# Isomorphism

An isomorphism is a loose sense of equality.

"A equals to B" means A and B share all their properties in common
. Formally, "A=B" <=> "For any proposition P(x), P(A) <=> P(B)". T
his is Leibniz's law.

However, this definition is too strict. You and what you were in 1
0 years ago are not identical, because they don't share all their
properties in common. We need a loose sense of equality. That's wh
ere an isomorphism come in.

If A and B share SOME properties in common, we say A and B are "is
omorphic". A relation which A and B have is called an "isomorphism
". Since they share SOME properties, you should say WHAT propertie
s they share in common. If they have the same color, they are "col
or isomorphic". If they have the same firmness, they are "firmness
isomorphic". We denote "A and B are isomorphic" as "A ~= B".

A morphism is a weak version of an isomorphism. For firmness, "A i
s more firm than B" is a morphism. For propositions, "A implies B"
is a morphism. We denote a morphism as "A -> B". When a morphism h
as its inverse, it becomes an isomorphism. If "A is more firm than
B" and "B is more firm than A", then A and B have the same firmnes
s, i.e., they are "firmness isomorphic". If "A implies B" and "B i
mples A", then they are logically equivalent, i.e., they are "logi
cally isomorphic". Note that these examples are simple, because th
ey take the form of "[A->B & B->A] <=> [A~=B]". But if a morphism
is a function, f:A->B and g:B->A do not need to be each other's in
verse.

We call objects, morphisms, the composition together as a "categor
y". It makes a domain of the property, like "Color category", "Fir
mness category", &c. In a category, we can't distinguish an isomor
phism with the identity, because we restricted the domain. Thus,

> Isomorphic objects are identical.
>
> The Principle of Structuralism

We can also make a relation between categories. A relation of this
kind is called a "Functor", like a "function" but between categori
es. By a functor, objects map to objects, morphisms map to morphis
ms, a composition maps to another composition. You can think of a
category of functors too. In that category, an isomorphism is a "n
atural isomorphism". It means the two are REALLY the same.

There are many general results by category theory. Think of the fi
rmness category. [A->B] <=> [A is more firm than B]. If there is a
A->X for every object X, then it means A is the most firm object.
If we think of X->B instead, we get the least firm object B. We ca
ll these kind of objects, the initial object and the terminal obje
ct. These objects are the optimal solutions with a type of functor
s called "Adjoint functor". With categories, We can think of the o
ptimality from a more general point of view.

For more information, I recommend "Category theory" by Steve Awode
y. "Categories for the Working Mathematician" by Saunders Mac Lane
is the bible because HE made the theory (Awodey is a student of hi
m), but it requires mathematical maturity. If you can, read the ma
ster and not his pupil. In contrast, Awodey didn't write for math
students only, so it is better for noobs.