Troy's relationship with both his sons, Lyons and Cory, are dominated by his own relationship with his father.
His father was brutal and controlling, and although Troy loves Cory, he knows of no other way to bring up a son. Thus, he repeats the mistakes of the previous generation.
There is some irony in this, and perhaps even some tragedy, as Troy deeply resents his father and never forgives him for the way he was treated, as chattel and then as a rival.
Much of the irony here stems from the idea that Troy treats his sons as he does, especially Cory, due to a desire to avoid a repetition of history. Troy does not want Cory to grow up to haul garbage like he does. This is why Troy refuses to let Cory play football, despite the fact that this sport would allow Cory to attend college.
Worried that Cory would sacrifice an opportunity to gain real, work-related skills by pursuing football, Troy also worries that Cory will suffer from the same disappointment he did as a baseball player (kept out of the big leagues because of race). Troy wants Cory to have a better life than he had and this leads him to curb Cory's dreams.
Cory represents all the possibilities his father never had, but he also represents Troy's unmet dreams.
By squashing Cory's dreams of playing football and going to college, Troy believes he is setting his son up for success. Cory sees things differently, interpreting Troy's behavior as mere jealousy and rivalry. Cory's view is accurate but incomplete.
Troy is so bitter over his own lack of opportunity that he holds his son back from any success he might achieve.
Troy's intentions are to provide opportunities for his sons to become men with an understanding of duty and a chance at success. Regrettably, he enacts these intentions without kindness.
Though Cory comes to a moral maturity in the end, taking responsibility for his own emotional development, Troy dies before doing so, caught up in the battle against death and against himself, which he cannot possibly win.