  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.
