# A Cubical Language for Bishop Sets

@article{Sterling2020ACL, title={A Cubical Language for Bishop Sets}, author={Jonathan Sterling and Carlo Angiuli and Daniel Gratzer}, journal={Log. Methods Comput. Sci.}, year={2020}, volume={18} }

We present XTT, a version of Cartesian cubical type theory specialized for
Bishop sets \`a la Coquand, in which every type enjoys a definitional version
of the uniqueness of identity proofs. Using cubical notions, XTT reconstructs
many of the ideas underlying Observational Type Theory, a version of
intensional type theory that supports function extensionality. We prove the
canonicity property of XTT (that every closed boolean is definitionally equal
to a constant) using Artin gluing.

## 7 Citations

### Strict universes for Grothendieck topoi

- MathematicsArXiv
- 2022

Hofmann and Streicher famously showed how to lift Grothendieck universes into presheaf topoi, and Streicher has extended their result to the case of sheaf topoi by sheafification. In parallel, van…

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- Computer Science, MathematicsLog. Methods Comput. Sci.
- 2022

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### Logical Relations as Types

- Computer Science
- 2021

A fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction.

### Normalization for Cubical Type Theory

- Mathematics2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021

The normalization result is reduction-free, in the sense of yielding a bijection between equivalence classes of terms in context and a tractable language of β/η-normal forms.

### Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

- Computer ScienceJ. ACM
- 2021

A fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction.

### Synthetic topology in Homotopy Type Theory for probabilistic programming

- Mathematics, Computer ScienceMath. Struct. Comput. Sci.
- 2021

This paper proposes the use of synthetic topology to model continuous distributions for probabilistic computations in type theory and shows how the Lebesgue valuation, and hence continuous distributions, can be constructed.

### Objective Metatheory of Cubical Type Theories

- Computer Science
- 2020

The semantic methods of the objective metatheory enable the design and implementation of correct-by-construction elaboration algorithms, providing a principled interface between real proof assistants and ideal mathematics.

## 142 References

### Unifying Cubical Models of Univalent Type Theory

- MathematicsCSL
- 2020

It is formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom.

### A General Framework for the Semantics of Type Theory

- MathematicsArXiv
- 2019

We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-Lof type theory, two-level type theory and cubical type theory. We establish basic…

### Foundations of Constructive Analysis

- Mathematics
- 2012

This article has no associated abstract. (fix it)

### Notes on Clans and Tribes

- Linguistics
- 2017

The purpose of these notes is to give a categorical presentation/analysis of homotopy type theory. The notes are incomplete as they stand (October 2017). The chapter on univalent tribes is missing.…

### Extensional concepts in intensional type theory

- Mathematics
- 1995

The best ebooks about Extensional Concepts In Intensional Type Theory that you can get for free here by download this Extensional Concepts In Intensional Type Theory and save to your desktop. This…

### Canonicity for Cubical Type Theory

- MathematicsJournal of Automated Reasoning
- 2018

This paper proves canonicity for cubical type theory: any natural number in a context build from only name variables is judgmentally equal to a numeral.

### Natural models of homotopy type theory

- MathematicsMathematical Structures in Computer Science
- 2016

It is shown that a category admits a natural model of type theory if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: They should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class.

### Universes in Toposes

- PhilosophyFrom sets and types to topology and analysis
- 2005

We discuss a notion of universe in toposes which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH) that allows one to construct families of types…

### Univalence for inverse diagrams and homotopy canonicity

- MathematicsMathematical Structures in Computer Science
- 2014

A homotopical version of the relational and gluing models of type theory that uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory.