This book provides an introduction to modern homotopy theory through the lens of higher
categories after Joyal and Lurie, giving access to methods used at the forefront of research …

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 …

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 …

We construct a left semi-model structure on the category of intensional type theories
(precisely, on CxlCat Id, 1, Σ (, Π ext)). This presents an∞-category of such type theories; we …

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 define an elementary $\infty $-topos that simultaneously generalizes an elementary
topos and Grothendieck $\infty $-topos. We then prove it satisfies the expected topos …

We prove that the homotopy theory of Joyal's tribes is equivalent to that of fibration
categories. As a consequence, we deduce a variant of the conjecture asserting that Martin …

There are so many different notions of “space”(topological spaces, manifolds, schemes,
stacks, and so on, as discussed in various other chapters of this book and its companion …

We introduce the notion of a “category with path objects”, as a slight strengthening of
Kenneth Brown's classical notion of a “category of fibrant objects”. We develop the basic …

The aim of this book is to introduce the basic aspects of the theory of∞-categories: a
homotopy theoretic variation on Category Theory, designed to implement the methods of …