Category Theory for Computing Science
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Category Theory for Computing Science is a textbook in basic category theory, written specifically to be read by researchers and students in computing science.
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.
In line with our emphasis on understanding, we frequently recommend one or another way of thinking about a concept. It is typical of most of the useful concepts in mathematics that there is more than one way of perceiving or understanding them. It is simply not true that everything about a mathematical concept is contained in its definition. Of course it is true that in some sense all the theorems are inherent in its definition, but not what makes it useful to mathematicians or to scientists who use mathematics. We believe that the more ways you have of perceiving an idea, the more likely you are to recognize situations in your own work where the idea is useful.
We have acted on the belief just outlined with many sentences beginning with phrases such as `This concept may be thought of as …'. We have been warned that doing this may present difficulties for a non-mathematician who has only just mastered one way of thinking about something, but we feel it is part of learning about a mathematical topic to understand the contextual associations it has for those who use it.
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