I'm a computer scientist, not a mathematician (but I've taken around 25 college math courses spread over many years). Nevertheless, I do have a couple of suggestions and observations that I will address to the author that I hope is seeing this.
First, it would be a very sophisticated high school student to tackle topology and some of the other areas of abstract mathematics. I really like the topics you've picked for your book, but they do seem to require quite a bit of mathematical sophistication (e.g. Topology).
Secondly, I feel that there are a few important fields that you might consider adding to your napkin: Combinatorics, Statistics, Differential Equations, and Logic.
The usefulness and the importance of understanding statistics is pretty obvious in today's data dominated world. Statistics seems to fall outside of Mathematics at some (most?) universities, but I keep my statistics books right next to my math books.
Combinatorics is full of interesting results some esoteric (the friendship theorem) and some practical (stars and bars). The proof techniques of combinatorics are also worth studying for their own sakes (like the probabilistic method).
I've always felt a love hate relationship with Differential Equations. Theoretically, they are disappointing ("oh hey, let's try this, surprise its the solution!") but practically they are needed everywhere.
One of the best math experiences that I had in high school was a logic course that I took one summer with two other students. What fun and it always served me well in course 18.
The point of the "napkin" isn't to be a generic Maths textbook; it's to trace up the prerequisite chain from category theory until it connects with high-school-level maths. Do you need statistics or differential equations to understand category theory?
I like the book but the contents seem to go well beyond the prerequisites of category theory—-it’s almost 900 pages.
I got the impression that the author was not simply attempting to connect high school math to category theory but was providing a broader survey of higher math. I interpreted the author’s remarks about the path to category theory as the inspiration for embarking on the project that has turned out to be a wide survey of higher math that might benefit young mathematicians.
First, it would be a very sophisticated high school student to tackle topology and some of the other areas of abstract mathematics. I really like the topics you've picked for your book, but they do seem to require quite a bit of mathematical sophistication (e.g. Topology).
Secondly, I feel that there are a few important fields that you might consider adding to your napkin: Combinatorics, Statistics, Differential Equations, and Logic.
The usefulness and the importance of understanding statistics is pretty obvious in today's data dominated world. Statistics seems to fall outside of Mathematics at some (most?) universities, but I keep my statistics books right next to my math books.
Combinatorics is full of interesting results some esoteric (the friendship theorem) and some practical (stars and bars). The proof techniques of combinatorics are also worth studying for their own sakes (like the probabilistic method).
I've always felt a love hate relationship with Differential Equations. Theoretically, they are disappointing ("oh hey, let's try this, surprise its the solution!") but practically they are needed everywhere.
One of the best math experiences that I had in high school was a logic course that I took one summer with two other students. What fun and it always served me well in course 18.