well, as always, I initially took a look at what wikipedia says. the idea of talking about general mathematical objects and arrows between them sounds pretty impressive and quite exciting to me, but just like any other math stuff, the idea looks quite simple and the examples that wikipedia gives are quite understandable but when you get seriously engaged with the subject you find it challenging. so I wanna know what are the prerequisites for studying category theory. I'm already familiar with group theory and have solved a considerable portion of Herstein's problems (except the ones that it describes them as hard or very hard) and I feel comfortable understanding group theory main topics now. I don't know much about rings but I've studied Linear Algebra. I haven't taken Topology in the university yet but I'm taking a Real Analysis course this semester and it seems easy to me. Can I start reading about category theory or I have to wait to get more experienced and informed in mathematics?
Category theory doesn't have any prerequisites. The theory can be understood by a (good) freshman in mathematics. The point, however, is that you need many examples to see why category theory is useful or to see why certain concepts are defined the way they are. Right now, you've seen three categories: - Sets and functions - Groups and homomorphisms - Vector spaces and Linear maps Category theory generalizes this situation. But you will maybe not see why certain generalization are useful... If you want to learn category theory, then it is useful to learn the examples concurrently with the categorical language. Here are some books to help you: - "Arrows structures and functors. The categorical imperative" by Arbib This book presents many examples in category theory. And it also presents the categories in a way an undergraduate should understand. - "Algebra: Chapter 0" by Aluffi This book presents algebra from the view of category theory. It's worth to read the categorical definitions behind things like group theory. Certainly worth a read!! More advanced books: (not advisable for studying right now, but do give them a look) - "Abstract and concrete categories" by Adamek, Herrlich Strecker The book is freely available at katmat.math.uni-bremen.de/acc/acc.pdf The book is rather encyclopedic, and might not be suited for self-study. Do check it out occasionally, if only for the rich examples in the book!! -"Category theory for the working mathematician" by MacLane The bible of category theory. Many people learn categories from this very book. It's a bit advanced though -"Handbook of categorical algebra" by Borceux It contains almost everything you every want to know about categories. It is a 3 volume book and really, really nice.
Thanks. I knew about 'category theory for the working mathematician' by Saunders McLane but I read somewhere that It was a graduate textbook. Will knowing category theory help me to understand advanced algebraic structures easier? the idea looks pretty clear and impressive but the question is -as you said- why it's useful? what things will I learn if I study category theory?
You mustn't really see category theory as a mathematical theory, it's more a kind of language. It's a very handy and cool kind of language in which most of mathematics can be told. Many students have trouble with the concept of "tensor product". They don't quite see what it is and how to handle it. But once you've seen categories, then you can grasp the tensor product quite easily: it's the coproduct in the category of algebra's. And it's an adjoint to the hom-functor. Category theory will allow you to see connections between different branches of mathematics and it will make these connections rigorous. For example, why is the products of groups defined as it is?? Categories (concrete categories actually) will answer these questions very neatly. Knowing categories won't help you with group theory specifically or topology or etc. But it will help you see some connections between the subjects. On an undergraduate level, category theory can be eliminated (however, I actually prefer not to do that), but on a graduate level categories are very necessary. Most of algebraic geometry (for example) has been done with category theory. Homology must be done by categories. Algebraic topology must be done with categories.
it helps if you are a masochist. i wouldn't bother much with category theory, or maybe i should say i wouldn't bother with much category theory. it is more useful to learn some mathematics. the basic moral of category theory is that the maps are more important than the objects, so every time you learn a new definition, like vector space, spend even more time studying the maps between them, i.e. linear transformations. and when you memorize the definition of a topological space, spend much more time making up examples of spaces and of continuous maps between them. And when you learn the definition of a functor, like the fundamental group, or homology, practice computing the induced transformation of maps, from continuous maps to group homomorphisms, and compute as many as possible. E.f. try to understand when the induced homomorphism is an isomorphism and when it is zero. on the other hand, almost nobody needs to know the definition of a category.
well, I'm not a masochist. I just find the idea of category theory very interesting, to me it can somehow unify a significant portion of mathematics that I'll face in undergraduate level. I was very comfortable studying set theory on my own, so would it be so hard for me to study category theory? because I remember that later when I was taking set theory as a course in the university, most students had problems with cardinals, axiom of choice, Zorn's lemma and other set theoretic theorems and definitions but I liked the course very much. the thing is that this semester I'm taking linear algebra, abstract algebra, calculus III, differential equations and number theory and I'm afraid that studying category theory would distract me from my schooling (it surely will help my education though). if I know that knowing category theory will help me learn and understand mathematics better I'll definitely start studying it now because I've had a very hard time understanding tensors since years ago to even now, although I was in high school when I studied them out of curiosity about general relativity but I remember that I never fully understood tensors no matter how hard I tried and even today I can hardly understand advanced topics about tensors if I can at all. so, what do you suggest? Is it a wise decision to study category theory now?
I'd say yes. Categories will give you a unified understanding of mathematics. It will help you (a bit) in abtract algebra, topology and linear algebra. And it will certainly help you to see connections between those topics. Then again, I'm a category theory-lover
I've found this book: Category Theory, by Steve Awodey by Oxford science publications. It covers these topics. It covers these topics: Chapter 1. Categories 1.1 Introduction 1.2 Functions of sets 1.3 Definition of a category 1.4 Examples of categories 1.5 Isomorphisms 1.6 Constructions on categories 1.7 Free categories 1.8 Foundations: large, small, and locally small 2. Abstract structures 2.1 Epis and monos 2.2 Initial and terminal objects 2.3 Generalized elements 2.4 Sections and retractions 2.5 Products 2.6 Examples of Products 2.7 Categories with products 2.8 Hom-sets 3. Duality 3.1 Duality principle 3.2 Coproducts 3.3 Equalizers 3.4 Coequalizers 4 Groups and categories 4.1 Groups in a category 4.2 The category of groops 4.3 Groups as categories 4.4 Finitely presented categories 5. Limits and colimits 5.1 Subobjects 5.2 Pullbacks 5.3 Properties of Pullbacks 5.4 Limits 5.5 Preservation of limits 5.6 Colimits 6. Exponentials 6.1 Exponential in a category 6.2 Cartesian closed categories 6.3 Heyting algebras 6.4 Equational definition 6.5 Lambda calculus 7. Functors and naturality 7.1 Category of categories 7.2 Representable structures 7.3 Stone duality 7.4 Naturality 7.5 Examples of natural transformations 7.6 Exponentials of categories 7.7 Equivalence of categories 7.8 Examples of equivalence 8. Categories of diagrams 8.1 Set valued functor categories 8.2 The Yoneda embedding 8.3 The Yoneda Lemma 8.4 Applications of the Yoneda Lemma 8.5 Limits in categories of diagrams 8.6 Colimits in categories of diagrams 8.7 Exponentials in categories of diagrams 8.8 Topoi 9. Adjoints 9.1 Preliminary definition 9.2 Hom-set definition 9.3 Examples of adjoints 9.4 Order adjoints 9.5 Quantifiers as adjoints 9.6 RAPL 9.7 Locally cartesian closed categories 9.8 Adjoint functor theorem 10. Monads and algebras 10.1 The triangle identities 10.2 Monads and adjoints 10.3 Algebras for a monad 10.4 Comonads and coalebgras 10.5 Algebras for endofunctors Is it suitable for self-studying?
to quote miles reid, in his historical survey of algebraic geometry: "the study of category theory for its own sake [is] surely one of the most sterile of all intellectual pursuits". [reid, undergraduate algebraic geometry, p. 116.] on the other hand miles reid and everyone else knows what a product is, and a sum, and what duality means, and what Hom and tensor are, and what projectives and injectives are, and initial and final objects, and kernels and cokernels, and quotients, and fibered products and inverse and direct limits, and sheaves, and what is an adjoint functor, and what derived functors are. its not entirely useless, but it is sort of like studying grammar instead of literature. I would suggest the little book by peter freyd, abelian categories, as a nice short introduction. for starters you might read section I.10 of my free algebra course notes at: http://www.math.uga.edu/~roy/843-1.pdf called "categories and functors: what are they?" I used the concept in that course to motivate the definition of normal field extensions. I.e. once we defined the notion of galois group of a field extension, categorical thinking made us ask what are the induced maps between galois groups induced by maps of fields. it turns out there aren't any unless the map of fields induces a normal extension., so we were naturally led to that concept as well. so it is helpful to have the most rudimentary ideas of category theory to guide many other studies, but obsessing over detailed concepts such as seem to be belabored in that book you reference, is for most of us a recipe for extreme boredom. other opinions do exist however. e.g. yoneda's lemma motivates a philosophy for how to define things in mathematics. it says that if you know all the maps in or out of an object, then you know the object. I.e. if Hom(.,A) and Hom(.,B) define naturally equivalent functors, then A ≈ B. This is proved by noting that the assumed isomorphism between Hom(A,A) ≈ Hom(A,B) sends the identity map 1A, to an isomorphism from A to B. Big deal. This was a homework exercise in my algebra class. [Hint: how do you suppose you would go about finding its inverse? If you think of looking at the isomorphism Hom(B,A)≈ Hom(B,B), you get the idea.] But the implications are that in a subject like algebraic geometry, one way to define the moduli space, or universal parameter space< of all algebraic curves is to tell what morphisms into it should be. These should be parametrized families of algebraic curves, i.e. maps X-->B whose fibers are curves, i.e. a family of curves parametrized by a space B, should define a morphism B-->M from the parameter space B into the universal parameter space M. whether or not such an M exists, and what are its properties, is another matter, and more interesting.
It seems to be freely available at teaguesterling.com/category-theory.pdf It seems like a very good book which introduces the most important topics in category theory and from a modern point-of-view. Starting with this book will certainly be ok. You might miss some basic examples. Most examples come from abstract algebra/algebraic geometry/topology. Since you only seen groups, it means that you can only refer to groups for examples and insights. Concepts like limits, Yoneda embeddings, etc. might not be easy to understand in the beginning without much examples. It is worth knowing them because they are a real handy tool sometimes. I would occasionaly check the "joy of cats" at katmat.math.uni-bremen.de/acc for interesting examples. I strongly disagree that category theory is boring or useless. But such opinions do exist. I think most people have black-and-white visions on categories: they love it or they hate it. I think it's obvious what kind of person I am
to a geometer, the basic problem is representability of functors: i.e. given a functor F, how do you find an object M such that F(X) = Hom(X,M) or perhaps Hom(M,X).
There's a very good book for Category theory that has no prerequisites to speak of. Conceptual mathematics: A First Introduction to Categories. Very fun book. It was kind of on the easy side for me, since I had already seen lots of Algebraic topology, so I didn't finish the book, but I hear at some point, they turn the tables on you and it gets deeper, so one day I would like to finish reading it. And that's really the trick, I think, if you want to learn category without a lot of previous background. Yes, the subject itself will be very dull, if you don't have a lot of good examples and things to make it fun. The way that usually happens is when you study enough algebraic topology or maybe some other subjects. But it is possible to come up with more elementary examples. That just isn't the usual way of motivating category theory. I think it may depend a lot on what you end up doing. Categories actually play a key role in my own work. Maybe one of the main points of my PhD thesis is that you can make things in the subject a lot more elegant and understandable if you put them in terms of category theory. Many papers in the subject are just a mess compared to my (well, not just mine) neat way of phrasing things, which is all made possible by using categories. This might sound like a bunch of hot air, but if you were to actually read the papers in question, the difference is just night and day, I think. I wasn't particularly a fan of category theory when I first encountered it. But, I have found it has proved its worth. Mind you, this is coming from someone who is an extreme conceptual/intuitive/visual thinker--way out there on the fringes of it. Not what you might expect for such an abstract subject. Also, at some point, you'll want to look at John Baez's website (I particularly recommend the seminar notes). Very impressive category theory ideas there, and much more.
Really? I can't think of anything more worthy of obsession. It sounds absolutely lovely. Hmm, category theory...
While it looks like you are going to spend a lot of effort on category theory, most mathematics I have seen only uses what it needs of it. I have never seen category theory to be essential in itself - not to say that that can't happen.
Another book that pulls together almost all of Mathematics is "Mathematics: Form and Function" by Saunders Mac Lane. This book groups and shows the relationships among various mathematical areas. It also has a chapter on "Category Theory" in the context of these areas. By the way I am a Mechanical Engineer and Software Engineer, and after some research that the best way to tie these concrete and abstract fields of engineering is through is Category