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)

Reading Book: chap 10

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