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.
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