Harley, please.
Scripture and the Fathers command prayers and oblations for the departed, and the Council of Trent (Sess. XXV, "De Purgatorio") in virtue of this tradition not only asserts the existence of purgatory, but adds "that the souls therein detained are aided by the suffrages of the faithful and principally by the acceptable sacrifice of the altar." That those on earth are still in communion with the souls in purgatory is the earliest Christian teaching, and that the living aid the dead by their prayers and works of satisfaction is clear from the tradition above alleged. That the Holy Sacrifice was offered for the departed was received Catholic Tradition even in the days of Tertullian and Cyprian, and that the souls of the dead, were aided particularly "while the sacred victim lay upon the altar" is the expression of Cyril of Jerusalem quoted above. Augustine (Serm.. clxii, n. 2) says that the "prayers and alms of the faithful, the Holy Sacrifice of the altar aid the faithful departed and move the Lord to deal with them in mercy and kindness, and," he adds, "this is the practice of the universal Church handed down by the Fathers." Whether our works of satisfaction performed on behalf of the dead avail purely out of God's benevolence and mercy, or whether God obliges himself in justice to accept our vicarious atonement, is not a settled question. Suarez thinks that the acceptance is one of justice, and alleges the common practice of the Church which joins together the living and the dead without any discrimination (De poenit., disp. xlviii, 6, n. 4). - New Advent on Purgatory
It does raise the question how does the Church knows who is in heaven and who is in purgatory? Did Pope John Paul go straight to heaven? Did he spend some time in purgatory? When does he become an "official" saint? Actually, I think the Mormons are a little easier to understand.