Category Theory for Functional Programming

Category Theory is called “The Mathematics of Mathematics“, also the “Abstract Nonsense“.

Key Motivations for Category Theory 范畴 :
1. Programming is Math.
2. Object-Oriented is based on Set Theory which has 2 weaknesses:
◇ Set has contradiction: The famous “Russell’s Barber Paradox”.

◇ Data Immutability for Concurrent Processing : OO can’t control the mutable state of objects, making debugging impossible.

Category (“cat“) has 3 properties:
1. Objects
eg. Set, List, Group, Vector Space, anything…
2. Arrows (“Morphism”, between Objects) which are Associative
eg. function, sheaf, etc
3. Identity Object

Note: If the Identity is “0” or “Nothing”, then it is called Free Category.

Extensions :
1. “Cat” = Category of categories, is also a category.
2. Functor 函子 = Arrows between Categories.
3. Monoid = A Category with ONLY 1 Object.

Monoid (么半群) is a very powerful concept (used in Natural Language Processing) — basically it is a Group with No ‘I‘nverse (Mo‘No‘-‘I‘-d).

Note: In the pure Functional Programming Language Haskell:
Monad (单子) = Monoid with endo-functor.
Endo- = to itself (自己) eg. “f : A -> A”

Recommended :
1. The Best Book on Category Theory : Lawvere et al, Cambridge (2009, 2nd Edition)

2. A humorous book on Category Theory by Dr. Eugenia Cheng (2016): (found in Singapore NLB Loan book #501.1 CHE)