When the five-act inset play opens, it first appears that Kate's role in the story will be secondary to that of Bianca, who is pursued by a string of suitors. Kate, it seems, is a mere afterthought (literally, no man in his right mind even considers Kate as an option) until her father says that Bianca cannot marry until Kate finds a husband. This propels Kate to the status of "Protagonist," as the main storyline revolves around her battle with Petruchio and her transformation from a "shrew" to a seemingly obedient wife. We don't really have access to Kate's interiority. Unlike characters such as Hamlet, Kate doesn't deliver dramatic monologues that give us any insight into her oh-so-complicated soul. (This is a comedy, after all.) We must depend on her actions and behavior to determine what makes Kate tick. This makes it difficult to pin her down, especially at the play's end when she delivers a long speech about how women should be obedient to their husbands.
Even though Petruchio behaves in an antagonistic fashion (like a jerk) toward Kate, this doesn't make him the play's "Antagonist." (Besides, Kate acts like a jerk, too.) Petruchio shares the title of "Protagonist" with Kate as he struggles with Katherine to "tame" her into submission – that is, make her into an obedient wife. Unlike Kate, Petruchio is a bit easier to figure out. He tells us exactly how he plans to whip Kate into shape and it appears, outwardly at least, that his actions are successful.