I am having a hard time understanding the concepts of a universal property :\ What are they used for? Do they abstract the notion of constructing mathematical objects? Can the concept of a group be described by a universal property or is this not how it works?
Universal properties are used to classify a certain type of object in a category up to unique isomorphism, usually together with a certain set of morphisms. The construction of the object in question is separate from this. The construction is just an existence proof of the object. The universal property states that it is unique up to unique isomorphism. Many objects of various categories are classified by a universal property. Examples are tensor products of modules, and colimits/limits in general categories. Also products and coproducts (which are special cases of limits and colimits). The usage of universal properties highlights the fact that we are not always interested in the specific construction of the object, but rather in the properties it has as an object satisfying a certain universal property. Note that while we may state a universal property, it is not given that this object always exists. For example, in the category of schemes both limits and colimits may fail to exist. A group might be described as an object satisfying a universal property in a certain context, but this is unlikely to have any real usage. Satisfying a universal property makes it unique up to isomorphism, hence you would only end up with a single isomorphism class of this group.
Universal properties make it a lot easier to find suitable mappings into our out of the space. I'm not sure what kind of mathematics you know, so it's hard to give an example. But tensor products is a really neat example of this. So let ##V## and ##W## be two vector spaces over ##\mathbb{R}##. We can find the following map: [tex]V\otimes W\rightarrow W\otimes V:v\otimes w\rightarrow w\otimes v[/tex] The vectors ##v\otimes w## form a spanning set, so it would certainly define a mapping. However, the mapping is not well-defined since the set is not linear independent. It's easy to prove in this case that the mapping is well-defined, but you might as well work through the universal property. The universal property says that any bilinear mapping ##g:V\times W\rightarrow A## with A any vector space gives rise to a linear mapping ##g^\prime:V\otimes W\rightarrow A## with ##f(v,w) = g(v\otimes W)##. So in this case, we can consider [tex]V\times W\rightarrow V\otimes W:(v,w)\rightarrow w\otimes v[/tex] This is perfectly well-defined since the ##(v,w)## are a basis of ##V\times W##. The universal property thus says exactly that the map [tex]V\otimes W\rightarrow W\otimes V:v\otimes w\rightarrow w\otimes v[/tex] is well-defined. So any time you have to find a certain mapping for the tensor product, it is worth it to check the universal property. And this holds for anything that has the universal property. It might certainly be possible to find a mapping without it, but it will almost always be messier. If you tell me what kind of math you already know, I might be able to give an example. Also, where exactly did the concept of universal property come up? In relation to which structure?
It didn't come up in relation to any structure.. I'll be honest, I recently discovered this whole aspect of mathematics that I didn't know existed and I am very eager and excited to learn all about it, but my undergraduate school doesn't offer any courses on the subject, so I've been trying to learn it by myself. But without guidance I always find myself stuck.. Lately I've been trying to learn category theory, proof theory and theory of types all at the same time because I sincerely don't know what I should be doing first. I also have been thinking of adding lambda calculus to the stack since it keeps popping up all over the place. I guess what I really need is for someone to point me in the right direction :) Thanks!
Category theory is really not all too difficult formally, but it might make little sense without a vast collection of examples. So I would certainly learn some point-set topology and some abstract algebra (groups and commutative rings) first. This will give a lot of motivation for the concepts of category theory. The book "Joy of Cats" is freely available: http://katmat.math.uni-bremen.de/acc/acc.pdf It's one of my favorite reference works. However, it's not really suitable as an intro text for beginners. Still, it contains a massive amount of examples. So I would study from another book and each time you encounter a new topic, check the Joy of Cats to see some motivating examples. The classic on category theory is of course MacLane, so you might want to consider working through it: http://www.amazon.com/Categories-Working-Mathematician-Graduate-Mathematics/dp/0387984038 It seems from the other topics you mentioned that you're into foundational stuff such as logic. In that case, you also might want to consider Goldblatt: http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260 This is probably easier than MacLane too.
An extremely good book to read before tackling pure category theory is Aluffi's "Algebra: Chapter 0": http://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813 It treats algebra very nicely, but from the categorical point of view. It really shows the connections between different algebraic structures very well.
Easy book on category theory: Conceptual Mathematics: A First Introduction to Category Theory. It's written in such a way as to teach even non-math people, so it doesn't assume you know algebraic topology. Some people consider it too easy, but I'm not sure that's really the case. I didn't work through the whole thing due to lack of time, but they say at some point it gets a lot harder if you haven't done enough of the exercises. So, it appears to have some depth if you make it through the whole book.
And one of the worst math books ever written too. Stay far away from it. And don't study category theory without knowing a good host of examples. Otherwise it will be extremely unmotivated, a lot like this book.
I'm not sure why you dislike it so much. I thought it was kind of a fun book. And you can't argue with fun. John Baez recommends it. Of course, I already knew some algebraic topology and category theory, so it was more motivated for me already.
I don't think it's necessary nor very common to learn category theory from a book, at least not for the working mathematician.
I would tend to agree, actually. It's something that you can pick up as needed. If you attempt to learn it from most books, it is just too boring and there are not enough examples. I used some category theory in my thesis and didn't really rely on category theory books for it. Specifically, I used symmetric monoidal categories, and I had very concrete examples from topological quantum field theory that made it really come alive (and there are a lot of great motivating examples of category theory in that field, my thesis stuff aside). Very intuitive, visual stuff that meshed very well with the small amount of category theory that I used. Some material I used can be found in MacLane, but it seemed more like a reference book that you could take what you needed from and just too boring to sit down and read through the whole thing. I might have to re-evaluate my opinion of Conceptual Mathematics, since it has been so long since I looked at it and I have changed as a mathematician since then, but the main virtue I saw in it was that it was sort of light-hearted and informal, in contrast to the usual very formal, deadly tomes of boredom, like MacLane. It maybe somewhat flawed, though. It helps to at least have a few examples of mathematical structures, like groups and topological spaces, and maybe the fundamental group before the idea of category theory will make sense.
Right. My main issue with Conceptual Mathematics is that I don't really see the point of reading about category theory until you really need it (or at least until you've seen enough examples to really appreciate its use). There is this other book by Lawvere called "Set for mathematics" which is also quite horrible. It tries to explain set theory from a categorical point of view, which I think is totally unnecessary and nonintuitive (at least for the level of the reader that Lawvere aims at). The only use I see of category theory at the undergrad level is to provide some kind of unifying language of mathematics and to motivate constructions such as the product topology (although other motivations should of course be given). For this purpose, you can learn the necessary category theory in matter of hours. You certainly don't need to go read books where they define the inverse image of sets as some kind of pullback diagram or where they define a group to be a groupoid with one object. At the undergrad level (and for most mathematicians) such things are really unnecessary. And I'm certainly not a category-hater since I use category theory all the time. I just think that pedagogically, the student should never go too abstract without good reason. Mathematics shouldn't be about mindless abstraction of concepts, the abstractions should actually serve some purpose.
I'm just not convinced that it should be completely dismissed as worthless, though. It could be that it has an audience problem, in terms of being too abstract for beginners, but with a lot of superfluous, elementary junk to wade through for the people who might gain something from it. Specifically, if anyone is crazy enough to want to learn topos theory, it appears to be a good place to start. Here's what Baez said about it: "It may seem almost childish at first, but it gradually creeps up on you. Schanuel has told me that you must do the exercises - if you don't, at some point the book will suddenly switch from being too easy to being way too hard! If you stick with it, by the end you will have all the basic concepts from topos theory under your belt, almost subconsciously." So, I don't think the point is just abstraction for its own sake. I don't know topos theory, so I'm not sure how much of a point there is to it (in fact, it seems like too abstract nonsense to me, but what do I know?), but it does appear to be something that some logicians like, so it might be eventually relevant to the OP, but it sounds like that will be much later down the line.
I agree with this. But there are better books to learn topos theory from, for example, the Goldblatt book I posted is really nice and emphasizes the links with logic very well. That said, I don't see how you could really go far into topos theory without knowing stuff like topology and sheafs. Topos theory really forms a very nice alternative set theory, but it's also known to be quite difficult. Books like that of Lawvere make the topos theory accessible, but you'll still need to do a lot of studying before you can actually get to the good interesting stuff.
Another nice book that deals with some (concrete) categories is "Mathematical Physics" by Geroch. Don't worry, the book contains no physics and is only math: http://www.amazon.com/Mathematical-Physics-Chicago-Lectures/dp/0226288625
Thanks for all the advice and book recommendations! I'll take it all under consideration when looking deeper into this subject :) I have a way more specific request at this point.. Can you give me another example of a universal property? Possibly the definition of an ordered pair?
I don't know one for ordered pairs, but I do know one for the cartesian product. So let ##A## and ##B## and ##C## be sets. We have maps ##p_A:A\times B\rightarrow A:(a,b)\rightarrow a## and similar for ##p_B##. Now if ##C## is some set and if ##f_A:C\rightarrow A## and ##f_B:C\rightarrow B## are maps, then there exists a unique map ##p:C\rightarrow A\times B## such that ##p_A\circ p = f_A## and ##p_B\circ p = f_B##. An analogous definition works for products of arbitrary (even infinitely many) factors. http://en.wikipedia.org/wiki/Product_(category_theory)