My course of Formal Methods at Santa Clara University, Winter 2014.

1. Formal Methods in
Software
Lecture 7. Category Theory
Vlad Patryshev
SCU
2014

2. In This Lecture
• database example
• terminal object, initial object;
• products, unions
• equalizers
• pullbacks

3. Database Example
create 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)
);

4. “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}

5. “More Conceptual Model”
Person
name
Company
name
Job
Relationship
rels
{spouse,
child,
partner}
jobs
{ceo, cto,
eng, sales}

6. “More Conceptual Model”
Person
name
Company
name
Job
Relationship
rels
{spouse,
child,
partner}
jobs
{ceo, cto,
eng, sales}
It’s a category!

7. Pet Database
create 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));

8. “More Conceptual Model”
Person
name
Animal
kind
{cat, dog,
fly, hamster,
e.coli}
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’)));

9. 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.

10. 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!

11. 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.

12. 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:

13. Initial and Terminal Object in our DB
terminal

14. 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?)

15. 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)

17. 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∘iX and g=h∘iY for some
unique h. (this is its “universal property”)
Disjoint union is unique up to an isomorphism. (proof?)

18. 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)

19. 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?)

20. 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∘pY = f∘pX, 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?)

21. 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.

22. References
http://www.amazon.com/Category-Computer-Scientists-Foundations-Computing/dp/0262660717
Wikipedia