Canterbury cap
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The Canterbury cap is a square cloth hat with sharp corners found in the Anglican communion, similar to the Counter-Reformation's biretta, the notable exception being that a Canterbury cap has four ridges, compared to the biretta's three. It is also soft and foldable, whereas the biretta is rigid. The Canterbury cap is the medieval birettum, descended from the ancient pileus headcovering. It is sometimes called the "catercap."
It should be noted that a similar cap called the Oxford soft cap is worn today as part of academic dress by some women undergraduates at Oxford University instead of the mortarboard. It has a flap at the back which is held up with buttons unlike the Canterbury cap. The Tudor bonnet is also a similar academic cap worn by a person who holds a doctorate.
In the Anglican Church, clergy are entitled to wear the cap. Priests and deacons wear a black version of the headgear; bishops purple.