Conceptual Mathematics
In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics, Second Edition, introduces the concept of 'category' for the learning, development, and use of mathematics, to both beginning students and general readers, and to practicing mathematical scientists. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories.
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- F. William Lawvere - Conceptual Mathematics: A First Introduction to Categories
Eugenia Cheng - The Joy of Abstraction: An Exploration of Math, Category Theory, and Life
She builds up to the category theory chapters.
Book club: https://youtube.com/playlist?list=PLhgq-BqyZ7i7tEEQVG5rlOG8y...
- He has also done a youtube series on category theory for programmers [1]. I have personally been using a book recommended by someone here called "Conceptual Mathematics"[2] and have been finding it great so far in the sense that it only relies on high-school level maths concepts while introducing the categorical concepts in contexts where you can see the value of the approach [3].
It is really an interesting set of ideas and as soon as you start using it, some things start to change in how you approach problems. Just today in fact I was working on something and thought to myself "I need the opposite of a forgetful functor - I wonder what that is?" (turns out it's the "free group functor"). But because I knew about the existence of the concept it helped turn what would otherwise have been a maths problem that I would have really struggled with into something that was conceptually easy for me to deal with. I essentially could use the category theory to turn a problem that was hard to think about into something easy to think about (and also to show that this was valid) and then I could solve the problem in a conventional way.
[1] https://www.youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbm...
[2] https://www.cambridge.org/highereducation/books/conceptual-m...
[3] As opposed to say Emily Riehl's "Category Theory in Context" https://math.jhu.edu/~eriehl/161/context.pdf where the maths required to even get started with the examples in the intro is graduate level. To get a sense, this is literally the first sentence of the first chapter: "A group extension of an abelian group H by an abelian group G consists of a group E together with an inclusion of G ↪→ E as a normal subgroup and a surjective homomorphism E ->> H that displays H as the quotient group E/G. " It's pretty full on.
- _Conceptual Mathematics_ is basically category theory for high school students.
https://www.amazon.com/Conceptual-Mathematics-First-Introduc...
- I'm maybe too close to the problem to evaluate well (studied foundational math) but I know that Lawvere and Schanuel's book "Conceptual Mathematics" has been fairly well-regarded as a path into category theory.
- Books. Some books on my shelf covering these topics:
"Type Theory and Functional Programming" by Simon Thompson (PDF available via quick search)
First 120 pages of "Type Theory and Formal Proof" by Rob Nederpelt / Herman Geuvers
For some early stuff on dependent types (also covered in the above), check out "Intuitionistic Type Theory" by Per Martin-Lof
"Computation and Reasoning: A Type Theory for Computer Science" by Zhaouhui Luo
"Type-Driven Development with Idris" by Edwin Brady
"Gentle Introduction to Dependent Types with Idris" by Boro Sitnikovski
"Types and Programming Languages" by Pierce
"Practical Foundations for Programming Languages" by Robert Harper (or any of his YouTube videos re/Type Theory)
"The Little Typer" by Friedman and Christiansen
"Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy" by David Corfield (just took this one to OBX North Carolina to enjoy on my beach vacation last week! Got some salt water in those pages. Also speaks to Category Theory.)
For Category Theory:
"The Joy of Abstraction" (2023) by Euginia Cheng
"Basic Category Theory for Computer Scientists" by Benjamin C. Pierce
"Conceptual Mathematics: A first introduction to categories" by Lawvere and Schanuel
"Categories for the Working Philosopher" by Landry
- An amazing book that takes a clear and descriptive path to topos theory.
If you want to take it a little slower, you can start with Lawvere's
"Conceptual Mathematics"
https://api.pageplace.de/preview/DT0400.9780511590092_A23569...
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