Jaco Ruit: A double ∞-categorical approach to formal ∞-category theory

24th of October, 2024 ——— Formal ∞-category theory starts with the observation that there are many variants of ∞-category theory, for example, enriched ∞-categories, internal ∞-categories, and monoidal ∞-categories, which come with specialized notions of adjunctions, point-wise Kan extensions, and so on. It is natural to ask whether one can give a uniform and synthetic treatment of the foundational concepts and theorems for these different flavors of ∞-category theory. In this talk, we propose an extension of the ideas from formal (strict) category theory of Street-Walters, Wood, Verity, and Shulman, to the ∞-categorical context, and give a leisurely introduction to the theory of ∞-equipments. These ∞-equipments are certain double ∞-categories in which many concepts of category theory may be developed and expressed (using only the double categorical structure). We will present an overview and highlight some of these aspects. Now, since this approach yields category theories for the objects of these ∞-equipments, developing a category theory for a flavor of ∞-categories is a question of constructing the right suitable ambient ∞-equipment. Throughout the talk, we discuss some of these examples of ∞-equipments.