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

    ..

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    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)

    Reading Book: chap 10