Category Theory (Steven Roman) – (Part II)

[Continued from (Part I)…]

Category Theory (范畴学) is the “lingua franca” (通用语) of mathematicians, used commonly by the 2 different major Math branches : Algebra & Analysis.

比喻:
武术分”外家拳”少林派, “内家拳” 武当派。
两家的 lingua franca (通用语)是 “” – 硬气功(少林), 柔气功(武当)。

In essence: A Category consists of
1. Objects
2. Relationship among objects (Morphism = a set denoted by Hom (A,B), where A, B are objects)
3. Structure: preserved by Morphism
4. An Identity morphism

Examples:
1. SET Category:
◇ Objects (Sets),
◇ Structure (Cardinality)
◇ Morphism (Set Functions: which preserve Set Structure)
◇ Identity morphism (Set to itself)

2. GROUP Category
◇ Objects (groups)
◇ Structure (Set, 1 closed binary operation)
◇ Morphism (group homomorphisms)
◇ Identity (neutral element ‘e’)

3. SINGAPOREAN Category
◇ Objects (Singapore citizens)
◇ Structure (multi-racial)
◇ Morphism (kiasu-ism)
◇ Identity morphism (I = ME-ism = 令伯 ‘lim-peh-ism’)

Lecture 2: [Video]
◇ Functor: morphism between Categories
◇ Diagrams: Morphisms uses arrows –> (Functors use double arrows ==>)
◇ Commute
Special Types of Functors:
Full (Surjective)/ Faithful (Injective) /Fully Faithful (Bijective) / Embedding (Fully Faithful + Objects injective)

Yoneda’s Lemma (Deep and Trivial Philosophy)

◇ Intuitively, All objects are defined by its actions (or by all the actions acting upon it).

◇ Hence, it is more important to study the actions / relationships (Functors) than the object (Category) itself.

Formal definition explained below (last 5 mins of Tom Galatta’s video,

image

Special Type of Morphisms
Epi(c): right cancellable f
gof = hof => g = h
Mono (monics) : left cancellable f
fog = foh => g = h

Concrete Categories: Objects are Sets, morphisms are set functions.
Otherwise, they are called Abstract Categories

Notes: Analogy of Yoneda’s Lemma

This corresponds to the old saying, 

“A man is judged by (all) his actions“.

《易·系辞上》:“言行,君子之枢机。

圣经里也是如此说:

image


原来最高深的数学( Category Theory) 回归 到 哲学(易经, 宗教)。
数学只是把它 符号化 / 规律 化, 一个tool, 容易推理, 从人类看得见的空间到看不见的无限大(或无限小)的大千世界。

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