As Major said, purgatory is not in the Bible. I believe that the idea was taken from something in the Apocrypha, but I can't tell you where. What you will find in the Bible is the concept of a temporary "holding place" for the unrighteous dead (see the story of Lazarus and the rich man, Luke 16:19-31). This is not the "lake of fire", their ultimate destiny, but a place of torment where they are held until the time of the Great White Throne judgment. It is possible that someone developed this into the idea of purgatory, but the Biblical concept is quite different. Purgatory is supposedly a place where people are punished for any sins for which they have not received "absolution" (unnecessary, because Jesus has already taken ALL our punishment) and from which they will be released once they are "cleansed." The Biblical concept is of a place where the unrighteous dead are held until the day of judgment, at which time they will be thrown into the lake of fire (no "get out of jail" card.)
Again as Major said, the "catching away" is in the Bible. What is NOT in the Bible is the timing of the "catching away." For 1800 years the church taught that this event would occur at the return of Christ, immediately before He establishes His millennial Kingdom. In 1830 a young woman in Scotland gave a "prophecy" that said that some of God's people would be "caught away" before the 7 year tribulation period. That was then developed into the teaching of "the rapture" which says that the whole church will be caught away pre-trib. That teaching has spread through the church, but some of us still reject it and hold to the traditional teaching that God's people will be caught up to meet Christ as He returns to establish the millennial Kingdom.