Peter was not bishop of Rome.

There is no contemporary document that says that Peter was bishop of Rome, still less, potatoes, simply because it did not happen. Irenaeus, bishop of Lyon (178-200), wrote about the year 180, a work to refute Gnosticism. It included the oldest list of Roman bishops is preserved. In all there were the first twelve to time. Peter's name does not appear.
The first is Lino, successor and calls it the "founding apostles" in the plural, and there is no mention of the Apostle Peter in particular about it.  
What is this Irenaeus writes:

"The blessed apostles founders Lino transmitted to episcopal ministry continues Lino Irenaeus mentions that Paul's letters to Timothy. Anacleto followed. And after this, ranking third after the apostles, Clement gets the episcopal ministry, which also saw the blessed apostles personally, and frequented their treatment. As a revolt broke out under him not least among the brethren at Corinth, the church sent a letter to the Corinthians. "

Note that in this paragraph from a man of faith Century III say interesting things: First, it was not an apostle, Peter, who call it the presbytery transmitted by succession Lino, but all the "founding apostles." Second, as to Lino, Anacleto, and even Clement, all of them were treated equally with the "blessed apostles," ie, there was mention of someone special exalted. Third, when the revolt mentioned in Corinth, the Corinthians not called faithful, but brothers, ie makes them the same height as Clement and himself. Fourth, and not least important, is not saying that Clement as Bishop of Rome he wrote to the Corinthians, but "the church wrote to the Corinthians," that is, the church in Rome, writes to the church at Corinth, ie treated as equals.
Returning to Peter, Eusebius of Caesarea, the author of the "Ecclesiastical History", never mentions him as bishop of Rome. She could not, because Peter never was. As usual more or less generalized, before V century, the bishops of every city, loved and appreciated by Christians, they were called "potatoes" as a pet name, a title not as hierarchical as understood today day, and even less as vicars of Christ. The latter neither had crossed his mind to those men.
Habemus papam (We Have a Pope)

However, from the emperor Constantine (fourth century), things went wrong, and began to notice more and more the difference between two social classes: the clergy and the laity. Neither one nor the other taught the Lord Jesus nor his apostles (see 1 Peter 2: 4-10). Over time, the nickname "Papa" became a title, and was given to the bishop was politically more important of the Empire, the bishop of the city of Rome, at the time, Siricio in the late fourth century. This simply was because Rome was the capital of the Empire. That designation was agreed at the Council of Toledo that year, although so far posed no exclusivity, it came much later, in 1073, by the imposition of Gregory VII. However, the pope, in that year, by decree prohibits named "Papa" to anyone other than himself.
So we find that it is not until the Middle Ages when he finally understood the pope of Rome Pope exclusively, and by imposing a determined Roman pope himself. Write Anton Casariego very aptly:

"In times of Christianity was the beginning ... Hebrew tradition inherited from the apostles. Then ... abandoned this principle and begin instituting the separation between laity and priests (consecration theory). This group is further divided into categories, and which is strengthened the power of the bishops, who become head of a community or church, as successors of the apostles, so to that address ... (Above), a monarchical episcopate happens influenced by Romanism. The hierarchy is becoming the repository of the doctrine of salvation, and believers see their role reduced to that of faithful. Moreover, during the first three centuries, the church functioned as a federation of local churches united by a common faith, but relatively free and autonomous in their area. "