As I stated in an answer to an entirely different question, the Catholic Church distinguishes between three types of beliefs which Catholics must hold. In 1998, the Congregation for the Doctrine of the Faith (CDF) issued a document stating and clarifying these types of beliefs, and noting the consequences for failing to assert them.
Briefly, it notes that the various statements about Peter and the papacy are formal dogmas, divinely revealed truths officially declared by the Church:
To the truths of the first paragraph [formal dogmas] belong ... the doctrine on the primacy and infallibility of the Roman Pontiff ...
And these doctrines of course rest on the position of Peter among the apostles, and the tradition that Rome was the see of Peter. To deny these or any formal dogmas is to commit heresy.
The CDF document provides footnotes for each item it lists as a formal dogma. For the beliefs about the papacy, the footnote reads
Cf. DS 3059–3075.
DS is a commonly-used reference for dogmas of the Church; it stands for the Denzinger-Schönmetzer Enchiridion symbolorum, definitionum et declarationum de rebus fidei et morum (Handbook of creeds, definitions, and declarations on matters of faith and morals), a reference book periodically updated to reflect statements of faith to which Catholics are expected to give some degree of assent. Unfortunately, I don't have access to the modern version of DS, only to an older version, with very different numbering, with texts solely in Latin or Greek.
From this version, however, some material can be drawn: the most apposite is probably item 163, an excerpt from a letter attributed to Pope Gelasius I in 495 AD (the Decretal on books to be received, and not to be received):
Addita est etiam societas beatissimi Pauli Apostoli, vasis electionis, qui non diverso, sicut haeretici garriunt, sed uno tempore, uno eodemque die glorioso morte cum Petro in urbe Roma sub Caesare Nerone agonizans coronatus est; et pariter supradictam sanctam Romanam Ecclesiam Christo Domino consecrarunt aliisque omnibus urbibus in universo mundo sua praesentia atque venerando triumpho praetulerunt.
(Note: It appears that this letter is now generally considered to have been written by an anonymous scholar, not at the end of the 5th century but perhaps 20 to 50 years later. Nonetheless it is of interest as representing an accepted view of the Church at that time, as well as a statement which all Catholics are required to believe.)
I have found no official translation of this portion of the Decretal; following is my own:
In addition [to the presence of Peter in Rome] there was the fellowship of the most blessed Apostle Paul, the vessel of [God's] choice, who (not separately, as heretics gossip, but at the same time, even on the same day) as part of his struggle was crowned with a glorious death with Peter in the city of Rome, under Nero Caesar; and the two, equally, consecrated the before-said holy Roman Church to Christ the Lord, and offered it up before all other cities in the whole world, by reason of their presence and their revered triumph.
This is one of the bases for the Church's official declaration that Peter was in Rome, and died in Rome.
For the explicit statement that Peter was the bishop of Rome, or a bishop in Rome, I can find no direct sources. The official statements of the Church, however, state that the bishops of Rome are the successors of Peter, which of course could only be the case if Peter was indeed a bishop:
Si quis ergo dixerit, non esse ex ipsius Christi Domini institutione seu iure divino, ut beatus PETRUS in primatu super universum Ecclesiam habeat perpetuos successores; aut Romanum Pontificem non esse beati PETRI in eodem primatu successorem: anathema sit.
[My translation, as I cannot seem to find an authorized one] And so, if anyone says that it was not by divine law, and the institution of Christ the Lord Himself, that the blessed PETER has perpetual successors as primate over the whole Church, or that the Roman Pontiff is not the successor of the blessed PETER in that same primacy: let him be anathema.
(Pastor Aeternus, Documents of the First Vatican Council)
Thus, from these excerpts, we can deduce the following:
- The Church teaches as dogma that Peter was in Rome, and died in Rome.
- The Church teaches as dogma that Peter was the bishop of Rome.
- Specifically, the Church teaches as dogma that the scope of Peter's episcopacy was by divine institution set over the whole Church in all the world—this is, if you like, what "bishop" means in Peter's case.
- There is no mention in Church dogma of a college of presbyters ruling at Rome. As far as Church dogma is concerned, this is false.
- If a Catholic derogates from (disagrees with) any of these points, he or she commits heresy.