A little Geography
The play within the play takes place in Italy. Baptista and his daughters, Katherina and Bianca, are from Padua. Petruchio reigns from Verona, which is not far from Padua. The pedant who poses as Vicentio of Verona, is from Mantua (which is located in Lombary). Lucentio and Tranio are from Pisa, which is a city in Tuscany, Italy.
Even though the play is occurring in Italy, Christopher Sly is actually English [hence saying "...we came in with Richard Conqueror."]