Lectures on Higher Topos Theory

Abstract

This course surveys the theory of \(\infty\)-topoi. After introducing the basic notions (descent, univalence, modalities, ...), we discuss general constructions (limits and colimits, congruences, parametrization, Grothendieck topologies) and a range of examples, including Goodwillie calculus, finitary functors, geometric structures and schemes in Lurie's framework, and analytic stacks in the sense of Clausen–Scholze.

Preface

These are lecture notes based on the course Higher Topos Theory taught by Marc Hoyois at the University of Regensburg in the winter term 2025/2026, written by Bastiaan Cnossen. The presentation is at times different from the original lectures. The notations and terminology follow the preferred conventions of the note-taker, which usually match those of the lecturer. The following are exceptions:

• We say `animae' for the plural of `anima'.

• The presheaf category of a category \(C\) is denoted \(\PSh(C)\) rather than \(P(C)\).

• Given topoi \(S\) and \(T\), we refer to a left exact colimit-preserving functor \(f^*\colon S \to T\) as a morphism of logoi rather than a geometric morphism. We refer to the right adjoint \(f_*\colon T \to S\) as a morphism of topoi. We write \(\Logos\) and \(\Topos\) for the resulting (non-full) subcategories of \(\Cat\), so that \(\Logos \simeq \Topos\catop\). This terminology is due to Anel and Joyal (2021).