Is there a somewhat established term for one culture (not state) having a comparatively large influence over surrounding/neighboring cultures?
I’m not talking about the volitional political or military influence or control of a country, where the terms hegemony or soft power (“when one country gets other countries to want what it wants”) come to mind. Rather, I mean when surrounding cultures voluntarily imitate, adapt, import and include the fashion, architecture, technology and ways of thinking and writing of the other culture, even if this is not desired or especially promoted by said culture. (Of course, military power may very well preceede the phase I’m looking for. And in turn the phase I’m looking for may very well preceede becoming a soft power. But neither are necessary.)
Examples in the West are probably the Roman Empire (vis-à-vis the Germanics), the Catholic Church and later the French and then British high societies, as well as the American Way of Life after WW2. Or the influence of Chinese culture over surrounding peoples.