The Catholic Catechism makes the relationship of the church dependant solely upon a bishop of bishops... the Pope.
881: The Lord made Simon alone, whom he named Peter, the "rock" of his Church. He gave him the keys of his Church and instituted him shepherd of the whole flock. "The office of binding and loosing which was given to Peter was also assigned to the college of apostles united to its head." This pastoral office of Peter and the other apostles belongs to the Church's very foundation and is continued by the bishops under the primacy of the Pope.http://www.scborromeo.org/ccc/para/881.htm
"The claim of the Roman pope to have universal ordinary jurisdiction over the worldwide Church is dependent upon that the notion that the universal Church — rather than the diocese — is the Catholic Church."
Carlton, C., (1999) “The Truth: What Every Roman Catholic Should Know about the Orthodox Church”, (Regina Orthodox Press; Salisbury, MA), p120.
The Catholic notion is at odds with the Orthodox idea that all bishops are equal, and that all churches are complete. We can examine the Church Fathers to see how they used the term ‘catholic’.
The first time the term is used in a Christian sense it refers to the local church (remember here that the Orthodox approach is that the church always held that each bishop is equal in The Epistle of Ignatius to the Smyrnaeans "Chapter VIII.-Let Nothing Be Done Without the Bishop.
See that ye all follow the bishop, even as Jesus Christ does the Father, and the presbytery as ye would the apostles; and reverence the deacons, as being the institution of God. Let no man do anything connected with the Church without the bishop. Let that be deemed a proper Eucharist, which is [administered] either by the bishop, or by one to whom he has entrusted it. Wherever the bishop shall appear
, there let the multitude [of the people] also be; even as, wherever Jesus Christ is, there is the Catholic Church
. It is not lawful without the bishop either to baptize or to celebrate a love-feast; but whatsoever he shall approve of, that is also pleasing to God, so that everything that is done may be secure and valid"St. Ignatius “The Epistle to the Smyrnaeans" Chapter VIII.-Let Nothing Be Done Without the Bishop quoted at http://www.ccel.org/fathers2/ANF-01/anf01-21.htm#P2123_357530
Further, the Catholic notion of the Trinity makes the Holy Spirit a 'junior partner', dependant on the other two.