I believe you meant to say the church eradicated the Cathar heresy. The essential characteristic of the Catharist faith was Dualism, i.e. the belief in a good and an evil principle, of whom the former created the invisible and spiritual universe, while the latter was the author of the material world. The Catharist system was a simultaneous attack on the Catholic Church and the then existing State. The Church was directly assailed in its doctrine and hierarchy. The denial of the value of oaths, and the suppression, at least in theory, of the right to punish, undermined the basis of the Christian State. But the worst danger was that the triumph of the heretical principles meant the extinction of the human race. This annihilation was the direct consequence of the Catharist doctrine, that all intercourse between the sexes ought to be avoided and that suicide or the Endura, under certain circumstances, is not only lawful but commendable. The assertion of some writers, like Charles Molinier, that Catholic and Catharist teaching respecting marriage are identical, is an erroneous interpretation of Catholic doctrine and practice. Among Catholics, the priest is forbidden to marry, but the faithful can merit eternal happiness in the married state. For the Cathari, no salvation was possible without previous renunciation of marriage. Mr. H.C. Lea, who cannot be suspected of partiality towards the Catholic Church, writes: "However much we may deprecate the means used for its (Catharism) suppression and commiserate those who suffered for conscience' sake, we cannot but admit that the cause of orthodoxy was in this case the cause of progress and civilization. Had Catharism become dominant, or even had it been allowed to exist on equal terms, its influence could not have failed to prove disastrous." source
See also, The Cathar Heresy
The question is did the Catholic church sanction the killing of Cathars? Unless you believe that the Church was right in killing them and that it was part of eradication of the heresy?