We present Voevodsky's construction of a model of univalent type theory in the category of
simplicial sets. To this end, we first give a general technique for constructing categorical …

We prove the conjecture that any Grothendieck $(\infty, 1) $-topos can be presented by a
Quillen model category that interprets homotopy type theory with strict univalent universes …

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 …

Univalent homotopy type theory (HoTT) may be seen as a language for the category of ∞-
groupoids. It is being developed as a new foundation for mathematics and as an internal …

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 …

Despite the plausible necessity of topological protection for realizing scalable quantum
computers, the conceptual underpinnings of topological quantum logic gates had arguably …

We prove normalization for MTT, a general multimodal dependent type theory capable of
expressing modal type theories for guarded recursion, internalized parametricity, and …

We report on the development of the HoTT library, a formalization of homotopy type theory in
the Coq proof assistant. It formalizes most of basic homotopy type theory, including …

We present a new coherence theorem for comprehension categories, providing strict models
of dependent type theory with all standard constructors, including dependent products …