Theories, Sites, Toposes : Relating and Studying Mathematical Theories Through Topos-theoretic 'bridges'

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 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 Cover 1 Preface 6 Contents 10 Notation and terminology 14 Introduction 16 1 Topos-theoretic background 22 1.1 Grothendieck toposes 22 1.1.1 The notion of site 23 1.1.2 Sheaves on a site 25 1.1.3 Basic properties of categories of sheaves 27 1.1.4 Geometric morphisms 30 1.1.5 Diaconescu's equivalence 33 1.2 First-order logic 35 1.2.1 First-order theories 36 1.2.2 Deduction systems for first-order logic 40 1.2.3 Fragments of first-order logic 41 1.3 Categorical semantics 42 1.3.1 Classes of `logical' categories 44 1.3.2 Completions of `logical' categories 47 1.3.3 Models of first-order theories in categories 48 1.3.4 Elementary toposes 52 1.3.5 Toposes as mathematical universes 56 1.4 Syntactic categories 58 1.4.1 Definition 58 1.4.2 Syntactic sites 61 1.4.3 Models as functors 62 1.4.4 Categories with `logical structure' as syntactic categories 63 1.4.5 Soundness and completeness 64 2 Classifying toposes and the `bridge' technique 66 2.1 Geometric logic and classifying toposes 66 2.1.1 Geometric theories 66 2.1.2 The notion of classifying topos 68 2.1.3 Interpretations and geometric morphisms 73 2.1.4 Classifying toposes for propositional theories 76 2.1.5 Classifying toposes for cartesian theories 77 2.1.6 Further examples 78 2.1.7 A characterization theorem for universal models in classifying toposes 80 2.2 Toposes as `bridges' 82 2.2.1 The `bridge-building' technique 82 2.2.2 Decks of `bridges': Morita equivalences 83 2.2.3 Arches of `bridges': site characterizations 87 2.2.4 Some simple examples 89 2.2.5 A theory of `structural translations' 93 3 A duality theorem 96 3.1 Preliminary results 96 3.1.1 A 2-dimensional Yoneda lemma 96 3.1.2 An alternative view of Grothendieck topologies 97 3.1.3 Generators for Grothendieck topologies 99 3.2 Quotients and subtoposes 101 3.2.1 The duality theorem 101 3.2.2 The proof-theoretic interpretation 107 3.3 A deduction theorem for geometric logic 118 4 Lattices of theories 120 4.1 The lattice operations on Grothendieck topologies and quotients 120 4.1.1 The lattice operations on Grothendieck topologies 121 4.1.2 The lattice operations on theories 125 4.1.3 The Heyting implication in ThT∑ 128 4.2 Transfer of notions from topos theory to logic 132 4.2.1 Relativization of local operators 132 4.2.2 Open, closed, quasi-closed subtoposes 139 4.2.3 The Booleanization and DeMorganization of a geometric theory 143 4.2.4 The dense-closed factorization of a geometric inclusion 145 4.2.5 Skeletal inclusions 146 4.2.6 The surjection-inclusion factorization 147 4.2.7 Atoms 148 4.2.8 Subtoposes with enough points 151 5 Flat functors and classifying toposes 152 5.1 Preliminary results on indexed colimits in toposes 153 5.1.1 Background on indexed categories 153 5.1.2 E-filtered indexed categories 157 5.1.3 Indexation of internal diagrams 158 5.1.4 Colimits and tensor products 159 5.1.5 E-final functors 163 5.1.6 A characterization of E-indexed colimits 170 5.1.7 Explicit calculation of set-indexed colimits 182 5.2 Extensions of flat functors 189 5.2.1 General extensions 189 5.2.2 Extensions along embeddings of categories 191 5.2.3 Extensions from categories of set-based models to syntactic categories 194 5.2.4 A general adjunction 200 5.3 Yoneda representations of flat functors 207 5.3.1 Cauchy completion of sites 208 6 Theories of presheaf type: general criteria 210 6.1 Preliminary results 211 6.1.1 A canonical form for Morita equivalences 211 6.1.2 Universal models and definability 212 6.1.3 A syntactic criterion for a theory to be of presheaf type 215 6.1.4 Finitely presentable = finitely presented 217 6.1.5 A syntactic description of the finitely presentable models 221 6.2 Internal finite presentability 223 6.2.1 Objects of homomorphisms 223 6.2.2 Strong finite presentability 225 6.2.3 Semantic E-finite presentability 229 6.3 Semantic criteria for a theory to be of presheaf type 230 6.3.1 The characterization theorem 230 6.3.2 Concrete reformulations 236 6.3.3 Abstract reformulation 252 7 Expansions and faithful interpretations 254 7.1 Expansions of geometric theories 254 7.1.1 General theory 254 7.1.2 Another criterion for a theory to be of presheaf type 259 7.1.3 Expanding a geometric theory to a theory of presheaf type 260 7.1.4 Presheaf-type expansions 262 7.2 Faithful interpretations of theories of presheaf type 264 7.2.1 General results 264 7.2.3 Injectivizations of theories 269 7.2.3 Finitely presentable and finitely generated models 271 7.2.4 Further reformulations of condition (iii) of Theorem 6.3.1 274 7.2.5 A criterion for injectivizations 281 8 Quotients of a theory of presheaf type 286 8.1 Studying quotients through the associated Grothendieck topologies 287 8.1.1 The notion of J-homogeneous model 287 8.1.2 Axiomatizations for the J-homogeneous models 293 8.1.3 Quotients with enough set-based models 297 8.1.4 Coherent quotients and topologies of finite type 301 8.1.5 An example 305 8.2 Presheaf-type quotients 306 8.2.1 Finality conditions 306 8.2.2 Rigid topologies 308 8.2.3 Finding theories classified by a given presheaf topos 312 9 Examples of theories of presheaf type 318 9.1 Theories whose finitely presentable models are finite 318 9.2 The theory of abstract circles 320 9.3 The geometric theory of finite sets 323 9.4 The theory of Diers fields 325 9.5 The theory of algebraic extensions of a given field 330 9.6 Groups with decidable equality 331 9.7 Locally finite groups 333 9.8 Vector spaces 334 9.9 The theory of abelian l-groups with strong unit 335 10 Some applications 338 10.1 Restrictions of Morita equivalences 338 10.2 A solution to the boundary problem for subtoposes 339 10.3 Syntax-semantics `bridges' 340 10.4 Topos-theoretic Fraïssé theorem 344 10.5 Maximal theories and Galois representations 353 10.6 A characterization theorem for geometric logic 358 10.7 The maximal spectrum of a commutative ring 359 10.8 Compactness conditions for geometric theories 367 Bibliography 372 Index 376 This book is devoted to a general study of geometric theories from a topos-theoretic perspective. After recalling the necessary topos-theoretic preliminaries, it presents the main methodology it uses to extract ‘concrete’ information on theories from properties of their classifying toposes—the ‘bridge’ technique. As a first implementation of this methodology, a duality is established between the subtoposes of the classifying topos of a geometric theory and the geometric theory extensions (also called ‘quotients’) of the theory. Many concepts of elementary topos theory which apply to the lattice of subtoposes of a given topos are then transferred via this duality into the context of geometric theories. A second very general implementation of the ‘bridge’ technique is the investigation of the class of theories of presheaf type (i.e. classified by a presheaf topos). After establishing a number of preliminary results on flat functors in relation to classifying toposes, the book carries out a systematic investigation of this class resulting in a number of general results and a characterization theorem allowing one to test whether a given theory is of presheaf type by considering its models in arbitrary Grothendieck toposes. Expansions of geometric theories and faithful interpretations of theories of presheaf type are also investigated. As geometric theories can always be written (in many ways) as quotients of presheaf type theories, the study of quotients of a given theory of presheaf type is undertaken. Lastly, the book presents a number of applications in different fields of mathematics of the theory it develops. According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures." It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things." The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics. The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis. The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision of the notion of topos. This has been accomplished by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics. The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis