Formal category theory in $\infty$-equipments II: Lax functors, monoidality and fibrations

TL;DR

This paper develops the theory of $ abla$-equipments, focusing on lax functors, monoidal structures, and fibrations, to establish a uniform framework for internal $ abla$-category theory.

Contribution

It introduces a formalism for $ abla$-equipments that unifies various aspects of $ abla$-category theory, including lax functors and fibrations, in a synthetic setting.

Findings

Framework for $ abla$-equipments developed.

Internal $ abla$-category theory foundations established.

Formalism unifies different generalizations of $ abla$-categories.

Abstract

We study the framework of $\infty$-equipments which is designed to produce well-behaved theories for different generalizations of $\infty$-categories in a synthetic and uniform fashion. We consider notions of (lax) functors between these equipments, closed monoidal structures on these equipments, and fibrations internal to these equipments. As a main application, we will demonstrate that the foundations of internal $\infty$-category theory can be readily obtained using this formalism.