Categorical introduction to Algebra and Topology

Solution 1:

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

• Sketches of an Elephant: First read Section D1. This is about what first-order logic looks like in categorical language. You want to understand the syntactic (bi-)category of predicates and functional (predicates) classes associated to a first-order theory. Second, read sections A1 and A2 in order to understand what a topos (hence a "set theory") looks like categorically.

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

• Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory: read chapters V and VI on Aspects of Monads and on Algebraic Categories, respectively.

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

• Basic Concepts of Enriched Category Theory

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.

Solution 2:

Paolo Aluffi's Algebra: Chapter 0 is just what you're looking for, I think, for the algebra part.

As for the topological part, I don't know of any introductions to -general- topology that are all that categorical, but I think point set topology, as it is so close to set theory, is not really fit for interesting and useful categorical thinking in general. But that is my opinion. Algebraic topology, on the other hand, is something entirely different but it is also off topic.

Solution 3:

Ronald Brown's text Topology and Groupoids is probably what you want from a topology text. He gives an introduction to general topology and the fundamental groupoid using the language of category theory throughout. It's an excellent textbook. I would also second other posters' recommendations of Aluffi's algebra textbook; it's very well-written and beginner-friendly. For homological algebra, I would recommend Weibel's text.

Solution 4:

If you can understand German (and according to your original post I assume so), for an introduction to topology with a touch of category theory I'd avice you to have a look at "Grundkurs Topologie" by Gerd Laures and Markus Szymik. It might just be what you are looking for. Although I doubt you will see Yoneda "in action".