The genesis of this book, which focuses on geometric theories and their classifying
toposes, dates back to the author’s Ph.D. thesis The Duality between Grothendieck
Toposes and Geometric Theories [12] defended in 2009 at the University
of Cambridge.
The idea of regarding Grothendieck toposes from the point of view of the
structures that they classify dates back to A. Grothendieck and his student M.
Hakim, who characterized in her book Topos annelés et schémas relatifs [48]
four toposes arising in algebraic geometry, notably including the Zariski topos,
as the classifiers of certain special kinds of rings. Later, Lawvere’s work on the
Functorial Semantics of Algebraic Theories [59] implicitly showed that all finite
algebraic theories are classified by presheaf toposes. The introduction of geometric
logic, that is, the logic that is preserved under inverse images of geometric
functors, is due to the Montréal school of categorical logic and topos theory active
in the seventies, more specifically to G. Reyes, A. Joyal and M. Makkai. Its
importance is evidenced by the fact that every geometric theory admits a classifying
topos and that, conversely, every Grothendieck topos is the classifying
topos of some geometric theory. After the publication, in 1977, of the monograph
First Order Categorical Logic [64] by Makkai and Reyes, the theory of
classifying toposes, in spite of its promising beginnings, stood essentially undeveloped;
very few papers on the subject appeared in the following years and, as
a result, most mathematicians remained unaware of the existence and potential
usefulness of this fundamental notion.
One of the aims of this book is to give new life to the theory of classifying
toposes by addressing in a systematic way some of the central questions that
have remained unanswered throughout the past years, such as:
The problem of elucidating the structure of the collection of geometric
theory-extensions of a given geometric theory, which we tackle in Chapters
3, 4 and 8;
The problem of characterizing (syntactically and semantically) the class of
geometric theories classified by a presheaf topos, which we treat in Chapter
6;
The crucial meta-mathematical question of how to fruitfully apply the
theory of classifying toposes to get ‘concrete’ insights on theories of natural
mathematical interest, to which we propose an answer by means of the
‘bridge technique’ described in Chapter 2.
It is our hope that by the end of the book the reader will have appreciated that
the field is far from being exhausted and that in fact there is still much room for
theoretical developments as well as great potential for applications.
Pre-requisites and reading advice
The only pre-requisite for reading this book is a basic familiarity with the language
of category theory. This can be achieved by reading any introductory
text on the subject, for instance the classic but still excellent Categories for the
Working Mathematician [62] by S. Mac Lane.
The intended readership of this book is therefore quite large: mathematicians,
logicians and philosophers with some experience of categories, graduate students
wishing to learn topos theory, etc.
Our treatment is essentially self-contained, the necessary topos-theoretic background
being recalled in Chapter 1 and referred to at various points of the book.
The development of the general theory is complemented by a variety of examples
and applications in different areas of mathematics which illustrate its scope
and potential (cf. Chapter 10). Of course, these are not meant to exhaust the
possibilities of application of the methods developed in the book; rather, they
are aimed at giving the reader a flavour of the variety and mathematical depth
of the ‘concrete’ results that can be obtained by applying such techniques.
The chapters of the book should normally be read sequentially, each one being
dependent on the previous ones (with the exception of Chapter 5, which only
requires Chapter 1, and of Chapters 6 and 7, which do not require Chapters 3 and
4). Nonetheless, the reader who wishes to immediately jump to the applications
described in Chapter 10 may profitably do so by pausing from time to time to
read the theory referred to in a given section to complement his understanding.
Acknowledgements
As mentioned above, the genesis of this book dates back to my Ph.D. studies
carried out at the University of Cambridge in the years 2006-2009. Thanks are
therefore due to Trinity College, Cambridge (U.K.), for fully supporting my
Ph.D. studies through a Prince of Wales Studentship, as well as to Jesus College,
Cambridge (U.K.) for its support through a Research Fellowship. The one-year,
post-doctoral stay at the De Giorgi Center of the Scuola Normale Superiore di
Pisa (Italy) was also important in connection with the writing of this book,
since it was in that context that the general systematization of the unifying
methodology ‘toposes as bridges’ took place. Later, I have been able to count
on the support of a two-month visiting position at the Max Planck Institute
for Mathematics (Bonn, Germany), where a significant part of Chapter 5 was
written, as well as of a one-year CARMIN post-doctoral position at IHÉS, during
which I wrote, amongst other texts, the remaining parts of the book. Thanks are
Preface vii
also due to the University of Paris 7 and the Università degli Studi di Milano,
who hosted my Marie Curie fellowship (cofunded by the Istituto Nazionale di
Alta Matematica “F. Severi”), and again to IHÉS as well as to the Università
degli Studi dell’Insubria for employing me in the period during which the final
revision of the book has taken place.
Several results described in this book have been presented at international
conferences and invited talks at universities around the world; the list is too
long to be reported here, but I would like to collectively thank the organizers of
such events for giving me the opportunity to present my work to responsive and
stimulating audiences.
Special thanks go to Laurent Lafforgue for his unwavering encouragement
to write a book on my research and for his precious assistance during the final
revision phase.
I am also grateful to Marta Bunge for reading and commenting on a preliminary
version of the book, to the anonymous referees contacted by Oxford
University Press and to Alain Connes, Anatole Khelif, Steve Vickers and Noson
Yanofsky for their valuable remarks on results presented in this book.
Como
October 2017 Olivia Caramello
Author(s): Olivia Caramello
Publisher: Oxford UP
Year: 2018
Language: English
Pages: 381
City: Oxford
Notation and terminology 1
Introduction 3
1 Topos-theoretic background 9
1.1 Grothendieck toposes 9
1.1.1 The notion of site 10
1.1.2 Sheaves on a site 12
1.1.3 Basic properties of categories of sheaves 14
1.1.4 Geometric morphisms 17
1.1.5 Diaconescu’s equivalence 20
1.2 First-order logic 22
1.2.1 First-order theories 23
1.2.2 Deduction systems for first-order logic 27
1.2.3 Fragments of first-order logic 28
1.3 Categorical semantics 29
1.3.1 Classes of ‘logical’ categories 31
1.3.2 Completions of ‘logical’ categories 34
1.3.3 Models of first-order theories in categories 35
1.3.4 Elementary toposes 39
1.3.5 Toposes as mathematical universes 43
1.4 Syntactic categories 45
1.4.1 Definition 45
1.4.2 Syntactic sites 48
1.4.3 Models as functors 49
1.4.4 Categories with ‘logical structure’ as syntactic
categories 50
1.4.5 Soundness and completeness 51
2 Classifying toposes and the ‘bridge’ technique 53
2.1 Geometric logic and classifying toposes 53
2.1.1 Geometric theories 53
2.1.2 The notion of classifying topos 55
2.1.3 Interpretations and geometric morphisms 60
2.1.4 Classifying toposes for propositional theories 63
2.1.5 Classifying toposes for cartesian theories 64
x Contents
2.1.6 Further examples 65
2.1.7 A characterization theorem for universal models in
classifying toposes 67
2.2 Toposes as ‘bridges’ 69
2.2.1 The ‘bridge-building’ technique 69
2.2.2 Decks of ‘bridges’ : Morita equivalences 70
2.2.3 Arches of ‘bridges’ : site characterizations 74
2.2.4 Some simple examples 76
2.2.5 A theory of ‘structural translations’ 80
3 A duality theorem 83
3.1 Preliminary results 83
3.1.1 A 2-dimensional Yoneda lemma 83
3.1.2 An alternative view of Grothendieck topologies 84
3.1.3 Generators for Grothendieck topologies 86
3.2 Quotients and subtoposes 88
3.2.1 The duality theorem 88
3.2.2 The proof-theoretic interpretation 94
3.3 A deduction theorem for geometric logic 105
4 Lattices of theories 107
4.1 The lattice operations on Grothendieck topologies and quotients
107
4.1.1 The lattice operations on Grothendieck topologies 108
4.1.2 The lattice operations on theories 112
4.1.3 The Heyting implication in ThT
115
4.2 Transfer of notions from topos theory to logic 119
4.2.1 Relativization of local operators 119
4.2.2 Open, closed, quasi-closed subtoposes 126
4.2.3 The Booleanization and DeMorganization of a geometric
theory 130
4.2.4 The dense-closed factorization of a geometric inclusion 132
4.2.5 Skeletal inclusions 133
4.2.6 The surjection-inclusion factorization 134
4.2.7 Atoms 135
4.2.8 Subtoposes with enough points 138
5 Flat functors and classifying toposes 139
5.1 Preliminary results on indexed colimits in toposes 140
5.1.1 Background on indexed categories 140
5.1.2 E-filtered indexed categories 144
5.1.3 Indexation of internal diagrams 145
5.1.4 Colimits and tensor products 146
5.1.5 E-final functors 150
5.1.6 A characterization of E-indexed colimits 157
Contents xi
5.1.7 Explicit calculation of set-indexed colimits 169
5.2 Extensions of flat functors 176
5.2.1 General extensions 176
5.2.2 Extensions along embeddings of categories 178
5.2.3 Extensions from categories of set-based models to
syntactic categories 181
5.2.4 A general adjunction 187
5.3 Yoneda representations of flat functors 194
5.3.1 Cauchy completion of sites 195
6 Theories of presheaf type: general criteria 197
6.1 Preliminary results 198
6.1.1 A canonical form for Morita equivalences 198
6.1.2 Universal models and definability 199
6.1.3 A syntactic criterion for a theory to be of presheaf
type 202
6.1.4 Finitely presentable = finitely presented 204
6.1.5 A syntactic description of the finitely presentable
models 208
6.2 Internal finite presentability 210
6.2.1 Objects of homomorphisms 210
6.2.2 Strong finite presentability 212
6.2.3 Semantic E-finite presentability 216
6.3 Semantic criteria for a theory to be of presheaf type 217
6.3.1 The characterization theorem 217
6.3.2 Concrete reformulations 223
6.3.3 Abstract reformulation 239
7 Expansions and faithful interpretations 241
7.1 Expansions of geometric theories 241
7.1.1 General theory 241
7.1.2 Another criterion for a theory to be of presheaf type 246
7.1.3 Expanding a geometric theory to a theory of presheaf
type 247
7.1.4 Presheaf-type expansions 249
7.2 Faithful interpretations of theories of presheaf type 251
7.2.1 General results 251
7.2.2 Injectivizations of theories 256
7.2.3 Finitely presentable and finitely generated models 258
7.2.4 Further reformulations of condition (iii) of Theorem
6.3.1 261
7.2.5 A criterion for injectivizations 268
8 Quotients of a theory of presheaf type 273
8.1 Studying quotients through the associated Grothendieck
topologies 274
xii Contents
8.1.1 The notion of J-homogeneous model 274
8.1.2 Axiomatizations for the J-homogeneous models 280
8.1.3 Quotients with enough set-based models 284
8.1.4 Coherent quotients and topologies of finite type 288
8.1.5 An example 292
8.2 Presheaf-type quotients 293
8.2.1 Finality conditions 293
8.2.2 Rigid topologies 295
8.2.3 Finding theories classified by a given presheaf topos 299
9 Examples of theories of presheaf type 305
9.1 Theories whose finitely presentable models are finite 305
9.2 The theory of abstract circles 307
9.3 The geometric theory of finite sets 310
9.4 The theory of Diers fields 312
9.5 The theory of algebraic extensions of a given field 317
9.6 Groups with decidable equality 318
9.7 Locally finite groups 320
9.8 Vector spaces 321
9.9 The theory of abelian `-groups with strong unit 322
10 Some applications 325
10.1 Restrictions of Morita equivalences 325
10.2 A solution to the boundary problem for subtoposes 326
10.3 Syntax-semantics ‘bridges’ 327
10.4 Topos-theoretic Fraïssé theorem 331
10.5 Maximal theories and Galois representations 340
10.6 A characterization theorem for geometric logic 345
10.7 The maximal spectrum of a commutative ring 346
10.8 Compactness conditions for geometric theories 354
Bibliography 359
Index 363
