Category Theory in Context

by Emily Riehl

Category theory provides a cross-disciplinary language for mathematics designed to delineate general phenomena, which enables the transfer of ideas from one area of study to another.

Category theory has provided the foundations for many of the twentieth century's most significant advances in pure mathematics. This concise, original text for a one-semester course on the subject is derived from courses that author Emily Riehl taught at Harvard and Johns Hopkins Universities.

The treatment introduces the essential concepts of category
 
Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in the
 
Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas.

Dover is widely recognized for a magnificent mathematics list featuring such world-class theorists as Paul J. Cohen ( Set Theory and the Continuum Hypothesis ), Alfred Tarski ( Undecidable Theories ), Gary Chartrand ( Introductory Graph Theory ), Hermann Weyl ( The   Concept of a Riemann Surface ), Shlomo Sternberg ( Dynamical Systems ), and multiple works by C. R. Wylie in geometry, plus Stanley J. Farlow's  Partial Differential Equations for Scientists and Engineers.

Genres: Mathematics, Textbooks, Science, Logic, Nonfiction, Computer Science, Algebra

272 pages, Paperback

First published January 1, 2015

Reviews

[Andrew Foote · Rating 3 out of 5]

This book has been "read" in the sense that I have frequently consulted it as a reference while trying to learn category theory. Of all the category theory textbooks I've looked at, Riehl's has turned out to be the one I've consulted most frequently (although I have also consulted just about every category theory textbook out there, as well as Wikipedia and the nLab). This has surprised me, as my first impression that the book would be unsuitable for a beginner like me. Riehl does write quite tersely and formally, with an intimidating effect. But if you avoid succumbing to the intimidation, you'll find that the exposition is admirably clear and enlightening. In this respect the book is similar to Rudin's books on analysis.

Perhaps one could say the same for any other book on category theory, though. I think possibly the main reason I have kept coming back to Riehl is that it has an epilogue where it has a list of "theorems in category theory". This is a very valuable resource for getting to grips with the subject. Category theory has a lot of horrendously complicated definitions and not a lot of enlightening theorems and without knowing which theorem the exposition is building up to, it's easy to feel disinclined to keep following along. Since I found this list I've been tackling the theorems in it one at a time, and this has allowed me to feel like I'm making reasonable progress, whereas for a while I felt quite bogged down in the subject.

[Jeff · Rating 5 out of 5]

If you have any sort of background in higher-level mathematics, this is *the* book to read on Category Theory. The structure is "concise exposition followed by tons of familiar examples." Loved it.

[Mady · Rating 5 out of 5]

A ver, non o lin enteiro, pero si un bo cacho. A intención desta review é evidentemente celebrar a fin da carreira e o tfg. As categorías molan. E o nome deste libro é o mellor posible para un libro sobre elas.

PD. Les go lesbians

[Tomáš Ševček · Rating 5 out of 5]

I read the first four chapters of the book for my category theory course. I find the exposition to be very clear. While category theory per se is quite easy to learn and understand (no more challenging than an introductory abstract algebra class), the complexity lies in the many examples from advanced fields such as algebraic topology, algebraic geometry or higher abstract algebra, which demonstrate the true power of the theory. Therefore, I do not recommend this book to anyone who is not familiar with at least one of those fields (unless you enjoy abstract mathematics solely for its neatness and beauty), otherwise you might end up thinking that category theory really is just abstract nonsense. Being an analyst, I was happy to see at least a few examples from analysis even though I do not think that the concepts are made clearer if you look at them through the lens of category theory. Still, I think this book successfully demonstrates how category theory revolutionized certain fields such as algebraic topology.

[Ayaz · Rating 5 out of 5]

Don't get me wrong, this is a tough book and Riehl's suggestion of mathematical maturity is not just a suggestion; if you really want to appreciate the subject and everything Riehl discusses, you indeed need to be comfortable with the abstract and to quickly fill in details as it goes. Like any textbook for self study, a lot will be demanded on your part. The first couple chapters are not very involved; indeed, you do not need to lift your perspective outside that of the set theoretical view of more "classical" mathematics. But Riehl's invitation to do so is not a fly-by-night suggestion - without truly embracing a categorical viewpoint chapter 3-5 will be very challenging, especially the interesting exercises of those chapters. Chapter 6 on Kan extensions is a must-read, though it is yet another "step above" abstraction into higher categories.

[Chris · Rating 3 out of 5]

readable, with examples. However, category theory. And monads. Riehl notes that many mathematicians have said "There are no theorems in category theory". After reading her book I find myself agreeing with them, not the author.

I like the presentation around adjunctions and would recommend chapter 6 (Kan Extensions) to everyone who cares about Kan extensions.


What "context" category theory exists in … might disappoint the applied theory crowd.

[mishu_2432 · Rating 5 out of 5]

great book you just have to read it 3 times

[0xd34df00d · Rating 4 out of 5]

This book could've been 50% more accessible if it was merely 5% longer, by expanding on some implied or omitted steps in some theorems proofs (I spent maybe a third of the study time retracing those), and perhaps adding some more illustrative examples (like, the intuition behind what it means for a category to be cartesian closed instead of the mere definition, and so on).

Also, definitely not a first book on category theory — I have some prior exposure to the subject from the topos theory (and the general osmosis from the computer science/type theory noosphere), and yet I struggled in the main text (so, not the exercises) more often than I'd like to.

[Siddharth · Rating 5 out of 5]

I'm reviewing this two years after I finished reading it. With my current perspective, I can see that it was an excellent book. When I originally read it, I thought it was a good introduction. But it actually motions towards lots of fruitful generalizations in the exercises that one encounters later!

[Francis · Rating 5 out of 5]

I actually really enjoyed this.

[Rick · Rating 5 out of 5]

The best introduction to Category Theory that I'm familiar with. I recommend it over any of the other well known ones.

[Jacob Getch · Rating 3 out of 5]

"There appear no common values, yet each object is related to the other."

The writing shows how topological spaces relate through continuous functions. A brilliant exercise in abstract mathematics.

[Lulu]

Category Theory