Separate comment for references to classical category theory books and resources. I don’t think any of these are exactly what you are looking for, but they each propose a different perspective on these concepts, which might be the best we have now.
The best textbook I know of is Category Theory by Awodey. It is both rigorous and intuitive, at least at the level of maths textbook. There are a lot of examples, as concrete as possible, and the differences between them and how that inform the abstract definitions are treated in details.
Do not go to Maclane’s Category Theory for Working Mathematician. Not that it is a bad book, just that it is the most honest title of a book I ever saw. Maclane writes for the working mathematician, so not even the graduate student in maths fits exactly his standards.
For a glimpse of the structuring power of category theory and its links to physics and computer science, Physics, Topology, Logic and Computation: A Rosetta Stone by Baez and Stay is the place to go. This paper also argues eloquently that the most important categories are not the one similar to the category of sets.
A short one that I like is Basic Category Theory for Computer Scientists by Pierce. It is short, to the point, and goes deeper into the applications to theoretical computer science. One caveat is that Pierce is the kind of computer scientist that studies proof theory and teaches Coq and theorem proving. So it might be slightly too abstract for some people.
Dis-recommending Awodey’s book. I tried reading it as an introduction, and it was tremendously confusing. I’m going through Harold Simmons’ An Introduction to Category Theory now, and from a beginner’s standpoint, it’s far superior.
I didn’t know of Simmons book, will take a look. To be honest, I never went very far in Awodey, but it still worked better for me than McLane (not hard, I know).
I just found another interesting reference: Categories for the practising physicist. Although this is not exactly about discarding undue abstraction, it does present many concepts in terms of concrete examples, and there are even real-world categories defined in it!
Separate comment for references to classical category theory books and resources. I don’t think any of these are exactly what you are looking for, but they each propose a different perspective on these concepts, which might be the best we have now.
The best textbook I know of is Category Theory by Awodey. It is both rigorous and intuitive, at least at the level of maths textbook. There are a lot of examples, as concrete as possible, and the differences between them and how that inform the abstract definitions are treated in details.
Do not go to Maclane’s Category Theory for Working Mathematician. Not that it is a bad book, just that it is the most honest title of a book I ever saw. Maclane writes for the working mathematician, so not even the graduate student in maths fits exactly his standards.
For a glimpse of the structuring power of category theory and its links to physics and computer science, Physics, Topology, Logic and Computation: A Rosetta Stone by Baez and Stay is the place to go. This paper also argues eloquently that the most important categories are not the one similar to the category of sets.
A short one that I like is Basic Category Theory for Computer Scientists by Pierce. It is short, to the point, and goes deeper into the applications to theoretical computer science. One caveat is that Pierce is the kind of computer scientist that studies proof theory and teaches Coq and theorem proving. So it might be slightly too abstract for some people.
Dis-recommending Awodey’s book. I tried reading it as an introduction, and it was tremendously confusing. I’m going through Harold Simmons’ An Introduction to Category Theory now, and from a beginner’s standpoint, it’s far superior.
Thanks for the feedback!
I didn’t know of Simmons book, will take a look. To be honest, I never went very far in Awodey, but it still worked better for me than McLane (not hard, I know).
Cool. Let me know if you have opinions on it.
I just found another interesting reference: Categories for the practising physicist. Although this is not exactly about discarding undue abstraction, it does present many concepts in terms of concrete examples, and there are even real-world categories defined in it!