An Invitation to Applied Category Theory: Seven Sketches in Compositionality

A huge improvement over all the other category theory books where all the examples are from 20th century pure math and all the morphisms are functions. Especially the last chapter.

Category Theorists often joke that they work on Abstract nonsense but really what they work on is the study of compositionality which shows up computational graphs like circuits and neural networks and mathematical proofs. Learning Category Theory is a lot like learning programming, you’ll be able to express proofs in a much more succinct way after investing some time learning the new opaque terminology.

The book is wonderful but, why do even pages have the margin on the left making the lines in those pages to almost get into the spine?

Engaging and well written. I'm learning and enjoying as I (slowly) go along. I was attracted to this book because it has lots of exercises with solutions. As an adult learner who has rediscovered math, this book is encouraging me to keep going. I really appreciate that. Layout is easy on the eyes, too.

This is the best introductory text on Applied Category I have seen. I am a software developer with a background in functional languages and without a strong maths education. This book has been extremely informative and helpful for me in connecting to Category Theory. The print edition looks great and is a high quality printing and binding. Highly recommended!

Oh, precious book! You functor my heart! Into a land of crisp pedagogy Right from the start! I needn't profunctors To decipher a mess That lurk inside yellow books tucked next to my desk My head is a tumult My soul soaked beyond Many stones flipped and turn'd ov'r By spivak and fong

Even as a math person, pure category theory seemingly always went over my head. Fong & Spivak provide great examples that are "concrete" enough for me to get it.

I love math, but I am not a mathematician. I excelled in math in school, acing the highest high-school level class, and have been a mathematical tourist and to a lesser degree, an applied mathematician, ever since. I'm continually trying to broaden and deepen my math understanding. In life, I research biological pattern formation. Enough about me. The only relevance here is as a warning to those potential readers who love math but are not mathematicians. This book was advertised as high-school level; in the preface, the authors claim this book is suitable for motivated high school students who haven't seen calculus. I've studied multivariate calculus, dynamical systems theory, probability and measure theory. This book makes measure theory seem like a piece of cake. So, my main gripe is not the content of the book itself. It is the pitching of the book to an audience that, like myself, will find much of this book inaccessible. Or, it may be accessible, but the investment is likely to be huge for the uninitiated. The authors sell category theory as a "central hub" that provides a way to find commonalities and facilitate communication - a tool for thinking. Moreover, one of the book jacket reviewers commented that if your work requires thinking, this book is for you. These statements were why I bought the book. However, it is unclear to me from a first pass how the formalism of category theory provides advantages over other formalisms. For example, the authors state that Galois connections were first considered by Galois who called them field extensions and automorphism groups. OK, I'm familiar with that terminology. But the book recasts all the terminology into a new language. What is gained by doing this? Effectively, this book is a "new language book" that requires substantial effort to connect the terminology that you might have learned from high school mathematics to the language of category theory. The first chapter, for example, has 117 definitions, theorems, and exercises. Each, on average, presents at least 1 new concept, and, for me, most of the concepts involved new language. A snapshot from the first chapter includes: preorders (discrete, codiscrete, product, etc.), meets and joins, generative effects, total order, (left / right) adjoints, functors, adjunctions, closure operators, etc. The ideas are presented, then exercises requiring proofs follow suit. For the committed, there is enough information to learn the material, but for me, the book is not fulfilling its advertised promises. At least, not yet. Moreover, many of the examples (applications) seem contrived and over simplified. Being a biologist, the "tree of life" example was oversimplified and borderline false due to processes such as reticulate evolution. Applications are discussed in a few pages, then the standard math textbook format ensues: definitions, examples, theorems, proofs, ... The application ultimately takes backseat to the pure math, and connections from the pure math back to the application are often not made explicit in this book. The power of the pure math for application is not immediately evident, although advertised as an applied math book. Despite my concerns about the overwhelming content and the grandiose statements of what a reader will achieve by learning category theory, this book has many strengths. It is systematic in its presentation. Exposition is clear. The book provides answers to the exercises. The stated goals are amazing. Despite my misgivings, I plan to continue working and reworking (and reworking) through this book because even if there is a chance that I might achieve the advertised goals, then it will be worth it. I'm very far from that outcome.

This book has a very nice gradual progression building from basic ideas. The tone and pace are quite pleasant, which is a blessing since its so easy to make the presentation of definitions dull and dry. I've found it very straight forward to follow the explanations and at the end of them I feel like I truly understand the concepts. I'm half way through it and don't feel overwhelmed or lost. A proof of its good pedagogy is that I have been able to successfully complete all of the exercises. It was a great decision by the authors to intersperse the exercises with the content instead of adding them at the end of the chapters. It's the first Maths book in which I'm doing absolutely all of the exercises. The authors do a good job in linking the concepts with actual useful stuff, like relating Monoids to concurrency and Categories, Functors and Natural transformations to databases. Even though there are quite a few of this kind of examples, I think the work would benefit from even more of them. In the Haskell community there is a lot of talk about Category Theory, but every time I tried to learn a concept it would be explained in terms of other category-theoretical concepts in a definition loop. This book is filling an important void and I am finally being able to understand the relation between these ideas and their implementation in Haskell.

This has lots of explanation and nice exercises. Just did 5 questions in the first chapter. 3/5. After that the notation became very dense.

Livro muito bem escrito, claro, intuitivo e correto

The book is wonderful but, why do even pages have the margin on the left making the lines in those pages to almost get into the spine?

The book itself is made with attention to many small details that make it a pleasure to read. To mention a few: there is a margin broad enough for my own annotations, the references to other resouces go back and forth (showing you where it was cited), there are lots of cross-references in the book itself, the paper is of good quality and allows me to use text-markers the way I like it, there are plenty of illustrations and so on. The topics are chosen to give am impression of the wide applicability of category theory, thus connecting many other mathematical and practical applications with categories. The tone varies from casual notes that explain quite clearly the ideas and intentions behind certain constructions and rigid mathematical definitions and proofs. For me this is much easier to digest then a bare definition-proposition-proof style. There are many examples and exercises (inlcuding solutions) in the book and one is well adviced to put in the work to use ones own brain. Many examples and exercises come up again in later chapters. All chapters start easy and then slowly pick up speed until some become quite demanding. If you are looking for a book that is not only about category theory, but shows the ramifications into other sciences, is well written and takes you by the hand through different interesting topics, this is the book I would recommend!