Categories and Computer Science

http://cis.k.hosei.ac.jp/~yukita/

2

What is category theory?

• Algebra of functions– Composition is the principal operation on functions

• Abstract structure– Collection of objects– Collection of arrows between them

• Invented by S. Eilenberg and S. Mac Lane in 1945– originated from algebraic topology– influence on

• Algebraic geometry A. Grothendieck• Logic F.W. Lawvere• Computer science

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How does category theory appear in Computer Science?

• Construction of functions out of a given set of simple functions – using various operations on functions such as:

• composition and repeated composition

– An important aspect of CS

• Dynamical systems– having states which vary over time– Computing is concerned with machines

• An algebra of functions which really does not consist of functions– Programs and languages are formal things that specify actual

functions– Syntactical side of CS

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Topics we choose

• Grammars and Languages• Data types• Boolean algebra• Circuit theory• Flow charts• Imperative programming• Specification• Lambda calculus

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CategoriesThe Algebra of Functions

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Note. The notion of category is axiomatically defined and formal. The objects of a category need not be actual sets and the arrows need not be actual functions. As is usual with axiomatic definitions this allows great flexibility.

CS note. An object of a category does not necessarily have methods such as union, intersection, Cartesian producs, etc.

7

A Bf

1A 1B

8

Ex 1. Sets: The category of sets

,: functions all arecategory thisof morphisms The

,,, sets all arecategory thisof objects The

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CBA

Note. If you are unfamiliar with the notions of sets and functions, you should consult with some elementary textbooks on set theory. Can you clearly tell the differences between surjection, injection, and bijection?

9

Small examples

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(ii) Case (i) Case

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arrows. no objects; No

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Ex.4.

Ex.3.

Ex.2.

We should check if all the axioms are satisfied in these examples.

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Ex.5. Two objects and three arrows

defined.partially isn compositio The required. is 11

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Ex. 6 One object with four arrows

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12

Is Ex.6 really a category?

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13

Ex. 7 The category of power set 2X

• The subsets of X = {0,1,2} is the set of objects.

• The arrows are inclusions.

• In the figure, identities are omitted.

{0,1,2}

{0,1} {1,2}{0,2}

{0} {1} {2}

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Ex. 8 Identities are only arrows.

A

1A

B

1B

C

1C

E

1E

D

1D

15

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such that : arrowan is there,: arroweach to

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monoid

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Remark

Def.

10. Ex.

9. Ex.

Monoids and Groups

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Examples of monoids

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,,,,,1,,, arrows ;object One

additionunder Integers

.by given isn Compositio

,,,,,1 arrows ;object One

additionunder numbers Natural

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17

Order

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since satisfiedlly automatica are laws eassociativ theand lawsidentity The

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object.other any object to oneany from arrow onemost At

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X14. Ex.

Def.

Notation.

13. Ex.

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Presenting a category by its generators and relations

• Generators– All of the objects– Some of the arrows

• Relations– Some equations between composites of given

arrows

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Ex. 15. Modulo 4 addition

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322

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32

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4

1

1

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category a ofon presentati relation with :arrow One

:object One

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20

Ex. 16. A category of words

relations. no with },,,,,,,,{ arrows ofset The

* :object One

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Def. Free monoid

• A category with one object and several arrows with no relations is called a monoid.

• The set of arrows is call the alphabet of the monoid.

• Languages are subsets of a free monoid.• We usually think of a language as a set of

well-formed sentences with respect to some grammar.

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Ex. 17.

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.: and : arrows two(2)

and , , objects two(1)

generators as Take

.

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Def. Directed graphs

• A (directed) graph is a set of objects and a set of arrows, each with a prescribed domain object and codomain object.

• There’s no composition. This is the difference between graphs and categories.

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Def. The free category on a graph G

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paths. ofion concatenat isn Compositio

.

i.e. up, maching codomains and domains with to

from arrows of paths are object toobject from Arrows

. of thoseas same thearecategory free theof objects The

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category a formcan werelations,given no and ,graph aGiven

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Ex. 18. A Regular Language

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exponent. negative-non a with numbers Signed

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26

Ex. 19. Function f(n)=2n

.8)4( meaning

,

following; thehave We

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relations with :,:,:

arrows and and objects graph withA

f

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ssffsofo

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The dual of a category

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and

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reversed. being arrows theallwith

and as objects same ith thecategory w a is category. a be Let

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28

Ex. 23. Setsfinite and BoolAlgfinite

Highly advanced topic

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Isomorphic categories

. writeWe

.)identities preserves ( 1)1(

A thenin identity an is 1 if (iii)

ns.)compositio preserves ( )()()(

A thenin : and : if (ii)

)codomains. and domains preserves (

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then in : if (i)

such that ly,respective , of arrows and objects the to of arrows

and objects thefrom bijection a is to from misomorphisAn

)(

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30

.2 toisomorphic is )2( 27. Ex. op XX

{0,1,2}

{0,1} {1,2}{0,2}

{0} {1} {2}

{0,1,2}

{0,1}{1,2} {0,2}

{0}{1}{2}

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Def. Product of categories

component.by component performed isn Compositio

.: and :

arrows of ),( pairs are ),( to),( from arrows The

. ofobject an

is and ofobject is where),( pairs are of objects The

following. by the defined

is , and of the, then categories are , If

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product

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32

arrows. nine and objectsfour has Then

.: and ,1,1 arrows threeand , objects twohas

.: and ,1,1 arrows threeand , objects twohas

figure.) in the omitted are s(Identitie

2121

2121

21

21

BA

B

A

28. Ex.

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),( 12 BA ),( 22 BA

),( 11 BA ),( 21 BA

)1,(2B

f

),1(2

gA

),1(1

gA

),( gf)1,(

1Bf

33

Remark

• In any category,

• given any two objects in the diagram, say P and Q, any composite of arrows from P to Q yields the same result.

• We then say that the diagram commutes.