Category Theory

A Category is:

Some Objects
Some Functions
Some Laws

Who cares?

Motivations

People in the scala, f# and especially haskell world seem to use powerful but strange terminology and abstractions to build software.

What is that all about?

A comical example

1990 - A committee formed by Simon Peyton-Jones, Paul Hudak, Philip Wadler, Ashton Kutcher, and People for the Ethical Treatment of Animals creates Haskell, a pure, non-strict, functional language. Haskell gets some resistance due to the complexity of using monads to control side effects. Wadler tries to appease critics by explaining that "a monad is a monoid in the category of endofunctors, what's the problem?"
- from a brief and mostly wrong history of programming languages

A digression

1972 - Dennis Ritchie invents a powerful gun that shoots both forward and backward simultaneously. Not satisfied with the number of deaths and permanent maimings from that invention he invents C.

Who Cares

Scala + ScalaZ

  a Scala library for functional programming

Sum types
Product types
Functor
Applicative
Id
Monoid
so on

Examples

Category Theory is:

All about composition

Composition:

f(g(x)) = f o g x
o = composition operator

Some Laws:

The Associative property
- f . (h . g) == (f . h) . g
The Identity property
- There is a function that does nothing
- fi(a) = a

Addition Of Ints Category

1 Object: Set of integers
1 function: Addition

Addition Of Ints Category

Is this associative?
Is there an identity function?

Addition Of Ints Category

Is this associative?
- Yes 1 + (2+3) + (1+2) + 3
Is there an identity function?
- it can be with the addzero function
- addzero(x) = x + 0

A Category is:

Objects:
- A scala type ( and all its possible values)
- integers, doubles ...
Functions
- These are normal functions and represent the arrows in our diagrams
Laws
- some assumptions about the way things work
- ie. assositivity

A Category is:

~~Objects~~ Types
Functions
Laws

A Category is:

A way to reason about software using only the data types and function signatures

Monoids

monoids are a category sometimes described as "string like"
Once you learn about them, you will see them come up everywhere

Examples

Lists
Strings
Numbers With Plus
Numbers With Multiplication

Abstractions

In c#, java, scala we have an iterator interface, which exposes hasNext() and next() functions.

Common Interface

All things implementing the iterator interface can now be used equivalently, in for loops, and other library functions

Abstractions

A moniod interface can similarly be written. With 2 methods:

mappend (monoidal append)
zero (our identity function)

Common Interface

All things implementing the monoid interface can now be used equivalently.
The various monoids are quite differnt things, but can be treated the same.
This is the power of abstraction Category Theory gives us.
It is the power to notice related things and name them and abstract over them

Some cool monoid stuff

Some Examples

(go down)

Category Theory

By Adam Bell