"Paul's" is the possessive form of the name Paul, showing ownership or association. It can also refer to a place or business owned or run by someone named Paul.
The power of citizenship as a shield against oppression was widely known from the example of Paul's Roman citizenship, which sent the centurion scurrying to his higher-ups with the message: "Take heed what thou doest: for this man is a Roman." Robert H. Jackson
Our little drawing Room [Constable's lodgings at Hamptstead with a view on London] commands a view unequalled in Europe - from Westminster Abbey to Gravesend - the dome of St Paul's in the Air - realizes Michael Angelo's Idea on seeing that of the Pantheon - 'I will build such a thing in the Sky. John Constable
Next morning I went over to Paul's for coffee and told him I had finished. "Good for you,” he said without looking up. "Start the next one today. Steven Pressfield
Why do they cover Paul's songs but never mine? Yoko Ono
St. Paul's arose like some huge mountain above the enormous mass of smaller buildings. Karl Philipp Moritz
The church of St. Peter at Berlin, notwithstanding the total difference between them in the style of building, appears in some respects to have a great resemblance to St. Paul's in London. Karl Philipp Moritz