Knowing Monads Through The Category Theory

While Mathematicians like to talk non-sensical abstract idea, Informaticians want to know how to apply the idea concretely:

Mathematical Parlance:

Monad = Monoid +Endofunctor

Monoid = Identity + Associative

Endo-functor = functor between 2 same categories

IT Parlance:

Monad is a ‘function’ to wrap the ‘side effects’ (exception errors, I/O,… ) so that function composition in ‘pipeline‘ chained operation sequence is still possible in pure FP (Functional Programming, which forbids side-effects).

Some common Monads: ‘Maybe’, ‘List’, ‘Reader’…
This allows monads to simplify a wide range of problems, like handling potential undefined values (with the Maybe monad), or keeping values within a flexible, well-formed list (using the List monad). With a monad, a programmer can turn a complicated sequence of functions into a succinct pipeline that abstracts away additional data management, control flow, or side-effects.[2][3]

Exploring Monads in Scala Collections

Alejandro Serrano: Category Theory Through Functional Programming

(Part 1/3) – λC 2017

What is Category ?


Morphism (Arrows )

Rule1: Associative

Rule 2: Identity

A <– C –> B

Product of Categories : A x B


Sum of Categories: A + B


(Either a b)


Reverse all arrows.


Functor F: C-> D

Mapping of all objects (A, B) in categories C,D

Mapping of arrows f
f : A -> B
Ff : FA -> FB (preservation)
F Id = Id
F (f。g) = Ff。Fg


Constant C -> F
FC = k
Ff = Id

Arrow Functor F: C -> D

For any object A in C,

F A = D -> A
(Functional Type is also Type)

Functors compose !

Category of categories:

Objects: categories

Arrows : Functors

Haskell Category (Hask) is always Endo-Functor, ie Category Hask to itself.

Mapping of arrows.

Mapping of Objects = predefined

(Part 2/3) – λC 2017

(part 3/3) – λC 2017