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.

# BM 5&6 : Category Adjunctions 伴随函子

Adjunction is the “weakening of Equality” in Category Theory.

Equivalence of 2 Categories if:

5.2  Adjunction definition: $(L, R, \epsilon, \eta )$ such that the 2 triangle identities below ( red and blue) exist.

6.1 Prove: Let C any category, D a Set.

$\boxed {\text {C(L 1, -)} \simeq \text {R}}$

${\text {Right Adjoint R in Set category is }}$ ${\text {ALWAYS Representable}}$

1 = Singleton in Set D

From an element in the singleton Set always ‘picks’ (via the function) an image element from the other Set, hence :
$\boxed {\text{Set (1, R c) } \simeq \text{Rc }}$

Examples : Product & Exponential are Right Adjoints

Note: Adjoint is a more powerful concept to understand than the universal construction of Product and Exponential.

6.2

$\boxed {\text {Every Adjunction gives a Monad}}$

vice-versa.

$\boxed {\text {Left Adjoint: L } \dashv \text { R }}$

$R \circ \ L = m = \text { Monad}$

$L \circ \ C = \text { Co-Monad}$

With Product (Left Adjoint) and Exponential (Right Adjoint) => $\text {State Monad}$

# Category Theory II 2: Limits, Higher order functors

1.2 Introduction to Limit

Analogy : Product to Cones (Limit)

2.1 Five categories used to define Limit:

1. Index category (I)
2. Category C: Functors (constant ,  D)
3. Cones (Lim D)
4. Functor Category [I, C ]:objects are (constant, D ), morphisms are natural transormations
5. Set category of Hom-Set Cones [I,C] to Hom-Set C (c , Lim D )

2.2 Naturality

3.1 Examples: Equalizer

CoLimit = duality of Limit (inverted cone = co-Cone)

Functor Continuity = preserve Limit

# BM Category Theory 10: Monad & Monoid

Analogy

Function : compose “.“, Id

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

# Category Theory 9: Natural Transformations, BiCategories

In essence, in all kinds of Math, we do 3 things:

1) Find Pattern among objects (numbers, shapes, …),
2) Operate inside the objects (+ – × / …),
3) Swap the object without modifying it (rotate, flip, move around, exchange…).

Category  consists of :
1) Find pattern thru Universal Construction in Objects (Set, Group, Ring, Vector Space, anything )
2) Functor which operates on 1).
3) Natural Transformation as in 3).

$\boxed {\text {Natural Transformation}}$
$\Updownarrow$

$\boxed {\text {Morphism of Functors}}$

Analogy:

Functors (F, G) := operation inside a container
$\boxed { F :: X \to F_{X}, \: F :: Y \to F_{Y}}$

$\boxed {G :: X \to G_{X}, \: G :: Y \to G_{Y}}$

Natural Transformation  (${\eta_{X}, \eta_{Y}}$) := swap the content ( $F_{X} \text { with } G_{X}, F_{Y} \text { with } G_{Y}$) in the container without modifying it.
$\boxed{\eta_{X} :: F_{X} \to G_{X} , \: \eta_{Y} :: F_{Y} \to G_{Y}}$

9.2 Bicategories

“Diagram Chasing”:

2- Category

Cat = Category of categories (C, D)

The functors {F, G} instead of being a Set (“Hom-Set”) – like functions  form a function object “Exponentialfunctors also form a category, noted : $\boxed {[C,D] = D^{C} }$

BiCategory (different from 2-Category): the Associativity and Identity are not equal (as in 2-Category),  but only up to Isomorphism.
Note : when n is infinity,  n-Category & Groupoid (HOTT: Homotopy Type Theory)