So, I recently learned that the Cathedral Church of the Pope isn't, as I would have thought, St. Peter's Basilica, but rather is the Archbasilica of St. John Lateran in Rome. This even though most/all of the Pope's official services take place at St. Peter's. So how/when/why did this happen, and why has the Pope's seat remained at St. John Lateran instead of moving? Have a Pope ever tried to "officially" move to St. Peter's?
They constructed five basilicas to represent the five traditional Patriarchs. St. John Lateran was the Pope in Rome, St. Peter's was assigned to the Patriarch of Constantinople, St. Paul's was emblematic of Patriarch of Alexandria, St. Mary Major referred to the Patriarch of Antioch, and St. Lawrence-outside-the-Walls is specially attributed to the Patriarch of Jerusalem. So symbolically a move to St. Peter's would be problematic. Practically, ever since the Pope moved into the Vatican there really hasn't been a practical reason to change anything.
There's also the question of tradition. St. John Lateran was the site of a church dating back to at least 311 and might be the oldest church site in Rome. It is and remains the Cathedral of Rome, despite no longer being the residence of the Pope after the Avignonese captivity and a couple of destructive fires during 14th Century.
From the Catholic Encyclopedia entries on St. John Lateran and Basilica.