# How should a good course and textbook in category theory be constructed?

I think I am not the only person to feel frustrated with category theory - or perhaps more with the textbooks. The ones I have come across - like Saunders MacLane and Emily Riehl - are certainly insightful and quite well-written, but I feel that I am drowning in exotic examples that I can't follow. This may be because I'm poorly educated or a bit dim, of course, but I feel that since category theory is such a powerful tool, and increasingly important, efforts ought to be made to make it accessible to people without a background in advanced, higher mathematics. As an example of the kind of difficulties I keep running up against (from Emily Riehl, "Category Theory in Context"):

The isomorphisms $A \cong T A \oplus (A/T A)$ are not natural in $A \in Ab_{fg}$

Here, $A$ is Abelian, $TA$ is "the torsion subgroup of $A$" and $Ab_{fg}$ the category of freely generated, abelian groups. I understand Abelian group, and I can just about remember that I've heard about freely generated ones, but in the hassle of having to understand the premises of the example, I completely lose sight of the thing I'm supposed to learn here: how natural transformations work.

Are there no resources that treat the theory, while relying only on relatively elementary mathematics (naive set theory, elementary group theory, - vector spaces, - topology etc)?

EDIT: I have left the original last section in, since it is being referred to in the comments, but here is perhaps a better final section.

I am of course aware of the many, freely available texts about category theory - I've recently come across Algebra and Topology by Pierre Schapira, which looks promising. But what I have in mind is - if someone wanted to write "the ultimate introduction", targeted at, say the entry-level at university, is that even possible in a meaningful way? It ought to be - after all, we do teach naive set theory very early, and vector spaces and abstract algebra loom large as soon as you enter a maths department, I believe. The actual category theory isn't really that much harder, I think - what makes it hard is the examples.

• Your title asks for something completely different than your last paragraph. Do you want examples of category theory textbooks that would more accessible than MacLane's or Riehl's? And if so, could you clarify what is your background? – Arnaud D. Aug 29 '18 at 9:31
• Both - however, I see the question in the final paragraph as a narrowing down of the question in the title. I will think about how I can improve the last paragraph. – j4nd3r53n Aug 29 '18 at 9:34
• It would help if you first made precise why you would like to study category theory. – Tobias Kildetoft Aug 29 '18 at 9:35
• Maybe you should go for a book that is not completely focused on categories but in many cases treats them as underlying material and also explains about them. Something like Algebra chapter 0 of Aluffi. – drhab Aug 29 '18 at 9:45
• I found An Introduction to Category Theory by Harold Simmons (published by CUP) to be very accessible. The Preface says "The book is aimed primarily at the beginning graduate student ... I have designed the book so that it can be used by a single student or small group of students to learn the subject on their own ... The book does not assume the reader has a broad knowledge of mathematics. Most of the illustrations use rather simple ideas ..." – gandalf61 Aug 29 '18 at 9:55

## 1 Answer

Other than Simmons, a good suggestion from the comments, Leinster's Basic Category Theory aims to be more, well, basic than either example you give. Lawvere and Schanuel's Conceptual Mathematics has even, I think, been used with some success with high-school age students. You might also look into David Spivak and Brendan Fong's recent Seven Sketches in Compositionality, which aims to introduce category theory to the non-mathematician with applications in mind. John Baez has been leading an online discussion forum full of relative novices working through the latter.

• Thanks a lot to everyone! Once again I must marvel at the kind and patient way people always seem to answer and try to understand my often stupid questions. – j4nd3r53n Aug 31 '18 at 10:50