Internalizing Intensional Type Theory

Homotopical interpretations of Martin-Löf type theory lead toward an interpretation of equality as a richer, more extensional notion. Extensional or axiomatic presentations of the theory with principles based on such models do not yet fully benefit from the power of dependent type theory, that is its computational character. Reconciling intensional type theory with this richer notion of equality requires to move to higher-dimensional structures where equality reasoning is explicit, which explains the emerging interest on simplicial or cubical models. However, such models are still based on set theory, which is somehow in opposition with the goal to replace set theory by homotopy type theory in the foundations of mathematics. Therefore, it is important to be able to express those models directly in type theory. In this paper, we follow this idea and pursue the internalization of a setoid model of Martin-Löf type theory based on an internalization of groupoids. Our work shows that a (proof-relevant) setoid model of type theory constitutes an internal category with families, as introduced by Dybjer, but with the advantage that the (unsolved) coherence problem mentioned by Dybjer now becomes a lemma parametrized by a 2-dimensional proof of confluence of the conversion rule (which can not be internalized). Our formal development relies crucially on ad-hoc polymorphism to overload notions of equality and on a conservative extension of Coq’s universe mechanism with polymorphism.