What purpose does The Taming of the Shrew's Induction serve?
The Induction with Christopher Sly is in fact the only "Induction" of its type in all of Shakespeare's works. The Induction shows Sly, a beggar, falling asleep. He is tricked by some Lords into thinking that he too is a "Lord", and, dressed as a Lord, he sits to watch a play presented by some travelling actors.
The Induction bookmarks the story of Petruchio and Katharina with another story: and makes the story of the "Shrew" simply a story acted out by actors. It reminds you, in other words, that it's all pretend, that it's all theatre.
Except that it doesn't, exactly - because there's only an "induction" of Sly, and not an epilogue. Sly never reappears at the end to be transformed back into a beggar. It's not clear whether the epilogue was written, and is now missing, or whether Shakespeare (for some reason) intended the play to finish without one.
Some critics have argued over the years that the whole play is meant to be seen as a male dream of wish-fulfilment: in which Sly, arguing with his wife as the play opens, "dreams" the taming of the shrew as a sort of revenge on her. Problem with this is that the Induction makes it quite clear that Sly's "lordship" isn't a dream, but a trick. What the Induction does, in my opinion, is to point up the idea of "clothes" and the name "Lord" (very important in the Tranio/Lucentio plot) as defining personality...