Substitudes, Bousfield localization, higher braided operads, and Baez-Dolan stabilization

TL;DR

This paper develops a unified framework using substitudes, Bousfield localization, and homotopical techniques to prove Baez-Dolan Stabilization Theorems across various models of higher categories, clarifying their connections to higher braided operads.

Contribution

It introduces a general stabilization machinery applicable to multiple models of higher categories, including new proofs for several recent stabilization theorems.

Findings

Proved Baez-Dolan Stabilization Theorems for Tamsamani weak n-categories

Extended stabilization results to higher Segal categories and n-quasi-categories

Connected stabilization machinery to higher braided operads

Abstract

This short note reports on joint work with Michael Batanin towards a general machine for proving Baez-Dolan Stabilization Theorems for various models of higher categories, based on substitudes, Bousfield localization, and homotopical Beck-Chevalley squares. I provide a road map to our recent papers, and include new results proving Baez-Dolan Stabilization Theorems for Tamsamani weak $n$-categories, higher Segal categories, Ara's $n$-quasi-categories, and cartesian models of Segal and complete Segal objects due to Bergner and Rezk. I also attempt to clarify the connection to higher braided operads, and our more general stabilization machinery.