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Categories for Types by Roy L. Crole

Abstract

This textbook explains the basic principles of categorical type theory and illustrates some of the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory. Categories, functors and natural transformations are covered, along with the Yoneda Lemma, cartesian closed categories, limits and colimits, adjunctions and indexed categories. Four kinds of formal system are presented in detail, namely algebraic, functional, second order polymorphic and higher order polymorphic type theories. For each of these type theories a categorical semantics is derived from first principles, and soundness and completeness results are proved. Correspondences between the type theories and appropriate categorical structures are formulated, along with a discussion of internal languages. Specific examples of categorical models are given, and in the case of polymorphism both PER and domain-theoretic structures are considered. Categorical gluing is used to prove results about type theories. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians, and mathematicians specialising in category theory.

Chapter Contents

  1. Order, lattices and domains;
  2. A category theory primer;
  3. Algebraic type theory;
  4. Functional type theory;
  5. Polymorphic functional type theory;
  6. Higher order polymorphism.

Availability

If you are interested in a copy, please see here .

Look out for this. You might like to look at a list of minor errata.

Author: Roy Crole.
© University of Leicester March 2018. Last modified: 21st March 2018, 09:24:54
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