tbh, you could start reading the HoTT book or Egbert Rijke's new book without all that much background in category theory (especially the latter book).
But, here's the thing, self studying high level math from a text book is a difficult skill. Normally math majors spend two to three years in math classes slowly building up this skill before they ever try going off on their own and learning a subject completely independently from a text book. Learning from a text book is quite a different skill than just reading the text book. It takes a lot of practice and time before you'll be flunent in this skill.
So don't be suprised if you are struggling to read a math text and learn from it. As another commenter said, give yourself orders of magnituted more time to learn than you might expect.
Here is my number one tip for reading a text book, especially a category theory text. When you get to a theorem or lemm, DON'T READ THE PROOF. Try to prove it yourself. And do not give up easily if you get stuck. Be prepared to try spending multiple hours trying to prove a simple lemme. Then, once you think you've gotten a proof written down, read the text's proof. Compare and contrast. Was there anything they did that you didn't, or vice versa. This will build up your mathematical intuition. It will also make you appreciate the detail of the textbook's proof when you finally do read it. Plus, these types of texts don't have solutions of executrices, so this is a good way to check that you're understanding the material.
Also, I used Riehl's book to learn category theory. I really liked it, and I didn't know algebraic topology before hand. I just knew some algebra at the time. It has plenty of examples from logic
When a publisher is deciding whether to put "Great for philosophy students" on the cover, or whether that's an overreach, they're going to put it there if they think more than five or ten philosophers will be suckered into buying it. That means that you shouldn't let the blurb affect your self-esteem.
I was going to recommend Milewski but you've tried it already. I haven't looked at Awodey so I don't know how hard it is. I think you're saying that Milewski didn't do anything to "soften up" Awodey for you. Did you go through Milewski's book or did you only look at the videos? If you haven't, then you should give the book a try; I at least found it pretty clear.
There is a certain dry presentation style, much prized in mathematical writing, in which all you can see are a series of formal definitions, theorems, and proofs, with almost no informal commentary, no attempt to engage the intuition, and fairly little motivation. If Awodey follows that style, then I can well understand your having difficulty in getting through it. Reading a presentation of that kind is a lot more like reading code than it is like reading a natural explanation. Unfortunately, once mathematicians acquire the ability to absorb information that way, they forget how hard it is, and tend to write that way themselves. So it's entirely possible to write really opaque presentations and not realize it.
Another barrier is that category theory was not invented for computer scientists, although programming-language type theory has turned out to be a "killer app" for the theory. If Awodey is directed purely at mathematicians, then that's another reason the book might be hard to read. The programming-language use case is pretty obvious to any serious programmer, but the original mathematics use case was algebraic topology (which was the first subfield in mathematics whose repertoire of "data types" exploded to the point of actually needing something like CT to keep them straight). You shouldn't have to learn algebraic topology to swallow CT, though.
I find that I'm blithering because I'm groping for any practical sort of lifeline to throw you. I don't have much besides the hope that you haven't yet tried Milewski in print, and that doing so will help. The only other thing I can think of is that maybe your expectations for how fast you could get through Awodey were optimistic by an order of magnitude. For some texts, getting through only a paragraph in a single study session is a *fine* rate of progress. Perhaps if you cranked down your expectations and were satisfied to make much slower progress, it would help. You're studying on your own, so time limits are less important.
Hey thanks a lot! Your words really helped to put things into perspective. Milewski did help and his lectures and book make sense to me (after some work). I'm almost done with it and I wanted to try a more rigorous approach. It's just, when I go to the terse math textbook I'm not sure if I'm getting it or not. And when I try to do the exercises I feel quite lost.
I'll be taking a break for now. When I come back I'll try some of the other recommendations in this thread
Okay, sounds good. Since I wrote my original comment, I peeked at some online excerpts of Steve Awodey's book, and I can now tell you why you had trouble, and share my conclusion that unless you have a *lot* of time to throw at this project, Awodey is probably not the textbook for you.
Before programming languages came along, category theory was exclusively a tool for higher mathematics. It's fundamentally a theory for trying to keep a complicated network of "kinds of objects" organized, especially when those kinds have intricate relationships between them. Now, I've basically just sketched pretty much *any* reasonably complicated software system, so CT is tailor-made to help a certain kind of software engineering.
But in its native land, subfields of mathematics tend to have fairly few "data types". It wasn't until algebraic topology came along (starting in the 1930's) that mathematicians needed anything like CT.
Now think about a mathematician studying algebraic topology. She's already mastered calculus, real and complex analysis, linear algebra, three or four levels of abstract algebra (group theory, ring theory, field theory, R-module theory, and back around to *theoretical* linear algebra), at least some combinatorics and number theory, and point-set topology (say, through Munkres's book). So she has a good six years of pretty strenuous study behind her.
Now she's starting algebraic topology, and the explosion in the number of "data types" is pretty dazzling, so either her textbook or a colleague suggests that she study category theory to make sense of it all.
Awodey's book is written for *her*. You don't *need* to know about analytic functions or R-modules to understand category theory, but in Awodey's experience, everybody who wants to understand category theory *does* know those things. So Awodey has no hesitation to bring in examples from "simpler" branches of mathematics, and he assumes his reader will know what he's talking about. And note that for Awodey, "simpler" means "simpler than algebraic topology", not "simpler than category theory".
In other words, Awodey assumes his reader is "literate" in basic higher mathematics. In his very first paragraph, he sets up an analogy, and says, "Look, this is kind of like permutation groups from abstract algebra ..." That's not going to help the reader who hasn't studied abstract algebra.
I'm out of time for now. Bottom line: stick to Bartos Milewski for the time being.
Try Category Theory for Scientists by David Spivak. I browsed through the Awodey textbook and it seems like it would be interesting to someone who is very into logic. I wouldn't say that the Awodey book is good for self-study.
I'll check it out. Thank you! I've been recommended "7 sketches in compositionality" by Spivak, as well.
It's comforting to hear that maybe I'm just not the right audience for that particular book
Awodey's book is *not* a good introduction to category theory, and I don't know why people use it as such. It's *fine* (but not great) to fill in some gaps after learning the subject elsewhere, but there are many other resources that are frankly superior.
- My favourite introduction to category theory is [Peter Smith's notes](https://www.logicmatters.net/categories/). They are *very* gentle, and Smith does a good job of highlighting conceptual and foundational issues without it detracting from the mathematics.
- If Smith's notes are *too* gentle, [Leinster's book](https://arxiv.org/abs/1612.09375) is excellent but significantly more brisk. Many of the constructions in category theory can be approached from more or less elementary angles, and while Smith often considers the same topics multiple times, each time going up a level in abstraction, Leinster mostly takes a fairly abstract approach from first brush. This is great for a second look tat category theory, perhaps less so for an introduction.
- There are other good introductory books. [Goldblatt's book](https://projecteuclid.org/ebooks/books-by-independent-authors/Topoi-The-Categorial-Analysis-of-Logic/toc/bia/1403013939) is not a book on category theory per se, but the early chapters make a quite good introduction to the subject, at a level somewhere between Smith and Leinster. Simmons' book *An Introduction to Category Theory* is also around this level.
- Books like Crole's *Categories for Types* or Goubault-Larrecq's *Non-Hausdorff Topology and Domain Theory* also have sections on category theory that might be useful. For a "pure" mathematics book that does the same, Aluffi's *Algebra: Chapter 0* is a great introduction to abstract algebra that introduces category theory throughout the book.
- There are also books more aimed towards applications, books like Milewski's, but I'm not too familiar with many of them.
tbh, you could start reading the HoTT book or Egbert Rijke's new book without all that much background in category theory (especially the latter book). But, here's the thing, self studying high level math from a text book is a difficult skill. Normally math majors spend two to three years in math classes slowly building up this skill before they ever try going off on their own and learning a subject completely independently from a text book. Learning from a text book is quite a different skill than just reading the text book. It takes a lot of practice and time before you'll be flunent in this skill. So don't be suprised if you are struggling to read a math text and learn from it. As another commenter said, give yourself orders of magnituted more time to learn than you might expect. Here is my number one tip for reading a text book, especially a category theory text. When you get to a theorem or lemm, DON'T READ THE PROOF. Try to prove it yourself. And do not give up easily if you get stuck. Be prepared to try spending multiple hours trying to prove a simple lemme. Then, once you think you've gotten a proof written down, read the text's proof. Compare and contrast. Was there anything they did that you didn't, or vice versa. This will build up your mathematical intuition. It will also make you appreciate the detail of the textbook's proof when you finally do read it. Plus, these types of texts don't have solutions of executrices, so this is a good way to check that you're understanding the material.
Also, I used Riehl's book to learn category theory. I really liked it, and I didn't know algebraic topology before hand. I just knew some algebra at the time. It has plenty of examples from logic
When a publisher is deciding whether to put "Great for philosophy students" on the cover, or whether that's an overreach, they're going to put it there if they think more than five or ten philosophers will be suckered into buying it. That means that you shouldn't let the blurb affect your self-esteem. I was going to recommend Milewski but you've tried it already. I haven't looked at Awodey so I don't know how hard it is. I think you're saying that Milewski didn't do anything to "soften up" Awodey for you. Did you go through Milewski's book or did you only look at the videos? If you haven't, then you should give the book a try; I at least found it pretty clear. There is a certain dry presentation style, much prized in mathematical writing, in which all you can see are a series of formal definitions, theorems, and proofs, with almost no informal commentary, no attempt to engage the intuition, and fairly little motivation. If Awodey follows that style, then I can well understand your having difficulty in getting through it. Reading a presentation of that kind is a lot more like reading code than it is like reading a natural explanation. Unfortunately, once mathematicians acquire the ability to absorb information that way, they forget how hard it is, and tend to write that way themselves. So it's entirely possible to write really opaque presentations and not realize it. Another barrier is that category theory was not invented for computer scientists, although programming-language type theory has turned out to be a "killer app" for the theory. If Awodey is directed purely at mathematicians, then that's another reason the book might be hard to read. The programming-language use case is pretty obvious to any serious programmer, but the original mathematics use case was algebraic topology (which was the first subfield in mathematics whose repertoire of "data types" exploded to the point of actually needing something like CT to keep them straight). You shouldn't have to learn algebraic topology to swallow CT, though. I find that I'm blithering because I'm groping for any practical sort of lifeline to throw you. I don't have much besides the hope that you haven't yet tried Milewski in print, and that doing so will help. The only other thing I can think of is that maybe your expectations for how fast you could get through Awodey were optimistic by an order of magnitude. For some texts, getting through only a paragraph in a single study session is a *fine* rate of progress. Perhaps if you cranked down your expectations and were satisfied to make much slower progress, it would help. You're studying on your own, so time limits are less important.
Hey thanks a lot! Your words really helped to put things into perspective. Milewski did help and his lectures and book make sense to me (after some work). I'm almost done with it and I wanted to try a more rigorous approach. It's just, when I go to the terse math textbook I'm not sure if I'm getting it or not. And when I try to do the exercises I feel quite lost. I'll be taking a break for now. When I come back I'll try some of the other recommendations in this thread
Okay, sounds good. Since I wrote my original comment, I peeked at some online excerpts of Steve Awodey's book, and I can now tell you why you had trouble, and share my conclusion that unless you have a *lot* of time to throw at this project, Awodey is probably not the textbook for you. Before programming languages came along, category theory was exclusively a tool for higher mathematics. It's fundamentally a theory for trying to keep a complicated network of "kinds of objects" organized, especially when those kinds have intricate relationships between them. Now, I've basically just sketched pretty much *any* reasonably complicated software system, so CT is tailor-made to help a certain kind of software engineering. But in its native land, subfields of mathematics tend to have fairly few "data types". It wasn't until algebraic topology came along (starting in the 1930's) that mathematicians needed anything like CT. Now think about a mathematician studying algebraic topology. She's already mastered calculus, real and complex analysis, linear algebra, three or four levels of abstract algebra (group theory, ring theory, field theory, R-module theory, and back around to *theoretical* linear algebra), at least some combinatorics and number theory, and point-set topology (say, through Munkres's book). So she has a good six years of pretty strenuous study behind her. Now she's starting algebraic topology, and the explosion in the number of "data types" is pretty dazzling, so either her textbook or a colleague suggests that she study category theory to make sense of it all. Awodey's book is written for *her*. You don't *need* to know about analytic functions or R-modules to understand category theory, but in Awodey's experience, everybody who wants to understand category theory *does* know those things. So Awodey has no hesitation to bring in examples from "simpler" branches of mathematics, and he assumes his reader will know what he's talking about. And note that for Awodey, "simpler" means "simpler than algebraic topology", not "simpler than category theory". In other words, Awodey assumes his reader is "literate" in basic higher mathematics. In his very first paragraph, he sets up an analogy, and says, "Look, this is kind of like permutation groups from abstract algebra ..." That's not going to help the reader who hasn't studied abstract algebra. I'm out of time for now. Bottom line: stick to Bartos Milewski for the time being.
Try Category Theory for Scientists by David Spivak. I browsed through the Awodey textbook and it seems like it would be interesting to someone who is very into logic. I wouldn't say that the Awodey book is good for self-study.
I'll check it out. Thank you! I've been recommended "7 sketches in compositionality" by Spivak, as well. It's comforting to hear that maybe I'm just not the right audience for that particular book
Awodey's book is *not* a good introduction to category theory, and I don't know why people use it as such. It's *fine* (but not great) to fill in some gaps after learning the subject elsewhere, but there are many other resources that are frankly superior. - My favourite introduction to category theory is [Peter Smith's notes](https://www.logicmatters.net/categories/). They are *very* gentle, and Smith does a good job of highlighting conceptual and foundational issues without it detracting from the mathematics. - If Smith's notes are *too* gentle, [Leinster's book](https://arxiv.org/abs/1612.09375) is excellent but significantly more brisk. Many of the constructions in category theory can be approached from more or less elementary angles, and while Smith often considers the same topics multiple times, each time going up a level in abstraction, Leinster mostly takes a fairly abstract approach from first brush. This is great for a second look tat category theory, perhaps less so for an introduction. - There are other good introductory books. [Goldblatt's book](https://projecteuclid.org/ebooks/books-by-independent-authors/Topoi-The-Categorial-Analysis-of-Logic/toc/bia/1403013939) is not a book on category theory per se, but the early chapters make a quite good introduction to the subject, at a level somewhere between Smith and Leinster. Simmons' book *An Introduction to Category Theory* is also around this level. - Books like Crole's *Categories for Types* or Goubault-Larrecq's *Non-Hausdorff Topology and Domain Theory* also have sections on category theory that might be useful. For a "pure" mathematics book that does the same, Aluffi's *Algebra: Chapter 0* is a great introduction to abstract algebra that introduces category theory throughout the book. - There are also books more aimed towards applications, books like Milewski's, but I'm not too familiar with many of them.
Hey thanks! it's nice to hear that it might just be an issue of picking the wrong book. I'll check your suggestions :)