I didn't call Bloomberg a Republican. When he first ran for the mayoralty, he was a ROC (Republican of convenience), not even a RINO, but he benefited from Giuliani's support. He was fortunate in a way that the election was two months after 9/11, when Giuliani was at the peak of his popularity.
You would be hard pressed to find any major Bloomberg appointees in the city administration who are known to be Republicans of any kind. Yet the GOP party bosses let him run twice more on their ticket for re-election because they felt there was no viable alternative.
The Republican party bosses in NYC have helped ensure that there would be no alternatives, even as they have financially benefitted from Bloomberg.
Bloomberg switched parties to win the office.
He was always a democrat. He lost before.
http://topics.nytimes.com/top/reference/timestopics/people/b/michael_r_bloomberg/index.html