At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book Algebra by Serge Lang. I have read the chapters on groups and rings, but then my motivation somehow disappeared and I turned to category theory.
More exactly, I started reading Categories for the Working Mathematician by Saunders MacLane. I now feel comfortable with all the concepts discussed in the first five Chapters, i.e. categories and functors and the usual formulations of universal properties.
I would really like to go on reading about algebra, but once I understood the strucutrual approaches to Mathematics, I can hardly imagine to continue doing all the awful calculations, basic Algebra books like Lang's are filled with, instead of using universal properties and so on.
So basically, my question is, if there are books on Algebra, not assuming any algebraic knowledge, but extensively using category-theoretic methods. Of course, it is very non-standard to cover all the basic category theory before turning to applications in Algebra, but I hope someone knows a book or some lecture notes satisfying my needs.
Furthermore, I would like to learn some topology. In this field I have even less knowledge than in Algebra, i.e. I don't even know the definition of a topological space. My question is the same as with Algebra: Is there a categorical/conceptional introduction to general topology?
Best Answer
Developing algebra categorically is unfortunately difficult because the necessary material is spread over a number of seemingly unrelated books and articles. Another difficulty is that the mixing of set-theoretic foundations with categorical language makes things somewhat difficult to understand. I do have a reading list from which you can learn the categorical perspective, but this is actually quite a lot of material, and none of it comes from textbooks, so it is quite slow-going to learn from. In particular, you will not actually learn algebra from this and will be better served in terms of acquiring working knowledge from, say, Aluffi. In any case, below is the list, arranged in a somewhat logical order, but you should really be reading all of the stuff simultaneously.
First, you need to understand the category of sets s being a well-pointed topos, internal to the syntactic (bi)-category of predicates and functional (predicates) classes. For this you want to look at
Second, algebra is really about monads, in the sense that any category of algebraic objects is a category of algebras for a monad. For this you should read
Next, you will need some knowledge of enriched category theory in monoidal closed categories since, since most of the monads of basic algebra comes from monoid objects in a monoidal closed category (e.g. group actions are algebras for the monad associated to a group object in Set, vector spaces are algebras for the simple objects in the category of monoid objects in the category of abelian groups, etc.). The relevant material for understanding the constructions of these categories are the first few chapters of
It is here, in considering the enriched category theory perspective, where having a good understanding of the category of sets as a well-pointed topos is crucial. Without well-pointedness, you cannot conclude much about the categories of functors that you are building, and which the various categories of algebraic objects ultimately are.
Finally, there are the papers "Monads on Symmetric Monoidal Categories" and "Closed Categories Generated by Commutative Monads" by Anders Kock that address the fact that algebras for commutative monads inherit a monoidal closed structure when the commutative monad is in a monoidal closed category. This is where tensor products, for example, really come from.
Regarding topology, the Categorical Foundations book also has Chapter III: A Functional Approach to General Topology, which is quite enlightening, but you should probably only read it concurrently with an actual topology book, like Munkres.