# Cubical Categories for Higher-Dimensional Parametricity

@article{Johann2017CubicalCF, title={Cubical Categories for Higher-Dimensional Parametricity}, author={Patricia Johann and Kristina Sojakova}, journal={ArXiv}, year={2017}, volume={abs/1701.06244} }

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…

## 7 Citations

### A General Framework for Relational Parametricity

- Computer Science, MathematicsLICS
- 2018

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

- Mathematics
- 2019

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

- Mathematics2021 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

- MathematicsArXiv
- 2019

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

- Mathematics
- 2019

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

- Mathematics, Computer Science
- 2019

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!

- MathematicsArXiv
- 2017

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.

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