# 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?

• 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
• "... my motivation somehow disappeared ..." Lang can have that effect. It's a more impressive book after you know the material and you're using it as a reference. For learning, it's a little austere for my taste and I don't think I'm the only one. Try Hungerford. Same material, but a bit more user friendly. – user4894 Jul 14 '16 at 6:13

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.

• 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. – Bruno Stonek 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. – Bruno Stonek Mar 18 '14 at 15:36
• "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."- That's probably false for the most part. The reality is more like: "We don't really know how it works" and that classical thinking becomes a huge hindrance (e.g. how subobjects are treated). Knowing that $\mathsf{Grp}$ is a concrete, strongly protomodular, pointed algebraic category already gives a huge chunk of group-theory theorems. That being said: Many things obviously haven't been worked out yet. – Stefan Perko Nov 29 '16 at 14:18
• @StefanPerko: I'm not convinced (to say the least). Do the adjectives that you characterized the category of groups with give any hint of statements as elementary as the Sylow theorems, for instance? – Bruno Stonek Nov 29 '16 at 19:26

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.

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.

• 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 '16 at 10:17

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".

I certainly asked myself about that some time ago (I'm still an undergraduate student), and I found Aluffi's: Algebra Chapter 0 to be the most exciting, interesting and categorical introduction to abstract algebra which I know of.

I must say that I even hated everything related to algebra during my first undergraduate months; then I took linear algebra and it was not that bad, but it was during an advanced linear algebra course where I suddenly learned several cool things about rings, modules, diagram chasing and canonical forms from an advanced point of view (such as how the Krull-Schmidt theorem is involved within the uniqueness of such descompositions). It was very hard for me as a second year student to grasp the course without being very exhausted, but it was absolutely awesome, so I drastically began to change my feelings for abstract algebra forever. At this point I discovered Aluffi's book and it became a timeless classic for me since the first moment. I think every serious student should take a look at this book, no matter if they like algebra or not. I wish I've came across this book earlier, and by the way, don't be intimitated by the fact that it is considered to be a graduate-level textbook, it might serve as an advanced and complete reference for undergraduate courses and students as well.

As for the topology book, I want to recommend a book which was written by my topology professor himself. As it was pointed out earlier, Aguilar, Gitler and Prieto's Algebraic Topology from an Homotopical Point of View is a very nice algebraic topology book which uses several gadgets from category theory whenever possible. Here at my university it is usually considered to be a graduate-level book, but actually there is another topology book written (in spanish) by Carlos Prieto, "Topología Básica" , which is aimed at the undergraduate students and is totally focused on developing point-set topology and basic algebraic topology with a strong categorical and geometric flavor, though not much heavy machinery is actually employed (In fact, it is aimed to serve as a solid stepping stone to more advanced algebraic topology books. It might be a very good book for the algebraically-minded student who is taking a first course on topology or anyone with interest on learning algebraic topology as it covers the basic point-set topology material as well)

Fortunately, there is an english translation of the book which is split into two parts: Elements of point-set topology and Elements of homotopy theory. Here one can find both courses as well as some unfinished notes dealing with fiber bundles and homology/cohomology from the point of view of homotopy [http://paginas.matem.unam.mx/cprieto/archivos/libros]

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 explicit.

Another very good reference about 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 applications of category theory to 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).

You should give a look to Categorical Foundations, by Pedicchio and Tholen.

• Note that this book is already mentioned in Vladimir Sotirov's answer. – Arnaud D. Sep 3 '18 at 16:17

I wrote a book "Introductory Algebra, Topology, and Category Theory" in 2006, exactly addressing these issues. It's currently available for free at www.hyperonsoft.com/algbk.pdf.

Chapters 2 and 3 cover universal algebra and order theory. Chapters 4-10 cover basic abastract algebra. Chapters 11-13 cover model theory, computability theory, and category theory through Abelian categories. Later chapters freely rely on chapter 13. The book has been reviewed by the MAA.

• Answers on MSE should be as self-contained as possible. It is not good practice to post answers which consist almost entirely of links to other resources---such links can suffer from "link rot," or go to places which are not accessible to everyone (e.g. to a physical book that a reader may not be able to access). While your book may contain the answer to this question, the answer that you posted here does not. – Xander Henderson Sep 2 '18 at 23:05
• You can do a Web search on the title. The text seems to be available via some other links. People interested in this topic should be aware of this text. – Martin Dowd Sep 3 '18 at 16:04
• @XanderHenderson For a "reference-request" question, this seems like a reasonable answer. Maybe the name of the author, the original publisher (if any) or a link to the aforementioned MAA review could be added to the answer in case he ever delete his account, but otherwise I don't see anything wrong with it. – Arnaud D. Sep 3 '18 at 16:13
• I agree with Arnaud D., this is an honest answer and deserves to stay. Please do not delete it. – Alex M. Sep 3 '18 at 18:43
• This is now available ad b-ok.org; search on the title. – Martin Dowd Sep 8 '18 at 23:31

MacLane himself wrote an algebra book https://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462, which makes heavily use of categorical tools. Highly recommended and it actually gets to some advanced (graduate-level) material.