Barr M., Wells C. Category Theory for Computing Science

Online ed. - Montreal: Université de Montréal, 2020. - 567 p.This book is a textbook in basic category theory, written specifically to be read by researchers and students in computing science. We expound the constructions we feel are basic to category theory in the context of examples and applications to computing science. Some categorical ideas and constructions are already used heavily in computing science and we describe many of these uses. Other ideas, in particular the concept of adjoint, have not appeared as widely in the computing science literature. We give here an elementary exposition of those ideas we believe to be basic categorical tools, with pointers to possible applications when we are aware of them. In addition, this text advocates a specific idea: the use of sketches as a systematic way to turn finite descriptions into mathematical objects. This aspect of the book gives it a particular point of view. We have, however, taken pains to keep most of the material on sketches in separate sections. It is not necessary to read to learn most of the topics covered by the book. As a way of showing how you can use categorical constructions in the context of computing science, we describe several examples of modeling linguistic or computational phenomena categorically. These are not intended as the final word on how categories should be used in computing science; indeed, they hardly constitute the initial word on how to do that! We are mathematicians, and it is for those in computing science, not us, to determine which is the best model for a given application. The emphasis in this book is on understanding the concepts we have introduced, rather than on giving formal proofs of the theorems. We include proofs of theorems only if they are enlightening in their own right. We have attempted to point the reader to the literature for proofs and further development for each topic.
This text assumes some familiarity with abstract mathematical thinking, and some specific knowledge of the basic language of mathematics and computing science of the sort taught in an introductory discrete mathematics course.Preliminaries.
Categories.
Functors.
Diagrams, naturality and sketches.
Products and sums.
Cartesian closed categories.
Finite product sketches.
Finite discrete sketches.
Limits and colimits.
More about sketches.
The category of sketches.
Fibrations.
Adjoints.
Algebras for endofunctors.
Toposes.
Categories with monoidal structure.Solutions to exercises.