DOCSLIB.ORG

Multimodal Dependent Type Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Multimodal Dependent Type Theory Multimodal Dependent Type Theory Daniel Gratzer G. A. Kavvos Aarhus University Aarhus University [email protected] [email protected] Andreas Nuyts Lars Birkedal imec-DistriNet, KU Leuven Aarhus University [email protected] [email protected] Abstract 1 Introduction We introduce MTT, a dependent type theory which sup- In order to increase the expressivity of Martin-Löf Type ports multiple modalities. MTT is parametrized by a mode Theory (MLTT) we often wish to extend it with new con- theory which specifies a collection of modes, modalities, nectives, and in particular with unary type operators that and transformations between them. We show that different we call modalities or modal operators. Some of these modal choices of mode theory allow us to use the same type theory operators arise as shorthands, while others are introduced as to compute and reason in many modal situations, includ- a device for expressing structure that appears in particular ing guarded recursion, axiomatic cohesion, and parametric models. Whereas the former class of modalities are internally quantification. We reproduce examples from prior work in definable [62], the latter often require extensive modifica- guarded recursion and axiomatic cohesion — demonstrating tions to the basic structure of type-theoretic judgments. In that MTT constitutes a simple and usable syntax whose in- some cases we are even able to prove that these changes are stantiations intuitively correspond to previous handcrafted necessary, by showing that the modality in question can- modal type theories. In some cases, instantiating MTT to not be expressed internally: see e.g. the ‘no-go’ theorems a particular situation unearths a previously unknown type by Shulman [67, §4.1] and Licata et al.[42]. This paper is theory that improves upon prior systems. Finally, we inves- concerned with the development of a systematic approach tigate the metatheory of MTT. We prove the consistency to the formulation of type theories with multiple modalities. of MTT and establish canonicity through an extension of The addition of a modality to a dependent type theory is a recent type-theoretic gluing techniques. These results hold non-trivial exercise. Modal operators often interact with the irrespective of the choice of mode theory, and thus apply to context of a type or term in a complicated way, and naïve a wide variety of modal situations. approaches lead to undesirable interplay with other type formers and substitution. However, the consequent gain in CCS Concepts: • Theory of computation ! Modal and expressivity is substantial, and so it is well worth the effort. temporal logics; Type theory; Proof theory. For example, modalities have been used to express guarded recursive definitions [10, 15, 16, 33], parametric quantifi- Keywords: Modal types, dependent types, categorical se- cation [54, 55], proof irrelevance [3, 54, 57], and to define mantics, gluing, guarded recursion global operations which cannot be localized to an arbitrary context [42]. There has also been concerted effort towards ACM Reference Format: the development of a dependent type theory correspond- Daniel Gratzer, G. A. Kavvos, Andreas Nuyts, and Lars Birkedal. ing to Lawvere’s axiomatic cohesion [41], which has many Proceedings of the 2020. Multimodal Dependent Type Theory. In interesting applications [32, 40, 64, 65, 67]. 35th Annual ACM/IEEE Symposium on Logic in Computer Science Despite this recent flurry of developments, a unifying ac- (LICS ’20), July 8–11, 2020, Saarbrücken, Germany. ACM, New York, NY, USA, 15 pages. https://doi.org/10.1145/3373718.3394736 count of modal dependent type theory has yet to emerge. Faced with a new modal situation, a type theorist must hand- craft a brand new system, and then prove the usual battery of metatheorems. This introduces formidable difficulties on Permission to make digital or hard copies of part or all of this work for two levels. First, an increasing number of these applications personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies are multimodal: they involve multiple interacting modalities, bear this notice and the full citation on the first page. Copyrights for third- which significantly complicates the design of the appropri- party components of this work must be honored. For all other uses, contact ate judgmental structure. Second, the technical development the owner/author(s). of each such system is entirely separate, so that one can- LICS ’20, July 8–11, 2020, Saarbrücken, Germany not share the burden of proof even between closely related © 2020 Copyright held by the owner/author(s). systems. To take a recent example, there is no easy way to ACM ISBN 978-1-4503-7104-9/20/07. https://doi.org/10.1145/3373718.3394736 transfer the work done in the 80-page-long normalization LICS ’20, July 8–11, 2020, Saarbrücken, Germany Daniel Gratzer, G. A. Kavvos, Andreas Nuyts, and Lars Birkedal proof for MLTTµ [30] to a normalization proof for the modal and we can prove very little about it without additional 2- dependent type theory of Birkedal et al.[14], even though cells. these systems are only marginally different. Put simply, if If we add a 2-cell n : ` ) 1 to M, we can define a function one wished to prove that type-checking is decidable for the extract : h` j 퐴i ! 퐴 latter, then one would have to start afresh. 퐴 We intend to avoid such duplication in the future. Rather inside the type theory. If we also add a 2-cell X : ` ) ` ◦ ` than designing a new dependent type theory for some preor- then we can also define dained set of modalities, we will introduce a system that is duplicate : h` j 퐴i ! h` j h` j 퐴ii parametrized by a mode theory, i.e. an algebraic specification 퐴 of a modal situation. This system, which we call MTT, solves Furthermore, we can control the precise interaction between both problems at once. First, by instantiating it with different duplicate퐴 and extract퐴 by adding more equations that relate mode theories we will show that it can capture a wide class n and X. For example, we may ask that M be the walking of situations. Some of these, e.g. the one for guarded recur- comonad [63] which leads to a type theory with a dependent sion, lead to a previously unknown system that improves S4-like modality [27, 57, 67]. We can be even more specific, upon earlier work. Second, the predictable behavior of our e.g. by asking that ¹`, n, Xº be idempotent. rules allows us to prove metatheoretic results about large Thus, a morphism ` : = ! < introduces a modality classes of instantiations of our system. For example, our h` j −⟩, and a 2-cell U : ` ) a of M allows the definition of canonicity theorem applies irrespective of the chosen mode a function of type h` j 퐴i ! ha j 퐴i @ <. theory. As a result, we only need to prove such theorems Relation to other modal type theories. Most work on once. Returning to the previous example, careful choices of modal type theories still defies classification. However, we mode theory yield two systems that closely resemble the can informatively position MTT with respect to two qualita- calculi of Birkedal et al.[14] and MLTT [30] respectively, µ tive criteria, viz. usability and generality. so that our proof of canonicity applies to both. Much of the prior work on modal type theory has fo- In fact, we take things one step further: MTT is not just cused on bolting a specific modality onto a type theory. The multimodal, but also multimode. That is, each judgment of benefit of this approach is that the syntax can be designed MTT can be construed as existing in a particular mode. All to be as convenient as possible for the application at hand. modes have some things in common—e.g. there will be depen- For example, spatial/cohesive type theory [67] features two dent sums in each—but some might possess distinguishing modalities, £ and ¥, and is presented in a dual-context style. features. From a semantic point of view, different modes cor- This judgmental structure, however, is applicable only be- respond to different context categories. In this light, modal- cause of the particular properties of £ and ¥. Nevertheless, ities intuitively correspond to functors between those cate- the numerous pen-and-paper proofs in op. cit. demonstrate gories: in fact, they will be structures slightly weaker than that the resulting system is easy to use. dependent right adjoints (DRAs) [14]. At the other end of the spectrum, the framework of Licata- Mode theories. At a high level, MTT can be thought of Shulman-Riley (LSR) [44] comprises an extremely general as a machine that converts a concrete description of modes toolkit for simply-typed, substructural modal type theory. and modalities into a type theory. This description, which Its dependent generalization, which is currently under de- is often called a mode theory, is given in the form of a small velopment, is able to handle a very large class of modalities. strict 2-category [43, 44, 61]. A mode theory gives rise to the However, this generality comes at a price: its syntax is com- following correspondence: plex and unwieldy, even in the simply-typed case. MTT attempts to strike a delicate balance between those object ∼ mode two extremes. By avoiding substructural settings and some morphism ∼ modality kinds of modalities we obtain a noticeably simpler apparatus. 2-cell ∼ natural map between modalities These restrictions imply that, unlike LSR, we do not need to annotate our term formers with delayed substitutions, The equations between morphisms and between 2-cells in a and that our system straightforwardly extends to dependent mode theory can be used to precisely specify the interactions types.
Load more

Recommended publications
  • Sets in Homotopy Type Theory
    Predicative topos Sets in Homotopy type theory Bas Spitters Aarhus Bas Spitters Aarhus Sets in Homotopy type theory Predicative topos About me I PhD thesis on constructive analysis I Connecting Bishop's pointwise mathematics w/topos theory (w/Coquand) I Formalization of effective real analysis in Coq O'Connor's PhD part EU ForMath project I Topos theory and quantum theory I Univalent foundations as a combination of the strands co-author of the book and the Coq library I guarded homotopy type theory: applications to CS Bas Spitters Aarhus Sets in Homotopy type theory Most of the presentation is based on the book and Sets in HoTT (with Rijke). CC-BY-SA Towards a new design of proof assistants: Proof assistant with a clear (denotational) semantics, guiding the addition of new features. E.g. guarded cubical type theory Predicative topos Homotopy type theory Towards a new practical foundation for mathematics. I Modern ((higher) categorical) mathematics I Formalization I Constructive mathematics Closer to mathematical practice, inherent treatment of equivalences. Bas Spitters Aarhus Sets in Homotopy type theory Predicative topos Homotopy type theory Towards a new practical foundation for mathematics. I Modern ((higher) categorical) mathematics I Formalization I Constructive mathematics Closer to mathematical practice, inherent treatment of equivalences. Towards a new design of proof assistants: Proof assistant with a clear (denotational) semantics, guiding the addition of new features. E.g. guarded cubical type theory Bas Spitters Aarhus Sets in Homotopy type theory Formalization of discrete mathematics: four color theorem, Feit Thompson, ... computational interpretation was crucial. Can this be extended to non-discrete types? Predicative topos Challenges pre-HoTT: Sets as Types no quotients (setoids), no unique choice (in Coq), ..
    [Show full text]
  • Types Are Internal Infinity-Groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau
    Types are internal infinity-groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau To cite this version: Antoine Allioux, Eric Finster, Matthieu Sozeau. Types are internal infinity-groupoids. 2021. hal- 03133144 HAL Id: hal-03133144 https://hal.inria.fr/hal-03133144 Preprint submitted on 5 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Types are Internal 1-groupoids Antoine Allioux∗, Eric Finstery, Matthieu Sozeauz ∗Inria & University of Paris, France [email protected] yUniversity of Birmingham, United Kingdom ericfi[email protected] zInria, France [email protected] Abstract—By extending type theory with a universe of defini- attempts to import these ideas into plain homotopy type theory tionally associative and unital polynomial monads, we show how have, so far, failed. This appears to be a result of a kind of to arrive at a definition of opetopic type which is able to encode circularity: all of the known classical techniques at some point a number of fully coherent algebraic structures. In particular, our approach leads to a definition of 1-groupoid internal to rely on set-level algebraic structures themselves (presheaves, type theory and we prove that the type of such 1-groupoids is operads, or something similar) as a means of presenting or equivalent to the universe of types.
    [Show full text]
  • Sequent Calculus and Equational Programming (Work in Progress)
    Sequent Calculus and Equational Programming (work in progress) Nicolas Guenot and Daniel Gustafsson IT University of Copenhagen {ngue,dagu}@itu.dk Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums. 1 Programming with Equations Functional programming has proved extremely useful in making the task of writing correct software more abstract and thus less tied to the specific, and complex, architecture of modern computers. This, is in a large part, due to its extensive use of types as an abstraction mechanism, specifying in a crisp way the intended behaviour of a program, but it also relies on its declarative style, as a mathematical approach to functions and data structures. However, the vast gain in expressivity obtained through the development of dependent types makes the programming task more challenging, as it amounts to the question of proving complex theorems — as illustrated by the double nature of proof assistants such as Coq [11] and Agda [18]. Keeping this task as simple as possible is then of the highest importance, and it requires the use of a clear declarative style.
    [Show full text]
  • On Modeling Homotopy Type Theory in Higher Toposes
    Review: model categories for type theory Left exact localizations Injective fibrations On modeling homotopy type theory in higher toposes Mike Shulman1 1(University of San Diego) Midwest homotopy type theory seminar Indiana University Bloomington March 9, 2019 Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Some caveats 1 Classical metatheory: ZFC with inaccessible cardinals. 2 Classical homotopy theory: simplicial sets. (It's not clear which cubical sets can even model the (1; 1)-topos of 1-groupoids.) 3 Will not mention \elementary (1; 1)-toposes" (though we can deduce partial results about them by Yoneda embedding). 4 Not the full \internal language hypothesis" that some \homotopy theory of type theories" is equivalent to the homotopy theory of some kind of (1; 1)-category. Only a unidirectional interpretation | in the useful direction! 5 We assume the initiality hypothesis: a \model of type theory" means a CwF.
    [Show full text]
  • Proof-Assistants Using Dependent Type Systems
    CHAPTER 18 Proof-Assistants Using Dependent Type Systems Henk Barendregt Herman Geuvers Contents I Proof checking 1151 2 Type-theoretic notions for proof checking 1153 2.1 Proof checking mathematical statements 1153 2.2 Propositions as types 1156 2.3 Examples of proofs as terms 1157 2.4 Intermezzo: Logical frameworks. 1160 2.5 Functions: algorithms versus graphs 1164 2.6 Subject Reduction . 1166 2.7 Conversion and Computation 1166 2.8 Equality . 1168 2.9 Connection between logic and type theory 1175 3 Type systems for proof checking 1180 3. l Higher order predicate logic . 1181 3.2 Higher order typed A-calculus . 1185 3.3 Pure Type Systems 1196 3.4 Properties of P ure Type Systems . 1199 3.5 Extensions of Pure Type Systems 1202 3.6 Products and Sums 1202 3.7 E-typcs 1204 3.8 Inductive Types 1206 4 Proof-development in type systems 1211 4.1 Tactics 1212 4.2 Examples of Proof Development 1214 4.3 Autarkic Computations 1220 5 P roof assistants 1223 5.1 Comparing proof-assistants . 1224 5.2 Applications of proof-assistants 1228 Bibliography 1230 Index 1235 Name index 1238 HANDBOOK OF AUTOMAT8D REASONING Edited by Alan Robinson and Andrei Voronkov © 2001 Elsevier Science Publishers 8.V. All rights reserved PROOF-ASSISTANTS USING DEPENDENT TYPE SYSTEMS 1151 I. Proof checking Proof checking consists of the automated verification of mathematical theories by first fully formalizing the underlying primitive notions, the definitions, the axioms and the proofs. Then the definitions are checked for their well-formedness and the proofs for their correctness, all this within a given logic.
    [Show full text]
  • Constructivity in Homotopy Type Theory
    Ludwig Maximilian University of Munich Munich Center for Mathematical Philosophy Constructivity in Homotopy Type Theory Author: Supervisors: Maximilian Doré Prof. Dr. Dr. Hannes Leitgeb Prof. Steve Awodey, PhD Munich, August 2019 Thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in Logic and Philosophy of Science contents Contents 1 Introduction1 1.1 Outline................................ 3 1.2 Open Problems ........................... 4 2 Judgements and Propositions6 2.1 Judgements ............................. 7 2.2 Propositions............................. 9 2.2.1 Dependent types...................... 10 2.2.2 The logical constants in HoTT .............. 11 2.3 Natural Numbers.......................... 13 2.4 Propositional Equality....................... 14 2.5 Equality, Revisited ......................... 17 2.6 Mere Propositions and Propositional Truncation . 18 2.7 Universes and Univalence..................... 19 3 Constructive Logic 22 3.1 Brouwer and the Advent of Intuitionism ............ 22 3.2 Heyting and Kolmogorov, and the Formalization of Intuitionism 23 3.3 The Lambda Calculus and Propositions-as-types . 26 3.4 Bishop’s Constructive Mathematics................ 27 4 Computational Content 29 4.1 BHK in Homotopy Type Theory ................. 30 4.2 Martin-Löf’s Meaning Explanations ............... 31 4.2.1 The meaning of the judgments.............. 32 4.2.2 The theory of expressions................. 34 4.2.3 Canonical forms ...................... 35 4.2.4 The validity of the types.................. 37 4.3 Breaking Canonicity and Propositional Canonicity . 38 4.3.1 Breaking canonicity .................... 39 4.3.2 Propositional canonicity.................. 40 4.4 Proof-theoretic Semantics and the Meaning Explanations . 40 5 Constructive Identity 44 5.1 Identity in Martin-Löf’s Meaning Explanations......... 45 ii contents 5.1.1 Intensional type theory and the meaning explanations 46 5.1.2 Extensional type theory and the meaning explanations 47 5.2 Homotopical Interpretation of Identity ............
    [Show full text]
  • Proof Theory of Constructive Systems: Inductive Types and Univalence
    Proof Theory of Constructive Systems: Inductive Types and Univalence Michael Rathjen Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England [email protected] Abstract In Feferman’s work, explicit mathematics and theories of generalized inductive definitions play a cen- tral role. One objective of this article is to describe the connections with Martin-L¨of type theory and constructive Zermelo-Fraenkel set theory. Proof theory has contributed to a deeper grasp of the rela- tionship between different frameworks for constructive mathematics. Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky’s univalence axiom. A further goal is to investigate the strength of intuitionistic theories of generalized inductive definitions in the framework of intuitionistic explicit mathematics that lie beyond the reach of Martin-L¨of type theory. Key words: Explicit mathematics, constructive Zermelo-Fraenkel set theory, Martin-L¨of type theory, univalence axiom, proof-theoretic strength MSC 03F30 03F50 03C62 1 Introduction Intuitionistic systems of inductive definitions have figured prominently in Solomon Feferman’s program of reducing classical subsystems of analysis and theories of iterated inductive definitions to constructive theories of various kinds. In the special case of classical theories of finitely as well as transfinitely iterated inductive definitions, where the iteration occurs along a computable well-ordering, the program was mainly completed by Buchholz, Pohlers, and Sieg more than 30 years ago (see [13, 19]). For stronger theories of inductive 1 i definitions such as those based on Feferman’s intutitionic Explicit Mathematics (T0) some answers have been provided in the last 10 years while some questions are still open.
    [Show full text]
  • Introduction to Synthese Special Issue on the Foundations of Mathematics
    Introduction to Synthese Special Issue on the Foundations of Mathematics Carolin Antos,∗ Neil Barton,y Sy-David Friedman,z Claudio Ternullo,x John Wigglesworth{ 1 The Idea of the Series The Symposia on the Foundations of Mathematics (SOTFOM), whose first edition was held in 2014 in Vienna, were conceived of and organised by the editors of this special issue with the goal of fostering scholarly interaction and exchange on a wide range of topics relating both to the foundations and the philosophy of mathematics. The last few years have witnessed a tremendous boost of activity in this area, and so the organisers felt that the time was ripe to bring together researchers in order to help spread novel ideas and suggest new directions of inquiry. ∗University of Konstanz. I would like to thank the VolkswagenStiftung for their generous support through the Freigeist Project Forcing: Conceptual Change in the Foun- dations of Mathematics and the European Union’s Horizon 2020 research and innova- tion programme for their funding under the Marie Skłodowska-Curie grant Forcing in Contemporary Philosophy of Set Theory. yUniversity of Konstanz. I am very grateful for the generous support of the FWF (Austrian Science Fund) through Project P 28420 (The Hyperuniverse Programme) and the VolkswagenStiftung through the project Forcing: Conceptual Change in the Foun- dations of Mathematics. zKurt Godel¨ Research Center for Mathematical Logic, University of Vienna, Au- gasse 2-6, A-1090 Wien. I wish to thank the FWF for its support through Project P28420 (The Hyperuniverse Programme). xUniversity of Tartu. I wish to thank the John Templeton Foundation (Project #35216: The Hyperuniverse.
    [Show full text]
  • CS 6110 S18 Lecture 32 Dependent Types 1 Typing Lists with Lengths
    CS 6110 S18 Lecture 32 Dependent Types When we added kinding last time, part of the motivation was to complement the functions from types to terms that we had in System F with functions from types to types. Of course, all of these languages have plain functions from terms to terms. So it's natural to wonder what would happen if we added functions from values to types. The result is a language with \compile-time" types that can depend on \run-time" values. While this arrangement might seem paradoxical, it is a very powerful way to express the correctness of programs; and via the propositions-as-types principle, it also serves as the foundation for a modern crop of interactive theorem provers. The language feature is called dependent types (i.e., types depend on terms). Prominent dependently-typed languages include Coq, Nuprl, Agda, Lean, F*, and Idris. Some of these languages, like Coq and Nuprl, are more oriented toward proving things using propositions as types, and others, like F* and Idris, are more oriented toward writing \normal programs" with strong correctness guarantees. 1 Typing Lists with Lengths Dependent types can help avoid out-of-bounds errors by encoding the lengths of arrays as part of their type. Consider a plain recursive IList type that represents a list of any length. Using type operators, we might use a general type List that can be instantiated as List int, so its kind would be type ) type. But let's use a fixed element type for now. With dependent types, however, we can make IList a type constructor that takes a natural number as an argument, so IList n is a list of length n.
    [Show full text]
  • Models of Type Theory Based on Moore Paths ∗
    Logical Methods in Computer Science Volume 15, Issue 1, 2019, pp. 2:1–2:24 Submitted Feb. 16, 2018 https://lmcs.episciences.org/ Published Jan. 09, 2019 MODELS OF TYPE THEORY BASED ON MOORE PATHS ∗ IAN ORTON AND ANDREW M. PITTS University of Cambridge Department of Computer Science and Technology, UK e-mail address: [email protected] e-mail address: [email protected] Abstract. This paper introduces a new family of models of intensional Martin-L¨oftype theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types. 1. Introduction Homotopy Type Theory [Uni13] has re-invigorated the study of the intensional version of Martin-L¨oftype theory [Mar75]. On the one hand, the language of type theory helps to express synthetic constructions and arguments in homotopy theory and higher-dimensional category theory. On the other hand, the geometric and algebraic insights of those branches of mathematics shed new light on logical and type-theoretic notions.
    [Show full text]
  • Reasoning in an ∞-Topos with Homotopy Type Theory
    Reasoning in an 1-topos with homotopy type theory Dan Christensen University of Western Ontario 1-Topos Institute, April 1, 2021 Outline: Introduction to homotopy type theory Examples of applications to 1-toposes 1 / 23 2 / 23 Motivation for Homotopy Type Theory People study type theory for many reasons. I'll highlight two: Its intrinsic homotopical/topological content. Things we prove in type theory are true in any 1-topos. Its suitability for computer formalization. This talk will focus on the first point. So, what is an 1-topos? 3 / 23 1-toposes An 1-category has objects, (1-)morphisms between objects, 2-morphisms between 1-morphisms, etc., with all k-morphisms invertible for k > 1. Alternatively: a category enriched over spaces. The prototypical example is the 1-category of spaces. An 1-topos is an 1-category C that shares many properties of the 1-category of spaces: C is presentable. In particular, C is cocomplete. C satisfies descent. In particular, C is locally cartesian closed. Examples include Spaces, slice categories such as Spaces/S1, and presheaves of Spaces, such as G-Spaces for a group G. If C is an 1-topos, so is every slice category C=X. 4 / 23 History of (Homotopy) Type Theory Dependent type theory was introduced in the 1970's by Per Martin-L¨of,building on work of Russell, Church and others. In 2006, Awodey, Warren, and Voevodsky discovered that dependent type theory has homotopical models, extending 1998 work of Hofmann and Streicher. At around this time, Voevodsky discovered his univalance axiom.
    [Show full text]
  • Higher Algebra in Homotopy Type Theory
    Higher Algebra in Homotopy Type Theory Ulrik Buchholtz TU Darmstadt Formal Methods in Mathematics / Lean Together 2020 1 Homotopy Type Theory & Univalent Foundations 2 Higher Groups 3 Higher Algebra Outline 1 Homotopy Type Theory & Univalent Foundations 2 Higher Groups 3 Higher Algebra Homotopy Type Theory & Univalent Foundations First, recall: • Homotopy Type Theory (HoTT): • A branch of mathematics (& logic/computer science/philosophy) studying the connection between homotopy theory & Martin-Löf type theory • Specific type theories: Typically MLTT + Univalence (+ HITs + Resizing + Optional classicality axioms) • Without the classicality axioms: many interesting models (higher toposes) – one potential reason to case about constructive math • Univalent Foundations: • Using HoTT as a foundation for mathematics. Basic idea: mathematical objects are (ordinary) homotopy types. (no entity without identity – a notion of identification) • Avoid “higher groupoid hell”: We can work directly with homotopy types and we can form higher quotients. (But can we form enough? More on this later.) Cf.: The HoTT book & list of references on the HoTT wiki Homotopy levels Recall Voevodsky’s definition of the homotopy levels: Level Name Definition −2 contractible isContr(A) := (x : A) × (y : A) ! (x = y) −1 proposition isProp(A) := (x; y : A) ! isContr(x = y) 0 set isSet(A) := (x; y : A) ! isProp(x = y) 1 groupoid isGpd(A) := (x; y : A) ! isSet(x = y) . n n-type ··· . 1 type (N/A) In non-homotopical mathematics, most objects are n-types with n ≤ 1. The types of categories and related structures are 2-types. Univalence axiom For A; B : Type, the map id-to-equivA;B :(A =Type B) ! (A ' B) is an equivalence.
    [Show full text]