The Joy of Abstraction: An Exploration of Math, Category Theory, and Life

by Eugenia Cheng

Mathematician and popular science author Eugenia Cheng is on a mission to show you that mathematics can be flexible, creative, and visual. This joyful journey through the world of abstract mathematics into category theory will demystify mathematical thought processes and help you develop your own thinking, with no formal mathematical background needed. The book brings abstract mathematical ideas down to earth using examples of social justice, current events, and everyday life – from privilege to COVID-19 to driving routes. The journey begins with the ideas and workings of abstract mathematics, after which you will gently climb toward more technical material, learning everything needed to understand category theory, and then key concepts in category theory like natural transformations, duality, and even a glimpse of ongoing research in higher-dimensional category theory. For fans of How to Bake Pi, this will help you dig deeper into mathematical concepts and build your mathematical background.

Genres: Mathematics, Nonfiction, Science, Popular Science, Technology, Computer Science, Technical

424 pages, Hardcover

Published October 13, 2022

About the author

Eugenia Cheng is a mathematician, pianist, and lecturer. She is passionate about ridding the world of math-phobia. Eugenia’s first book, How to Bake Pi, has been an international success. Molly’s Mathematical Adventure is her first children's book.

Reviews

[Sandy Maguire · Rating 4 out of 5]

This is a hard to review book. But the fact that I got through it, unlike any of the other ten category theory books on my shelf, I suppose is a good endorsement. I was keeping up entirely until full/faithful functors, and the remainder of the book washed over me, but I persisted on for at least a tour of the vibes.

However, the book itself is perplexing. Who is it written for? The first hundred pages give off a vibe of "math for people who have been traumatized by math" while the last hundred go hard into the paint. While I appreciated that the book doesn't use examples from math, the examples it chooses are POLITICALLY CHARGED and therefore EXTREMELY DISTRACTING. For the most part they're fine, but it does seem wildly unnecessary to add the parenthetical in excerpts like this:

> Now consider some other object x in the category. We're going to show that x "can't tell the difference" between a and b because whatever relationships x has with a, it has the same system of relationships with b. This is quite a deep idea, and is a bit like how I tell people apart, if I'm going to be honest. A lot of people look the same as each other to me in terms of physical appearance (especially white men) and I can only tell them apart via personal interaction with them.

I dunno man. There's a lot of this sort of thing and it all feels unnecessary and a bit gross. It's not enough to prevent me from recommending this book, but it's certainly not a strong recommendation.

[Bugzmanov · Rating 4 out of 5]

Nice intro to category theory. The first part is pretty laid back and can be bed time reading, the second part is noticeably more challenging and requires attention and googling for alternative explanations.

To be honest by the time I got to yoneda I was completely lost and the big reveal did not have knock off effect on me. Overall I think it's a good companion to Bartosz Milewski cause this one is all about math and functional programming hasn't been mentioned even once. This book helps to wrap your head around the fact that category theory is an abstraction that by itself has nothing to do with any applications (like FP)

I found this youtube channel https://www.youtube.com/@RichardSouth... to be helpful wile reading this book.

To conclude: I don't know why I've read this book but I don't regret reading it.

[J Earl · Rating 5 out of 5]

Eugenia Cheng, in The Joy of Abstraction, shows how the pragmatic uses of mathematical thinking does not preclude the fun of it.

This is the third of her books that I have read and, as I have come to appreciate, it both taught me a great deal as well as entertained me. No, not like a film might, more like when your mind begins to grasp something and it is entertained by how many ways there are to use that new information. While Cheng says that How to Bake Pi isn't necessary for understanding this book, I can say I was glad I had read it. I do agree, it isn't necessary, but it sure helped with my comfort level while reading this one.

This is mathematics, but not like we learned in school. This is engaging, creative, and, yes, fun. This leads the reader to some new ways of thinking, mathematical thinking, that can affect every aspect of your life. Her enthusiasm is contagious.

Since this real math made more accessible for non-mathematicians, there will be sections that may require rereading. The reward, however, is well worth the extra time to make sure you get the big picture. That said, because the big picture is probably what most readers want, some sections can, I think, be skimmed without losing the view. I also, at least for myself, think this will be a work that will get richer with subsequent readings. My plan is to give it a couple of months to ferment in my mind then revisit it.

Highly recommended for everyone from creative types to those in the mathematics-based fields (science and engineering as well as math). Since this is as much about mathematical thinking as it is category theory itself, the benefits go far beyond just learning about a field of mathematics.

Reviewed from a copy made available by the publisher via NetGalley.

[Anusha Datar · Rating 3 out of 5]

This book provides non-mathematicians like myself with an overview of how category theory works and why we should care about and appreciate it. The author starts with definitions and carefully prepares the reader to understand the symbolic notation and abstract concepts she relies on in the second half of the book.

I found this book hard to get through. I’m not a mathematician, and in reading this book cover-to-cover (maybe it would have been easier over a couple more days?) I found myself lost and constantly re-referencing other chapters during the second half of the book. That also may have just been a consequence of its survey-like nature. Separately, while I appreciated her effort to connect category theory to social issues, a lot of her notes from everything from COVID-19 compliance to white male privilege felt a little surface-level.

I did get through it and I learned a bit about a topic I’ve been eluded by in the past, so I still feel like this book deserves a net positive review from me. That being said, I’d just recommend it as a primer and not suggest it introduced any new appreciation of beauty or overall synthesis for me.

[Preston Malen · Rating 2 out of 5]

The political and social commentary is unbearable. If she would stfu and just focus on the math it would be way better. She constantly makes jabs at white men when a majority of mathematicians are white men. Its just hard to take her seriously even as a researcher because of this unprofessionalism. Criiiiinge

[Josh · Rating 4 out of 5]

Nice intro, lively prose

[Julia · Rating 3 out of 5]

I am not a maths nerd, I just like maths. Hence I enjoyed the first part of this book more than the second half. I needed to check online for extra information to understand the content. That is not a reflection of the writing, the writing is good. It's just a niche book that does not make for bedtime reading.

[danielle; ▵ · Rating 1 out of 5]

extremely embarrassing publication; not joyful

[Justin · Rating 4 out of 5]

I've never taken a class in category theory, but it's something that regularly showed up in my research. From that point of view, it was very difficult to understand what motivates category theory. It often gives the impression of abstraction for the sake of abstraction. This text does a great job at building up category theory from the ground up, giving lots of motivation along the way. Having said that, I really had trouble finishing this book. The deeper you get into it, the more it felt like a textbook.

[Manuel Del Río Rodríguez · Rating 5 out of 5]

I have made a short video review, which you can watch here:

https://www.youtube.com/watch?v=o6sYP...

[Witt Igahluk · Rating 4 out of 5]

This is an interesting exploration of the world of abstract mathematics for non-mathematicians, a kind of a tourist guide for those who want to embark on an adventure in the category theory land. Eugenia Cheng makes complexity an accessible challenge for the non professional mathematician.

The book is divided in two parts, the first introduces the reader into categorical thinking and the re-examination of concepts as sets, functions and orders. Cheng uses examples and analogies from everyday life to explain abstract concepts and also delves into social and privilege issues under categorical lenses.

The second part is quite more technical, where Cheng does a great job of explaining concepts and proofs in an accessible way (although maybe long and repetitive for experienced or professional readers). Cheng keeps examples as simple as possible, giving enough context and analogies to understand the underlying concepts without overwhelming details nor giving too much for granted.

While I thoroughly enjoyed the book, I do find it challenging to recommend it to a specific audience. Experienced readers may find the first part of the book too basic and repetitive, while the second part may be rewarding with interesting insights. On the other hand, for newcomers to this journey the first part will offer a nice view of the scenery, while keeping the second one as optional if it becomes too technical or hard to follow.

"The Joy of Abstraction" is an enlightening exploration of abstract mathematics and an excellent introduction to category theory.

[Kyle Wright · Rating 3 out of 5]

Made it about ~60% of the way through this, think I'm gonna stop there. I think this book is attempting to answer an ill-posed question. The target audience for this (non-mathematicians who are willing and able to learn category theory) doesn't really exist, so it ends up simultaneously too broad and too specific. Someone with an advanced STEM or quant-phil background outside of pure math might get something out of it, but if it didn't work for me, I struggle to imagine the kind of person for whom it would. Maybe ideal for students who already need to learn category theory and need a marginally less textbook-y way to get them there.

I expect there will be a lot of complaints about the "social justice" angle of the real-world examples Cheng uses throughout this, so I feel like I should specifically say: I don't dislike their inclusion nor their message (God forbid a single math thing not be made for white guys), but I do unfortunately think they are done pretty poorly here. If the word "intersectionality" means something to you, you already have a more complex understanding of social issues than the examples used in this book. Again, this book imagines a reader who can simultaneously understand monoids and functors with no math background, but is just now finding out that rich people are more privileged than non-rich people.

In every way, a very perplexing concept for a book.

[Mohammad Tanviruzzaman]

Though sounds a bit sweeping, liked the observation: "Every academic discipline is seeking a particular type of truth, and develops a method or framework to decide what counts as true."

"Abstraction is about digging deep into a situation to find out what is at its core making it tick." The statement, at first, feels like looking deeper into say atomic structure, subatomic particles like russian doll. I think what was meant is by abstracting we take a step back, see the essence of stuff, to be able to see the woods instead of only trees all around.

Loved how abstraction helped reveal the two types of disagreements that happen around an analogy. Say someone claims "a" is similar to "b" but does not sufficiently clarify of which x are a and b specific examples.

x
/ \
a b

Someone else might disagree in the below two ways:
1) They may find that a is actually an example of w, an intermediate level whereas at that intermediate level b isn't
x
/ \
w b
/
a

2) They may go up one level and find a distant and irrelevant example c and then say "if a and b are similar, as you claim, then a and c are also similar" essentially setting up what is called a straw man.
w
/ \
x \
/\ \
a b c

Helps to clarify at the onset which level of abstraction makes a and b analogus.

Is 1+1 = 2? It depends. Does circle always look like hula hoop? It depends. Clarify or at least be conscious about context.

Factor lattice is amazingly rich!

This is the fourth book by Eugenia Cheng I am reading and loving. By this time, when I come across the phrase "the sense in which" it feels like seeing a friend, makes me chuckle with delight :)

aRb is a statement, either true or false; a+b on the otherhand is a number given a and b are also numbers. The three properties that makes a relation behave like the familiar "=" relation are: reflexivity, symmetry, and transitivity. These three properties invole considering more and more elements at once. Reflexivity considers one element, symmetry considers two, and transitivity considers three. Like modus ponens for composing implications, it is transitivity that helps create long chains and thus discover more and more aRb pairs. While testing if a property holds for a given relation, the elements we consider do not have to be all distinct. So, while testing, letting some or all of the elements under consideration the same, is useful. Like the equality relation, equivalence relation involves the concept of "sameness", perhaps in disguise. Logically, equivalnce relation is same as partitioning the set neatly and call two elements in any partition related. Due to the rigidity of an equivalence relation, many otherwise interesting relations do not qualify as equivalence relations. To study such interesting ones, in category theory, we would relax one or more of the three required properties: reflexivity, symmetry, and transitivity.

Given that the earth is spherical, does it make sense to say there is The East and there is The West? East or West of what?

About "is congruent modulo n" she says: """I hope you *feel* that this relation ought to be an equivalence relation as we're talking about things being "the same" on an n-hour clock. A feeling isn't enough, but I think it's an important start.""" This in my opinion is something that makes her books more alive.

Loved: "We sometimes have to pick which problem to live with though, and usually the problem of anomalies with rounding is a bit irritating, where the problem of non-transitivity is catastrophic." Specially the part about choosing the lesser evil, even in math!

A category C has a collection of objects called obC. For any pair of objects a and b in obC, there is a collection of arrows called C(a,b) like a->b. For each object a, there is an identity arrow a--1_a-->a. For a pair of arrows a--f-->b--g-->c we have a composite arrow a--gof-->c. There are two axioms or properties: (1) for an arrow a--f-->b we have f o 1_a = f = 1_a o f. Also f o 1_b = f = 1_b o f. (2) for arrows a--f-->b--g-->c--h-->d, we have (h o g) o f = h o (g o f). The two axioms are characterizing some arrows as the same.

With categories there is an identity arrow (like reflexivity) and two arrows can be composed into a unique composite arrow (like transitivity). Symmetry is not a property anymore, it comes if there is an inverse arrow. Reminds of directed graph. Although in digraph all arrows represent the same relation, in category, arrows can represent different relations (or morphisms). For example, a --is sister of--> b and b --is mother of--> c produce a--is aunt of-->c

A totally ordered set (toset) is a category with "exactly" one arrow between a pair of objects. The natural number line represents a toset with a <= b as the relation. A partially ordered set (poset) is a category with "at most" one arrow between a pair of objects. Factor lattice is a poset. Poset is a generalization of toset.

[Tanner Duve · Rating 5 out of 5]

I usually find pop-science pop-math type of stuff pretty unattractive but this is legitimately such an important book. If you want to learn category theory then look elsewhere, but this isn’t some category theory textbook, it is a larger commentary on the importance and utility of abstraction and formal reasoning in life and society; category theory just happens to be the vehicle of choice (no better one if you ask me). It is partly written to show the general public the beauty of abstraction, and expose how most people’s recoiling at the word “math” is simply a failure of the education system.

I am admittedly skeptical of the attempt to teach in depth category theory to a non-mathematical audience, although she does say that her students at art school seem to pick it up well. In any case it’s really not the math content that I find great about this book. Cheng’s ability to explain the essence of abstraction and give examples as to how mathematical-like reasoning can and should be applied to how we discuss the important problems in real life honestly gave me a perspective on things that has been really important to me since coming across it. Some people (exactly those who need it most) bash this for being “woke”. She will freely pull examples from topics like gender, privilege, or racial oppression and explain how having a formal understanding of concepts like analogy, “sameness”, context, relations, and patterns is crucial to properly discussing these social issues. I’ve never seen anything like it. I often find myself analyzing political and ethical arguments using the frameworks she talks about in here, and I think a major cause of disagreement and distress in our world is simply a lack of the ability to reason in this manner. I gifted this to my mom as her first exposure to abstract math and we talk about it regularly to this day. Very few books I’d say “everybody should read” but this is one of them. Great talk by the author here, I’m a huge fan: https://www.youtube.com/watch?v=48VqW...

[Arthur Ryman · Rating 3 out of 5]

I applaud the author for this attempt to create an accesible book on category theory and further the general public understanding of mathematics. She does a good job defining categories and giving small detailed examples. However, she did not deliver on her goal of showing that category theory can be appreciated on its own as a thing of pure abstract beauty. Although the modern approach to teaching mathematics is to present it as a collection of somewhat isolated topics, the true power and beauty of mathematics only emerges when one topic is used to illuminate another topic, showing that at its core mathematics is an intricate tapestry of ideas. The true joy of category theory is that it provides a framework for describing how apparently separate topics are intimately related, the primary and motivating example of which was the application of algebra to topology, aka algebraic topology. However, the author did not want to assume any prior knowledge of other mathematical topics, resulting in a standalone treatment of categories. The author also uses this book as an opportunity to discuss her research in higher category theory. However, her treatment of higher category theory is far too technical to be understand or appreciated by her stated target audience. Finally, I found her use of category theory to discuss social issues such as white male priviledge to be both unilluminating and oddly out of place.

[Cod Er · Rating 4 out of 5]

Disclaimer: I only read part 1 and the interlude because I’m not a professional mathematician or a serious student of mathematics. I quickly gave up after starting part 2 because I wasn’t looking for a technical read - I largely picked this up because I’ve seen category theory mentioned in various places and wanted to sate my curiosity.
The parts which I did read were great - easy to follow but informative and I love that the author included thinking points for the reader throughout to help the material sink in.

[Petru Cervac · Rating 4 out of 5]

Every sane reader I listen to says to me that you shouldn't continue reading a book that doesn't vibe with you. Maybe it is time for me to listen.
I didn't finish the book - I've only reached page 343. It is a good introductory book on category theory. The most important thing the book provides is the motivation for category theory. It is less challenging than Category Theory for programmers, but still requires effort to fully comprehend. Don't expect this book to be easy!
I'll come back to it one day to give it a second run.

[Michael · Rating 5 out of 5]

A highly satisfying and surprisingly accessible introduction to category theory. Cheng's genuine enthusiasm and love for the theory is there to encourage through the more strenuous sections and present the results with the significance they deserve and their relevance to the world outside mathematics.
Category theory can also act as a basis for systems design and this book would serve as a great starting point for anyone interested in the field.

[Adam · Rating 3 out of 5]

Good text as intro to category theory. Five stars for that. But the frequent racist references and misguided 'social justice' made it a very frustrating read. I don't want Yoneda lemma mixed with talk about white priviledge. Please stick to the math, don't polute it with the activist postmodern nonsense. I take two stars off for that.

[Kory · Rating 2 out of 5]

Cheng's political views pervade her narrative and she makes some logical leaps in attempting to tie mathematical ideas to societal issues. Perhaps this was not an accident, but it was moderately annoying to me. I may return to the book again - the mathematics was interesting and worth reading, at least as much as I read.

[Paul · Rating 4 out of 5]

For those interested in math -- learning more and doing a deeper dive into the topic. So, the audience for this is pretty niche, but for those that pick it up it will probably be of interest and value.

Thanks very much for the free ARC for review!!

[Jack Davidson · Rating 3 out of 5]

For a non-mathematician I can imagine this as a great introduction to the categorical way of doing maths. However, for someone who knows (some) category theory already it is perhaps slightly dull. It’s more of a textbook than it is made out to be.

[Kirsten Hill]

The first time I tried to read this, I had a copy from the library and just didn't have enough time. I now own a copy and look forward to taking it slow and working my way through it (Maybe summer 2024?).

[Zach Beever · Rating 5 out of 5]

Genuinely one of the best books I've ever read. Extremely playful and fun, yet with actual substance. In my opinion, this is the book for any non-mathematician looking to jump into category theory

[Gordon Ironside · Rating 5 out of 5]

A very readable book about a non-trivial branch of maths. I will be looking for other books by Eugenia.

[Jagatheesan Jag · Rating 5 out of 5]

It was a pure joy reading this book, a journey into abstractions over abstractions. :)

[monderin · Rating 5 out of 5]

eugenia u literally saved me from being a software engineer for the rest of my life ilysm !!!!

[Lucas · Rating 4 out of 5]

I would like to say that there is indeed much truth in what others have written, but I find myself diverging on a key aspect, and I would like to highlight a few positive things, and generally express a fully positive opinion about this type of books. I completely agree that some of the comments, such as those about white men, were very inappropriate and undermined my trust in the author as a kind, warm guide. This does not detract from the fact that it is a relatively serious book. There is a distinct lack of exactly this type of publication which provides real knowledge while simultaneously being intended for people uninitiated in the intricacies of science or philosophy. I remember a work by Süskind that was something like this and actually surprised me those few years ago by challenging the reader accustomed to works that are more introductory, citing anecdotes about the field rather than delving into the field itself.

It is a fact that the target audience is limited from both sides – the uninitiated will be disappointed because it will be too difficult, while those who are initiated will choose more serious works. As I belong to that small group of people in between, and because I believe that one should value it when a certain niche is catered to, I wanted to emphasize that this is very good. I think that didactics and the art of teaching consist in entering territories challenging for laypeople in a subtle, yet demanding way. On the other hand, such a tactic is also a slap in the face to specialists because it shows that teaching and disseminating knowledge is not an offer directed only to other initiated, talented people, or those similar to oneself, but requires building bridges between completely different worlds. I think the author, having experience working with people who think very differently from computer scientists or mathematicians, knows this; even if this challenge is posed both to experts, who will see too much superfluous text and certain simplifications, and also to beginners, who at certain moments will get stuck and think "but this is difficult". There is a risk of exposure to criticism from both sides.

I wish, however, that there were more books of this sort available in general-interest bookstores, so that perhaps it would become normal, just as we sometimes buy crossword puzzles or Sudoku for intellectual challenge, to come across a book that is designed to be a little more difficult than most, but still easier than a textbook. Nevertheless, I would advise the author to take a step back and give examples, or even pretend that someone else thinks this way, because in several places I felt that the author's tendency towards personal disclosure overshadows the core message. Some people will like this very much, but many will not, and perhaps this is precisely a risk that doesn't need to be taken in this type of literature?

Really, instead of talking about how the author sees something and always had a problem with this or that, she could insert the perspective of a person who is learning and present it using an example like, "many of my students seem to prefer arrows pointing left or right". Transitions from these types of remarks to certain social issues provide too much opportunity to enter into excessive intimacy, which can undermine the authority or trust in the teacher. Overall, the book is a net positive.

[mkfs · Rating 1 out of 5]

Off-putting in a way that math books really shouldn't be, particularly ones about category theory.

This one talks about all sorts of irrelevant nonsense, as if it were a pop-science book, then just sorta dumps the actual category theory on you without any explanation as to why this is useful or necessary when we already have, say, set theory and abstract algebra and all that.

Going off memory here - chucked the book about halfway through, as I was thoroughly repulsed. And I ain't the sort to repulse easily, so maybe that should get the book an extra star. Nah, looks like other readers were far to charitable to this mess.

[Mingus Bingus · Rating 4 out of 5]

Loved the book club!