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 …

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 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 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 …

The goal of this thesis is to prove that $\pi_4 (S^ 3)\simeq\mathbb {Z}/2\mathbb {Z} $ in
homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof …

Higher inductive types (HITs) in Homotopy Type Theory allow the definition of datatypes
which have constructors for equalities over the defined type. HITs generalise quotient types …

Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by
category theory and homotopy theory. With these extensions, type theory can be used to …

This is an introductory textbook to univalent mathematics and homotopy type theory, a
mathematical foundation that takes advantage of the structural nature of mathematical …

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 show that Voevodsky's univalence axiom for intensional type theory is valid in categories
of simplicial presheaves on elegant Reedy categories. In addition to diagrams on inverse …