theories, sites, toposes Theories, Sites, Toposes Relating and studying mathematical theories through topos-theoretic ‘bridges’ OLIVIA CARAMELLO Università degli Studi dell’Insubria – Como 3 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©OliviaCaramello2018 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2018 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2016947978 ISBN978–0–19–875891–4 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. PREFACE Scope of the book Thegenesisofthisbook,whichfocusesongeometrictheoriesandtheirclassifying toposes, dates back to the author’s Ph.D. thesis The Duality between Grothen- dieck 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 algebraictheoriesareclassifiedbypresheaftoposes.Theintroductionofgeomet- ric 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 ac- tive 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 clas- sifying topos and that, conversely, every Grothendieck topos is the classifying topos of some geometric theory. After the publication, in 1977, of the mono- graph First Order Categorical Logic [64] by Makkai and Reyes, the theory of classifying toposes, in spite of its promising beginnings, stood essentially unde- veloped; 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 Chap- ters 3, 4 and 8; • Theproblemofcharacterizing(syntacticallyandsemantically)theclassof geometrictheoriesclassifiedbyapresheaftopos,whichwetreatinChapter 6; • The crucial meta-mathematical question of how to fruitfully apply the theoryofclassifyingtoposestoget‘concrete’insightsontheoriesofnatural vi Preface 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 thefieldisfarfrombeingexhaustedandthatinfactthereisstillmuchroomfor 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 lan- guage 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. Theintendedreadershipofthisbookisthereforequitelarge:mathematicians, logiciansandphilosopherswithsomeexperienceofcategories,graduatestudents wishing to learn topos theory, etc. Ourtreatmentisessentiallyself-contained,thenecessarytopos-theoreticback- groundbeingrecalledinChapter1andreferredtoatvariouspointsofthebook. The development of the general theory is complemented by a variety of exam- ples 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. Thechaptersofthebookshouldnormallybereadsequentially,eachonebeing dependent on the previous ones (with the exception of Chapter 5, which only requiresChapter1,andofChapters6and7,whichdonotrequireChapters3and 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.studiesthroughaPrinceofWalesStudentship,aswellastoJesusCollege, 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,aswellasofaone-yearCARMINpost-doctoralpositionatIHÉS,during whichIwrote,amongstothertexts,theremainingpartsofthebook.Thanksare 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 pre- liminary 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 CONTENTS 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