PRELIMINARY ANNOUNCEMENT ------------------------------- The following book will be available shortly: CATEGORIES FOR TYPES Cambridge Mathematical Textbooks, Cambridge University Press Roy L. Crole, Imperial College, University of London. 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. Contents: (1) Order, Lattices and Domains [1--36] (2) A Primer on Category Theory [37--119] (3) Algebraic Type Theory [120--153] (4) Functional Type Theory [154--196] (5) Polymorphic Functional Type Theory [197--268] (6) Higher Order Polymorphism [269--308] Bibliography [309--313] Index [314--335] ISBN 0521 450926 (HB) The Edinburgh Building, Shaftesbury Road, CAMBRIDGE, CB2 2RU, England, UK. 40 West 20th Street, New York, NY 10011-4211, USA. 10 Stamford Road, Oakleigh, Victoria 3166, Australia. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++