We introduce MTT, a dependent type theory which supports multiple modalities. MTT is
parametrized by a mode theory which specifies a collection of modes, modalities, and …

Proof assistants based on dependent type theory provide expressive languages for both
programming and proving within the same system. However, all of the major …

We prove normalization for (univalent, Cartesian) cubical type theory, closing the last major
open problem in the syntactic metatheory of cubical type theory. Our normalization result is …

The implementation and semantics of dependent type theories can be studied in a syntax-
independent way: the objective metatheory of dependent type theories exploits the universal …

The homotopical approach to intensional type theory views proofs of equality as paths. We
explore what is required of an object I in a topos to give such a path-based model of type …

Cubical type theory provides a constructive justification to certain aspects of homotopy type
theory such as Voevodsky's univalence axiom. This makes many extensionality principles …

Modalities are everywhere in programming and mathematics! Despite this, however, there
are still significant technical challenges in formulating a core dependent type theory with …

We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory
which combines two different type theories. We refer to them as the 'inner'and the 'outer'type …

Directed type theory is an analogue of homotopy type theory where types represent
categories, generalizing groupoids. A bisimplicial approach to directed type theory …

We propose an abstract notion of a type theory to unify the semantics of various type
theories including Martin–Löf type theory, two-level type theory, and cubical type theory. We …