What is Montag's relationship with Beatty in Fahrenheit 451?
Montag and Beatty have a relationship that is both antagonistic and reflective. Beatty is often characterized as "playing cat-and-mouse" with Montag, and Montag ultimately kills him, which depicts them as classical enemies. However, Montag and Beatty are also very much alike in many ways, and Beatty sees himself as an older version of Montag, wiser and less optimistic but also more pragmatic.
Beatty reveals a great deal of his own past, as well as the history of society, at various times to Montag. If we choose to view Montag as going through the steps of the "Hero's Journey" motif, Beatty is a complex figure because he is both the source of conflict, and also one of the guides, prophets and gatekeepers along the way, revealing things to Montag at crucial junctions in his development. For example, Beatty reveals that books were not taken away because of a hatred of knowledge, but in order to promote social peace and unity, and that it's entirely common for firemen to get curious about them, though that doesn't make it acceptable. The fact that Montag's constant suspicions of Beatty often turn out to be well-founded suggest that Beatty is always one step ahead of head, and could have him punished or killed at any time, but instead wants to guide Montag to the same mindset that Beatty has.
Later, Beatty reveals himself as a thoroughly and mysteriously well-read person, even moreso than we were led to believe, and his taunting of Montag is, from a relationship perspective, confusing and complex. At the surface, Beatty may simply be boasting, based on his confidence and his perception of Montag as being weaker in willpower. However, he may also be speaking from a place of envy or loneliness, wishing that he could have committed himself to these ideals in the way Montag has, or perhaps even allowing himself to be a sacrifice on Montag's behalf that will allow Montag to transcend this conflict. Returning to the "Hero's Journey" motif, Beatty may be the true "death" that is required for Montag to be reborn, rather than the anonymous person settled on during the later manhunt.