This book develops abstract homotopy theory from the categorical perspective with a
particular focus on examples. Part I discusses two competing perspectives by which one …

The language of∞-categories provides an insightful new way of expressing many results in
higher-dimensional mathematics but can be challenging for the uninitiated. To explain what …

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 define and develop two-level type theory (2LTT), a version of Martin-Löf type theory
which combines two different type theories. We refer to them as the 'inner'and the 'outer'type …

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy
coherent adjunction and characterise its n-arrows using a graphical calculus that we …

Directed type theory is an analogue of homotopy type theory where types represent
categories, generalizing groupoids. A bisimplicial approach to directed type theory …

We use the terms∞-categories and∞-functors to mean the objects and morphisms in an∞-
cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched …

In this paper we re-develop the foundations of the category theory of quasi-categories (also
called∞-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi …

We develop further the theory of weak factorization systems and algebraic weak factorization
systems. In particular, we give a method for constructing (algebraic) weak factorization …

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 …