Face/Off is set in the Star Trek universe.
Sean Archer is Jonathan Archer's great-grandfather who fought in the Eugenics Wars. Castor Troy and his brother are among the few remaining Augments who survived the wars.- Following this line of logic, for the surname to carry on, it is likely Adam (Castor’s illegitimate son which the Archer family adopts) is John’s grandfather. Biologically speaking, this means Castor Troy is John’s great-grandfather. Could explain some of his inconsistent (read: crazy) behavior over the course of Enterprise.
Sean Archer and Castor Troy used to be college friends
If one ignores the fact that John Travolta is actually ten years older than Nicolas Cage and puts Archer and Castor as being aged no more than two to three years apart, it's hypothetically possible that they could have gone to college together, and may even have been roommates. This might explain why Castor considers Archer a Worthy Opponent throughout the movie and possibly explain why Castor addresses Archer by first name multiple times especially in the first action sequence.- The novelization actually gave Castor a backstory, that he is a rogue CIA agent-turned-terrorist, so there is a good chance that maybe Archer and Castor worked together at some point.
Who comprises Castor Troy's extended family?
Think about it. Castor's younger brother is named Pollux. They're both named for Greek mythology. So, theoretically, they might have a sister named Helen Troy, maybe even another sister named Clytemnestra. Their father was named Tyndareus Troy, and their mother's name is Leda. If they ever make a sequel to this movie, Archer will hunt down whoever employed Castor.
And it will be a mafia boss with a surgery addiction, keeping up with the surgery theme from the first movie.