# 范畴学和物理 Category Theory & Physics

1. 代数拓扑Algebraic Topology,
2. 代数几何 Algebraic Geometry,
3. 范畴学 Category Theory

1. Newton Physics =>他发明 New Math “Calculus”

2. Maxwell Physics => Magnetic Field => New Math : Fibre Bundle

3. Einstein General Relativity => Gravitational-Space-Time => New Math : Riemann Non-Euclidean Geometry

4. Quantum Physcis => Particles + Wave => Linear Algebra (Quantum Mechanics)

5. Quantum IT Revolution => 量子信息和它们的量子纠缠 (Quantum Entanglement)

https://m.toutiaocdn.com/group/6786864463778677252/?app=news_article_lite&timestamp=1580324149&req_id=202001300255480101941000191343BA89&group_id=6786864463778677252

# A Programmer’s Regret: Neglecting Math at University – Adenoid Adventures

Video Game Animation: Verlet Integration

AI: Stats, Probability, Calculus, Linear Algebra

Search Engine : PageRank: Linear Algebra

Abstraction in Program “Polymorphism” : Monoid, Category, Functor, Monad

Program “Proof” : Propositions as Types, HoTT

https://awalterschulze.github.io/blog/post/neglecting-math-at-university/

Abstraction: Monoid, Category

Category

# New Math applied in Modern Physics

Keywords:

• Quantum Entanglement (QE) : 量子纠缠
• Topological Order (Long-range QE) : 拓扑序
• Topological Insulator (Short-range QE) : 拓扑绝缘体
• Rigid State : 凝聚态
• Cohomology : 上同调
• Tensor : 张量
• Category Theory : 范畴学
• Group Theory : 群论
• Algebraic Topology : 代数拓扑

….

1. Four “Physics” Revolutions aided by Math

Math is applied to explain the Nature (Physics) :

Calculus (Differential Equation) : Newtonian Physics

Riemann Geometry : Einstein General Relativity

Partial Differential Equation: Maxwell Electro-Magnetic Field

Linear Algebra (Matrix) : Quantum Mechanics

Group Theory: Physics / Chemistry Symmetry

Next…

Category Theory : Topological Order of the long-range Quantum Entanglement

Algebraic Topology: Topological Insulator (2016 Nobel Prize Physcis)

2. The 5th Physics Revolution aided by New Math tool : Category Theory

3. Category Theory = “Structural Relationship” Math

4. Algebraic Topology

(想看更多合你口味的内容，马上下载 今日头条)

# 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

..

# Category Theory III Part 3

Tip:

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

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

R: D – > C

Tip: Think Monad as “List” container

# Category Theory : String diagrams

String diagrams (1 – 5)

2. Natural Transformation:

# 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

https://youtu.be/lqq9IFSPp7Q

Refs:

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

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

# David Spivak – Category Theory

Part 1 of 6 – λC 2017

1. Set & function
2. Category – Initial & Terminal Objects (may not exist in some category)
3. Product AxB
4. CoProduct (Sum) A+B
5. Exponential : (Type A-> B) or Function Object or Currying ${B^A}$

# 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.

# Crash course in Category Theory

Key Point:

Haskell & any FP compiler don’t check the Category Theory proof if your codes (eg. fmap) follow Functor’s Laws (eg. Preserve structure, identity) or Monad’s Laws !

# Category Theory in Life

Category Theory can be approved from 2 directions: 1) Pure Math, 2) IT Functional Programming (FP) .

Dr. Eugenia Cheng doesn’t know 2), she comes from 1).

The second video (below) approaches Category Theory from 2): Scala – FP language.

# Category Theory for Typescript

Key Points:

1. BCCC (Bi-Cartesian Closed Category): PRODUCT (tuple) , SUM (either) , EXPONENTIAL (function)

2. UNIT, ZERO, Absurd

3. Equivalences

View at Medium.com

# Programmers need Advanced Math ?

As software becomes more complicated for high-speed trains, driverless cars, missile weapons. .. and AI deeplearning algorithm, we can’t depend our life safety on the programmers who don’t understand the advanced maths behind these algorithms.

The Advanced Math is the Category Theory – the most advanced math foundation above “Set Theory” since WW2. Functional Programming is based on Category Theory with mathematical functions – always output correctly with no “side-effects”.

https://www.extremetech.com/computing/259977-software-increasingly-complex-thats-dangerous

# Category Theory – Purest of pure mathematical disciplines may also be a cornerstone of applied solutions in computational science

There exists in almost all Universities a clear division between pure and applied mathematics. A friendly (and sometimes not so friendly) rivalry exists between both sides of the divide, with separate conferences, separate journals and in many cases even a whole separate language. Category Theory was seen as such an abstract area of research that even pure mathematicians started to refer to it as “abstract nonsense“, and until the mid 1980’s almost all category theorists occupied a place hidden somewhere up above the ‘cloud level’ in the highest reaches of the peaks that defined “pure” maths.

By the mid 1990’s and then by the turn of the millenium, a whole world of computer programmers were learning basic category theory as part of their induction into functional programming. The best known product of these efforts is the Haskell language, but even in the past 7 or 8 yrs, workshops on category theory for computer programmers of all types have flourished and proliferated. It is almost as if there are two separate communities masquerading as one – mathematical category theory and computer programming category theory – and never the twain would meet. Or so it seemed, until now.

# Monads – FP’s answer to Immutability

Introduction:

• The curse of Immutability in Functional Programming – no “Looping” (recursion ok), no Date, no Random, …no I/O …
• Monad is the Savior of “No Side Effect: IO Monads

Promise of Monads (A)

Promise of Monads (B)

# Alejandro Serrano: Category Theory Through Functional Programming

(Part 1/3) – λC 2017

What is Category ?

Objects

Morphism (Arrows )

Rule1: Associative

Rule 2: Identity

A <– C –> B

Product of Categories : A x B

Unique

Sum of Categories: A + B

Unique

(Either a b)

Co-Product

Reverse all arrows.

Unique

Functor F: C-> D

Mapping of all objects (A, B) in categories C,D

Mapping of arrows f
f : A -> B
Ff : FA -> FB (preservation)
F Id = Id
F (f。g) = Ff。Fg

Example:

Constant C -> F
FC = k
Ff = Id

Arrow Functor F: C -> D

For any object A in C,

F A = D -> A
(Functional Type is also Type)

Functors compose !

Category of categories:

Objects: categories

Arrows : Functors

Haskell Category (Hask) is always Endo-Functor, ie Category Hask to itself.

Mapping of arrows.

Mapping of Objects = predefined

(Part 2/3) – λC 2017

(part 3/3) – λC 2017

# Category Theory II 9: Lenses

Lens = {get, set}

w = whole tuple, p = a field

get :: w -> p
set :: w -> p -> w

Example: take a pair (tuple)

get1 (x, y) = x

get1 :: (a, b) -> a

set1 (x, y) x’ = (x’,y)

set1 :: (a, b) -> a -> (a,b)

Polymorphic Types: change type of field

set :: w -> p’ -> w’
set1 :: (a, b) -> a’ -> (a’,b)

Lens Laws:

set w (get w) = id

get (set w p) = p

get (set w p) p’ = set w p’

Combine get & set (co-Algebra):

$\boxed {w \to(p, p \to w)}$

data Store p w = Store p (p-> w)

fmap g (Store p f ) = Store p (g.f)
g: w-> v
pf : p -> w
g.f : p-> v

Store = functor from w to Store = coalgebra = comonad [W a ->a]

instance Comonad (Store p) where

extract (Store p f) = f p

duplicate  (Store x f) = Store x ( λy -> Store y f)

[Think of p as key, f is the lookup function
f p = retrieve current value]

Comonad & coalgebra – compatible?

(Coalgebra) coalg ::a -> w a
(Comonad) extract :: w a -> a
duplicate :: w a ->w (w a)

coalg w = Store (get w) (set w)

set w (get w) = id [Lens Law]

$\boxed {\text {Lens = comonad + coalgebra}}$

If Type change (Set = Index Store : a = p, b = p’, t = w)

IStore a b t = IStore a (b ->t)

Object-oriented: eg. school.class.student

<=>”.” = functional programming

9.2 Lenses

Type Lense s t a b = forall f . Functor f => (a -> f b) -> s -> f t

$s \to \forall f. (a \to f b ) \to f t$

s  ->IStore a b t

forall  = polymorphism (?  Natural Transformation)

Yoneda Embedding Lemma

[C, Set](C (a -> – ), C (b -> – )) ~ C(b, a)

Adjunction: C (Ld,C) ~ D (D,  Rc)

Reference: “Understanding Yoneda”

https://bartoszmilewski.com/2013/05/15/understanding-yoneda/

adjoint functor in nLab

Yoneda lemma in nLab

维基百科: Hom functor

# The Yoneda Lemma

Representable Functor F of C ( a, -):

$\boxed {(-)^{a} = \text {F} \iff a = \text {log F}}$

Video 4.2 Yoneda Lemma

Prove :

Yoneda Lemma :
$\text {F :: C} \to \text {Set}$

$\boxed {\alpha \text { :: [C, Set] (C (a, -),F) } \simeq \text {F a}}$

$\alpha : \text {Natural Transformation}$
$\simeq : \text {(Natural) Isomorphism, "naturally" } \forall ( a, F )$

Proof: By “Diagram chasing” below, shows that

Left-side
: $\alpha \text { :: [C, Set] (C (a, -),F) }$ is indeed a (co-variant) Functor. (Higher-order Function)

Right-side: Functor “F a“. (Data Structure)

$\boxed {\forall x, (a \to x) \to \text { F } x \simeq \text { F } a }$

Note: When talking about the natural transformations, always mention their component “x”: $\alpha_{x}, \beta_{x}$

Video 5.1 Yoneda Embedding

Example 1: F = List functor [x]

$\boxed {\alpha \text { :: } (a \to x) \to [x] \simeq [a] }$

Example 2: F = C (b, -)

$\boxed {\alpha \text { :: [C, Set] (C (a, -), C (b, -) ) } \simeq \text {C (b, a)}}$

Note: check a , b is in co-variant or contra-variant position.

Example 3: F = Id

$\boxed {\alpha \text { :: } (a \to x) \to x \simeq a }$

Right-hand-side: a (data structure)

Left-hand-side : “(a -> x) “is a function called “handler” (or “continuation“) which takes the argument “a” to provide it as output : “(a -> x ) -> x“.

eg. handler to database query, over internet…(technique used in Compiler)

Co-Yoneda Lemma : (Contra-variant a , F)

$\boxed {\alpha \text { :: [C, Set] (C(-, a),F) } \simeq \text {F a}}$

Yoneda Embedding: Full and Faithful
$\alpha \text { :: [C, Set] (C (a, -), C (b, -) ) } \simeq \text {C (b, a)}$

Note: Instead of proving a , b are isomorphic, sometimes it is easier to prove the functors C(a, -) & C(b, -) are isomorphic.[Proof Trivial: functors preserve composition and identity]

ApplicationCo-Yoneda Lemma : (Contra-variant a , F)
$\boxed {\text {[C, Set] (C(- , a), C(- , b) } \simeq \text {C(a ,b)}}$

Pre-order Category $\boxed {a \leq b}$

$\forall x, \text {C(x , a)} \to \text {C(x , b)} \simeq \text {C(a , b)}$

$C(x , a) = x \leq a = \{\varnothing, 1 \}$
$C(x , b) = x \leq b =\{\varnothing, 1 \}$

3 possibilities: (“1” = singleton)
$id_{\varnothing} :: \varnothing \to \varnothing$
$absurd :: \varnothing \to 1$
$id_{1} :: 1 \to 1$
Note: $1 \to \varnothing$ impossible (function must have an image)

Verify

Right-Hand Side: $a \leq b$

Left-Hand Side: $\forall x,( x \leq a \implies x \leq b) \implies a \leq b$

Yoneda Embedding (Lagatta)

# BM Category Theory II 1.1: Declarative vs Imperative Approach

Excellent lecture using Physics and IT to illustrate the 2 totally different approaches in Programming:

1. Imperative (or Procedural) – micro-steps or Local 微观世界 [eg. C / C++, Java, Python…]
2. Declarative (or Functional) – Macro-view or Global 大千世界 [eg. Lisp / Clojure, Scala, F#, Haskell…]

In Math

1. Analysis (Calculus)
2. Algebra (Structures: Group, Ring, Field, Vector Space, Category …)

In Physics:

1. Newton (Laws of Motion), Maxwell (equations)
2. Fermat (*)  (Light travels in least time), Feynman (Quantum Physics).

In IT: Neural Network (AI)  uses both 1 & 2.

More examples…

In Medicine:

1. Western Medicine: germs/ viruses, anatomy, surgery
2. Traditional Chinese Medicine (中医): Accupunture, Qi, Yin-Yang.

Note (*): Fermat : My alma mater university in Toulouse (France) named after this 17CE amateur mathematician, who worked in day time as a Chief Judge of Toulouse City, after works spending time in Math and Physcis. He co-invented Analytic Geometry (with Descartes), Probability (with Pascal), also was the “Father of Number Theory” (The Fermat’s ‘Little’ Theorem and The Fermat’s ‘Last’ Theorem). He used Math to prove light travels in straight line (before Newton) due to “Least Time taken” (explained here in this BM video).

https://tomcircle.wordpress.com/2013/04/05/lay-cables-at-least-cost/

# BM Category Theory : Motivation and Philosophy

Object-Oriented  has 2 weaknesses for Concurrency and Parallel programming :

1. Hidden Mutating States;
2. Data Sharing.

Category Theory (CT): a higher abstraction of all different Math structures : Set , Logic, Computing math, Algebra… =>

$\boxed {\text {CT reveals the way how our brain works by analysing, reasoning about structures !}}$

Our brain works by:  1) Abstraction 2) Composition 3) Identity (to identify)

What is a Category ?
1) Abstraction:

•  Objects
• Morphism (Arrow)

2) Composition: Associative
3) Identity

Notes:

• Small  Category with “Set” as object.
• Large Category without Set as object.
• Morphism is a Set : “Hom” Set.

Example in Programming
:

• Object : Types Set
• Morphism : Function “Sin” converts degree to R: $\sin \frac {\pi}{2} = 1$

Note: We just look at the Category “Types Set” from external Macroview, “forget ” what it contains, we only know the “composition” (Arrows) between the Category “Type Set”, also “forget” what these Arrows (sin,cosin, tgt, etc) actually are, we only study these arrows’ behavior (Associativity).

2.1 : Function of Set, Morphism of Category

Set: A function is

• Surjective (greek: epic / epimorphism 满射),
• Injective (greek : monic / monomorphism 单射)

Category:  [Surjective]

g 。f = h 。f
=> g = h (Right Cancellation )

2.2 Monomorphism

f 。g = f 。h
=> g = h
(Left cancellation)

$\boxed { \text {Epimorphism + Monomorphism =? Isomorphism }}$

NOT Necessary !! Reason ( click here):

In Haskell, 2 foundation Types: Void, Unit

Void = False
Unit ( ) = True

Functions : absurd, unit
absurd :: Void -> a (a = anything)
unit :: a -> ()

[to be continued 3.1 ….]

# 代 数拓扑 Algebraic Topology (Part 1/3)

Excellent Advanced Math Lecture Series (Part 1 to 3) by 齊震宇老師

（2012.09.10) Part I:

History: 1900 H. Poincaré invented Topology from Euler Characteristic (V -E + R = 2)

Motivation of Algebraic Topology : Find Invariants [1]of various topological spaces (in higher dimension). 求拓扑空间的“不变量” eg.

• Vector Space (to + – , × ÷ by multiplier Field scalars);
• Ring (to + x) in co-homology
• etc.

then apply algebra (Linear Algebra, Matrices) with computer to compute these invariants  (homology, co-homology, etc).

A topological space can be formed by a “Big Data” Point Set, e.g. genes, tumors, drugs, images, graphics, etc. By finding (co)- / homology – hence the intuitive Chinese term (上) /同调 [2] – is to find “holes” in the Big Data in the 10,000 (e.g.) dimensional space the hidden information (co-relationship, patterns, etc).
Note: [1] Analogy of an”Invariant” in Population: eg. “Age” is an invariant can be added in the “Population Space” as the average age of the citizens.

Side Reading (Very Clear) : Invariant and the Fundamental Group Primer

Note [2]: Homology 同调 = same “tune”.

南朝 刘宋 谢灵运山水诗:
“谁谓古今殊，异代可同调
同调 = Homology
(希腊 homo = 同, -logy = 知识 / 调)

– “Reading an ancient text  allows us to think “in tune” (or resonant) with the ancient author.”

[温习] Category Theory Foundation – 3 important concepts:

• Categories
• Functors
• Natural Transformation

[Skip if you are familiar with Category Theory Basics: Video 16:30 mins to 66:00 mins.]

[主题] Singular Homology Groups 奇异同调群  (See excellent writeup in Wikipedia) (Video 66:20 mins to end)

1. Singular Simplices 奇异 单纯
2. Singular Chain Groups 奇异 链 群
3. Boundary Operation 边界
4. Singular Chain Complex 奇异 单纯复形

Part 1/3 Video (Whole) :

# Category Theory in Computing Languages

Yes, lots.

Just one example: a function with 2 inputs from A and B and results from C would have the type A x B -> C but in functional languages like Haskell we are using A -> (B -> C), i.e. a function that returns a function. This “currying” is exactly a the categorical definition of a cartesian closed category as one where Hom(AxB,C) is isomorphic to Hom(A,B -> C) and in this case you can replace Hom(X,Y) with X -> Y.

It is well known that effects in functional programming can be modelled by monads which is a concept from category theory. Nowadays a weaker structure called applicative functors has become very popular – needless to say also a concept from Category Theory.

Not all languages are functional (yet) but even non-functional languages can be understood using concepts from category theory.

Free Categories, Free MonoIds, Monads = MonoId+Endofunctor

# Category Theory by Steven Roman (Part I)

Excellent Category Theory lectures by retired Prof Steven Roman  from Uni. California: he used pen and A4 – paper with iPhone camera. Simple & good. (Only lighting could be brighter.)

Category Theory is one level higher abstraction, above the Abstract Algebra (Group, Ring, Field, Vector Space, Set…). It is the “Math of Math”, to make difficult math easy by studying the ‘Relationship’ (or Morphism, drawn as Arrows).

Lecture 1: study five topics
◇ Category
◇ Functor
◇ Natural Transformatiom
◇ Universal Ptoperty