# Exercises in category theory for a non-working mathematican (undergrad)

I'm trying to learn category theory pretty much on my own (with some help from a professor). My main information source is the good old Categories for the working mathematician by Mac Lane. I find the book very good and and I don't have very much trouble understanding the theory, proofs and motivations and so on.

But many examples fly over my head as do the exercises.

The thing is that I'm far from a working mathematician or a grad student, which the book seems to be aimed towards. I know that one have to do learn math by doing math and I find it almost impossible when exercises involve things I'm not yet familiar with (modules, algebras etc). I simply don't have time to learn these things just so I can solve my exercises but on the other hand I don't want to miss out on learning things just because the exercises are on a too high level for me.

So I want to ask you for references on exercises in category theory aimed at someone with limited knowledge of abstract algebra, suitable for concepts in Categories for the working mathematician but with more basic objects but still meaningful and challenging.

(I consider myself to have fairly good basic knowledge of "elementary"-style abstract algebra (monoids, groups, rings, fields, linear algebra, some galois theory, related number theory, some algebraic graph theory etc))

• "I've read the book and done the exercises before going to the university. The exercises are easy, and when you don't know a notion, just look it up" Good for you. I'm 2 years into my studies and I find the book quite intellectually challenging, but then I'm pretty slow and stupid compared to other math students I know. But is it really weird that I find it harder to work with structures I know next to nothing about rather than things I'm familiar with? I don't get it. Jul 21, 2014 at 18:28
• Martin, your comment essentially amounts to 'it was easy for me so I don't see why you should be having any problems' Jul 21, 2014 at 21:59

Check out the new book (amazon-link)

Tom Leinster, Basic Category Theory, Cambridge Studies in Advanced Mathematics, Vol. 143, 2014

• +1. Just from what I can see off the Amazon preview, the book looks excellently written at an elementary level. A comment for beginners: Do not be put off that Leinster's book "only" covers (Co)Limits, Representables and Adjoints. Even though Mac Lane covers much more, a solid foundation in just these three concepts will take you surprisingly far. It's similar to Set Theory: In an elementary text of 10 chapters, you'll use things from the first 5 chapters every day of your life, and things from the last 5 chapters only on occasion (depending on what you study, of course). Jul 22, 2014 at 17:03
• There is a free copy of the book on arXiv. May 4, 2018 at 15:32

A more elementary text is Conceptual Mathematics: A First Introduction to Categories. Also interesting is the list given here.

I can recommend Category Theory by Awodey for your situation. It is more elementary but not too slow for an average undergraduate.

Introduction to Category Theory by Harold Simmons is a nice and gentle way to get into category theory with plenty of exercises (and full solutions!). I'm an undergrad as well, and I worked through this book before moving on to Categories for the Working Mathematician because it is more leisurely. More to the point of your question, Intro to Category Theory always describes the structures that it uses as examples (except for topological spaces) and describes their morphisms. It also works out many examples explicitly and diagrammatically, which I found really helpful.

On the other hand, I usually just think of $R$-modules as "vector spaces but over a ring" and algebras as "vector spaces, but you can multiply vectors". I've only worked through chapter 3, so I don't know if I will need more knowledge about this stuff later on, but these vagaries (and brief trips to wikipedia) have sufficed for me so far.

Sets for mathematicians by Lawvere is very good and pitched at a fairly elementary but sophisticated level.

Also the catsters videos by Simon Willerton & Eugenia Cheng are excellent; and also short (5-8 minutes long!). Her notes on the subjects are also detailed and cover the basics.

These notes by Knighten are a bit scrappy, but usefully elementary and detailed.

An alternative: Category theory for applied scientists: You find links and commentary here: http://johncarlosbaez.wordpress.com/2013/05/23/category-theory-for-scientists/

(A version is also posted on arXiv)

I would wholeheartedly recommend "Seven Sketches in Compositionality: An Invitation to Applied Category Theory by David Spivak and Brendan Fong"

In their own words,

*The purpose of this book is to offer a self-contained tour of applied category theory. It is an invitation to discover advanced topics in category theory through concrete real-world examples. Rather than try to give a comprehensive treatment of these topics—which include adjoint functors, enriched categories, proarrow equipments, toposes, and much more—we merely provide a taste of each. We want to give readers some insight into how it feels to work with these structures as well as some ideas about how they might show up in practice. The audience for this book is quite diverse: anyone who finds the above description intriguing. This could include a motivated high school student who hasn’t seen calculus yet but has loved reading a weird book on mathematical logic they found at the library. Or a machine-learning researcher who wants to understand what vector spaces, design theory, and dynamical systems could possibly have in common. Or a pure mathematician who wants to imagine what sorts of applications their work might have. Or a recently-retired programmer who’s always had an eerie feeling that category theory is what they’ve been looking for to tie it all together, but who’s found the usual books on the subject impenetrable. *

And they have made their complete book available through arXiv