# Mindset to understand category theory

I am a 17 years old student and I am really interested in category theory due to its abstraction and beauty. I wanted to know if you'd have any advices to approach this theory and if you have papers to begin with this.

• This question is likely to be closed as not a good fit for the site. With that said, and with the proviso that I am not at all a category theorist or even close to one, one piece of advice might be: learn some of the math that motivated the introduction of categories (e.g. algebraic topology/geometry). Emily Riehl once pithily described category theory as "the mathematics of mathematics"- it seeks to recognize patterns common to many mathematical contexts. Studying category theory before other math would be like studying abstract theories of sociology without having experienced any cultures. – Sam Hopkins Apr 2 at 22:38
• I think this is a good question. The evolved culture at MO suggests to me that this question should be Community Wiki. (I would not expect a new user to be aware of such conventions.) This (nonobvious) convention is consistent with getting answers in the comments, including from me. – Theo Johnson-Freyd Apr 3 at 0:01
• You might benefit from checking out the books Sets for mathematics by Lawvere and Rosebrugh, and Conceptual mathematics by Lawvere and Schanuel. – Mike Shulman Apr 3 at 1:09
• I vote against closing. While the question may appear to be superficially unsuitable, ask yourself the following question: where else one could ask the same question and hope to get nontrivial answers from professional mathematicians? MSE will not work, and I do not know of any other resources with remotely comparable audience. – Dmitri Pavlov Apr 3 at 2:43
• As someone who got interested in category theory before learning much “serious” mathematics I echo Mike’s recommendation of Sets for mathematics. Awodey’s Category theory is also good. But I think you have to learn “ordinary” mathematics alongside in order to really appreciate category theory. – Zhen Lin Apr 3 at 3:25

An approach to studying that works with many branches of mathematics is to learn the prerequisites first, then study a good textbook on the subject. For instance, one could study subjects like algebra, measure theory, smooth manifolds, functional analysis in this manner.

Although a textbook on the subject can include examples from other subjects that one has not studied yet, the presence of such examples typically does not prevent one from successfully mastering the particular subject under consideration. (I would say that a good textbook should certainly try to include connections to other areas and not portray its subject as an isolated field.)

For category theory, taking such a route can be quite difficult. Indeed, the (formal) prerequisites are almost nonexistent: elementary logic and set theory will do. But such simplicity can be deceiving.

For instance, the notion of a Kan extension can be very puzzling until one learns a few specific nontrivial examples, such as pushforward and pullback maps for sheaves, various geometric realization functors, or quantization of Dijkgraaf–Witten theories.

Even the definitions of relatively simple notions, such as equivalence of categories may convey a rather deceiving picture of their importance, until one learns some really deep examples of equivalences of categories, such as the Gelfand duality, Hahn–Banach theorem, spectral theorem for normal operators, Pontrjagin duality, Serre–Swan theorem, GAGA, or second and third Lie theorems.

As may be obvious from the above examples, it may be difficult to present category theory with sufficient motivation without making considerable use of other areas. This immediately presents a serious problem: from what areas one should draw examples? Trying to cover all of mathematics is too hard. Concentrating on examples from a specific area (e.g., algebra, like some books on category theory do) immediately creates its own problems: an analyst reading a category theory textbook with examples from algebra may not see much value in it, since it does not seem to relate directly to analysis.

Thus, one point of view is that the best way to learn category theory is to study other branches of mathematics that actively use categorical concepts.

Here are some textbooks that present their subject using categorical tools where appropriate:

• Algebra: Paolo Aluffi. Algebra. Chapter 0. Graduate Studies in Mathematics 104 (2009).

• Algebraic topology: Tammo tom Dieck. Algebraic Topology. EMS Textbooks in Mathematics (2008).

• Functional analysis: Alexander Helemskii. Lectures and Exercises on Functional Analysis. Translations of Mathematical Monographs 233 (2006).

• Elementary set theory: F. William Lawvere, Robert Rosebrugh. Sets for Mathematics. Cambridge University Press (2003).

• General topology: Tai-Danae Bradley, Tyler Bryson, John Terilla. Topology. A Categorical Approach. MIT Press (2020).

• Elementary topology: Ronald Brown. Topology and Groupoids. BookSurge (2006).

Once you are familiar with a few subjects that use category theory, you may want to solidify your knowledge by studying categories more systematically. Of the available books, the better one in terms of size and selection of material appears to be

• I would add: start with Lawvere-Rosebrugh, then proceed to Bradley-Bryson-Terilla and after that you can get into the deep waters of the first three references. Also the videos by the "Catsters" as suggested by @mozibur ullah in another answer should be useful to start. – Leo Alonso Apr 3 at 7:53
• These sound reasonable for a strong university student, or a professional mathematician who somehow has missed out on category theory, but how realistic is it to expect a 17 year old student to learn functional analysis from Helemskii and use it a springboard into category theory? – Mark Wildon Apr 3 at 11:43
• @MarkWildon: What is it that you find unrealistic here? I am not suggesting to learn functional analysis without first learning real analysis. Concerning your comment about age, I started studying functional analysis when I was 18 years old and real analysis when I was 17 years old, so studying real analysis for a year and then moving on to functional analysis worked just fine for me and it can also work for others. – Dmitri Pavlov Apr 3 at 16:26
• Definitely going to give a +1 over Paolo Aluffi's book. I was 18 and in my first semester of community college when I picked it up -- it was my first exposure to algebra and category theory and caused me to fall in love with both areas. – Andrew Tawfeek Apr 3 at 19:02
• @CalvinMcPhail-Snyder: Some references I am aware of (with various degree of explicitness): Jeffrey C. Morton's Cohomological Twisting of 2-Linearization and Extended TQFT Section 2 and 3 of Gijs Heuts and Jacob Lurie's Ambidexterity, Section 3 of Freed–Hopkins–Lurie–Teleman's Topological quantum field theories from compact Lie groups. – Dmitri Pavlov Apr 5 at 16:19

Unfortunately, there aren't many textbooks available to learn Category Theory from for the bright undergraduate or college student, unlike set theory or calculus.

Although, it is characterised as being abstraction heavy, it is no more so than set theory or calculus. Personally, what sold it to me is that category theory showed that the multiplication (product) was dual to addition (coproduct). Another slogan is that set theory emphasises the membership relation, whilst category theory emphasises the functional relation. The concrete application that showed how it was concretely useful, was to notice that the chain rule in differential calculus in vector spaces generalises to over manifolds and there it is seen as a functor.

One textbook is Lawvere & Schanuel's, Conceptual Mathematics. This was designed for high-school students.

At a more advanced level, check out Eugenia Chengs online lecture notes on category theory. I can recommend this highly - it's where I learnt category theory from. Eugenia is also known as a presenter of The Catsters, a series of short videos on category theory, and available on YouTube, and again, highly recommended.

I have an advice: Don't do it. I was once like you, and went into lectures about CT barely knowing any mathematics at all. The result: I didn't understand anything and failed. Learning CT requires a lot of background so that the ideas sound more "natural". If you're asking how to learn CT because of It's "abstraction and beauty", you're probably romanticizing for no good reason.

I am being blunt because that failure almost took my motivation to study any mathematics. That could also happen to you.

• +1: I agree with you simply because there aren't any good books on teaching Category Theory to the informed layman, unlike calyclus, geometry, logic and so on. I think it can actially be made more engaging than it tends to be by tbose fond of abstraction. – Mozibur Ullah Apr 5 at 21:36
• Romanticization is a very big temptation indeed. At least it was for me, and I have seen it in others as well. Very good to warn against this. – RP_ Apr 6 at 11:27

I think Fong and Spivak's Seven Sketches in Compositionality is the ideal answer to this question. You phrased your question as being about the "mindset" that you need to learn category theory, and if I were to put that mindset into a single sentence it would be: "It usually pays off to think about the structure created by the relationships between the objects that you're studying". "Seven sketches" takes that attitude and runs with it, applying it to examples coming from chemistry, informatics, computer science, and so on.

It's also structured in such a way that you can get a broad feel for the subject without getting overwhelmed by the theory; the authors present a nice visualization of the difficulty of the book:

If you were like me at 17, you might be better served starting off with an introductory text on the language of mathematics itself. I'd recommend the book "A transitition to advanced mathematics" available free here, or get an ereader version here.

A list of other free math textbooks can be found here.

You may want to take a look at Applied Compositional Thinking for Engineers (ACT4E), an ongoing graduate course at ETH Zürich:

In many domains of engineering it would be beneficial to think explicitly about abstraction and compositionality, to improve both the understanding of problems and the design of solutions.

A kind of mathematics particularly well-suited for thinking about compositionality is applied category theory. However, at present, this mathematics is relatively inaccessible to the average engineer. This is due in part to the inertia of the education system: outside of computer engineering, only little algebra is taught, in favor of analysis and related fields. This made sense some decades ago, but does not reflect today's real-world needs.

Recently, many good expositions of category theory for applications have appeared (see resources) yet to date, none are oriented explicitly toward engineering. This course will fill this gap. We will introduce the framework of monotone co-design as a means to teach basic concepts of category theory alongside principles of compositional engineering. Special care will be taken to illustrate the ideas with concrete examples (especially from autonomous robotics), and to indicate applications.

A nice resource (a learning roadmap) for exploring introductory papers, books, and vids on category theory is Bradley's "What is Category Theory Anyway?"

You like to see connections, so I would suggest, to provide a little focus yet an avenue for exploring diverse crossroads in math and physics (maybe even with some like-minded friends/schoolmates), that you try this challenge at a leisurely pace;

Given a real function $$f(x)$$ of a real variable represented as a power series about the origin that vanishes at the origin and its compositional inverse $$f^{(-1)}(x)$$, sketch the relationships among the two functions and the compositions $$e^{t \; f(x)}$$ and $$e^{t \; f^{(-1)}(x)}$$ elucidated by

1. category theory

2. Hopf algebra

3. differential geometry

4. geometric combinatorics

5. Lie theory

6. umbral / finite operator calculus

7. matrix algebra

8. statistical and classical mechanics.

I think you'll come to realize the strengths and shortcomings of the different approaches and how good Feynman's advice to constantly expand your bag of tricks / toolbox is.

Not trying to self-promote, but I have about 100 pages of basic category theory notes I was planning on eventually pushing towards internal/two-dimensional category theory and putting on the arxiv as a survey article. They've been sitting around for two years now, though, so I uploaded them in their current (very unfinished) form in case they might help you. All the content on $$1$$-category theory is solid (sections 1-7), and sections 8-9 are serviceable, but please ignore the later sections for now.

They assume no familiarity with anything besides what a set and function are, although being familiar with the naturals/integers/rationals as distinct entities will help understand an early example of a non-surjective epimorphism. If you need to develop familiarity with sets please feel free to grab any standard text on set theory and read it, or if you like my notes you can try this arxiv paper I uploaded a few years back which builds up the set theory you'll need from scratch. Again, the research at the end is shaky and shouldn't be used, but pages 1-6 give a 'from-scratch' development of the axioms of MK class theory and the stuff through page 19 offers some basic lemmas to try and prove (don't read my proofs unless you need to!)

Best of luck, and remember not to beat yourself up if things get challenging! The level of abstraction in either of the above note sets is likely to be a signifigant step up from the kinds of mathematics you've encountered thus far, so be sure to come back to MSE and ask further questions! (MO is a bit too high level for this stuff generally speaking, as evidenced by the users trying to close the question, but these are completely reasonable questions over at MSE)

• thank you so much, I just started reading your papers ! – Matias T-C Apr 6 at 9:39
• @MatiasT-C No problem, I'm glad they're of assistance :). – Alec Rhea Apr 6 at 9:40