Category Theory. & Functional Data Abstraction. Brandon Shapiro. Math 100b
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1 & Functional Data Abstraction Math 100b
2 Categories A category C is a collection of objects with arrows (often called morphisms) pointing between them Hom C (X, Y ) is the set of morphisms in C from X to Y If f Hom C (X, Y ) and g Hom C (Y, Z ), then there exists a morphism f g in Hom C (X, Z ) (composition is associative) For every object X in C, there is an identity morphism 1 X Hom C (X, X) (f 1 X = f and 1 X g = g)
3 Categories A category C is a collection of objects with arrows (often called morphisms) pointing between them Hom C (X, Y ) is the set of morphisms in C from X to Y If f Hom C (X, Y ) and g Hom C (Y, Z ), then there exists a morphism f g in Hom C (X, Z ) (composition is associative) For every object X in C, there is an identity morphism 1 X Hom C (X, X) (f 1 X = f and 1 X g = g)
4 Examples Set is the category of all sets, with functions between sets as the morphisms All groups also form a category, Grp, with group homomorphisms as its morphisms Ring and R-mod for some ring R can be formed with ring and module homomorphisms as morphisms A subcategory of category C is a category with all of its objects and morphisms contained in C Finite sets and the functions between them form a subcategory of Set, and abelian groups are a subcategory of Grp. Fields form a subcategory of the category of commutative rings, which is itself a subcategory of Ring
5 Examples Set is the category of all sets, with functions between sets as the morphisms All groups also form a category, Grp, with group homomorphisms as its morphisms Ring and R-mod for some ring R can be formed with ring and module homomorphisms as morphisms A subcategory of category C is a category with all of its objects and morphisms contained in C Finite sets and the functions between them form a subcategory of Set, and abelian groups are a subcategory of Grp. Fields form a subcategory of the category of commutative rings, which is itself a subcategory of Ring
6 Functors A functor is a structure preserving map between categories For categories C and D, a covariant functor F : C D sends the objects of C to objects in D, and sends the morphisms in C to morphisms in D If f Hom C (X, Y ), F(f ) Hom D (F(X), F(Y )) F(1 X ) = 1 F(X), F(f g) = F(f ) F(g)
7 Functors A functor is a structure preserving map between categories For categories C and D, a covariant functor F : C D sends the objects of C to objects in D, and sends the morphisms in C to morphisms in D If f Hom C (X, Y ), F(f ) Hom D (F(X), F(Y )) F(1 X ) = 1 F(X), F(f g) = F(f ) F(g)
8 Examples The identity functor from C to C sends every object and morphism in C to itself. Let F be a map from Grp to Set sending groups and homomorphisms in Grp to themselves in Set. F is a functor from Grp to Set called the forgetful functor Similarly, forgetful functors exist from Ring and R-mod to Grp and to Set A functor from a category to itself is called an endofunctor The identity functor is an endofunctor
9 Examples The identity functor from C to C sends every object and morphism in C to itself. Let F be a map from Grp to Set sending groups and homomorphisms in Grp to themselves in Set. F is a functor from Grp to Set called the forgetful functor Similarly, forgetful functors exist from Ring and R-mod to Grp and to Set A functor from a category to itself is called an endofunctor The identity functor is an endofunctor
10 Date Types In computer programming languages, a data type is a set of elements that can be represented by a computer (finitely in binary) in the same way Two of the most common data types are Z and R Real-world computing has constraints on memory, etc. Mathematically, a data type can be treated just as a set
11 Maybe Set has sets as objects and functions as morphisms Maybe : Set Set Maybe(A) = A {Nothing} Maybe lets us define safe versions of partial functions f : R Maybe(R) f (0) = Nothing f (x) = 1/x (x 0)
12 Maybe Set has sets as objects and functions as morphisms Maybe : Set Set Maybe(A) = A {Nothing} Maybe lets us define safe versions of partial functions f : R Maybe(R) f (0) = Nothing f (x) = 1/x (x 0)
13 Maybe Maybe is a functor from Set to Set (endofunctor) Needs a mapping for the morphisms (functions) Mmap : Hom(A, B) Hom(Maybe(A), Maybe(B)) Mmap(f )(Nothing) = Nothing Mmap(f )(x) = f (x) (x Nothing) Mmap(1 A ) = 1 Maybe(A) Mmap(f g) = Mmap(f ) Mmap(g)
14 Maybe Maybe is a functor from Set to Set (endofunctor) Needs a mapping for the morphisms (functions) Mmap : Hom(A, B) Hom(Maybe(A), Maybe(B)) Mmap(f )(Nothing) = Nothing Mmap(f )(x) = f (x) (x Nothing) Mmap(1 A ) = 1 Maybe(A) Mmap(f g) = Mmap(f ) Mmap(g)
15 List List sends a set A to the set of lists of elements in A List : Set Set List(A) = {()} {(x, xlist) x A, xlist List(A)} () is called the empty list (1, (2, (3, (4, ())))) List(Z) (1/2, (Nothing, (1/4, ()))) List(Maybe(Q)) (1, 2, 3, 4) List(Z)
16 List List sends a set A to the set of lists of elements in A List : Set Set List(A) = {()} {(x, xlist) x A, xlist List(A)} () is called the empty list (1, (2, (3, (4, ())))) List(Z) (1/2, (Nothing, (1/4, ()))) List(Maybe(Q)) (1, 2, 3, 4) List(Z)
17 List List sends a set A to the set of lists of elements in A List : Set Set List(A) = {()} {(x, xlist) x A, xlist List(A)} () is called the empty list (1, (2, (3, (4, ())))) List(Z) (1/2, (Nothing, (1/4, ()))) List(Maybe(Q)) (1, 2, 3, 4) List(Z)
18 List List is an endofunctor on Set Needs a mapping for the morphisms (functions) Lmap : Hom(A, B) Hom(List(A), List(B)) Lmap(f )(()) = () Lmap(f )((x, xlist)) = (f (x), Lmap(f )(xlist)) For f (x) = x 2, Lmap(f )((1, 2, 3, 4)) = (1, 4, 9, 16) Clearly satisfies functor laws (identity and composition)
19 List List is an endofunctor on Set Needs a mapping for the morphisms (functions) Lmap : Hom(A, B) Hom(List(A), List(B)) Lmap(f )(()) = () Lmap(f )((x, xlist)) = (f (x), Lmap(f )(xlist)) For f (x) = x 2, Lmap(f )((1, 2, 3, 4)) = (1, 4, 9, 16) Clearly satisfies functor laws (identity and composition)
20 List List is an endofunctor on Set Needs a mapping for the morphisms (functions) Lmap : Hom(A, B) Hom(List(A), List(B)) Lmap(f )(()) = () Lmap(f )((x, xlist)) = (f (x), Lmap(f )(xlist)) For f (x) = x 2, Lmap(f )((1, 2, 3, 4)) = (1, 4, 9, 16) Clearly satisfies functor laws (identity and composition)
21 List List is an endofunctor on Set Needs a mapping for the morphisms (functions) Lmap : Hom(A, B) Hom(List(A), List(B)) Lmap(f )(()) = () Lmap(f )((x, xlist)) = (f (x), Lmap(f )(xlist)) For f (x) = x 2, Lmap(f )((1, 2, 3, 4)) = (1, 4, 9, 16) Clearly satisfies functor laws (identity and composition)
22 Applicative Functors What does an endofunctor on Set to do a set of functions? An applicative functor is a functor with a splat function Fsplat : F(Hom(A, B)) Hom(F(A), F(B)) Fsplat can also be defined as a binary function Fsplat : F(Hom(A, B)) F(A) F(B) There are rules applicative functors must follow
23 Applicative Functors What does an endofunctor on Set to do a set of functions? An applicative functor is a functor with a splat function Fsplat : F(Hom(A, B)) Hom(F(A), F(B)) Fsplat can also be defined as a binary function Fsplat : F(Hom(A, B)) F(A) F(B) There are rules applicative functors must follow
24 Applicative Functors What does an endofunctor on Set to do a set of functions? An applicative functor is a functor with a splat function Fsplat : F(Hom(A, B)) Hom(F(A), F(B)) Fsplat can also be defined as a binary function Fsplat : F(Hom(A, B)) F(A) F(B) There are rules applicative functors must follow
25 Applicative Functors What does an endofunctor on Set to do a set of functions? An applicative functor is a functor with a splat function Fsplat : F(Hom(A, B)) Hom(F(A), F(B)) Fsplat can also be defined as a binary function Fsplat : F(Hom(A, B)) F(A) F(B) There are rules applicative functors must follow
26 Applicative Functors Maybe is an applicative functor Msplat : Maybe(Hom(A, B)) Maybe(A) Maybe(B) Msplat(Nothing)(_) = Msplat(_)(Nothing) = Nothing Msplat(f )(x) = f (x) List is an applicative functor Lsplat : List(Hom(A, B)) List(A) List(B) Lsplat 1 (())(_) = Lsplat 1 (_)(()) = () Lsplat 1 ((f, flist))((x, xlist)) = (f (x), Lsplat 1 (flist)(xlist)) Could List be an applicative functor in any other ways?
27 Applicative Functors Maybe is an applicative functor Msplat : Maybe(Hom(A, B)) Maybe(A) Maybe(B) Msplat(Nothing)(_) = Msplat(_)(Nothing) = Nothing Msplat(f )(x) = f (x) List is an applicative functor Lsplat : List(Hom(A, B)) List(A) List(B) Lsplat 1 (())(_) = Lsplat 1 (_)(()) = () Lsplat 1 ((f, flist))((x, xlist)) = (f (x), Lsplat 1 (flist)(xlist)) Could List be an applicative functor in any other ways?
28 Applicative Functors Maybe is an applicative functor Msplat : Maybe(Hom(A, B)) Maybe(A) Maybe(B) Msplat(Nothing)(_) = Msplat(_)(Nothing) = Nothing Msplat(f )(x) = f (x) List is an applicative functor Lsplat : List(Hom(A, B)) List(A) List(B) Lsplat 1 (())(_) = Lsplat 1 (_)(()) = () Lsplat 1 ((f, flist))((x, xlist)) = (f (x), Lsplat 1 (flist)(xlist)) Could List be an applicative functor in any other ways?
29 Sources & Images Sources Abstract Algebra by Dummit and Foote Lectures by and conversations with Kenny Foner Images kz6vwx9ntzrs2fcduapsq0e-ysddsporfnwykp2j- 49Uu8_5ahulOEfHLmT7w2lZMvQ_vhDGxCkqHlMo1C_0VCkrCFeSzfvtW4PjD =w426-h310
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