Categorical Logic and Type Theory

779 Pages·2005·28.631 MB·English

by  Bart Jacobs

CATEGORICAL LOGIC
AND TYPE THEORY
STUDIES IN LOGIC
AND
THE FOUNDATIONS OF MATHEMATICS
VOLUME 141
Honorary Editor:
P. SUPPES
Editors:
S. ABRAMSKY, London
S. ARTEMOV, Moscow
R.A. SHORE, Ithaca
A.S. TROELSTRA, Amsterdam
ELSEVIER
AMSTERDAM • LAUSANNE • NEW YORK • OXFORD • SHANNON • SINGAPORE • TOKYO
CATEGORICAL LOGIC
AND TYPE THEORY
Bart JACOBS
Research Fellow of the
Royal Netherlands Academy of Arts and Sciences
Computing Science Institute^
University of Nijmegen, P.O. Box 9010, 6500 GL Nijmegen,
The Netherlands
1999
ELSEVIER
AMSTERDAM • LAUSANNE • NEW YORK • OXFORD • SHANNON • SINGAPORE • TOKYO
ELSEVIER SCIENCE B.V.
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First edition 1999
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Preface
This book has its origins in my PhD thesis, written during the years 1988 -
1991 at the University of Nijmegen, under supervision of Henk Barendregt.
The thesis concerned categorical semantics of various type theories, using
fibred categories. The connections with logic were not fully exploited at the
time. This book is an attempt to give a systematic presentation of both logic
and type theory from a categorical perspective, using the unifying concept of
a fibred category. Its intended audience consists of logicians, type theorists,
category theorists and (theoretical) computer scientists.
The main part of the book was written while I was employed by NWO,
the National Science Foundation in The Netherlands. First, during 1992 -
1994 at the Mathematics Department of the University of Utrecht, and later
during 1994 - 1996 at CWI, Center for Mathematics and Computer Science, in
Amsterdam. The work was finished in Nijmegen (where it started): currently, I
am employed at the Computing Science Institute of the University of Nijmegen
as a Research Fellow of the Royal Netherlands Academy of Arts and Sciences.
This book could not have been written without the teaching, support, en-
couragement, advice, criticism and help of many. It is a hopeless endeavour
to list them all. Special thanks go to my friends and (former) colleagues at
Nijmegen, Cambridge (UK), Utrecht and Amsterdam, but also to many col-
leagues in the field. The close cooperation with Thomas Streicher and Clau-
dio Hermida during the years is much appreciated, and their influence can
be felt throughout this work. The following persons read portions of the
manuscript and provided critical feedback, or contributed in some other way:
Lars Birkedal, Zinovy Diskin, Herman Geuvers, Claudio Hermida, Peter Lietz,
Jose Meseguer, Jaap van Oosten, Wesley Phoa, Andy Pitts, Thomas Streicher,
vi Preface
Hendrik Tews and Krzysztof Worytkiewicz. Of course, the responsibility for
mistakes remains entirely mine.
The diagrams in this book have been produced with Kristoffer Rose's X<^-pic
macros, and the proof trees with Paul Taylor's macros. The style files have
been provided by the publisher.
Bart Jacobs,
Nijmegen, August 1998.
Contents
Preface v
Contents vii
Preliminaries xi
0. Prospectus 1
0.1. Logic, type theory, and fibred category theory 1
0.2. The logic and type theory of sets 11
1. Introduction to fibred category theory 19
1.1. Fibrations 20
1.2. Some concrete examples: sets, ct/-sets and PERs 31
1.3. Some general examples 40
1.4. Cloven and split fibrations 47
1.5. Change-of-base and composition for fibrations 56
1.6. Fibrations of signatures 64
1.7. Categories of fibrations 72
1.8. Fibrewise structure and fibred adjunctions 80
1.9. Fibred products and coproducts 93
1.10. Indexed categories 107
2. Simple type theory 119
2.1. The basic calculus of types and terms 120
2.2. Functorial semantics 126
viii Contents
2.3. Exponents, products and coproducts 133
2.4. Semantics of simple type theories 146
2.5. Semantics of the untyped lambda calculus as a corollary 154
2.6. Simple parameters 157
3. Equational Logic 169
3.1. Logics 170
3.2. Specifications and theories in equational logic 177
3.3. Algebraic specifications 183
3.4. Fibred equality 190
3.5. Fibrations for equational logic 201
3.6. Fibred functorial semantics 209
4. First order predicate logic 219
4.1. Signatures, connectives and quantifiers 221
4.2. Fibrations for first order predicate logic 232
4.3. Functorial interpretation and internal language 246
4.4. Subobject fibrations I: regular categories 256
4.5. Subobject fibrations II: coherent categories and logoses 265
4.6. Subset types 272
4.7. Quotient types 282
4.8. Quotient types, categorically 290
4.9. A logical characterisation of subobject fibrations 304
5. Higher order predicate logic 311
5.1. Higher order signatures 312
5.2. Generic objects 321
5.3. Fibrations for higher order logic 330
5.4. Elementary toposes 338
5.5. Colimits, powerobjects and well-poweredness in a topos 346
5.6. Nuclei in a topos 353
5.7. Separated objects and sheaves in a topos 360
5.8. A logical description of separated objects and sheaves 368
6. The effective topos 373
6.1. Constructing a topos from a higher order fibration 374
6.2. The effective topos and its subcategories of sets, u;-sets, and PERs .... 385
6.3. Families of PERs and u;-sets over the effective topos 393
6.4. Natural numbers in the effective topos and some associated principles . . 398
7. Internal category theory , . 407
7.1. Definition and examples of internal categories 408
7.2. Internal functors and natural transformations 414
7.3. Externalisation 421
7.4. Internal diagrams and completeness 430
Contents ix
8. Polymorphic type theory 441
8.1. Syntax 444
8.2. Use of polymorphic type theory 454
8.3. Naive set theoretic semantics 463
8.4. Fibrations for polymorphic type theory 471
8.5. Small polymorphic fibrations 485
8.6. Logic over polymorphic type theory 495
9. Advanced fibred category theory 509
9.1. Opfibrations and fibred spans 510
9.2. Logical predicates and relations 518
9.3. Quantification 535
9.4. Category theory over a fibration 547
9.5. Locally small fibrations 558
9.6. Definability 568
10. First order dependent type theory 581
10.1. A calculus of dependent types 584
10.2. Use of dependent types 594
10.3. A term model 601
10.4. Display maps and comprehension categories 609
10.5. Closed comprehension categories 623
10.6. Domain theoretic models of type dependency 637
11. Higher order dependent type theory 645
11.1. Dependent predicate logic 648
11.2. Dependent predicate logic, categorically 653
11.3. Polymorphic dependent type theory 662
11.4. Strong and very strong sum and equality 674
11.5. Full higher order dependent type theory 684
11.6. Full higher order dependent type theory, categorically 692
11.7. Completeness of the category of PERs in the effective topos 707
References 717
Notation Index 735
Subject Index 743