Category Theory (Steven Roman) – (Part II)

Math Online Tom Circle

[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)
3. Structure: preserved by Morphism
4. An identity (Self)

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

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

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

Lecture 2:
◇ Functor: morphism between Categories
◇ Diagrams: arrows
◇ Commute
Special Types of Functors:

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