Cubical models of homotopy type theory an internal approach Richard Ian Orton November 2018 This dissertation is submitted for the degree of Doctor of Philosophy CHURCHILL COLLEGE Abstract This thesis presents an account of the cubical sets model of homotopy type theory using an internal type theory for elementary topoi. Homotopy type theory is a variant of Martin-L¨of type theory where we think of types as spaces, with terms as points in the space and elements of the identity type as paths. We actualise this intuition by extending type theory with Voevodsky’s univalence axiom which identifies equalities between types with homotopy equivalences between spaces. Voevodsky showed the univalence axiom to be consistent by giving a model of homotopy type theory in the category of Kan simplicial sets in a paper with Kapulkin and Lumsdaine. However, this construction makes fundamental use of classical logic in order to show certain results. Therefore this model cannot be used to explain the computational content of the univalence axiom, such as how to compute terms involving univalence. This problem was resolved by Cohen, Coquand, Huber and M¨ortberg, who presented anewmodeloftypetheoryinKancubicalsetswhichvalidatedtheunivalenceaxiom using a constructive metatheory. This meant that the model provided an understanding of the computational content of univalence. In fact, the authors present a new type theory, cubical type theory, where univalence is provable using a new “glueing” type former. This type former comes with appropriate definitional equalities which explain how the univalence axiom should compute. In particular, Huber proved that any term of natural number type constructed in this new type theory must reduce to a numeral. This thesis explores models of type theory based on the cubical sets model of Cohen et al. It gives an account of this model using the internal language of toposes, where we present a series of axioms which are sufficient to construct a model of cubical type theory, and hence a model of homotopy type theory. This approach therefore generalises the original model and gives a new and useful method for analysing models of type theory. We also discuss an alternative derivation of the univalence axiom and show how this leads to a potentially simpler proof of univalence in any model satisfying the axioms mentioned above, such as cubical sets. Finally, we discuss some shortcomings of the internal language approach with respect to constructing univalent universes. We overcome these difficulties by extending the internal language with an appropriate modality in order to manipulate global elements of an object. Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. It is not substantially the same as any that I have submitted, or am concurrently submitting, for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. Ifurtherstatethatnosubstantialpartofmydissertationhasalreadybeensubmitted,or is being concurrently submitted, for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. This dissertation does not exceed the prescribed limit of 60 000 words. Richard Ian Orton January 23, 2019 Publications and collaborations: The work presented in this thesis is largely based on the publications listed below and is the result of work done in collaboration with my coauthors. Ian Orton, Andrew Pitts, Axioms for modelling cubical type theory in a topos, Logical • Methods in Computer Science (LMCS), 2018. Special issue for CSL 2016, to appear; arXiv:1712.04864. Ian Orton, Andrew Pitts, Decomposing the univalence axiom, Leibniz International • Proceedings in Informatics (LIPIcs), 2018. Post-proceedings for TYPES 2017, to appear; arXiv:1712.04890. Dan Licata, Ian Orton, Andrew Pitts, Bas Spitters, Internal universes in models • of homotopy type theory, Leibniz International Proceedings in Informatics (LIPIcs), 2018. Proceedings for FSCD 2018. arXiv:1801.07664. Acknowledgements First and foremost I would like to thank my supervisor Andrew Pitts for his guidance and collaboration over the past three years. Our many meetings and discussions were invaluable and I have learned an incredible amount from him over the course of my PhD. Andy was always available to o↵er advice and guidance while leaving me free to tackle interesting problems as I saw fit. He has been both a fantastic supervisor and coauthor. I would also like to thank my other coauthors, Dan Licata and Bas Spitters, for the chance to collaborate, as well as for their insights and discussions, on our paper and many other things. I also owe thanks to countless other members of the HoTT/UF community for detailed discussions, both over email and over co↵ee at various workshops and conferences, including (but not limited to): Thorsten Altenkirch, Carlo Angiuli, Steve Awodey, Guillaume Brunerie, Paolo Capriotti, Thierry Coquand, Floris van Doorn, Mart´ın Escard´o, Kuen-Bang Hou (Favonia), Jonas Frey, Simon Huber, Nicolai Kraus, Peter LeFanu Lumsdaine, Anders M¨ortberg, Egbert Rijke, Christian Sattler and Jon Sterling. In addition, I would like to thank my colleagues at the Computer Lab. My cohort of PhD students: Matthew Daggitt, Dylan McDermott, Hugo Paquet and Philip Saville, for many hours of discussion, many lunches at the Cavendish and generally for being excellent friends over the past three years. I would also like to thank the various other CL members, ex-members and visitors for their insights and friendship: Mohammad Abdulaziz, Aurore Alcolei, Simon Castellan, Pierre Clairambault, Marcelo Fiore, Tamara von Glehn, Tim Griffin, Ohad Kammar, Tamas Kispeter, Guillaume Munch-Maccagnoni and Jean Pichon-Pharabod. I would also like to thank my friends in the computer architecture group, Sam Ainsworth and Colin Rothwell, for their friendship over the past seven years and for, in Colin’s words, their “entertainingly singular take on life in the Computer Lab.” Over the course of my PhD I have had fantastic opportunities to travel and collaborate with researchers from across the world. In particular I would like to thank Steve Awodey and Carnegie Mellon University for hosting me in Pittsburgh, and Bas Spitters and Aarhus University for hosting me in Aarhus. I would also like to thank the EPSRC for their generous funding and Lise Gough for dealing with my many travel requests. Finally I would like to thank my family and friends for their support, and particularly my wonderful partner Zoe Pillidge for her companionship over the past several years. Contents 1 Introduction 13 1.1 Layperson’s introduction . 13 1.2 Expert’s introduction . 16 1.3 Structure of this thesis . 18 2 Type theory: an introduction 19 2.1 Univalenttypetheory.............................. 19 2.1.1 Martin-L¨oftype theory . 20 2.1.2 IdentitytypesandUIP. 21 2.1.3 Voevodsky’s univalence axiom . 22 2.1.4 Function extensionality . 24 2.1.5 An alternative formulation of univalence . 25 2.1.6 ThefailureofUIP ........................... 26 2.2 Cubical type theory . 27 2.2.1 The formal interval I .......................... 27 2.2.2 Path types . 28 2.2.3 The face lattice F ............................ 28 2.2.4 Kan composition . 30 2.2.5 Glueing . 30 2.2.6 Univalence................................ 31 3 Models of Martin-L¨of type theory 33 3.1 CategorieswithFamilies(CwFs) . 33 3.1.1 AdditionalstructureonaCwF . 34 3.2 Presheaf models of type theory . 36 3.3 A model in cubical sets . 38 3.3.1 The interval and face lattice . 39 3.3.2 Fibranttypes .............................. 40 3.3.3 Path types . 41 3.3.4 Glueing . 42 3.3.5 A univalent universe . 42 4 The internal type theory of a topos 45 4.1 Elementary topoi . 45 4.1.1 Examples . 46 4.2 The internal type theory . 46 4.3 TheproofassistantAgda. 51 4.4 A remark on impredicativity . 52 5 Axioms for modelling cubical type theory in a topos 55 5.1 The axioms . 56 5.1.1 The interval I .............................. 56 5.1.2 Cofibrant propositions . 57 5.1.3 The strictness axiom . 62 5.2 Path types . 63 5.3 Cohen-Coquand-Huber-M¨ortberg (CCHM) fibrations . 65 5.3.1 Composition and filling structures . 65 5.3.2 Properties of fibrations . 69 5.3.3 Type formers and simple datatypes . 71 5.3.4 Path and identity types . 75 5.4 Glueing . 78 5.5 Univalence.................................... 84 5.6 Satisfying the axioms . 89 5.6.1 The interval object (ax1–ax4).....................89 5.6.2 Cofibrant propositions (ax5–ax8)andthestrictnessaxiom(ax9).. 90 5.7 Related work . 94 6 Decomposing the univalence axiom 97 6.1 The decomposition . 97 6.2 Applicationsinmodelsoftypetheory. 100 6.2.1 Strictification . 101 6.2.2 Misaligned paths between fibrations . 102 6.2.3 Function extensionality . 104 6.2.4 Axioms(1),(2),(4)and(5) . 104 6.2.5 Axiom(3)................................ 106 6.3 Conclusion . 108 7 Internal universes 111 7.1 The “no-go” theorem for internal universes . 111 7.2 Crisp type theory . 113 7.3 Universesfromtinyintervals. 116 7.4 Applications................................... 123 7.5 Conclusion . 125 8 Conclusion 127 8.1 Future work (higher inductive types) . 128 Bibliography 133 Chapter 1 Introduction This introduction is divided into two sections. The first o↵ers an introduction for a layperson. This is intended for friends, family, or anyone else with a school-level background in mathematics who happens to stumble across this thesis. My hope is that these few pages will o↵er at least some insight into what I’ve been doing for the past three years. The second introduction is aimed at experts, who may be more familiar with some of the technical background involved: type theory, category theory, etc.