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Word "kolpak" is widely used in Russian not only for Turkic kolpak, but for many types of caps that look like it. For others it is shapka, yes, and also shlyapa.

I often see Kirghiz people in their national white kolpak here. Other ethnic groups rarely wear their hats.

Also, do someone did any research about usage of hats in modern world? In photos and movies from past (especially more than 50 years ago), everyone had hat. As far as I know, going outside without hat may even be  considered strange in past. But nowadays (at least in European-like cultures) people rarely wear hats. Why this happened? Of course fashion matters, but it is pretty radical change.