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 III Part 3

    3.1 Adjunction and Monad

    Tip:

    L = Free,
    R = Conservative (get rid of structure / forgetful)

    Examples
    L: C- >D
    Monoid – > Free Monoid
    Algebras – > Free Algebras

    R: D – > C

    Adjunction gives “Monad”

    3.2 Monad Algebras

    Tip: Think Monad as “List” container

    Category Theory III for Programmers (Part 1 & 2)

    The most interesting “Category Theory” (范畴论) for Programmers course III by Dr. Bartosz Milewski , a follow-up of last year’s course II.

    Prerequisites:

    1. Fundamental of Category Theory: Functor, Natural Transformation, etc. (Course II Series)
    2. (Nice to have) : Basic Haskell Functional Programming Language. (Quick Haskell Tutorial)

    1.1: Overview Part 1

    Category Theory (CT) = Summary of ALL Mathematics

    Functional Programming = Application of CT

    Philosophical Background:

    • Math originated 3,000 years ago in Geometry by Greek Euclid with Axioms and deductive (演译) Proof-driven Logic.
    • Geometry = Geo (Earth) + Metry (Measurement).
    • Math evolved from 2-dimensional Euclidean Geometry through 17 CE French Descartes’s Cartesian Geometry using the 13CE Arabic invention “Algebra” in Equations of n dimensions: (x_1, x_2,..., x_n) , (y_1, y_2,..., y_n)
    • Use of Algebra: 1) Evaluation of algebraic equations (in CT: “Functor”) ; 2) Manipulation. eg. Substitution (in CT : “Monad” ), Container (in CT: “Endo-Functor” ), Algebraic Operations (in CT: “Pure, Return, Binding” ).
    • Lawvere Theories: unified all definitions of Monoids (from Set to CT)
    • Free Monoid = “List” (in Programming). Eg. Concatenation of Lists = new List (Composition, Associative Law) ; Empty List (Unit Law).
    • Advance of Math in 21CE comes back to Geometry in new Math branches like Algebraic Geometry, Algebraic Topology, etc.

    Note: The only 2 existing human Languages invented were derived from forms & shapes (images) of the mother Earth & Nature:

    1. Ancient Greek Geometry (3000 years) ;
    2. Ancient Chinese Pictogram Characters (象形汉字, 3000 years 商朝. 甲骨文 ) .

    https://youtu.be/F5uEpKwHqdk

    1.1: Overview Part 2

    Keypoints: (just a ‘helicopter’ view of the whole course syllabus)

    • Calculus: infinite Product, infinite Sum (co-Product), End, co-End.
    • Kan-extensions
    • Geometry in “Abstract” aka Topology: “Topos”
    • Enriched Category : (2-category) Analogy : complex number makes Trigonometry easy; same does Enriched Category.
    • Groupoid => “HTT” : Homotopic Type Theory

    https://youtu.be/CfoaY2Ybf8M

    2.1 String Diagrams (Part 1)

    Composing Natural Transformations (Vertical & Horizontal): \alpha \; \beta (assumed naturality)

    https://youtu.be/eOdBTqY3-Og

    2.2 Monad & Adjunction

    https://youtu.be/lqq9IFSPp7Q

    Refs:

    1. Download BM’s book “Category Theory for Programmers” :

    https://github.com/hmemcpy/milewski-ctfp-pdf

    Applied Category Theory Course by Prof John Baez

    Join John Baez’s Azimuth Math Forum 导读 (Study Tour Guide) in Applied Category Theory (CT):

    https://forum.azimuthproject.org/discussion/1717/welcome-to-the-applied-category-theory-course?from=timeline

    John Baez (1961-) is the world’s expert in Category Theory. He gave a talk on CT in Hokkaido University last year.

    The 导读 is using a book by 2 mathematicians Brenden Fong and David Spivak in “Applied Category Theory” – Download the free book of this course here.

    Note: Both John Baez and his wife Lisa Raphals (Professor in Chinese) work now in National University of Singapore – Center of Quantum Technologies & Philosophy, respectively.