Formal category theory in $\infty$-equipments I

Abstract: We introduce a generalization of proarrow equipments to the $\infty$-categorical setting, termed $\infty$-equipments. These equipments take the form of particular double $\infty$-categories that support a notion of internal formal higher category theory. We will explore several examples of $\infty$-equipments. In particular, we will study the prototypical example given by the $\infty$-equipment of $\infty$-categories, and construct $\infty$-equipments of $\infty$-categories internal to general $\infty$-toposes as well. The final objective of this article is to study the fundamental notions of formal higher category theory within an arbitrary $\infty$-equipment. To this end, we not only follow the classical approach but also expand upon it.