diagram commutes. Principle Commutative Diagrams . . . . Intro to Category Theory | Definitions and Concepts . 5 / 17 .. h ✲ B A f = g ◦ h g f ✲ ❄ C

. . . Mac Lane: .. . . “pasting together” of diagrams replacement for equations in category theory . Usage .. holds. This diagram is said to commute if the equation Show that every diagram commutes. Principle .. . . Intro to Category Theory | Definitions and Concepts 5 / 17 Commutative Diagrams . . . h ✲ B A f = g ◦ h g f ✲ ❄ C

. . . Mac Lane: .. . . “pasting together” of diagrams replacement for equations in category theory . Usage .. holds. This diagram is said to commute if the equation Show that every diagram commutes. Principle .. . . Intro to Category Theory | Definitions and Concepts 5 / 17 Commutative Diagrams . . . h ✲ B A f = g ◦ h g f ✲ ❄ C

. .. f b) (f a b) fmap :: (a class Functor f where . Haskell: .. . . The category Cat has small categories as objects and functors as arrows. . Category of Categories Type constructors are endofunctors! . Functors . . . . Intro to Category Theory | Definitions and Concepts 6 / 17 .. . where A Functor F between categories C and D is a pair of maps ( F 0 : Ob ( C ) → Ob ( D ) , F 1 : Hom ( C ) → Hom ( D )) a 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 , id F ( A ) = F ( id A ) a We let F ( A ) = F 0 ( A ) for objects A and F ( f ) = F 1 ( f ) for arrows f of C

. .. f b) (f a b) fmap :: (a class Functor f where . Haskell: .. . . The category Cat has small categories as objects and functors as arrows. . Category of Categories Type constructors are endofunctors! . Functors . . . . Intro to Category Theory | Definitions and Concepts 6 / 17 .. . where A Functor F between categories C and D is a pair of maps ( F 0 : Ob ( C ) → Ob ( D ) , F 1 : Hom ( C ) → Hom ( D )) a 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 , id F ( A ) = F ( id A ) a We let F ( A ) = F 0 ( A ) for objects A and F ( f ) = F 1 ( f ) for arrows f of C

. . class Functor f where . Haskell: .. . . The category Cat has small categories as objects and functors as arrows. . Category of Categories .. Type constructors are endofunctors! where 6 / 17 . . . . Intro to Category Theory | Definitions and Concepts Functors .. . A Functor F between categories C and D is a pair of maps ( F 0 : Ob ( C ) → Ob ( D ) , F 1 : Hom ( C ) → Hom ( D )) a 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 , id F ( A ) = F ( id A ) fmap :: (a → b) → (f a → f b) a We let F ( A ) = F 0 ( A ) for objects A and F ( f ) = F 1 ( f ) for arrows f of C