The Church of Rome (and its bishop) might have been coming to prominence as early as the early second century, bt that doesn't mean that this was a legitimate authority accepted by all. Polycarp and Polycrates (the former supposedly being directly connected to John!) opposed the Roman bishop over the issue of Easter. (Rome following a "tradition" of Sunday, and the two Asia Minor leaders favoring the continuation of the Passover date, while no longer mandated as a law, still had some sort of biblical basis over the tradition of Sunday, which was based on a generalization of references in the NT, and not any clear command).
And as others have said, "catholic" was a generic term in 110, not some sort of official institutional name.
And as others have said, "catholic" was a generic term in 110, not some sort of official institutional name.