Category Theory in Context (Aurora: Dover Modern Math Originals)

Riehl may well be the most talented category theorist of her generation - and as such, could well become one of the more influential mathematicians of the coming decade or so. Category theory is not just the foundation of Mathematics, but the bedrock upon which other foundations can support structures. As we move from a world where Moore's law and classical mechanics dominated our purview to one where quantum physics and quantum information start to directly influence our lives, Category theory will become an essential tool for navigating a synthesis between maths and physics that will become more more enabled than at any time in our past. For one thing, a simulation of even the most basic quantum systems will push the boundaries of what can and cannot be computed. In her recent work Riehl has gone from being a skilled and exciting young mathematician to one who challenges our understanding of the structure of reality, and in this book she exhibits a wonderful and refreshing ability to wear her talent lightly whilst providing a refreshing approach to the context of category theory. The book is not for beginners and it is not easy to read if you have not yet grasped the basics (or the specialised vocabulary). I am a big fan of Tom Leinster who is another category theorist from the maths side, and I am also very keen on Bob Coecke and Bartosz Milewski who approach CT from a different (theoretical physics and computer sciencefor Coecke and programming for Milewski) who I think also display an intuitive understanding of this beautiful subject, but Riehl stands out as the intellectual leader of the pack

Very good book on category theory, but certainly not the first one you should get. You probably need a good level in Algebra, and compared to Awodey's book it's much more difficult. That said, it's very cheap and incredibly well written.

Explaining a lot of mathematical concepts in category theory for a programmer with mathematical incline.

I have been working through various drafts of this book online, before it was published by Dover. It really is a fantastic guide to Category Theory; it is more through (alas longer) than Tom Leinster's book, and more accessible than Saunders MacLane's bible on Category Theory. As a graduate student in Mathematics, this really is the bible for an introduction to Category Theory and the book has so many examples and uses of Category Theory that it is accessible and of use to research in almost any area of mathematics. This should be top on the list for many mathematics research students!

This is probably a great book if you have the right background, but be warned, it is certainly not the case that "Prerequisites are limited to familiarity with some basic set theory and logic" as claimed in the blurb. The author has kindly provided a PDF of the book on her web site, and the very first sentence of chapter 1 there is: "A group extension of an abelian group H by an abelian group G consists of a group E together with an inclusion of G ,→ E as a normal subgroup and a surjective homomorphism E  H that displays H as the quotient group E/G." If you call that "basic", then go for it - this is the book for you. If not, Lawvere and Schanuel's "Conceptual Mathematics" will be a better choice.

Muy buen libro para profundizar en teoría de categorías, aunque no lo recomendaría para un primer contacto con la materia (Awodey es un buen comienzo). Si bien la impresión no es de buena calidad (bajo gramaje de las hojas), no se puede pedir más por un precio tan bajo. También es de destacar que la autora haya tenido la gentileza de dejar disponible el pdf gratis para descargar desde su página web.

Recebi hoje 29/04/24. Livro de capa comum, excelente impressão, porém letras são pequenas.

I returned to this book after some time passed and I appreciate it more. First of all, it wasn't all that hard to eventually fill in the gaps and be able to use the book. Second, ChatGPT and other LLMs are now available for interactively filling in gaps, and self-contained introductions seem less important now than they did a few years ago. After getting past my initial difficulties, I can also appreciate that the author's contextualization of the material with quotations, hints, and footnotes and the choice of references is very well considered and useful. This is how I felt initially (original review, 4 stars): I'm using this book for self-study and started out reading and working problems as I went along. It didn't work: the book is too difficult as a first text for this purpose. I do think this is a good book, but it's too advanced for a first introduction. There are some statements in the introduction to the effect that few prerequisites are required if you also have some "mathematical maturity". Don't be fooled: this is code for already knowing category theory, and some essential definitions are missing. I recommend that if you are acquiring category theory by self-study you should start with "Category Theory for the Working Mathematician," because that book does define everything with complete clarity. And don't let the title of that book scare you: it is perfectly sufficient for the beginner. Steve Awodey's book "Category Theory" is another good starting point or second reference. Essentially this is the problem: to understand basic category theory you need an unambiguous interpretations of commutative and non-commutative diagrams, abstract categories ("metacategories"), and concrete categories. Then as each auxiliary definition and tool is built up you need to understand its construction unambiguously and in its full generality. These things are all fairly simple. I'm observing that once people understand all of these things together as a "language" they enjoy the experience so much that they immediately forget how to explain the basic definitions to their readers and want to just write category theory at you. And to some extent that is a positive and promising thing. I'm hoping that once I get my basic grounding in category theory and come back to finish reading this book that the more advanced material is presented well enough that I can understand it. I expect that that will be true based on reviews by more advanced readers.

Es difícil de leer para estudiantes de licenciatura (como yo), pero presenta muchísimos ejemplos y la teoría se expone de manera muy interesante. Varias veces he quedado asombrado por los resultados que aparecen y cómo generalizan conceptos de diferentes ramas de la matemática simultáneamente. Me encanta!

Riehl always exhibits plenty of examples of mathematical phenomena BEFORE giving the categorical concept that subsumes them all. The effect is that category theory is seen as revelatory. In reponse to the usual saying that "there are no theorems in category theory," she also gives lots of categorical theorems, in other branches of math as well as in category theory itself. A fun and elucidating read!