**“Categories for Dummies”**

**(French)**

**Example 1:**

Paris (P) -> Rome (R) -> Amsterdam (A)

**Objects**: cities {P, R , A…}

**
Morphism (Arrow)**: railway

**Identity**: railway within the city**Associative**: (P -> R) -> A = P -> (R ->A)

=> **Category of “Euro-Rail” **

Remark:In the similar sense, China “One Belt One Road” (OBOR)” 一带一路” is a “Pan-China-Europe-Asian” Category “泛中-欧-亚” 范畴

Example 2:

A, B are categories

functor f : A -> B

f (B) has the “**information**” on A, with some loss of information since f *may not* be a MONOMORPHISM (单射 Injective).

**Remark**: Technical Drawing, the **views** from Top , Left, Right, or Bottom of the object are **Functors – **which provide **only** 1 view from 1 direction with **loss** of information from 3 other directions.

**Example 3: Natural Transformation**

**A = 0 1 2 3 **

f : A -> B

**B = Ladder steps:**

**f(0)|**

**f(1)|**

**f(2)|**

**f(3)|**

g : A -> B

**B = Staircase steps :**

**g(0)||**

**g(1)||**

**g(2)||**

**g(3)||**

**Natural Transformation: => ** α

**α : f (i) => g (i)**

**f(0)| => ****g(0)||**

**f(1)|**** => ****g(1)||**

**f(2)|**** => ****g(2)||**

**f(3)|**** => ****g(3)||**

α

transforms naturallytheLadderto theStaircase.