# Categorical introduction to Algebra and Topology

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?

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I think you'll be hard pressed to find a categorical framework for point set topology (algebraic topology is another story) - this is partly because point set topology is essentially based on set theory. Once you've got the category $Top$ you can start doing things with it using tools from category theory, but defining $Top$ itself requires a non-categorical framework (even if you want to define a topological space by its category of open sets, you still need an axiomatisation of a topology). – Dan Rust Mar 18 '14 at 15:52

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.

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I've already come across this book. But it does not go much beyond stating the universal properties for products and copoproducts, does it? Adjoint functors, Limits, Yoneda's Lemma and all that are entirely ignored, if I remember the table of contents correctly. All these things are so beautiful to me, I'd like to see them "in action". – user114885 Mar 18 '14 at 15:30
It does go beyond that. The mindset of the book is categorical, and as far as I remember every concept that can be introduced and explained categorically is explained in that way. Some concepts are introduced as needed (it is an algebra book, not a category theory book), such as adjoint functors which are introduced to explain the free-forgetful adjunction. It is already much more than can be said of a lot of algebra books that explain free objects (groups, modules...) Yoneda appears in the exercises, maybe because it doesn't appear all that obviously in a first course in algebra. – lentic catachresis Mar 18 '14 at 15:34
To see Yoneda's lemma in action you should go into algebraic geometry, for example :) but for that you need to learn your basic abstract algebra first! Also, it is important to remember that not everything can be done/explained categorically. For example, a lot of the material in a standard introduction to group theory. – lentic catachresis Mar 18 '14 at 15:36

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

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.

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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".

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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.

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This would also be my recommendation. Ronnie's book is wide ranging and very categorical in its approach. Not everything can be proved jst using category theoretic methods, but motivation for constructions can often be given that way. (Sometimes you have to get your hands `dirty' and not just manipulate things from a distance.) As a plus the groupoid stuff is great for understanding more algebra. – Tim Porter Jan 24 at 10:17

As far as I can tell I agree with lentic catachresis when he says that Aluffi's book is a very good introduction to algebra in a categorical setting, although I've found that Lang's text book is a good reference too, especially for more advanced topics.

Any book in homological-algebra makes intense use of category theory, which isn't that a surprise considering that category theory was born for solving problems in these fields.

From the topological point of view, I've studied from Manetti Topology. In my opinion it is a really good introductory book on general topology with a categorical perspective: many concepts are presented and emphatized from arrow-point-of-view. It also has the same limitation of Aluffi's book: it doesn't make use of anything more advanced of limits and universal properties.

If you want to see more advanced application of category theory Spanier's Algebraic Topology makes use of stuff like Yoneda lemma, although some time these application are not made esplicit.

Another very good reference too say application of category theory to algebraic topology is Aguilar, Gitler and Prieto's Algebraic Topology from an Homotopical Point of View: in this book you can really see a lot of application of category theory of topology (as an example, if I remember correctly, there is a proof of the fact that the category of compactly generated spaces is cartesian closed via the adjoint functor theorem).

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