On multiple ∞-categories and formal category theory via ∞-equipments

Abstract

This thesis expands our knowledge of multiple infinity categories, with a particular focus on double infinity categories. Its central objective is to develop a double infinity categorical framework for formal category theory via so-called infinity equipments, a higher categorical analog of the proarrow equipments from classical category theory. This framework is designed to provide the foundations of different generalizations of infinity categories in a uniform and systematic way. As an example, we demonstrate how the foundations of internal infinity categories may be readily recovered within this framework.