We begin by recalling the essentially global character of universes in various models of
homotopy type theory, which prevents a straightforward axiomatization of their properties …

In this thesis, we study abstract and concrete type theories. We introduce an abstract notion
of a type theory to obtain general results in the semantics of type theories, but we also …

We present a new constructive model of univalent type theory based on cubical sets. Unlike
prior work on cubical models, ours depends neither on diagonal cofibrations nor …

Cubical Assemblies, a Univalent and Impredicative Universe and a Failure of Propositional
Resizing Page 1 Cubical Assemblies, a Univalent and Impredicative Universe and a Failure of …

Abstract We show that Martin Hyland's effective topos can be exhibited as the homotopy
category of a path category EFF. Path categories are categories of fibrant objects in the …

This book introduces the notion of an effective Kan fibration, a new mathematical structure
which can be used to study simplicial homotopy theory. The main motivation is to make …

We say a universe L in dependent type theory is impredicative if it is closed under arbitrary
dependent products: for an arbitrary type A and a function B: A→ L, the dependent product …

This book starts to redevelop the foundations of simplicial homotopy theory, in particular
around the Kan-Quillen model structure on simplicial sets, in a more effective or “structured” …

We show that Martin Hyland's effective topos can be exhibited as the homotopy category of a
path category EFF. Path categories are categories of fibrant objects in the sense of Brown …

We introduce the notion of an effective Kan fibration, a new mathematical structure that can
be used to study simplicial homotopy theory. Our main motivation is to make simplicial …