Two-level type theory and applications - ERRATUM

Dependent pattern matching is an intuitive way to write programs and proofs in dependently typed languages. It is reminiscent of both pattern matching in functional languages and case analysis in on-paper mathematics. However, in general it is incompatible with new type theories such as homotopy type theory (HoTT). As a consequence, proofs in such theories are typically harder to write and to understand. The source of this incompatibility is the reliance of dependent pattern matching on the so-called K axiom - also known as the uniqueness of identity proofs - which is inadmissible in HoTT. The Agda language supports an experimental criterion to detect definitions by pattern matching that make use of the K axiom, but so far it lacked a formal correctness proof.In this paper, we propose a new criterion for dependent pattern matching without K, and prove it correct by a translation to eliminators in the style of Goguen et al. (2006). Our criterion both allows more good definitions than existing proposals, and solves a previously undetected problem in the criterion offered by Agda. It has been implemented in Agda and is the first to be supported by a formal proof. Thus it brings the benefits of dependent pattern matching to contexts where we cannot assume K, such as HoTT. It also points the way to new forms of dependent pattern matching, for example on higher inductive types.

Homotopy type theory is a version of Martin-Lof type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RPn, as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RPn by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take RP-1 to be the empty type. In the inductive step, we take RPn+1 to be the mapping cone of the projection map of the tautological bundle of RPn, and we use its universal property and the univalence axiom to define the tautological bundle on RPn+1.By showing that the total space of the tautological bundle of RPn is the n-sphere Sn, we retrieve the classical description of RPn+1 as RPn with an (n + 1)-disk attached to it. The infinite dimensional real projective space RP∞, defined as the sequential colimit of RPn with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z, 1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles.These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.

Recent innovation in the design of type theories—foundational systems of mathematics with a focus on constructivity—has produced the concept of interval variable, which can be used to capture relations between objects that carry computational content. We examine two such relationships in type theory: equality, in particularquotients, and arbitrary relations as applied in parametricity interpretations. Cubical type theory, a system using an interval-based formulation of equality, enables a permissive kind of content-carrying equality that includes in particularisomorphism. Cubical type theory provides a constructive interpretation of homotopy type theory and the Univalent Foundations, formalisms that introduced the idea of isomorphism as equality but which lack intrinsic computational meaning. The cubical approach to equality also rectifies long-standing deficiencies in the behavior ofquotients in type theory. We realize a system of generalized quotients for cubical type theory originally conceived in homotopy type theory, called higher inductive types, that merges the concepts of inductive type and quotient. Such a mutual generalization is particularly essential in the contentful equality setting, but also has significant applications to ordinary mathematics. Parametricity is, among other things, a proof technique for deriving propertiesof constructions performed in type theories, based on the idea that constructions preserve all relations between objects. Traditionally a meta-theoretical tool, recentwork has shown that parametricity properties can be integrated into a type theory itself using an interval-based system. We develop internal parametricity on top ofcubical type theory, examining the similarities and distinctions between the two applications of intervals, finding that a background of cubical equality improves thebehavior of internal parametricity, and applying internal parametricity as a tool to solve difficult problems in cubical type theory. We introduce a system of cohesionmodalities to express the interaction between parametric and non-parametric type theory, enabling the use of parametricity results in a non-parametric setting.

Definitional equality—or conversion—for a type theory with a decidable type checking is the simplest tool to prove that two objects are the same, letting the system decide just using computation. Therefore, the more things are equal by conversion, the simpler it is to use a language based on type theory. Proof-irrelevance, stating that any two proofs of the same proposition are equal, is a possible way to extend conversion to make a type theory more powerful. However, this new power comes at a price if we integrate it naively, either by making type checking undecidable or by realizing new axioms—such as uniqueness of identity proofs (UIP)—that are incompatible with other extensions, such as univalence. In this paper, taking inspiration from homotopy type theory, we propose a general way to extend a type theory with definitional proof irrelevance, in a way that keeps type checking decidable and is compatible with univalence. We provide a new criterion to decide whether a proposition can be eliminated over a type (correcting and improving the so-called singleton elimination of Coq) by using techniques coming from recent development on dependent pattern matching without UIP. We show the generality of our approach by providing implementations for both Coq and Agda, both of which are planned to be integrated in future versions of those proof assistants.

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened homotopy type theory . In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, called the Univalence Axiom , which has a number of striking consequences. He has subsequently advocated a program, which he calls univalent foundations , of developing mathematics in the setting of type theory with the Univalence Axiom and possibly other additional axioms motivated by the simplicial set model. Because type theory possesses good computational properties, this program can be carried out in a computer proof assistant. In this paper we give an introduction to homotopy type theory in Voevodsky’s setting, paying attention to both theoretical and practical issues. In particular, the paper serves as an introduction to both the general ideas of homotopy type theory as well as to some of the concrete details of Voevodsky’s work using the well-known proof assistant Coq. The paper is written for a general audience of mathematicians with basic knowledge of algebraic topology; the paper does not assume any preliminary knowledge of type theory, logic, or computer science. Because a defining characteristic of Voevodsky’s program is that the Coq code has fundamental mathematical content, and many of the mathematical concepts which are efficiently captured in the code cannot be explained in standard mathematical English without a lengthy detour through type theory, the later sections of this paper (beginning with Section \ref{sec2}) make use of code; however, all notions are introduced from the beginning and in a self-contained fashion.

In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an "outer" one, containing a strict equality type former, and an "inner" one, which is some version of HoTT. Our type theory is inspired by Voevosky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we no not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of "infinite structures" which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with "schematic" definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.

Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory, that offers an alternative to the foundations provided by ZFC set theory and category theory. This paper explains and motivates an account of how to define, justify and think about HoTT in a way that is self-contained, and argues that it can serve as an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be expected to answer, and find that the standard formulation of HoTT as presented in the ``HoTT Book'' does not answer many of them. More importantly, the way HoTT is developed in the HoTT Book suggests that it is not a candidate \emph{autonomous} foundation since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon sophisticated ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation). Our elaboration of HoTT is based on a new interpretation of types as \emph{mathematical concepts}, which accords with the intensional nature of the type theory.

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type is proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches—syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

Type theories with equality reflection, such as extensional type theory (ETT), are convenient theories in which to formalise mathematics, as they make it possible to consider provably equal terms as convertible. Although type-checking is undecidable in this context, variants of ETT have been implemented, for example in NuPRL and more recently in Andromeda. The actual objects that can be checked are not proof-terms, but derivations of proof-terms. This suggests that any derivation of ETT can be translated into a typecheckable proof term of intensional type theory (ITT). However, this result, investigated categorically by Hofmann in 1995, and 10 years later more syntactically by Oury, has never given rise to an effective translation. In this paper, we provide the first effective syntactical translation from ETT to ITT with uniqueness of identity proofs and functional extensionality. This translation has been defined and proven correct in Coq and yields an executable plugin that translates a derivation in ETT into an actual Coq typing judgment. Additionally, we show how this result is extended in the context of homotopy type theory to a two-level type theory.

This paper describes a fibrational categorical semantics for the modal necessity-only fragment of constructive modal type theory, both with and without dependent types. Constructive type theory does not usually discuss logical modalities, and modalities tend to be mostly studied within classical logic, not within type theory. But modalities should be very useful in type theory, as they are very useful in modelling theoretical computing systems. Providing constructive versions of modal logics and their associated Curry-Howard modal type theories is also a very productive program, e.g. helpful when dealing with computational effects, staged computation, and functional reactive types, for example. There seems to be renewed interest in the notion of constructive modal type theory (and in notions of linear type theory), in part because of the interest in homotopy type theory. The modal type theory presented here uses dependent types, in the style of Ritter's categorical models of the Calculus of Constructions. To build up to these, we first discuss the kinds of constructive modal type theory in the literature. Then we provide a non-dependent modal type theory, introduced in previous work, that we generalize to dependent types in the following section. Dependent type theories are usually but not always given categorical semantics in terms of fibrations. We provide semantics in terms of fibrations for both the non-dependent and the dependent type systems discussed and prove them sound and complete, thereby providing evidence that the type theory is meaningful. These fibrational models should be also applicable to the homotopy type theory setting.

Constructs that involve taking a quotient are commonplace in mathematics. Here I will consider how they are treated within dependent type theory. The notion of quotient type has its origins in the Nuprl theorem-proving system [R. L. Constable et al., 1986] for extensional type theory. Later Hofmann formulated a version for intensional type theory in his thesis [M. Hofmann, 1995]. This depends on having a pre-existing notion of intensional identity type. Hofmann used Martin-Löf’s notion, the indexed family inductively generated from proofs of reflexivity [B. Nordström et al., 1990]. The recent homotopical view of identity in terms of path types [The Univalent Foundations Program, 2013] gives a more liberal perspective and has brought with it the notion of higher inductive type (HIT) [P. L. Lumsdaine and M. Shulman, 2019], subsuming both inductive and quotient types. Inductively defined indexed families of types (in all their various forms) are perhaps the most useful concept that dependent type theory has contributed to the practice of computer assistance for formalizing mathematical proofs. However, it is often the case that a particular application of such types needs not only to inductively generate a collection of objects, but also to make identifications between the objects. In classical mathematics one can first generate and then identify, using the Axiom of Choice to lift infinitary constructions to the quotient. HITs can allow one to avoid such non-constructive uses of choice by inter-twining generation and identification. Perhaps more important than the constructive/non-constructive issue is that simultaneously declaring how to generate and how to identify can be a very natural way of defining some construct from the user’s point of view. This is why HITs promise to be so useful, once we have robust and convenient mechanisms in theorem-proving systems for defining HITs and defining functions out of HITs. Although some HITs have been axiomatized in various systems, at the moment the only system I know of with built in support for defining quite general forms of HIT and using them is the implementation of cubical type theory [C. Cohen et al., 2018] within recent versions of the Agda system [A. Vezzosi et al., 2019]. The higher dimensional aspect of identity in cubical type theory is fascinating; nevertheless, the simpler one-dimensional version of identity, in which one has uniqueness of identity proofs (UIP), is adequate for many applications. Although some regard UIP as a bug of early versions of Agda with it’s original form of dependent pattern matching [T. Coquand, 1992], it is by choice a feature of the Lean prover [J. Avigad et al., 2020]. Altenkirch and Kaposi [T. Altenkirch and A. Kaposi, 2016] have termed the one-dimensional version of HITs quotient inductive types (QITs) and they promise to be very useful even in the setting of type theory with UIP. In this talk I survey some of these developments, including a recent reduction of QITs to quotients [M. P. Fiore et al., 2020], and the prospects for better support in theorem-provers for quotient constructions.

We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an n-type can be delooped n + 2 times, then it is an infinite loop type. Most of the results have been formalized in the Lean proof assistant.

Capretta's delay monad can be used to model partial computations, but it has the "wrong" notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the "right" notion of equality, weak bisimilarity. However, recent work by Chapman et al. suggests that it is impossible to define a monad structure on the resulting construction in common forms of type theory without assuming (instances of) the axiom of countable choice. Using an idea from homotopy type theory - a higher inductive-inductive type - we construct a partiality monad without relying on countable choice. We prove that, in the presence of countable choice, our partiality monad is equivalent to the delay monad quotiented by weak bisimilarity. Furthermore we outline several applications.

Homotopy type theory proposes higher inductive types (HITs) as a means of defining and reasoning about inductively-generated objects with higher-dimensional structure. As with the univalence axiom, however, homotopy type theory does not specify the computational behavior of HITs. Computational interpretations have now been provided for univalence and specific HITs by way of cubical type theories, which use a judgmental infrastructure of dimension variables. We extend the cartesian cubical computational type theory introduced by Angiuli et al. with a schema for indexed cubical inductive types (CITs), an adaptation of higher inductive types to the cubical setting. In doing so, we isolate the canonical values of a cubical inductive type and prove a canonicity theorem with respect to these values.

We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the internal versions of the AB axioms from homological algebra. The main subtlety lies in proving AB4, which is that coproducts indexed by arbitrary sets are left-exact. To prove this, we replace a set X with the strict category of lists of elements in X. From showing that the latter is filtered, we deduce left-exactness of the coproduct. More generally, we show that exactness of filtered colimits (AB5) implies AB4 for any abelian category in HoTT. Our approach is heavily inspired by Roswitha Harting’s construction of the internal coproduct of abelian groups in an elementary topos with a natural numbers object. To state the AB axioms, we define and study filtered (and sifted) precategories in HoTT. A key result needed is that filtered colimits commute with finite limits of sets. This is a familiar classical result but has not previously been checked in our setting. Finally, we interpret our most central results into an $\infty$ -topos $ {\mathscr{X}} $ . Given a ring R in $ {\tau_{\leq 0}({{\mathscr{X}}})} $ – for example, an ordinary sheaf of rings – we show that the internal category of R-modules in $ {\mathscr{X}} $ represents the presheaf which sends an object $ X \in {\mathscr{X}} $ to the category of $ (X{\times}R) $ -modules in ${\mathscr{X}} / X$ . In general, our results yield a product-preserving left adjoint to base change of modules over X. When X is 0-truncated, this left adjoint is the internal coproduct. By an internalisation procedure, we deduce left-exactness of the internal coproduct as an ordinary functor from its internal left-exactness coming from HoTT.