Caps of various forms occur in heraldry and are used as marks of dignity, as distinct from crowns or coronets, which are generally borne with caps as below. Those more properly termed caps or hats, are as follow:

Cap in crown is the velvet cap which covers the head within the rim or circle of the crown. The caps of royal crowns are generally purple, whilst those of princes, princesses and peers crimson.
Cap of maintenance or of State.
Cardinal’s cap or hat is scarlet. Archbishops and bishops bear hats over their arms similar to the cardinals, but green and having only four rows of tassels. Abbots use the same, only black, with three rows of tassels. See Insignia of Office

Cap-copped or hat-copped is turned up in the form of a chapeau. This could be another term for the chapeau de fer (iron cap) which is an iron hat with a small brim, resembling the morion used by foot soldiers. Cap-copped was in fact sometimes by early writers called a morion, although the morion is better known as a form of helmet.

Cap-corner or four-cornered cap is that used by deans, doctors and all graduates and under-graduates of the universities, and might be placed over the arms of deaneries in the manner that the mitre surmounts Episcopal sees.

Cap three-cornered, as distinct from the 18th century tricorne or three cornered hat, is another form of academic cap.
Cap hood or old man’s cap is described by Randle Holme as turned up and open before, and turned down again below the neck and sides of the face; such he mentions as the arms of Winter.

infula
Cap tank, tanke or infula is a deep round cap with strings to tie under the chin used by the freed Roman slaves. It later became one of the symbols of the French revolution.
Note. The difference between a tank and an infula is that the first is of one width and round at the top, and the latter rises to a kind of point.