During the Michaelmas term of 1990, while at the University of Cambridge Computer Laboratory, the opportunity arose to lecture on categorical models of lambda calculi. The course consisted of sixteen lectures of about one hour's duration twice a week for eight weeks, and covered much of the material in this book, but excluded higher order polymorphism and some of the category theory. The lectures were delivered to an audience of computer scientists and mathematicians, with an emphasis on presenting the material to the former. It was kindly suggested by the Cambridge University Press that these lectures might form the core of a textbook, and the original suggestion has now been realised as “Categories for Types.”

What are the contents of “Categories for Types”? I will try to answer this question for those who know little about categorical type theory. In Chapter 1, we begin with a discussion of ordered sets. These are collections of things with an order placed on the collection. For example, the natural numbers form a set {1,2,3…} with an order given by 1 ≤ 2 ≤ 3 ≤ … where ≤ means “less than or equal to.” A number of different kinds of ordered set are defined, and results proved about them. Such ordered sets then provide a stock of examples of categories. A category is a very general mathematical world and various different sorts of mathematical structures form categories.