Antagonist
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Character Role Analysis
Sport, New York City
On a micro-level, you would probably say that Travis' biggest personal enemy is Sport. He's the one who disgusts Travis the most, the one who exploits a child and finally brings Travis' wrath down on him in full. Charles Palantine doesn't really do anything to oppose Travis in any way, even though Travis wants to kill him in an act of pure nihilism, before deciding to go after Sport instead.
However, in a deeper sense, New York City is Travis' major enemy. Its crime and sleaziness have the biggest impact on him, mentally, and he kills Sport because Sport personally crystallizes everything Travis already hates about the city.