Here’s a play that would have to be presented very carefully these days in order not to be shouted off the stage. Elizabethan audiences would presumably hold some traditional views about strong-minded and outspoken women, and be familiar with plays and stories about how they need to be brought back in line. Maybe Shakespeare is asking his audiences to reconsider their traditional views, but that’s not yet clear in Act 1.
And it’s not clear to me just why Shakespeare thought the Christopher Sly “induction” an important part of the play. It tends to make the “story” of Petruchio and Katharina just a “story” that we can watch as some distance. What does Sly — his character and his situation — have to do with the world of Petruchio and Katharina?
Maybe there’s no special reason. Maybe Shakespeare just likes to play with play-acting. I’m struck with how often Shakespeare inserts a sort of ‘play’ into his play. The most obvious ones are this play and Midsummer Night’s Dream, along with Hamlet and maybe The Tempest. But As You Like It contains a play-acting scene, as does King Lear. (Is Katharina stuck in the “role” of shrew that she is playing?)
It’s odd that Petruchio does not enter the play until the second scene of Act 1 (and that’s after the two Sly scenes). And we don’t meet Katharina in Act 1 at all. Why, I wonder, did Shakespeare make the point that Petruchio, who wants to marry a rich wife, is not penniless: he’s got money in his pocket. His Grumio is a rough and outspoken servant, but Petruchio knows how to deal with him, perhaps preparing us for the way he will deal with rough and outspoken Kate. The sweet younger sister, Bianca, has plenty of suitors: Lucentio (to whom we are introduced in 1.1 — when we might imagine he is going to be the central male figure in the play) Hortensio, and Gremio. (Why did Shakespeare give us a Gremio and a Grumio? Does he want to confuse us?)
I suppose it’s going to be amusing, and maybe even farcical, that two of Bianca’s suitors are going to disguise themselves as schoolmasters, to provide cover for their suit. But it seems a bit much that Petruchio too will be Bianca’s schoolmaster.
Petruchio’s boastful speech in 1.2, saying roughly “I can handle her,” obviously sets up the scene when he meets his match.
Act 1 is pretty plot-heavy. It’s headed for comic reversals and surprises. But so far it mostly seems “situation comedy” — Shakespeare’s “sit com”.
I have seen a couple of effective modern productions which make somewhat ironic the “taming” that Petruccio undertakes at the conclusion and emphasize Kate’s yielding as a strategic and joking strategy. It could be that the Sly induction suggests a topsy-turvy world that slides over into the main story; as you suggest, it makes the story of “taming” a story, a fiction that we can understand as such and not take as serious business. Interesting that the Sly induction isn’t concluded in the text we have, though there are scraps of text that suggest a possible continuing presence. In any case, I imagine that Sly and his companions are still off to the side and imagined as present.
I agree that “sit com” is a reasonable way to think of the play. Katherine’s ill temper and disinclination to deal with men seem exaggerated and strange — and the contrast to Bianca’s sweetness contrived. The beginning with Lucentio does make us think he’s going to be the center of things, and his changing clothes with Tranio only enforces that. But the opening exchange between Katherine and Baptista may hint at a reason for her irritability; she’s always been the less favored child, and her father has no compunction apparenly about disgracing her. And the Gremio piles on as well. Note that we do have brief, but consequential intro to Katherine at the beginning of Act I. The rest of the scene is devoted to the Lucentio/Tranio plot. Petruccio seems to take over in the next scene. He seems exaggeratedly interested in marrying money, even though he doesn’t seem to need it.