2-fold complete Segal operads

TL;DR

This paper extends the concept of $$-operads to the $(,2)$-categorical setting using Cartesian 2-fibrations and Barwick's operator categories, enabling the study of 2-rings in higher category theory.

Contribution

It introduces Cartesian 2-fibrations and extends $$-operads to the $(,2)$-categorical context, facilitating the study of 2-rings in higher categories.

Findings

Defined Cartesian 2-fibrations and proved a weak straightening analogue.

Extended $$-operads to the $(,2)$-categorical setting.

Applied the framework to define and analyze 2-rings in $$-categories.

Abstract

We define the notion of Cartesian 2-fibrations, and prove a weak analogue of straightening. Using Barwick's notion of operator categories and the notion of a Cartesian 2-fibration, we extend the notion of $\infty$-operads to the $(\infty,2)$-categorical context. These $(\infty,2)$-operads are used to define and study 2-rings in the $\infty$-categorical context.