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 …

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 …

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 …

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 …

In their usual form, representation independence metatheorems provide an external
guarantee that two implementations of an abstract interface are interchangeable when they …

Reasoning about weak higher categorical structures constitutes a challenging task, even to
the experts. One principal reason is that the language of set theory is not invariant under the …

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