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 propose foundations for a synthetic theory of $(\infty, 1) $-categories within homotopy
type theory. We axiomatize a directed interval type, then define higher simplices from it 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 combine homotopy type theory with axiomatic cohesion, expressing the latter internally
with a version of 'adjoint logic'in which the discretization and codiscretization modalities are …

We describe a homotopical version of the relational and gluing models of type theory, and
generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy …

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 …

The notion of a natural model of type theory is defined in terms of that of a representable
natural transfomation of presheaves. It is shown that such models agree exactly with the …

Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-
so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very …

In recent years, we have seen several new models of dependent type theory extended with
some form of modal necessity operator, including nominal type theory, guarded and clocked …