# An “Introduction of Introduction” to Category Theory

Category : 范畴 has 3 things: (hence richer than a Set 集合 which is only a collection of objects)

1. Objects 对象
2. Arrow (Morphism 态射) between Objects, includes identity morphism.
3. Associativity 结合性

Functor (函子) between 2 Categories (preserve structure)

Natural Transformation 自然变换

• Example :
Matrices -> Determinants

..

# 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 objectExponentialfunctors 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)