In the episode A Good Man Goes To War (season 6, episode 7), there is a mention of The Papal Mainframe.
What or who is this?
We finally see the actual Papal Mainframe in the latest Christmas Special, "the Time of the Doctor." It is an incredibly massive computer and the data-base it contains, much like "the Library." It safeguards the security of information "in this life and the next" across political lines for the Known Universe at some point, it is the organization encompassing the "Church of the Silence" and all of it's sects, including Madame Kovarian's branch that used time-travel and assassins, including an unrememberable T.A.R.D.I.S. bomber and a hybrid sociopath raised from childhood, to attempt to prevent the Doctor from ever reaching Trenzalore. The Mainframe itself seems to be the big blocky ship that can project massive holograms of the Mother Superioress' (the head-nun-in-charge, sort of a pope, a tough woman the Doctor has a history with, something of a sociopath ) head, though that ship, massive as it is, is never actually said to be the whole of the actual mainframe Itself. It turns out that we have been seeing many reference to things that are Papal Mainframe-related for quite some time.
It sounds like the Catholic Church (or whatever they’re calling themselves at that point, or whatever Catholic Church proxy this is) has replaced the Pope with a mainframe, that is, a computer. “Mainframe” implies a huge, room-sized or larger computer. I’m reasonably sure no more information is given.
The only clue we are given in 'A Good Man Goes to War' is that the Papal mainframe is female (in terms of some aspect of her personality, origin or honorific title, presumably). We are also shown that she retains considerable clout within the militarised structure of the Clerics. Thus she is able to give Colonel Manton special permission to unveil the identity of the Hooded Monks.
In terms of real world context this is a bit of a cheeky dig at the present day Catholic church. While other Christian denominations have made some moves towards introducing or considering female priests and bishops the Catholic church has remained conservatively opposed. Apparently in the future of Doctor Who this stance has altered to the extent that the Church allows its (real or ceremonial) leader to be both female and non-biological.