Program = Category 

2017 : (PhD Math made simple for IT programmers)

Keywords

  • Category 
  • Monad = Monoid + Endofunctor
  • Lewvere Theory

Category Theory is replacing Set Theory as the foundation of Math. Nowadays,  few Advanced Math papers are written without using Category to explain, and this trend is spreading to IT  through Functional Programming languages (Google’s Kotlin, Haskell, Clojure…)  – the latest paradigm to replace Object-Oriented languages like Java, C++, etc, as a safer “Strong Typed” languages for AI, BIG DATA…

\boxed {\text {Type = Category }}

Examples of “Types” in IT:

  • Integers
  • Real
  • Double
  • Boolean
  • String
  • etc

T-program defined in the following 6 examples: 

  • list (X), eg. {2, 5 , 3, 7, 1 }
  • partial (X), eg: +1 (error msg)
  • side-effect, eg: update a record
  • continuation (X),
  • non-det (X), 
  • prob-dist (X)

A Monad : a T-program which turns an arrow to a “Category” (ie + 2 properties:  Identity &  Associative).

Proof: List Computation is a Category

Proof: Partial operation is a Category

\boxed {\text {Monad = Lawvere Theory }}

Monad is for only one Category. Lawvere Theory is more general.


Ref
:

Slideshttp://www.math.jhu.edu/~eriehl/compose.pdf

Ref:
Excellent Intuitive Stackoverflow Answer:  What is a Monad ?

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