Section: New Results

Cubical type theory and univalent foundations

Participants : Cyril Cohen, Anders Mörtberg, Benedikt Ahrens [ASCOLA project-team, Inria and LINA Nantes] , Mark Bickford [Cornell Unversity, USA] , Thierry Coquand [Chalmers and Göteborg University, Sweden] , Ralph Matthes [CNRS, University of Toulouse] .

This work mainly concerns Univalent Foundations and Homotopy Type Theory which builds on recently discovered connections between type theory and abstract homotopy theory. The main question we have been working on lately is finding a computational interpretation for the univalence axiom, the main fruit of this work is a recent paper on, and implementation of, cubical type theory [23] which provides a constructive justification for this axiom. The code is visible at https://github.com/mortberg/cubicaltt. The last year Anders Mörtberg has been working together with Mark Bickford at Cornell University and Thierry Coquand at University of Gothenburg and Chalmers University of Technology on the formal verification of this model in the Nuprl proof assistant, this code is visible at http://www.nuprl.org/wip/Mathematics/cubical!type!theory/index.html.

Anders Mörtberg also recently visited Thierry Coquand to start a collaboration on the formalization of this model in the UniMath system implemented in Coq. Together with Benedikt Ahrens in the Ascola team at Inria Nantes and Ralph Matthes at IRIT in Toulouse, Anders Mörtberg also worked on the formalization of a translation from binding signatures to monads for representing languages with binders in UniMath [21]. This work uses the new possibilities for representing category theory in type theory that univalence provides.