You can see this in the Patriarchs, with Jerusalem, Antioch, and Rome as places Peter was at. Leaving Jerusalem for Antioch, and then Antioch for Rome. As to the Patriarchate of Alexandria, that was founded by Mark, who was Peter's secretary and commited Peter's oral gospel into writing. The Patriarchates are all significantly Petrine. At least the original four.
St. Peter may have dictated his
First Epistle of Peter from Antioch
c. 45-53 AD, after his Liberation from Prison (Acts 12) in
43-44 AD.
Church Tradition ascribes St. Peter 7 years as Bishop of Antioch, until Evodius was elevated in
53 AD. St. Peter residing in Antioch from
45-52 AD or so would accord well with Acts and St. Paul:
He was in Jerusalem again for the famine visit of Paul and Barnabas in 46 and for the Apostolic Council of 49. He visited Antioch (Galatians 2:11) and had associations with the churches in northern Turkey (I Peter 1:1). In 54 Paul can speak of Peter 'leading around a wife', presumably moving from place to place in missionary work (I Corinthians 9:5).
Luke was nearby during Paul's two-year stay at or near Caesarea about 57-59, and this would make a very suitable period for the final preparation of his material for publishing. Is it conceivable that Mark was written [from Rome] even earlier?
Sending the letter, from Antioch, up north to Asia Minor, to the same areas to which St. Paul also wrote around that time, he very reasonably also included word from major Christian figures from "Babylon", i.e. Jerusalem "where their Lord was Crucified" (Rev 11). The "exiles of the dispersion" would obviously have been very curious to know about the happenings there. As just observed, St. Peter frequently returned to Jerusalem while he was Bishop of Antioch.
It makes much more sense, that St. Peter wrote to Asia Minor from Antioch, than from Rome.
"Babylon" (Rev 11) signifies spiritually-oppressive 1st century
Jerusalem, then still dominating Christian communities. It
never meant "Rome". St. Peter wrote to Asia Minor, from Antioch, sharing with them news from Jerusalem as well.
Note that St. Peter plausibly first reach Rome around 54-55 AD, perhaps stopping in Corinth on the way. Mark then wrote his Gospel right around the time that Luke was writing his (and using Mark as a source). Paul was also writing prolifically at the time. And St. James, Bishop of Jerusalem, wrote his Epistle around 60 AD also. Plausibly, the years leading up to 60 AD were very prolific for early Christian literature. Perhaps one reason why the High Priest Ananias viewed James as such a threat, so as to martyr him in 62 AD?