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 …

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 propose foundations for a synthetic theory of $(\infty, 1) $-categories within homotopy
type theory. We axiomatize a directed interval type, then define higher simplices from it and …

To a “stable homotopy theory”(a presentable, symmetric monoidal stable∞-category), we
naturally associate a category of finite étale algebra objects and, using Grothendieck's …

We construct higher categories of iterated spans, possibly equipped with extra structure in
the form of higher-categorical local systems, and classify their fully dualizable objects. By the …

We provide new $\infty $-categorical models for unstable and stable global homotopy
theory. We use the notion of partially lax limits to formalize the idea that a global object is a …

We develop the theory of semi-orthogonal decompositions and spherical functors in the
framework of stable∞-categories. We study the relative Waldhausen S-construction S∙(F) of …

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 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 …