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lambek j introduction higher order categorical logic 1988

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Textbook in PDF format In this volume, Lambek and Scott reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher-order logic, and cartesian closed categories, are essentially the same. Part II demonstrates that another formulation of higher-order logic, (intuitionistic) type theories, is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludeds with a set of exercises. Preface Introduction to category theory Introduction to Part 0 Categories and functors Natural transformations Adjoint functors Equivalence of categories Limits in categories Triples Examples of cartesian closed categories Cartesian closed categories and λ-calculus Introduction to Part I Historical perspective on Part I Propositional calculus as a deductive system The deduction theorem Cartesian closed categories equationally presented Free cartesian closed categories generated by graphs Polynomial categories Functional completeness of cartesian closed categories Polynomials and Kleisli categories Cartesian closed categories with coproducts Natural numbers objects in cartesian closed categories Typed λ-calculi The cartesian closed category generated by a typed λ-calculus The decision problem for equality The Church-Rosser theorem for bounded terms All terms are bounded C-monoids C-monoids and cartesian closed categories C-monoids and untyped λ-calculus A construction by Dana Scott Historical comments on Part I Type theory and toposes Introduction to Part II Historical perspective on Part II Intuitionistic type theory Type theory based on equality The internal language of a topos Peano's rules in a topos The internal language at work The internal language at work II Choice and the Boolean axiom Topos semantics Topos semantics in functor categories Sheaf categories and their semantics Three categories associated with a type theory The topos generated by a type theory The topos generated by the internal language The internal language of the topos generated Toposes with canonical subobjects Applications of the adjoint functors between toposes and type theories Completeness of higher order logic with choice rule Sheaf representation of toposes Completeness without assuming the rule of choice Some basic intuitionistic principles Further intuitionistic principles The Freyd cover of a topos Historical comments on Part II Supplement to Section 17 Representing numerical functions in various categories Introduction to Part III Recursive functions Representing numerical functions in cartesian closed categories Representing numerical functions in toposes Representing numerical functions in C-monoids Historical comments on Part III Bibliography Author index Subject index

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lambek j introduction higher order categorical logic 1988