Unifying cubical and multimodal type theory

TL;DR

This paper introduces Cubical MTT, a unified type theory combining modalities from multimodal type theory with the computational features of cubical type theory, enabling more expressive and principled reasoning about modalities and recursion.

Contribution

It presents Cubical MTT, integrating modalities with cubical identity types, and develops an axiomatic semantics with models including guarded recursion in a cubical topos.

Findings

Validated desirable extensionality principles for modalities

Constructed presheaf models of Cubical MTT

Modeled guarded recursion and L"ob induction

Abstract

In this paper we combine the principled approach to modalities from multimodal type theory (MTT) with the computationally well-behaved realization of identity types from cubical type theory (CTT). The result -- cubical modal type theory (Cubical MTT) -- has the desirable features of both systems. In fact, the whole is more than the sum of its parts: Cubical MTT validates desirable extensionality principles for modalities that MTT only supported through ad hoc means. We investigate the semantics of Cubical MTT and provide an axiomatic approach to producing models of Cubical MTT based on the internal language of topoi and use it to construct presheaf models. Finally, we demonstrate the practicality and utility of this axiomatic approach to models by constructing a model of (cubical) guarded recursion in a cubical version of the topos of trees. We then use this model to justify an axiomatization of L\"ob induction and thereby use Cubical MTT to smoothly reason about guarded recursion.