## Section: New Results

### Homotopy Type Theory

Participants : Yves Bertot [correspondant] , Florent Bréhard.

Homotopy Type Theory is a domain born out of the conjuction of type theory, which serves as foundations for proof systems like Coq or Agda, and homotopy theory, and domain of mathematics which is concerned with equivalence classes of objects modulo continuous deformation. In particular, Homotopy Type Theory concentrates on paths (continuous substrate between various objects) and paths between paths: paths between points can be understood as lines, paths between lines can be understood as surfaces.

In particular, paths can be thought has having the same properties as the notion of equality that is usually defined inductively in type theory systems and homotopy type theory goes against the trend started in the 1990s where specialists thought an axiom should be added to express that all paths between paths should be equal. On the contrary, if all paths between paths are not equal, type theory can be used to model homotopy theory and that domain of mathematics because a new area of applications for type theory-based theorem provers.

V. Voevodsky organized a special year at Institute for Advanced Study
in Princeton on this topic, and Yves Bertot participated to this
special year, during which many experiments were performed,
extensions to proof systems were designed, and a book was produced.
In particular, Yves Bertot devised an extension of the Coq system with
*private types* which makes it possible to simulate a new concept
known as *higher inductive types*. On top of this extension, the
members of the special year produced a collection of higher inductive
types, describing circles, spheres, truncations.

During his internship in the Marelle project, Florent Bréhard studied the equivalence between several presentations of higher-dimension spheres using higher inductive types.

Work on higher inductive types was pursued more precisely by Bruno Barras from Saclay. We expect that the result of this work will supersede the experiments made possible by Yves Bertot's implementation of private types, but the concept of private type may retain applications in other domains.