Conceptual Mathematics: A First Introduction to Categories

The theoretical physicist John Baez wrote, "[Conceptual Mathematics] may seem almost childish at first, but it gradually creeps up on you. Schanuel has told me that you must do the exercises--if you don't, at some point the book will suddenly switch from being too easy to being way too hard! If you stick with it, by the end you will have all the basic concepts from topos theory under your belt, almost subconsciously."Conceptual Mathematics has only two prerequisites: Basic high-school algebra, and willingness to work through the material carefully. In return, this book offers a solid introduction to Cartesian closed categories and topoi. Major topics include sections and retractions, initial and terminal objects, products and coproducts, exponentiation, and subobject classifiers.These topics are illustrated using a variety of basic categories, each of which the authors introduce from scratch. These categories include sets, dynamic systems, and graphs, plus many variations of these categories. The self-contained nature of these examples is the book's greatest strength--almost every other introduction to category theory assumes prior knowledge of either topology, logic, or theoretical computer science.But why take the time to study Cartesian closed categories and topoi? An example may help.In computer science, the best-known Cartesian closed category is the lambda calculus, which lies at the heart of functional programming languages like Haskell and Scheme. But Cartesian closed categories appear everywhere in mathematics, logic and theoretical physics. And these connections between subjects can be exploited: For example, there's a program named Djinn, which translates Haskell type signatures into statements in intuitionist logic (using the Curry-Howard-Lambek correspondence). From there, Djinn runs a theorem prover, and then translates the output back into Haskell functions satisfying the original type signatures. In other words, by exploiting the connection between type systems and logic, it becomes possible to use tools from one field to solve problems in another.A word of caution, however: Conceptual Mathematics omits several central topics in category theory, including functors, natural transformations, and adjoints. In many cases, it lays extensive groundwork for these topics, but never gets around to covering the topics themselves. So if you want to go beyond a basic introduction to closed Cartesian categories and topoi, you're going to need another book.Despite these limitations, however, Conceptual Mathematics is an enjoyable--and uniquely accessible--introduction to category theory.

This is the best introductory book on Category Theory that I've read.Not a simple read, but far gentler and more intuitive than the others. Uses illustration's and even at times an informal conversational style to highlight the concepts.It does use proofs, and even asks you to do them using proper notation. But the notation is reasonable, and the proofs logical, and can be skipped altogether if desired.I might like it to get to be shorter or get to the point quicker. You really do need to start at the beginning and work through the chapters. For the abstract groundwork laid by earlier chapters is essential to understanding the latter ones.Sure, it could be better. It could be clearer and have even better illustrations. But a survey of the alternatives reveals this author's love for the topic and so clearly shines above similar works, that I give it a 5 star rating.

Well, the first half is worth reading. But very much so.There are two major values to this book:(1) it is interesting as a pedagogic experiment. It does not hold as a masterpiece, but it does hold as a spark of genius worthy of iteration.More to talk about than I will. But the general layout is a few (5?) chapters that contain, alone, almost all the content of the book. An advanced reader could read them and be done. However, after each chapter there are several more breaking down the chapter in the form of a Socratic dialogue between teacher and student.People who “read math “ generally know you read it more than once. You read through it — get an idea of the matter then go back through to iron out particulars. This text introduces the student to that by giving an overview chapter followed by breakdowns.It also focuses on developing a few models the students can use to start approaching category theory.(2) The book, the first 1/2, is a great pre-category theory warm up. It’s basically about set composition and properties and playing with them. But it’s something many will find useful and illuminating.The book has flaws. Past the first half it starts to breakdown and it’s casualness gives way to ambiguity that is difficult to follow even if a reader was already familiar with the material. [Section 10 on Brouwer’s Theorem is the first example — largely incomprehensible. And a pity — it’s a beautiful theorem (one likely familiar to those who’ve learned some topology) and by the be guts well in the book. But it’s a disaster. Like a sketch of the chapter got published.The book continues to be strong/helpful for awhile with that section an outlier. But eventually the whole book goes that way. Perhaps useful in a specifically designed classroom setting to balance out the ambiguity of the later chapters.But. An excellent half a book is an excellent book in its own right. And many would do well to read this through.

The best introductory book to Category Theory I have found so far. The style is great, they present an article with a subject and follow it with several "sessions" where exercises and examples are presented in more detail. Although some times the sessions are repeating the same subjects that the main article, I found this really helpful: see the same concepts from different perspectives. From now on this will be the first book I'll recommend to anyone wanting to learn Category Theory.

Having read it myself when it first came out, I bought this copy for my niece, who is graduating the Danish gymnasium (high school equivalent) this summer, and as I had, she found it instructive and exciting.It is exactly what it says on the label: a first introduction to category theory, by one of the founders of the field. Lawvere simply imho does an exemplary job of teaching a different way of thinking about math and logic: this is how it is supposed to be done.