Formal Methods in Software

Lecture 7. Category Theory

Vlad PatryshevSCU2014

In This Lecture

• database example

• terminal object, initial object;

• products, unions

• equalizers

• pullbacks

Database Examplecreate type rels as enum (‘spouse’, ‘child’, ‘partner’);

create type jobs as enum (‘ceo’, ‘cto’, ‘eng’, ‘sales’);

create table Person (id bigint, name varchar(80), primary key (id));

create table Company (id bigint, name varchar(80), primary key (id));

create table Rel (from bigint, to bigint, kind rels,

constraint p1_fk foreign key (from references Person(id),

constraint p2_fk foreign key (yo references Person(id)

);

create table Job (company bigint, employee bigint, position jobs,

constraint c_fk foreign key (comp references Company(id),

constraint p_fk foreign key (pers references Person(id)

);

“Conceptual Model”

Person

id

name

Company

id

name

Job

position

company

employee

Relationship

kind

from

to

rels

{spouse, child, partner}

jobs

{ceo, cto, eng, sales}

“More Conceptual Model”

Person

name

Company

name

Job

Relationship

rels

{spouse, child, partner}

jobs

{ceo, cto, eng, sales}

‘from’

‘to’

‘kind’

‘employee’

‘position’

‘company’

“More Conceptual Model”

Person

name

Company

name

Job

Relationship

rels

{spouse, child, partner}

jobs

{ceo, cto, eng, sales}

‘from’

‘to’

‘kind’

‘employee’

‘position’

‘company’

It’s a category!

Pet Databasecreate type kind as enum (‘cat’, ‘dog’, ‘fly’, ‘hamster’, ‘e.coli’);

create table Person (id bigint, name varchar(80), pet bigint,

primary key (id),

constraint pet_fk foreign key (pet references Animal(id));

create table Animal (id bigint, kind kind, name varchar(80), owner bigint,

primary key (id),

constraint owner_pk foreign key owner references Person(id));

“More Conceptual Model”

Person

name

Animal

kind

{cat, dog, fly, hamster, e.coli}

‘owner’

‘pet’

‘kind’

Where’s composition?

kind(pet(owner(pet(“John”)))

select kind from Animal a where a.owner in (select id from Person where pet in (select id from Animal where owner in (select id from person where name=’John’)));

Initial Object in a Category

Definition. Given a category C, Initial Object is such an object 0 that there is a unique function iX:0 → X for any given X. (this is its “universal property”)

Note that if we take X=0, we see that there is just one function 0 → 0. Can you name a set (or two) with only one function S→S?

And what if there’s more than one such an object? Say 01 and 02, both have this interesting feature. Then, for 01, we have a unique a:01 → 02; and similarly we have a unique b:02 → 01. Composing a and b either way, we get an identity; so they are isomorphisms.

Initial object is unique up to an isomorphism.

Examples of Initial Objects

• Sets: ∅, and it is unique (by sets axioms)

• Monoids: {0} - all “such monoids” are isomorphic

• Categories: empty category (whether there’s a plurality of them…)

• In this category of three objects we see no initial object:

• In this category c is initial object:

• How about more than one

initial object?

They are

isomorphic!

Terminal Object in a Category

Definition. Given a category C, Terminal Object is such an object 1 that there is a unique function uX:X → 1 for any given X. (this is its “universal property”)

Note that if we take X=1, we see that there is just one function 1 → 1, same as with 0.

And what if there’s more than one such an object? Say 11 and 12, both have this interesting feature. Then, for 11, we have a unique a:11 → 12; and similarly we have a unique b:12 → 11. Composing a and b either way, we get an identity; so they are isomorphisms.

Terminal object, like initial one, is unique up to an isomorphism.

Examples of Terminal Objects

• Sets: any singleton {x} is terminal. They are not equal, but are isomorphic.

• Monoids: {0} - all “such monoids” are isomorphic

• Categories: empty category (whether there’s a plurality of them…)

• In this category of three objects we see no terminal object:

• In this category c is terminal object:

Initial and Terminal Object in our DB

terminal

Cartesian Product

Definition. Given a category C, and two objects in it, X and Y, their Cartesian Product is such an object Z=X×Y, together with two functions, pX:Z→X and pY:Z→Y, that for any pair f:A→X and g:A→Y, f=pX h∘ and g=pY h∘ for some unique h. (this is its “universal property”)

Cartesian product is unique up to an isomorphism. (proof?)

Examples of Cartesian Products

• Sets: AxB = {(a,b)|a A,b B}∈ ∈ , and it is unique (by sets axioms)

• Databases: select * from A,B;

• Good programming languages, e.g. Scala: (A,B)

val intWithString: (Int, String) = (42, “Hello 42”)

• In a monoid? We have just one object!

• In a poset? It’s min(a,b)

Disjoint Union (dual to Product)

Definition. Given a category C, and two objects in it, X and Y, their Disjoint Union is such an object Z=X+Y, together with two functions, iX:X→Z and iY:Y→Z, that for any pair f:X→A and g:Y→A, f=h i∘ X and g=h i∘ Y for some unique h. (this is its “universal property”)

Disjoint union is unique up to an isomorphism. (proof?)

Examples of Disjoint Unions

• Sets: A+B = A B∪ , and it is unique (by sets axioms)

• Databases: select * from A union B;

• Good programming languages, e.g. Scala: Either[A,B]

val intOrString: Either[Int, String] = Left(42)// Right(“Hello 42”)

• In a poset? It’s max(a,b)

Equalizers

Definition. Given a category C, two objects in it, X and Y, and two functions, f,g:X→Y, their Equalizer is such an object E=Eq(f,g), together with a function eq:E→X, such that that f eq=g eq∘ ∘ for any m:O→X, m=eq u∘ for some unique u. (this is its “universal property”)

An equalizer is unique up to an isomorphism. (proof?)

Pullbacks

Definition. Given a category C, objects, X, Y and Z, and two functions, f:X→Z and g:X→Z, their Pullback is such an object X×ZY, together with functions pX:X×ZY→X and pY:X×ZY→Y, such that g p∘ Y = f p∘ X, and for any pair x:U→X, y:U→Y, x=pX h∘ and y=pY h∘ for some unique h. (this is its “universal property”)

A pullback is unique up to an isomorphism. (proof?)

Examples of Pullbacks

• Sets: {(x,y)|x X,y∈ Y,∈ f(x)=g(y)}

• Databases: select * from A, B where A.f=B.g;

e.g. select * from Person A, Person B where A.pet=B.pet;

• In a poset? Same as min.

• Cartesian products are pullbacks.

Referenceshttp://www.amazon.com/Category-Computer-Scientists-Foundations-Computing/dp/0262660717

Wikipedia