The Inquisition aimed to monitor personal beliefs, and the Crusades had various objectives and causes.
The Inquisition was established to inquire about the personal morals and beliefs of individuals who were told to convert to Catholicism or leave Spain. The Crusades had multiple objectives including recapturing Jerusalem and aiding the Catholic Church's efforts against heresy. The sixth crusade in 1228 was initiated by Frederick II. Two causes that led to the Crusades were territorial greed and the blossoming of the Cathars contributing to regional unrest.
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