A Liberty Cap is the same as a Phrygian cap, like those worn by the Smurfs.
Originally worn by Mithraic initiates and the emancipated former slaves of the Roman Empire, it has been a symbol of revolution and freedom ever since.
It was used in ther American Revolution, and can even be found on the Seal of the US Senate (crowning and overruling the two fasces featured at the bottom of the seal):
A liberty cap is also featured on the seal of the US Department of the Army:
It’s also a type of psychedelic mushroom.
From Fulcanelli’s ‘Le Mystere des Cathedrales’:
The Phrygian cap, which was worn by the sans-culottes and acted as a sort of protective talisman in the midst of the revolutionary slaughter, was a distinctive sign of the Initiates. In the analysis, which he made of a work of Lombard (de Langres) entitled ‘Histoire des Jacobins, depuis 1789 jusqu’a ce jour, ou Etat de l’Europe en novembre 1820′ (Paris 1820), the scholar Pierre Dujols writes that for the grade of the Epopt (in the Eleusian Mysteries) the new member was asked whether he felt in himself the strength, the will and the devotion necessary for him to set his hand to the GREAT WORK. Then a red cap was put on his head, while this formula was pronounced: ‘Cover yourself with this cap, it is worth more than a king’s crown.’ Few suspected that this hat, called liberia in the Mithraic rituals and which formerly denoted the freed slaves, was a masonic symbol and the supreme mark of Initiation. It is not therefore surprising to see it represented on our coins and our public monuments.
“The Alchemist”, from Notre Dame cathedral in Paris, wearing a liberty cap.