The Compatibility of the Minimalist Foundation with Homotopy Type Theory

TL;DR

This paper demonstrates that both levels of the Minimalist Foundation are compatible with Homotopy Type Theory, bridging constructive mathematics with univalent foundations and enabling new computable models.

Contribution

It proves the compatibility of the Minimalist Foundation with Homotopy Type Theory, a novel result linking two foundational approaches in constructive mathematics.

Findings

Both levels of MF are compatible with HoTT.

Compatibility leverages HoTT's combination of intensional and extensional features.

MF gains new computable models from this compatibility.

Abstract

The Minimalist Foundation, for short MF, is a two-level foundation for constructive mathematics ideated by Maietti and Sambin in 2005 and then fully formalized by Maietti in 2009. MF serves as a common core among the most relevant foundations for mathematics in the literature by choosing for each of them the appropriate level of MF to be translated in a compatible way, namely by preserving the meaning of logical and set-theoretical constructors. The two-level structure consists of an intensional level, an extensional one, and an interpretation of the latter in the former in order to extract intensional computational contents from mathematical proofs involving extensional constructions used in everyday mathematical practice. In 2013 a completely new foundation for constructive mathematics appeared in the literature, called Homotopy Type Theory, for short HoTT, which is an example of Voevodsky's Univalent Foundations with a computational nature. So far no level of MF has been proved to be compatible with any of the Univalent Foundations in the literature. Here we show that both levels of MF are compatible with HoTT. This result is made possible thanks to the peculiarities of HoTT which combines intensional features of type theory with extensional ones by assuming Voevodsky's Univalence Axiom and higher inductive quotient types. As a relevant consequence, MF inherits entirely new computable models.