[**Continued** from previous BM Category Theory …]

**Intuition**: **[Artificial Intelligence**] You teach the computer like to a Primary 6 kid, that **Algebra** is a **type** of expression **(f**) which, after evaluation, returns a value.

**If a = i (initial)*** **[or u (terminal)]*,

**Intuition**: *Fix-point because, the Initial “i”, after evaluating the expression f, returns itself “i”.*

**Lambek’s Lemma**

**Note**: Endo-functor is a **functor** (equivalent to *function *in Set Theory) within the same Category (Endo = Self = 自)

Video 8.1 F-Algebras & Lambek’s Lemma

Video 8.2 Catamorphism & Anamorphism

**foldr ~ catamorphism (浅层变质) of a Fix-point endo-functor on a List.**

**Examples**: Fibonacci, Sum_List

Remark: **Cool Math**! the more advanced concept it is, the more closer to Nature (eg.Geology, Biology) : Catamorphism 浅层(风化)变质, or *“thin-layer change in nature” *(in Functional Programming languages: foldr or map) eg : **add1** to a list (1 5 3 8…)

= (2 6 4 9 …)

**Intuition: **Reverse of Algebra, given a value, Coalgebra returns an expression (f).

**Anamorphism (合成变质) ~ unfoldr**

**Example**: Prime numbers

**Remark**: Anamorphism (合成变质) or “*synthesised change in nature*“: eg. Start from a “**seed**” prime number “2” generates all other infinite prime numbers (3 5 7 9 11 13 17 …)

**Note: In Haskell, no difference** between Initial and Terminal Fix-points. However, since Fix-point is not unique, in **Category Theory **there is the **Least** Fix-point (Initial) and **Greatest** Fix-point (Terminal).

Ref:

Reading “**Understanding F-Algebra** ” by BM: https://bartoszmilewski.com/2013/06/10/understanding-f-algebras/

Catamorphism (下态) : https://www.zhihu.com/question/52180621/answer/129582557

Anamorphism : https://zhuanlan.zhihu.com/cofree/21354189

F-Algebra & F-coalgebra: http://stackoverflow.com/a/16022059/5822795