群论笔记 01

category is a bunch of objects

1. category
2. objects
   • what is an obj, you CAN NOT tell.
   • because object in this Cat is a primitive concept.
   • has no properties; has no internal structure
   • it like an atom, a point.
   • the reason Cat have objects, is to mark the begining and end of an arrow
   • don't server any other purpose
3. morphism(arrow)
   • something goes between 2 objects
   • also primitive concept.
   • has no properties; has no internal structure
   • but have begining and end
   • [0, infinite] arrows between 2 objects

when we talk about arrow, it is very like the hunter-gather in ancient time.

hunter-gather knows spatial.

In a mathmatical course, professor often draw some point on backboard, and they also have spatial realationship: above/below/left/right. Cat, object, also have spatial relationship , higher level of abstraction, lower level of abstraction.

hunter-gather knows movement.

Cat, arrow, has start point and end point, which is something like movement and relationship everytime you define a Cat, you specify what are the objects of ths Cat, and for each pair of objects you specify the arrows that between them.

you can have very arbitray manner of combination of arrows and objects:

1. [0, infinite] arrows from objects to itself 2. [0, infinite] arrows

from this to that 3. [0, infinite] arrows from that to this

tip: arrows can have different name, so, many arrows with same geginning and ending can have different name, they are differnet.

gºf — "g after f" which also illustrate the order of computation, computation of g come after computation of f, maybe, there are other composition of these two objects, but this "g after f" must exist.

if composable then there must be a composition.

what you need to describe a Cat, is a composition table, like a multiplication table, from which you can compute out many many numbers, Cat is some things "the whole objects computed form composition table, every composition is a arrow"

Axiom1 : Identity > every object has an circle arrow, this arrow is called Identity — notation:'id\_a'(if the object is called 'a'). why name it *'identity', because in an composition, it just play an role like "do nothing". Same with what 1 do in an production, like 3 * 1 = 3, where *=1= is the *identity

Axiom2 : Left/Right Identity > id\_aºf = f fºid\_a = f

Axiom3 : Association > hº(gºf) = (hºg)ºf

we must say that, axiom-3 maybe not "equal" but just "isomophic"

tip: keep in mind that, if some thing is a Cat, he must obey this 3 axiom, vice and versa.

why Cat is not a Set. set is just something just what it looks like.

{1,2,3}
// this is a set, has nothing implicit information.
// it has 3 elements: 1, 2, 3

Cat is something far more than what it looks like.

Cat{1,2,3}
// if no arrows in it, it is truely a set
// but if arrows appear, it exponentially grow like a production
// table.

isomophims and identity; isomophism is a weak identity.

Cat will be more complex in Haskell, because Haskell is a lazy language, type will contains this undefined value, the bottom value, this means if you try to evaluate it you will get into an infinite loop.

Cat rarely concern about time, because time is hard to describe in math, but for programming, ensentially it's a calculation, time is very important, infinite loop is meaningless in that scenario. So in Haskell, codes trap into a infinite loop will return an Integer — bottom value which menas it never terminate.

The simplest model for types is that they are just *sets*(not suited for Haskell, because of bottom value, but suit for ML) then, we can model programming as in a Cat of sets, functions are just functions between sets. Function from one set to another set.

Cat	Prog	ModelProg
object	type	set of values
arrow	function	function between sets