A "police forge hat" refers to a police forage cap, a formal peaked hat that is part of the dress uniform for many police services in Commonwealth countries. The term "forge" is a result of a common mispronunciation of "forage".
About the forage cap
Military origin: The forage cap originated as a comfortable, soft cap for military personnel to wear while performing everyday duties or "foraging" for supplies.
Police adoption: In Canada, police forces adopted a version of the forage cap, but dissatisfaction with its comfort and lack of sun protection has led to many forces switching to more practical baseball caps for regular duty.
Current use in policing: While no longer the standard for daily patrol, forage caps are still worn for more formal occasions. In Canada, you might see them as part of the ceremonial or administrative dress uniform for events like parades, funerals, or memorial services.
Ventilated Mesh Band, Removable Band, Black Shiny Peak, Single Black Strap