Just some info on how the early Church viewed Rome

(but remember it is special because of Peter, not because of the place.)
"With that church [of Rome], because of its superior origin, all the churches must agree, that is, all the faithful in the whole world, and it is in her that the faithful everywhere have maintained the apostolic tradition"
(Irenaeus)
"You cannot deny that you are aware that in the city of Rome the episcopal chair was given first to Peter; the chair in which Peter sat, the same who was head—that is why he is also called Cephas [‘Rock’]—of all the apostles; the one chair in which unity is maintained by all" (The Schism of the Donatists 2:2 [A.D. 367]).
(Optatus)
"Likewise it is decreed: . . . [W]e have considered that it ought to be announced that although all the Catholic churches spread abroad through the world comprise one bridal chamber of Christ, nevertheless, the holy Roman Church has been placed at the forefront not by the conciliar decisions of other churches, but has received the primacy by the evangelic voice of our Lord and Savior, who says: ‘You are Peter, and upon this rock I will build my Church, and the gates of hell will not prevail against it; and I will give to you the keys of the kingdom of heaven, and whatever you shall have bound on earth will be bound in heaven, and whatever you shall have loosed on earth shall be loosed in heaven’ [Matt. 16:18–19]. The first see, therefore, is that of Peter the apostle, that of the Roman Church, which has neither stain nor blemish nor anything like it.
"In addition to this, there is also the companionship of the vessel of election, the most blessed apostle Paul, who contended and was crowned with a glorious death along with Peter in the city of Rome in the time of Caesar Nero. . . . They equally consecrated the above-mentioned holy Roman Church to Christ the Lord; and by their own presence and by their venerable triumph they set it at the forefront over the others of all the cities of the whole world.
"The first see, therefore, is that of Peter the apostle, that of the Roman Church, which has neither stain nor blemish nor anything like it. The second see, however, is that at Alexandria, consecrated in behalf of blessed Peter by Mark, his disciple and an evangelist, who was sent to Egypt by the apostle Peter, where he preached the word of truth and finished his glorious martyrdom. The third honorable see, indeed, is that at Antioch, which belonged to the most blessed apostle Peter, where first he dwelt before he came to Rome and where the name Christians was first applied, as to a new people" (Decree of Damasus 3 [A.D. 382]).
"
f any bishop loses the judgment in some case [decided by his fellow bishops] and still believes that he has not a bad but a good case, in order that the case may be judged anew . . . let us honor the memory of the apostle Peter by having those who have given the judgment write to Julius, bishop of Rome, so that if it seem proper he may himself send arbiters and the judgment may be made again by the bishops of a neighboring province" (Canon 3 [A.D. 342]).
"[T]hey [the Novatian heretics] have not the succession of Peter, who hold not the chair of Peter, which they rend by wicked schism; and this, too, they do, wickedly denying that sins can be forgiven [by the sacrament of confession] even in the Church, whereas it was said to Peter: ‘I will give unto thee the keys of the kingdom of heaven, and whatsoever thou shalt bind on earth shall be bound also in heaven, and whatsoever thou shall loose on earth shall be loosed also in heaven’[Matt. 16:19]" (Penance 1:7:33 [A.D. 388]).
(Ambrose of Milan)
"Philip the presbyter and legate of the Apostolic See said: ‘There is no doubt, and in fact it has been known in all ages, that the holy and most blessed Peter, prince and head of the apostles, pillar of the faith, and foundation of the Catholic Church, received the keys of the kingdom from our Lord Jesus Christ, the Savior and Redeemer of the human race, and that to him was given the power of loosing and binding sins: who down even to today and forever both lives and judges in his successors. The holy and most blessed pope Celestine, according to due order, is his successor and holds his place, and us he sent to supply his place in this holy synod’" (Acts of the Council, session 3 [A.D. 431]).
"We exhort you in every respect, honorable brother, to heed obediently what has been written by the most blessed pope of the city of Rome, for blessed Peter, who lives and presides in his own see, provides the truth of faith to those who seek it. For we, by reason of our pursuit of peace and faith, cannot try cases on the faith without the consent of the bishop of Rome" (Letters 25:2 [A.D. 449]).
(Peter Chrysologus)