Formal category theory in $\infty$-equipments I

Abstract

We generalize proarrow equipments from strict category theory to the $\infty$-categorical setting, introducing the concept of $\infty$-equipments. These are specific double $\infty$-categories that support an internal higher category theory. This paper explores several examples of $\infty$-equipments, including the prototypical example of the $\infty$-equipment of $\infty$-categories and the more general $\infty$-equipments of internal $\infty$-categories. The ultimate objective of this article is to study the basic concepts of category theory within an arbitrary $\infty$-equipment, such as colimits and Kan extensions.

51 pages; v2: submitted version of Sept 2023; v3: major revision round