## Categories for TypesThis textbook explains the basic principles of categorical type theory and 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, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory. |

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### Contents

Order Lattices and Domains | 1 |

A Category Theory Primer | 37 |

Algebraic Type Theory | 120 |

Functional Type Theory | 154 |

Polymorphic Functional Type Theory | 201 |

Higher Order Polymorphism | 275 |

Bibliography | 315 |

### Common terms and phrases

2Ax-hyperdoctrine AC£atlr algebraic complete lattice algebraic theory arity Ax-theory axioms bijection binary product bool Boolean lattice cartesian closed category categorical semantics categorical type theory category theory category with finite Cl(Th classifying category commutes compact elements complete lattice components composition context continuous function dcpo define a functor definition diagram directed colimits DISCUSSION e-p pair equation-in-context equations example EXERCISE function symbol functional type theory functor F give given ground type homomorphism implies induction interpret judgement left adjoint Lemma model of Th monic monoid monotone function natural isomorphism natural numbers natural transformation non-empty notation Note notion operator poset preorder PROOF Proposition proved terms raw term raw types right adjoint satisfies Scott domain Sgop Sgtm Sgty signature specified structure Suppose syntax terminal object Th(C Theorem theory Th Thtm type theory type variables underlying set unique morphism verify write Yoneda lemma