• Corpus ID: 16709990

Cubical Categories for Higher-Dimensional Parametricity

  title={Cubical Categories for Higher-Dimensional Parametricity},
  author={Patricia Johann and Kristina Sojakova},
Reynolds' theory of relational parametricity formalizes parametric polymorphism for System F, thus capturing the idea that polymorphically typed System F programs always map related inputs to related results. This paper shows that Reynolds' theory can be seen as the instantiation at dimension 1 of a theory of relational parametricity for System F that holds at all higher dimensions, including infinite dimension. This theory is formulated in terms of the new notion of a p-dimensional cubical… 

A General Framework for Relational Parametricity

This work develops an abstract framework for relational parametricity that delivers a model expressing Reynolds' theory in a direct and natural way, and offers a novel relationally parametric model of System F (after Orsanigo), which is proof-relevant in the sense that witnesses of relatedness are themselves suitably related.

Higher Inductive Types and Parametricity in Cubical Type Theory

Cubical type theory is a novel extension of dependent type theory with a form of equality called a path. Path equality enjoys extensionality properties missing from traditional treatments of

Parametricity and Semi-Cubical Types

  • Hugo Moeneclaey
  • Mathematics
    2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2021
A model of type theory enjoying parametricity from an arbitrary one is constructed, and it is shown that a functor forgetting unary operations and equations defining them recursively in a generalized algebraic theory has a right adjoint.

Parametric Cubical Type Theory

We exhibit a computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of parametric type theory, drawing

J ul 2 01 9 Parametric Cubical Type Theory

We exhibit a computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of parametric type theory, drawing

Cubical Categories for Higher-Dimensional Parametricity Extended Version

A generalization of cubical sets is introduced, which is called cubical categories, and used to develop a framework for higher-dimensional parametricity, all the way up to and including infinity, which has the crucial property that if a model is p-parametric according to this definition, then it is l- parametric for every l < p.

Homotopies for Free!

It follows that every space defined as a higher inductive type has the same homotopy groups as some type of polymorphic functions defined without univalence orHigher inductive types.

Reflexive graphs and parametric polymorphism

It is proved that if the authors start with a non-parametric model which is left exact and which satisfies a completeness condition corresponding to Ma and Reynolds "suitability for polymorphism", then they can recover a parametric model with the same category of closed types.

Categorical Models for Abadi-Plotkin ’ s Logic for Parametricity

A new category-theoretic formulation of relational parametricity is proposed based on a logic for reasoning aboutParametricity given by Abadi and Plotkin (Plotkin and Abadi, 1993) and a way of constructing parametric models from given models of the second-order lambda calculus is described.

A Note on the Uniform Kan Condition in Nominal Cubical Sets

An analogue of the Yoneda Lemma for co-sieves that relates geometric open boxes bijectively to their algebraic counterparts, much as its progenitor for representables relates geometric cubes to theirgebraic counterparts in a cubical set is used to give a formulation of uniform Kan fibrations in which uniformity emerges as naturality in the additional dimensions.

State-dependent representation independence

This paper develops a possible-worlds model in which relational interpretations of types are allowed to grow over time in a manner that is tightly coupled with changes to some local state, and employs a step-indexed stratification of possible worlds, which facilitates a simplified account of mutable references of higher type.

Parametric limits

  • B. DunphyU. Reddy
  • Mathematics
    Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.
  • 2004
A categorical model of polymorphic lambda calculi is developed using the notion of parametric limits, which extend the idea of limits in categories to reflexive graphs of categories and axiomatize the structure of Reflexive graphs needed for modelling parametric polymorphism.

Constructive natural deduction and its ‘ω-set’ interpretation

The first part of this paper may be viewed as a short tutorial with a constructive understanding of the deduction theorem and some work on the expressive power of first and second order quantification and the presentation is meant to be elementary.

Proof-Relevant Parametricity

This paper shows how this can be done and, excitingly, the answer requires a trip into the world of higher dimensional parametricity.

A Model of Type Theory in Cubical Sets

A model of type theory with dependent product, sum, and identity, in cubical sets is presented, and is a step towards a computational interpretation of Voevodsky's Univalence Axiom.

Pasting Schemes for the Monoidal Biclosed Structure on

Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !-categories, which extends Gray's tensor product on 2-categories and which