No, högskola teaches the same level as university. Only institutions of education with a sufficient degree of own research may however be styled universities. I'm not sure as to what tertiary education means, but after brushing through its wikipedia article, I'd say maybe. Högskola/universitet refers to tertiary or higher education.
Swedish school is divided into grundskola years 1-9, gymnasium years 10-12 and then högskola/universitet from then on.
So is folkhögskola like a community college? (In case you don't know what that is, it's basically a college were you go to only earn a 2 year degree. They funded by the state so their normally cheaper but they don't usually offer as comprehensive a degree program as a college/university)
Former exchange student here, there seems to be a bit of confusion here, but a högskola is similar to a community college or vocational school, and a folkhögskola is for vocation, arts, etc. And these days also offer alot of courses and programs for swedish culture and language for immigrants and new residents. For example, I went to a wonderful language camp at a place called Ädelförs Folkshögskola which also seemed to offer things like elder care. So its almost like a combination college and community center, at least in this instance.