A bycoket is an ancient cap of state worn by the kings of England, the upper part of which was in the form of a double crown. The word is formed from an Old French term for a small castle at the top of a hill.
The word abacot has been mistakenly defined as this item since the mid-16th century.
A 16th century high crowned Italian hat, wide brimmed, peaked at the front and turned up at the back also goes by this name.