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).

Analogy:

Functors (F, G) := operation inside a container

Natural Transformation () := swap the content ( ) in the container without modifying it.

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 “Exponential“ – functors also form a category, noted :

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)

Reading Book: chap 10