Theories, Sites, Toposes cover image

Theories, Sites, Toposes

By Olivia Caramello
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Format
2018
Year
English
Language
Oxford UP
Publisher

Summary

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

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