Final answer:
The leaders of the Catholic Church, particularly Pope Innocent IV, were responsible for burning Albigensian heretics at the stake. Pope Gregory IX established the Inquisition to centralize the persecution of heresy and impose religious homogeneity on Western Europe.
Step-by-step explanation:
The burning of Albigensian heretics at the stake was carried out by the leaders of the Catholic Church, particularly Pope Innocent IV who authorized the use of torture on suspected heretics. The Inquisition was established by Pope Gregory IX in the 1230s to centralize the persecution of heresy. It aimed to impose religious homogeneity on Western Europe and used torture and execution to achieve this goal.