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.

Possibly a more apt subtitle for this book would be "A First Introduction to Ideas that Underlie Category Theory." Even after spending quite a bit of time with this book, I didn't really feel like I'd learned much category theory, per se. (Tom Leinster's Basic Category Theory seems like an excellent choice if you want to jump right into definitions of categories, functors and natural transformations, then start thinking in terms of adjoints, etc. He also makes that book available on arxiv.) But, early on, Lawvere/Schanuel's book introduced (quite clearly, I think) category-theoretic ideas like sections and retractions, which I hadn't even realized that I'd encountered before. (I'd spent some time with Tu's Intro to Manifolds before this book, and at first I wondered if his definition of a section in the discussion of vector bundles had typos in it or what; after some time with Lawvere/Schanuel, that section from Tu makes a lot more sense.)As others have mentioned, the books seems like it might be quite simple, near the beginning. At first, given my lack of familiarity with category theory, this book made me wonder if category theory was the study of the consequences of associativity of composition laws, as that's a bit of a recurring theme in this book. And speaking of composition laws, if one wants to come up with a list of prerequisites for this book (or to start reading it, at least), I'd dare say that a familiarity with the composition of functions might be all you really need. That said, I should say this: I recently took a first pass at Rotman's Intro to Algebraic Topology and, after reading his discussion of Brouwer's fixed point theorem, I went back to Lawvere/Schanuel to revisit their section of the same topic, but still didn't feel clear about the Lawvere/Schanuel version after re-reading that section. (Rotman, on the other hand, I found quite easy to understand.) So while one could start this book with minimal prerequisites, I don't expect to feel like I'd understood it all, any time soon (and I'm well past the minimal prerequisites I just offered). And that's sort of a drawback -- the difficultly level of the book doesn't exactly scale smoothly, once you're into the latter half or so of the book. But that's probably my only criticism, as I find the discussion-driven parts of the book generally quite lucid and insightful.

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.