Formal category theory in $\infty$-equipments I

TL;DR

This paper extends the concept of proarrow equipments to the $ abla$-categorical setting, introducing $ abla$-equipments that facilitate internal higher category theory, with applications to colimits and Kan extensions.

Contribution

It defines $ abla$-equipments as a new framework for internal higher category theory, generalizing strict category theory concepts to the $ abla$-categorical context.

Findings

Introduces the concept of $ abla$-equipments as double $ abla$-categories.

Provides examples including $ abla$-categories and internal $ abla$-categories.

Lays groundwork for studying colimits and Kan extensions in $ abla$-equipments.

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.