Unifying cubical and multimodal type theory
Frederik Lerbjerg Aagaard, Magnus Baunsgaard Kristensen, Daniel Gratzer, Lars Birkedal
TL;DR
Cubical MTT unifies Cubical Type Theory with Multimodal Type Theory to provide computational identity types in a modal setting. The approach builds a semantic foundation from modal context structures, cubical cosmoi, and cubical presheaves, enabling both univalence and guarded recursion through Löb induction. Key contributions include exchange and composition rules for modal types, modal extensionality, and a robust model theory that supports guarded recursion and practical programming. This framework offers a general, instantiable path for combining higher-dimensional type theory with a broad class of modalities, with promising implications for normalization and higher inductive types.
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 Löb induction and thereby use Cubical MTT to smoothly reason about guarded recursion.