Nominal presentation of cubical sets models of type theory

The cubical sets model of Homotopy Type Theory introduced by Bezem, Coquand and Huber uses a particular category of presheaves. We show that this presheaf category is equivalent to a category of sets equipped with an action of a monoid of name substitutions for which a finite support property holds. That category is in turn isomorphic to a category of nominal sets equipped with operations for substituting constants 0 and 1 for names. This formulation of cubical sets brings out the potentially useful connection that exists between the homotopical notion of path and the nominal sets notion of name abstraction. The formulation in terms of actions of monoids of name substitutions also encompasses a variant category of cubical sets with diagonals, equivalent to presheaves on Grothendieck's "smallest test category." We show that this category has the pleasant property that path objects given by name abstraction are exponentials with respect to an interval object.This is the final version of the article. It first appeared from Dagstuhl Publishing via http://dx.doi.org/10.4230/LIPIcs.TYPES.2014.20

Denotational semantics with nominal scott domains

When defining computations over syntax as data, one often runs into tedious issues concerning α -equivalence and semantically correct manipulations of binding constructs. Here we study a semantic framework in which these issues can be dealt with automatically by the programming language. We take the user-friendly “nominal” approach in which bound objects are named. In particular, we develop a version of Scott domains within nominal sets and define two programming languages whose denotational semantics are based on those domains. The first language, λν -PCF, is an extension of Plotkin’s PCF with names that can be swapped, tested for equality and locally scoped; although simple, it already exposes most of the semantic subtleties of our approach. The second language, PNA, extends the first with name abstraction and concretion so that it can be used for metaprogramming over syntax with binders. For both languages, we prove a full abstraction result for nominal Scott domains analogous to Plotkin’s classic result about PCF and conventional Scott domains: two program phrases have the same observable operational behaviour in all contexts if and only if they denote equal elements of the nominal Scott domain model. This is the first full abstraction result we know of for languages combining higher-order functions with some form of locally scoped names which uses a domain theory based on ordinary extensional functions, rather than using the more intensional approach of game semantics. To obtain full abstraction, we need to add two functionals, one for existential quantification over names and one for “definite description” over names. Only adding one of them is not enough, as we give counter-examples to full abstraction in both cases.This work is supported by a Gates Cambridge Scholarship and the ERC Advanced Grant Events, Causality and Symmetry (ECSYM)This version is the author accepted manuscript. The final version is available from ACM at http://dl.acm.org/citation.cfm?id=2629529

Axioms for modelling cubical type theory in a Topos

The homotopical approach to intensional type theory views proofs of equality as paths. We explore what is required of an interval-like object I in a topos to give a model of type theory in which elements of identity types are functions with domain I. Cohen, Coquand, Huber and Mörtberg give such a model using a particular category of presheaves. We investigate the extent to which their model construction can be expressed in the internal type theory of any topos and identify a collection of quite weak axioms for this purpose. This clarifies the definition and properties of the notion of uniform Kan filling that lies at the heart of their constructive interpretation of Voevodsky’s univalence axiom. Furthermore, since our axioms can be satisfied in a number of different ways, we show that there is a range of topos-theoretic models of homotopy type theory in this style.Engineering and Physical Sciences Research Council (Doctoral Training Award)This is the final version of the article. It first appeared from Schloss Dagstuhl via http://dx.doi.org/10.4230/LIPIcs.CSL.2016.2

Models of Type Theory Based on Moore Paths

This paper introduces a new family of models of intensional Martin-Löf type 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 on cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.EPSRC Studentshi

Decomposing the Univalence Axiom

This paper investigates Voevodsky's univalence axiom in intensional Martin-Löf type theory. In particular, it looks at how univalence can be derived from simpler axioms. We first present some existing work, collected together from various published and unpublished sources; we then present a new decomposition of the univalence axiom into simpler axioms. We argue that these axioms are easier to verify in certain potential models of univalent type theory, particularly those models based on cubical sets. Finally we show how this decomposition is relevant to an open problem in type theory

Axioms for modelling cubical type theory in a topos

The homotopical approach to intensional type theory views proofs of equality as paths. We explore what is required of an object $I$ in a topos to give such a path-based model of type theory in which paths are just functions with domain $I$. Cohen, Coquand, Huber and M\"ortberg give such a model using a particular category of presheaves. We investigate the extent to which their model construction can be expressed in the internal type theory of any topos and identify a collection of quite weak axioms for this purpose. This clarifies the definition and properties of the notion of uniform Kan filling that lies at the heart of their constructive interpretation of Voevodsky's univalence axiom. (This paper is a revised and expanded version of a paper of the same name that appeared in the proceedings of the 25th EACSL Annual Conference on Computer Science Logic, CSL 2016.

Models of Type Theory Based on Moore Paths

This paper introduces a new family of models of intensional Martin-Löf type 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.Orton was supported by a PhD studentship from the UK EPSRC funded by grants EP/L504920/1 and EP/M506485/1

A Dependent Type Theory with Abstractable Names

This paper describes a version of Martin-Löf's dependent type theory extended with names and constructs for freshness and name-abstraction derived from the theory of nominal sets. We aim for a type theory for computing and proving (via a Curry-Howard correspondence) with syntactic structures which captures familiar, but informal, ‘nameful’ practices when dealing with binders.Partially supported by the UK EPSRC program grant EP/K008528/1, Rigorous Engineering for Mainstream Systems (REMS). Supported by the UK EPSRC leadership fellowship (Peter Sewell) grant EP/H005633/1, Semantic Foundations for Real-World Systems.This is the final published version of the article. It was originally published in Electronic Notes in Theoretical Computer Science (Pitts AM, Matthiesen J, Derikx J, Electronic Notes in Theoretical Computer Science 2015, 312, 19–50, doi:10.1016/j.entcs.2015.04.003) http://dx.doi.org/10.1016/j.entcs.2015.04.00

Constructing Infinitary Quotient-Inductive Types

This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of inductive-inductive definitions involving strictly positive occurrences of Hofmann-style quotient types, and Abel's size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definitions of the elimination and computation properties of our encoding of QW-types. The development is formalized using the Agda theorem prover

Internal Universes in Models of Homotopy Type Theory

We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory