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}}

\boxed {\text {Morphism of Functors}}


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)

Reading Book: chap 10

One thought on “Category Theory 9: Natural Transformations, BiCategories

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s