I second this; I too would appreciate a Qiaochu’s-eye view. Also, are there hidden dependencies in the MIRI reading list? I feel like some subjects would be most profitably studied before or after others—category theory is mentioned as helping you understand the underlying structures of mathematics and generalize your understanding—so does that mean read early, or read near completion of the list?
As Gram Stone says, most people think category-theory-first is a bad idea. I agree that it won’t suit most people. My recommendation is to learn category theory simultaneously with everything else, and also to learn as much of it as seems helpful for understanding whatever else you’re learning, and no more; this is what I did. For example, I learned about adjoint functors as I was doing enough abstract algebra to run into interesting examples of adjoint functors, such as the induction and restriction functors on group representations. I never went through a category theory textbook (in general I mostly learn from blog posts, Wikipedia, the nLab, etc.) and so never learned things that didn’t seem useful for something else.
In general I’m a big fan of learning many fields simultaneously, so it’s easier to see connections between them. The relevant dependencies don’t parse into subjects for me; they’re smaller conceptual chunks like “understand, in any of the places where it appears, the concept of currying, or else you won’t understand a ton of things like how to pass between the two standard description of group actions, or why the double dual of a finite-dimensional vector space is naturally the same vector space again.” (It took me a few hours of frustrated thinking to really grok this and once I did I was able to use it smoothly everywhere it appeared forever.)
Re: category-theory-first approaches; I find that most people think this is a bad idea because most people need to see concrete examples before category theory clicks for them, otherwise it’s too general, but a few people feel differently and have published introductory textbooks on category theory that assume less background knowledge than the standard textbooks. If you’re interested, you could try Awodey’s Category Theory (free), or Lawvere’s Conceptual Mathematics. After getting some more basics under your belt, you could give either of those a shot, just in case you’re the sort of person who learns faster by seeing the general rule before the concrete examples. (These people exist, but I think it’s easy to fall into the trap of wishing that you were that sort of person and banging your head against the general rule when you really just need to pick up the concrete examples first. One should update if first-principles approaches are not working.)
I second this; I too would appreciate a Qiaochu’s-eye view. Also, are there hidden dependencies in the MIRI reading list? I feel like some subjects would be most profitably studied before or after others—category theory is mentioned as helping you understand the underlying structures of mathematics and generalize your understanding—so does that mean read early, or read near completion of the list?
As Gram Stone says, most people think category-theory-first is a bad idea. I agree that it won’t suit most people. My recommendation is to learn category theory simultaneously with everything else, and also to learn as much of it as seems helpful for understanding whatever else you’re learning, and no more; this is what I did. For example, I learned about adjoint functors as I was doing enough abstract algebra to run into interesting examples of adjoint functors, such as the induction and restriction functors on group representations. I never went through a category theory textbook (in general I mostly learn from blog posts, Wikipedia, the nLab, etc.) and so never learned things that didn’t seem useful for something else.
In general I’m a big fan of learning many fields simultaneously, so it’s easier to see connections between them. The relevant dependencies don’t parse into subjects for me; they’re smaller conceptual chunks like “understand, in any of the places where it appears, the concept of currying, or else you won’t understand a ton of things like how to pass between the two standard description of group actions, or why the double dual of a finite-dimensional vector space is naturally the same vector space again.” (It took me a few hours of frustrated thinking to really grok this and once I did I was able to use it smoothly everywhere it appeared forever.)
I like Conceptual Mathematics a lot.
Re: category-theory-first approaches; I find that most people think this is a bad idea because most people need to see concrete examples before category theory clicks for them, otherwise it’s too general, but a few people feel differently and have published introductory textbooks on category theory that assume less background knowledge than the standard textbooks. If you’re interested, you could try Awodey’s Category Theory (free), or Lawvere’s Conceptual Mathematics. After getting some more basics under your belt, you could give either of those a shot, just in case you’re the sort of person who learns faster by seeing the general rule before the concrete examples. (These people exist, but I think it’s easy to fall into the trap of wishing that you were that sort of person and banging your head against the general rule when you really just need to pick up the concrete examples first. One should update if first-principles approaches are not working.)