I'd like to dip my toes into some specific areas of mathematics (like Category Theory) and my problem is that I did not find a flow chart which displays how different branches of mathematics depend on each other so I can narrow down the topics I need to look into.
I've found this question but the links in the answers are either broken or useless.
Having a flowchart like that would be immensely helpful for a lot of people. Is there such a thing? If yes, where?
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It is my opinion that no such flow chart can exist. Plausibly, one could produce a directed graph of math subjects, but there is no chance it could be accurate and acyclic. The reality is that subjects build off each other.
Further, the main obstacle to learning math is not so much "not knowing the prerequisites" as that learning math takes a lot of effort. Learning math is a trainable skill, which you gain by... working hard to learn lots of math. If you haven't learned to learn math to a sufficient degree, you can technically know the prerequisites for, say, measure theory and still get nowhere trying to work through a book on it. This is in part (but not entirely) what people refer to when they speak of "mathematical maturity."
So if you want to learn about a specific area, what you should really do is find out what topics are somewhat connected to it that have textbooks at a level you currently find readable, learn from those, and every once in a while come back and try to learn your target subject.
Category theory, which you expressed interest in, provides a sharp example of this. There is a very, very good book "Category Theory in Context" by Emily Riehl. The technical prerequisites for C.T.C. are essentially zero, but the book is full of ideas and arguments that a mathematician might be able to pick up on with their first scan but which would cause fits in somebody without any math background. By far the best way to prepare to learn from this book is just to learn a bunch of vaguely algebraic and topological math until you can learn from it.
Summary: Figure out what is connected to your subject (which will be a lot) and learn whatever is interesting and accessible within those broad limits. Come back to the subject in question and see whether things are making more sense.
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If you're looking for a subject that is taught in Math Bachelor/Master, you can backtrack the dependencies of the subject by looking up the requirements in the module hand book of a university that's teaching this course.
Otherwise, maybe this link helps you:
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Here's an article with a diagram that can be interpreted as a DAG with implicit arrows going in the up direction towards more advanced areas with annotations:
http://www.brainmaker.net/blog/you-mean-theres-math-after-calculus
It's a pragmatic breakdown starting with Algebra I. As others have stated here, there is no definitive topology for a canonical mathematical curriculum.
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If you want to enter some area of mathematics, it would be better to say what you already know and ask what are the prerequisites (wrt to what you know) of such area.
That being said, you should take a look at the diagrams on Mac Lane's "Mathematics, Form and Function". I guess the diagrams with the rest of the material in the book are going to be the helpful and useful to you.
If you coarse grain your areas of mathematics like as broad as "category theory" then there will be parts were you need a certain prerequisite and parts where you don't. This means you will not really get a DAG anymore.
Say you have A is a prerequisite for B for C for D. Simple example. But A and C are both treated as parts of the subject 1 and B and D as parts of subject 2. You now have a loop where both subjects require each other.
Because these broad subjects have lots of parts that are interconnected with each other like this, you won't be able to have a directed system if you want such broad branch titles. If you go with the very specific parts, then you might be able to build a flowchart, but it will be too big to be useful.