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Joe Moeller (May 27 2020 at 19:15):This might have been said already, but this is a huge thread. I think a really important thing that has to happen for category theory to see a general acceptance by the broader mathematics community is to have undergrad/grad texts in NOT category theory written with the categories up front, present from the beginning and throughout. Most undergrad texts have a prechapter on set theory, it should just also have baby category theory as well.
Joe Moeller (May 27 2020 at 19:15):Books like this exist, Aluffi's Algebra Chapter 0 comes to mind.
Joe Moeller (May 27 2020 at 19:15):But I guess they actually have to get used.
Joe Moeller (May 27 2020 at 19:17):Also, we shouldn't be tricked by books that have "category" in the index. Hungerford is the book I learned modules and Galois theory from. It has a chapter on category theory, but it's complete garbage imo. I basically have to relearn everything from that class constantly because of the lack of categorical language that was used.
Joe Moeller (May 27 2020 at 19:20):And Hungerford's category theory chapter is the last one. What good is it after the fact?
John Baez (May 27 2020 at 19:32):That's weird. I never liked Hungerford's book enough to look at it for more than a minute. I guess it must have good points, since people use it a lot for algebra.
John Baez (May 27 2020 at 19:34):I really like Chris Isham's book Modern Differential Geometry for Physicists because it brings in a little category theory right at the start and then uses it.
Nikolaj Kuntner (May 27 2020 at 20:31):Do you know Corry Leo's work? https://www.springer.com/gp/book/9783764370022
John Baez (May 27 2020 at 20:31):No. It sounds interesting, since I like the history of these things.
John Baez (May 27 2020 at 20:32):Is that book good? If so, how?
Nikolaj Kuntner (May 27 2020 at 20:32):Yeah, it's nice and steps through some of Noethers work. The section on Hilbert is a bit longer.
Nikolaj Kuntner (May 27 2020 at 20:34):The jumping off point is Waerdens "Modern Algebra" text, when the work of Noether firstly entered introductory material
https://en.wikipedia.org/wiki/Moderne_Algebra
John Baez (May 27 2020 at 20:34):Great! Noether's theorem is about physics, not algebra... but everything Noether did is interesting to me.
Todd Trimble (May 27 2020 at 20:51):Nikolaj Kuntner said:
Do you know Corry Leo's work? https://www.springer.com/gp/book/9783764370022
Leo Corry. He's also written interesting things about Bourbaki.
Nikolaj Kuntner (May 27 2020 at 23:05):Todd Trimble said:
Leo Corry. He's also written interesting things about Bourbaki.
Copy-pasted the name from the website, I otherwise only know the Kellog's tiger going by that name.
The math historians community seems to be a tight one. I read just a few such books but they all speak of conferences and thank the same names. Like a science niche subfield. Maybe it is :)
Jacques Carette (May 28 2020 at 14:00):I'm a huge fan of the Smith & Romanowska book Post-Modern Algebra as it does Algebra in a "from what we know now" order instead of the 'history guides everything' order.
And, of course, Taylor's Practical Foundations of Mathematics is huge here. It really does take a lot of what we've learned from both mathematics and computer science (with a heavy emphasis on category theory and type theory), reorders it all, and presents it in a way that just makes a lot of sense. Still my favourite is that 'substitution' is the very first sub-section of the book, as it really is that foundational.
সায়ন্তন রায় (May 28 2020 at 15:03):@Jacques Carette Does the book take a categorical approach?
Jacques Carette (May 28 2020 at 15:07):Post-Modern Algebra is very aware of category theory, covers it in detail in later chapters, and so its development is coherent and influenced by CT, though it is hard to clearly say if their approach is 'categorical' per se.