﻿In this document I will talk about:
* Bijections and cardinality are directly related.


Sets are a category, and they’re structure is the empty set. In other words there is no direct structure.


Then, if sets were just made out of unreadable symbols, how would you be able to tell one set apart from another? The answer is cardinality. And the link exists through bijections.


Now that we have a category, we can talk about the structure preserving maps in that category. We can find a sufficient definition of what structure preserving means on sets doing some simple thinking.


Imagine you have a set A, and a map f. The question is: how must you restrict f such that the domain of the map f is A, and the target is a new set B.


** ugh it doesn’t work?? **


It has to do with invertibility for some reason, which is true for sets when the cardinality is equal, in other words when the map f is injective + surjective.


— I figured it out —


Linear maps are not the structure preserving maps for vector spaces. Rather, linear isomorphisms are (where linear isomorphism = linear map + bijection).


So my logic about what “structure preserving map” means is wrong. The first part is right: you have a space A and a map f, and you ask: what restrictions must you put on f such that the output of A and the structure on A is guaranteed to be also a structure.


The part that I’m missing is that you also need to ask the question: now having found a structure B, if you pull it back through the function inverse, is the new A equal to the old A, or in other words, make sure that you restrict f such that the inverse of B is A.


This fixes the issue with sets as a space, and presumably also topological spaces (since I remember having some issues).


So with sets, you must ask the questions: what restrictions must you put on f so that A passed through f is a set B, and B passed through f inverse is a set A. And the answer is that f must be injective + surjective, which equals bijective.


So the point is that when going forward, you must be one, when going backwards you must be the other, and when requiring all two, you must be bijective.


And the link to cardinality is also quite interesting. The tldr is that surjective means that domain is larger than target, and injective means target is larger than domain. So when you put both together, then the domain must be equal in size to the target. This is sort of like the classic if a is greater or equal to b. And a is smaller or equal to b, than a is equal to b.