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
- Order, lattices and domains;
- A category theory primer;
- Algebraic type theory;
- Functional type theory;
- Polymorphic functional type theory;
- Higher order polymorphism.
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