| dc.description.abstract | The goal of this thesis is to lay the foundations for a theory of ∞-categories internal to an ∞-topos B. Our model for such internal ∞-categories is based on the notion of a complete Segal object, but can equivalently be described by sheaves of ∞-categories on B. After setting up the basic framework of this theory, we study internal presheaf ∞-categories: we prove a version of Yoneda’s lemma in this context, and we show that internal presheaf ∞-categories can be characterized by a universal property: they provide a model for free cocompletions by internal colimits. As a prerequisite for the latter result, we develop the theory of adjunctions, limits and colimits and Kan extensions for internal ∞-categories.
We then move on to the study of accessibility and presentability of internal ∞-categories, which we use to define and study internal ∞-topoi. One of our main results is a correspondence between these internal ∞-topoi and geometric morphisms into the base ∞-topos B. We use this result to study relative aspects in higher topos theory from an internal point of view: we provide a formula for general pullbacks of ∞-topoi, and we characterise smooth and proper geometric morphisms in terms of properties of the associated internal ∞-topoi. We furthermore use the latter result to compare the notions of smooth and proper maps in topology and in higher topos theory. | en_US |
