Categorical Logic and Type Theory

B. Jacobs, Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141, North Holland, Elsevier, 1999. ISBN 0-444-50170-3 BibTex entry

This book gives a survey of categorical logic and type theory starting from the unifying concept of a fibration. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists. To get an impression, the prospectus from the book is made available.

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Contents

Chapter 0: Prospectus

1. Logic, type theory and fibred category theory
2. The logic and type theory of sets

Chapter 1: Introduction to fibred category theory

1. Fibrations
2. Some concrete examples: sets, omega-sets and PERs
3. Some general examples
4. Cloven and split fibrations
5. Change-of-base and composition of fibrations
6. Fibrations of signatures
7. Categories of fibrations
8. Fibrewise structure and fibred adjunctions
9. Products and coproducts
10. Indexed categories

Chapter 2: Simple type theory

1. The basic calculus of types and terms
2. Functorial semantics
3. Exponents, products and coproducts
4. Semantics of simple type theories
5. Semantics of the untyped lambda calculus as a corollary
6. Simple parameters

Chapter 3: Equational Logic

1. Logics
2. Specifications and theories in equational logic
3. Algebraic specifications
4. Fibred equality
5. Fibrations for equational logic
6. Fibred functorial semantics

Chapter 4: First order predicate logic

1. Signatures, connectives and quantifiers
2. Fibrations for first order predicate logic
4. Functorial interpretation and internal language
5. Subobject fibrations I: regular categories
6. Subobject fibrations II: coherent categories and logoses
7. Subset types
8. Quotient types
9. Quotient types, categorically
10. A logical characterisation of subobject fibrations

Chapter 5: Higher order predicate logic

1. Higer order signatures
2. Generic objects
3. Fibrations for higher order logic
4. Elementary toposes
5. Colimits, powerobjects and well-poweredness in toposes
6. Nuclei in a topos
7. Separated objects and sheaves in a topos
8. A logical description of separated objects and sheaves

Chapter 6: The effective topos

1. Constructing a topos from a higher order fibration
2. The effective topos and its subcategories of sets, omega-sets, and PERs
3. Families of PERs and omega-sets over the effective topos
4. Natural numbers in the effective topos and some associated principles

Chapter 7: Internal category theory

1. Definition and examples of internal categories
2. Internal functors and natural transformations
3. Externalisation
4. Internal diagrams and completeness

Chapter 8: Polymorphic type theory

1. Syntax
2. Use of polymorphic type theory
3. Naive set theoretic semantics
4. Fibrations for polymorphic type theory
5. Small polymorphic fibrations
6. Logic over polymorphic type theory

Chapter 9: Advanced fibred category theory

1. Opfibrations and fibred spans
2. Logical predicates and relations
3. Quantification
4. Category theory over a fibration
5. Locally small fibrations
6. Definability

Chapter 10: First order dependent type theory

1. A calculus of dependent types
2. Use of dependent types
3. A term model
4. Display maps and comprehension categories
5. Closed comprehension categories
6. Domain theoretic models of type dependency

Chapter 11: Higher order dependent type theory

1. Dependent predicate logic
2. Dependent predicate logic, categorically
3. Polymorphic dependent type theory
4. Strong and very strong sum and equality
5. Full higher order dependent type theory
6. Full higher order dependent type theory, categorically
7. Completeness of the category of PERs in the effective topos

References

Notation index

Subject index

Last modified: Tue Oct 24 11:15:58 MEST 2006