Reddit reviews Algebra: Chapter 0 (Graduate Studies in Mathematics)
We found 14 Reddit comments about Algebra: Chapter 0 (Graduate Studies in Mathematics). Here are the top ones, ranked by their Reddit score.
Used Book in Good Condition
This is from Paolo Aluffi's excellent Algebra: Chapter 0, which uses categories as a unifying theme.
A groupoid is a small category in which every morphism is an isomorphism. An automorphism of an object A of a category C is an isomorphism from A to itself. The set of automorphisms of A is denoted Aut_C(A).
Edit: added that groupoids are small categories (thank you cromonolith)
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
I have to second Dummit and Foote as a supplement to Lang's text, they're pretty much complete opposites; where Lang is very to the point (terse, some may say) and from a very abstract viewpoint, Dummit and Foote has a lot of exposition and examples and is done from, what at least what I would call, an appropriate level for a first graduate course in abstract algebra. It also has an appendix that deals with category theory, it's nothing extensive but it may help you become more familiar with the ideas of category theory. I am currently using this book for a graduate course in algebra so I have some familiarity with it; it is a bit too wordy for my tastes but that may be your thing.
A book with which I have limited experience but quite like so far is Mac Lane and Birkhoff's Algebra it's done with the same general perspective as Dummit and Foote but it has a bit more category theory (it is introduced at the end of the third chapter and the entire fifteenth chapter is dedicated to category theory), it isn't terse but it is less wordy than Dummit and Foote.
Another (very) popular choice (but one with which I have no experience) is Aluffi's Algebra: Chapter 0 it develops category theory pretty much from the start and supposedly is much less terse than Lang (I only say supposedly as I have no first hand experience with it).
If you want something that only deals with category theory, the classic text is Mac Lane's Category Theory for the Working Mathematician I have found looking at this book for a long period of time has helped me with understanding/getting used to categorical ideas. I also have experience with this book for which you can find on the internet (legally) for free and I find it rather good.
the "vanilla" books are IMO quite boring to read - especially when you don't know more than Set/Functions.
but I really enjoy P. Aluffi; Algebra: Chapter 0 that builds up algebra using CT from the go instead of after all the work
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remark I don't know if this will really help you understanding Haskell (I doubt it a bit) but it's a worthy intellectual endeavor all in itself and you can put on a knowing smile whenever you hear those horrible words after
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
Reading through Algebra: Chapter 0
If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):
Basic Algebra by Jacobson
Algebra by Lang
Algebra by MacLane/Birkhoff
Algebra by Herstein
Algebra by Artin
etc
But there are other books that are "essential" to modern readers:
Chapter 0 by Aluffi
Basic Algebra by Knapp
Algebra by Dummit/Foot
Gallian is basic undergrad stuff through Galois, right?
I can only recommend books that will start from scratch, so will cover many things you already know, but go much deeper than an undergrad text would. Mac Lane and Birkhoff is my favorite math text I've ever read. The only significant drawback is some of the terminology is awkward, the most significant example being that the word "homomorphism" is used once in the entire book, to note it as an alternative to their word "morphism". I'm also currently reading Aluffi to review for a qualifier, and while I personally don't like the exposition as much, it's definitely well-written, and is somewhat more modern. Both of them will cover things you know already, but they should have enough new stuff sprinkled in to keep you interested and help solidify your knowledge.
If you want a more direct transition I can't really be too helpful, sorry.
(edit: minor typo)
8 to 12 hours is really not that much, but it should be enough to learn something interesting! I would start with category theory if you can. I liked Emily Riehl's categories in context for an intro, but it will go a little slow for how little time you have to learn the basics. Maybe the first chapter of Algebra: Chapter 0 by Aleffi? [EDIT: you might want to find a "reasonably priced" pdf version of this book if you do decide to use it -- it's pretty expensive] If you can get through that, and understand a little about how types fit into the picture, you should be able to present the basic idea behind curry-howard-lambek. IIRC you do not need functors or natural transformations ("higher level" categorical concepts), as important as they usually are, to get through this topic; Aleffi doesn't go over them in his very first intro to categories which is why I'm recommending him. /u/VFB1210 has some very good recommendations above as well.
I am trying to think of a better introduction to type theory than HoTT -- if you can learn about types without getting infinity categories and homotopy equivalence mixed up in them, I would. Type theory is actually pretty cool and sleek.
Here's a selection of intro-to-type theory resources I found:
Programming in Martin-Löf's Type Theory is
pretty long, but you can probably put together a mini-course as follows: read chapters 1 & 2 quickly, skim 3, and then read 19 and 20.
The lecture notes from Paul Levy's mini-course on the typed lambda calculus form a pretty compact resource, but I'm not sure this will be super useful to you right now -- keep it in mind but don't start off with it. Since it is in lecture-note style it is also pretty hard to keep up with if you don't already kind of know what he's talking about.
Constable's Naïve Computational Type Theory seems to be different from the usual intro to types -- it's done in the style of the old Naive Set Theory text, which means you're supposed to be sort of guided intuitively into knowing how types work. It looks like the intuition all comes from programming, and if you know something functional and hopefully strongly typed (OCaml, SML, Haskell, or Lisp come to mind) you will probably get the most out of it. I think that's true about type theory in general, actually.
PFPL by Bob Harper is probably a stretch -- you won't find it useful right at the moment, but if you want to spend 2 semesters really getting to know how type theory encapsulates pretty much any modern programming paradigm (typed languages, "untyped" languages, parallel execution, concurrency, etc.) this book is top-tier. The preview edition doesn't have everything from the whole book but is a pretty big portion of it.
I don't know the first but I didn't really like the second book. Right now I do seem to make some good progress understanding stuff (not all but most) in Algebra Chapter 0 which is a lot bigger but introduces a lot of the Algebra I was missing (or forgot since school) along with the Category Theory terms.
There's no single book that's right for everyone: a suitable book will depend upon (1) your current background, (2) the material you want to study, (3) the level at which you want to study it (e.g., undergraduate- versus graduate-level), and (4) the "flavor" of book you prefer, so to speak. (E.g., do you want lots of worked-out examples? Plenty of exercises? Something which will be useful as a reference book later on?)
That said, here's a preliminary list of titles, many of which inevitably get recommended for requests like yours:
Good luck finding something useful!
I love Aluffi! It's a fun read, and more "modern" than texts like Dummit and Foote (in that it uses basic category theory freely). I like category theory, so I really enjoy Aluffi's approach.
Dummit (or just D&F), Artin, [Lang] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X), [Hungerford] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-v/dp/0387905189). The first two are undergraduate texts and the next two are graduate texts, those are the ones I've used and seen recommended, although some people suggest [Pinter] (https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178) and Aluffi. Please don't actually buy these books, you won't be able to feed yourself. There are free versions online and in many university libraries. Some of these books can get quite dry at times though. Feel free to stop by /r/learnmath whenever you have specific questions
I think category theory is best learned when taught with a given context. The first time I saw category theory was in my first abstract algebra course (rings, modules, etc.), where the notion of a category seemed like a necessary formalism. Given you already know some algebra, I'd suggest glancing through Paolo Aluffi's Algebra: Chapter 0. It is NOT a book on category theory, but rather an abstract algebra book that works with categories from the ground level. Perhaps it could be a good exercise to prove some statements about modules and rings that you already know, but using the language of category theory. For example, I'd get familiar with the idea of Hom(X,-) as a "functor"from the category of R-modules to the category of abelian groups, which maps Y \to Hom(X,Y). We can similarly define Hom(-,X). How do these act on morphisms (R-module homomorphisms)? Which one is covariant and which one is contravariant? If one of these functors preserves short exact sequences (i.e. is exact), what does that tell you about X?