# BM Category Theory : Motivation and Philosophy

Object-Oriented  has 2 weaknesses for Concurrency and Parallel programming :

1. Hidden Mutating States;
2. Data Sharing.

Category Theory (CT): a higher abstraction of all different Math structures : Set , Logic, Computing math, Algebra… =>

$\boxed {\text {CT reveals the way how our brain works by analysing, reasoning about structures !}}$

Our brain works by:  1) Abstraction 2) Composition 3) Identity (to identify)

What is a Category ?
1) Abstraction:

•  Objects
• Morphism (Arrow)

2) Composition: Associative
3) Identity

Notes:

• Small  Category with “Set” as object.
• Large Category without Set as object.
• Morphism is a Set : “Hom” Set.

Example in Programming
:

• Object : Types Set
• Morphism : Function “Sin” converts degree to R: $\sin \frac {\pi}{2} = 1$

Note: We just look at the Category “Types Set” from external Macroview, “forget ” what it contains, we only know the “composition” (Arrows) between the Category “Type Set”, also “forget” what these Arrows (sin,cosin, tgt, etc) actually are, we only study these arrows’ behavior (Associativity).

2.1 : Function of Set, Morphism of Category

Set: A function is

• Surjective (greek: epic / epimorphism 满射),
• Injective (greek : monic / monomorphism 单射)

Category:  [Surjective]

g 。f = h 。f
=> g = h (Right Cancellation )

2.2 Monomorphism

f 。g = f 。h
=> g = h
(Left cancellation)

$\boxed { \text {Epimorphism + Monomorphism =? Isomorphism }}$

In Haskell, 2 foundation Types: Void, Unit

Void = False
Unit ( ) = True

Functions : absurd, unit
absurd :: Void -> a (a = anything)
unit :: a -> ()

[to be continued 3.1 ….]