Monoid and Monad

Quora: What is the difference between monoid and monad? by Mort Yao

A well-said, perhaps the briefest ever answer is: monad is just a monoid in the category of endofunctors.

monoid is defined as an algebraic structure (generally, a set) M with a binary operation (multiplication) • : M × M → M and an identity element (unit) η : 1 → M, following two axioms:

i. Associativity
∀ a, b, c ∈ M, (a • b) • c = a • (b • c)

ii. Identity
∃ e ∈ M ∀ a ∈ M, e • a = a • e = a

When specifying an endofunctor T : X → X (which is a functor that maps a category to itself) as the set M, the Cartesian product of two sets is just the composition of two endofunctors; what you get from here is a monad, with these two natural transformations:

1. The binary operation is just a functor composition μ : T × T → T
2. The identity element is just an identity endofunctor η : I → T

Satisfied the monoid axioms (i. & ii.), a monad can be seen as a monoid which is an endofunctor together with two natural transformations. 

The name “monad” came from “monoid” and “triad”, which indicated that it is a triple (1 functor + 2 trasformations), monoidic algebraic structure.

In other words, monoid is a more general, abstract term. When applying it to the category of endofunctors, we have a monad.

State Monad: Milewski

Don’t fear the Monad

Brian Beckman: 

You can understand Monad without too much Category Theory.

Functional Programming = using functions to compose from small functions to very complex software (eg. Nuclear system, driverless car software…).

Advantages of Functional Programming:

  • Strong Types Safety: detect bugs at compile time. 
  • Data Protection thru Immutability: Share data safely in Concurrent / Parallel processing.
  • Software ‘Componentisation’  ie Modularity : Each function always returns the same result, ease of software reliability testing.

Each “small” function is a Monoid.
f : a -> a (from input of type ‘a‘ , returns type ‘a’)
g: a -> a

compose h from f & g : (strong TYPING !!)
h = f。g : a -> a

[Note] : Object in Category, usually called  Type in Haskell, eg. ‘a’ = Integer)

You already know a Monoid (or Category in general) : eg Clock

  1. Objects: 1 2 3 …12 (hours)
  2. Arrow (Morphism): rule “+”: 
    • 7 + 10 = 17 mod 12 = 5
  3. Law of Associativity:
    x + (y + z) = (x + y) + z
  4. Identity (or “Unit”):  (“12”):
    x + 12 = 12 + x = x

More general than Monoid is a “Monoidal” Category where: (instead of only single object ‘a’, now more “a b c…”)
f : a -> b
g: b -> c
h = f。g : a -> c

Function under composition Associative rule and with an Identity => Monoid

Monad (M): a  way to manage  the side-effects (I/O, exception , SQL Database, etc) within the Functional Programming way like monoidal categories: ie associative composition, identity.

Remark: For the last 60 years in Software, there have been 2 camps: 

  1. Bottom-Up Design: from hardware foundation,  build performance-based languages: Fortran, C, C++, C#, Java…
  2. Top-Down Design: from Mathematics foundation, build functional languages (Lambda-Calculus, Lisp, Algo, Smalltalk, Haskell…). 
  3. F# (Microsoft) is the middle-ground between 1 & 2.

Ref: What is a Monad ?

Monad = chaining operations with binding “>>=”

  • Possible use: allows to write mini-language, parser…

BM Category Theory 10: Monad & Monoid

10.1 Monad

Function : compose “.“, Id

Monad: compose “>>=“, return

Imperative (with side effects eg. state, I/O, exception ) to Pure function by hiding or embellishment in Pure function but return “embellished” result.

10. 2 Monoid

Monoid in category of endo functors = Monad

Ref Book : 

What is the significance of monoids in category theory? by Bartosz Milewski