Johan Lindberg Localic Categories of Models and Categorical Aspects of Intuitionistic Ramified Type Localic Categories of Models and Categorical Aspects of Intuitionistic Ramified Type Theory Type Aspects of Intuitionistic Ramified of Models and Categorical Localic Categories Theory Johan Lindberg ISBN 978-91-7911-350-6 Department of Mathematics Doctoral Thesis in Mathematics at Stockholm University, Sweden 2020 Localic Categories of Models and Categorical Aspects of Intuitionistic Ramified Type Theory Johan Lindberg Academic dissertation for the Degree of Doctor of Philosophy in Mathematics at Stockholm University to be publicly defended on Wednesday 16 December 2020 at 13.00 online via Zoom, public link is available at the department web site. Abstract This thesis contains three papers, all in the general area of categorical logic, together with an introductory part with some minor results and proofs of known results which does not appear to be (easily) available in the literature. In Papers I and II we investigate the formal system Intuitionistic Ramified Type Theory (IRTT), introduced by Erik Palmgren, as an approach to predicative topos theory. In Paper I we construct and study the category of "local sets" in IRTT, including an extension with inductive definitions. We there also give a model of IRTT in univalent type theory using h-sets. In Paper II we adapt triposes and hyperdoctrines to the ramified setting. These give a categorical semantics for certain formal languages ramified in the same way as IRTT. Paper III, which is part of a joint project with Henrik Forssell, concerns logical aspects of the localic groupoid/category representations of Grothendieck toposes that originate from the work of Joyal and Tierney. Working constructively, we give explicit logical descriptions of locales and localic categories used for representing classifying toposes of geometric theories. Aspects of these descriptions are related to work by Coquand, Sambin et al in formal topology, and we show how parts of their work can be captured and extended in our framework. Keywords: Topos theory, predicative topos theory, ramified type theory, type theory, localic groupoids. Stockholm 2020 http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-186353 ISBN 978-91-7911-350-6 ISBN 978-91-7911-351-3 Department of Mathematics Stockholm University, 106 91 Stockholm LOCALIC CATEGORIES OF MODELS AND CATEGORICAL ASPECTS OF INTUITIONISTIC RAMIFIED TYPE THEORY Johan Lindberg Localic Categories of Models and Categorical Aspects of Intuitionistic Ramified Type Theory Johan Lindberg ©Johan Lindberg, Stockholm University 2020 ISBN print 978-91-7911-350-6 ISBN PDF 978-91-7911-351-3 Printed in Sweden by Universitetsservice US-AB, Stockholm 2020 Abstract This thesis contains three papers, all in the general area of categorical logic, together with an introductory part with some minor results and proofs of known results which does not appear to be (easily) available in the literature. In Papers I and II we investigate the formal system Intuitionistic Ramied Type Theory (IRTT), introduced by Erik Palmgren, as an ap- proach to predicative topos theory. In Paper I we construct and study the category of local sets in IRTT, including an extension with induc- tive denitions. We there also give a model of IRTT in univalent type theory using h-sets. In Paper II we adapt triposes and hyperdoctrines to the ramied setting. These give a categorical semantics for certain formal languages ramied in the same way as IRTT. Paper III, which is part of a joint project with Henrik Forssell, con- cerns logical aspects of the localic groupoid/category representations of Grothendieck toposes that originate from the work of Joyal and Tierney. Working constructively, we give explicit logical descriptions of locales and localic categories used for representing classifying toposes of geometric theories. Aspects of these descriptions are related to work by Coquand, Sambin et al in formal topology, and we show how parts of their work can be captured and extended in our framework. Sammanfattning Denna avhandling innehåller tre artiklar, samtliga inom området kategori- teoretisk logik, tillsammans med en inledande del med några mindre re- sult samt bevis av tidigare kända result som inte (så enkelt) går att nna i litteraturen. I Artikel I och II undersöker vi det formella systemet Intuitionistic Ramied Type Theory (IRTT), vilket introducerades av Erik Palmgren, som ett sätt att närma sig predikativ toposteori. I Artikel I konstruerar vi och studerar en kategori av lokala mängder i IRTT och i en utvidgning av denna teori med induktiva denitioner. Vi ger där även en modell av IRTT med h-mängder i univalent typteori. I Artikel II anpassar vi triposar och hyperdoktriner till det ramierade fallet. Detta ger en kategoriteoretisk semantik för vissa formella språk som är ramierade på samma sätt som IRTT. Artikel III, som är en del av ett gemensamt projekt med Henrik Forssell, rör logiska aspekter av de representationer av Grothendieck topos med lokaliska kategorier/gruppoider som härstammar från Joyal och Tierneys arbete. Med ett konstruktivt tillvägagångssätt beskriver vi lokaler och lokaliska kategorier som kan användas för att representera de klassicerande toposen för geometriska teorier. Aspekter av dessa beskrivningar är relaterade till arbeten av Coquand, Sambin et al i formell topologi, och vi visar hur delar av deras arbete kan fångas in och utvidgas i vårt ramverk. List of Papers The following papers, referred to in the text by their Roman numerals, are included in this thesis. PAPER I: Intuitionistic Ramied Type Theory - Inductive Denitions and Categorical Properties J. Lindberg, 2020 PAPER II: Ramied Hyperdoctrines and Triposes J. Lindberg, 2020 PAPER III: Locales of Models for Geometric and First-order Theories H. Forssell and J. Lindberg, 2020 Contents Abstract iv Sammanfattning vi List of Papers ix Acknowledgements xiii 1 Introduction 15 1.1 Overview of the thesis . 16 2 Summary of Papers 17 2.1 Paper I . 17 2.2 Paper II . 20 2.3 Paper III . 21 3 Coverages and the generalized comparison lemma of Kock and Moerdijk 27 3.1 Coverages . 27 3.2 Sites and preservation and reection of covers . 29 3.3 A detailed proof of the generalized comparison lemma . 31 4 Some 2-categorical aspects of complete Heyting algebra- valued sets 43 4.1 The category Ω-Set(X) of O(X)-valued sets . 43 4.2 The equivalence between Ω-Set(X) and Sh(X) . 45 4.3 Ω-Set as a pseudofunctor . 47 4.4 Ω-Set(f) as a geometric morphism . 50 References 57 Acknowledgements First of all I would like dedicate a few words in memory of Erik Palmgren, my supervisor for roughly the rst four years of my PhD, who very sadly and completely unexpectedly passed away in November 2019 at the age of 56. It was in a course on computability and constructive mathematics that Erik gave at Stockholm University that introduced me to such topics. The ideas I encountered there aected me deeply and learning more about them was one of the primary reasons I wanted to pursue a PhD in mathematics. Erik was not only an insightful and widely knowledgeable researcher, but also a very kind and generous person. I am grateful for the many discussions we had on mathematics, logic and other topics in his oce on the fourth oor of house six at the Kräftriket campus. It was also Erik who introduced me to my co-supervisor and col- laborator Henrik Forssell. Thanks Henrik for tirelessly answering my questions and generally just being someone to talk to during this period. After Erik passed away, Peter LeFanu Lumsdaine became my (main) supervisor. I was fortunate that such an acute and knowledgeable re- searcher was able to step in more or less directly in Erik's place. Thanks Peter, not only for guiding me through the remaining and critical part of my studies, but also for the help you've provided over these years in numerous discussions. Next I would like to thank the PhD students in logic at SU, passed and present, for making my time there so pleasant : Christian Espíndola, Håkon Robbestad Gylterud, Jacopo Emmenegger, Anna Giulia Monta- rauli and Menno de Boer. Thanks also goes to the rest of the logic group for the same reasons: Per MartinLöf, Anders Mörtberg, Guil- laume Brunerie, Max Zeuner as well as the logic-minded students Oskar Berndal and Philipp Stassen. Last, but not least, I would like to thank my family and friends for their support during this period. Johan Lindberg Stockholm, 2020 1. Introduction This thesis contains three papers, all primarily in the area of categorical logic, together with an introductory part. Except for descriptions and summaries of each of the included papers, the introductory part contains some minor results and (new) proofs of known results which does not appear to be easily available in the literature. We next briey summarize the topics of the included papers, so the reader get a feel for what they contain. More extensive descriptions are given in the next chapter. The papers IIII are based on two more or less disjoint lines of re- search. On the hand, we develop an approach to predicative topos theory initiated by Erik Palmgren with the introduction of the formal system In- tuitionistic Ramied Type Theory (IRTT). This approach aims to char- acterize predicative toposes in terms of their internal set theories (with- out using dependent types). Characterizing the categorical counterpart of such theories amounts to nding a category-theoretic semantics for a kind of (proof-irrelevant) predicative higher-order logic. As a step in this direction, Paper I is primarily concerned with determining properties of a variant of the syntactic category which IRTT generates. However, we there also give a model of the theory in univalent type theory and study the relation between the axioms of IRTT with some of those of construc- tive set theory CZF.