A view of (right) the entrance to the complex, and the "tekke" part (centre). It felt like visiting a house, but one where the door stood always open and visitors were welcome. I never saw a soul.

The building which houses the tombs of Şeyh Edebali and his relatives can be characterised as a ‘zaviye’.
The actual construction is a 20th century restoration in 19th century late Ottoman (civil) style; the original zaviye was probably quite different.

Zaviye = a building designed specifically for gatherings of a Sufi brotherhood and was a place for spiritual retreat and character reformation. These places (‘hanegah’ or ‘zaviye’) often served as hospices for Sufi travelers and Islamic students. Important (and large) ‘zaviye’ are often referred to as ‘Tekke’.

Correspondent: J.M.Criel, Antwerpen.
Sources: Wikipedia & Personal visits (1997 – 1998).