Going back and reading prior posts, I had to connect the dots. It sounds like of the seven Churches of Asia Minor in Revelation, people are making the claim that Thyatira is the Catholic Church. (And I think I remember hearing this argument back in 2010 when a friend of mine made this case--though it's a fuzzy memory).
Though this claim falls apart very easily. To begin , it is a very modern perspective with no basis in the book itself. Nowhere does the Scripture indicates that these seven churches represent seven future periods of Church history, especially since even if I were to grant that the Catholic Church is wrong for argument's sake, it wouldn't be the ONLY wrong Church -- it would be weird to say "they're all right except for this one Church" when they all have different teachings.
Second, this is a pretty arbitrary division of Church history. It could be divided many other ways -- taking into account many more elements and movements than the limited ones presented here.
Moving forward, this is entering into the belief that we are now living in the last days (since Laodicea represents are current era). But that's not sound because we have no idea when Christ will return. God didn't reveal it to us because it is not for us to know (Matthew 24:36; Acts 1:7) -- But we do know that Christians in every era, since the time of Christ, have thought their era to be end times, and they were obviously wrong. Does it follow only the modern group saying this is right?
The case for Sardis era drops the subject of Catholicism altogether, but that would mean the counter-reformation movement, the Oxford movements, etc. never existed. I'm only suggesting that this isn't sound evidence. Also, Christ's statement that the false prophetess Jezebel told people to "practice immorality and to eat food sacrificed to idols" is said to signify the introduction of pagan practices into Christianity, a charge some non-Catholics bring against the Catholic Church. I don't think this is a fair assessment, Catholicism actively rejects paganism very strongly and always has. It does hold strongly to the Christian teaching however.
It's difficult to accept this notion. However, and this is just me stating what I believe in regards to early Church history, they would technically all have to represent the Catholic Church since there was only church in existence of that time.