List the facts Holmes presents to Watson about the battered hat in "The Adventure of the Blue Carbuncle."

From the hat, Holmes deduces that the owner was intelligent but has fallen on hard times and his wife has left him.

Watson walks in on Holmes one day, intending to wish him Merry Christmas.  He finds Holmes lounging around, examining a battered hat with a magnifying glass.

Beside the couch was a wooden chair, and on the angle of the back hung a very seedy and disreputable hard-felt hat, much the worse for wear,...

From the hat, Holmes deduces that the owner was intelligent but has fallen on hard times and his wife has left him.

Watson walks in on Holmes one day, intending to wish him Merry Christmas.  He finds Holmes lounging around, examining a battered hat with a magnifying glass.

Beside the couch was a wooden chair, and on the angle of the back hung a very seedy and disreputable hard-felt hat, much the worse for wear, and cracked in several places. 

This is not uncommon behavior for Holmes.  He is a very skilled observer, and he can tell a lot about a hat from observing it, and even more about its owner.  As a detective, Holmes can make correct deductions from the smallest details.

Holmes tells Watson that the matter of the hat is trivial, but that he is happy to be able to tell Watson his findings.  Holmes often uses Watson as a sounding board, because he likes to discuss his findings out loud.  Holmes comments that there is no crime involved this time.

The “battered billycock” was found by a man named Peterson, along with a goose.  It was knocked off a man’s head in a fight with “a little knot of roughs.”  Holmes wants to use the hat to deduce who it belongs to.

Watson considers the hat ordinary but battered.  Despite this, Holmes comes up with a very specific list of deductions.

That the man was highly intellectual is of course obvious upon the face of it, and also that he was fairly well-to-do within the last three years, although he has now fallen upon evil days. He had foresight, but has less now than formerly, pointing to a moral retrogression, which, when taken with the decline of his fortunes, seems to indicate some evil influence, probably drink, at work upon him. This may account also for the obvious fact that his wife has ceased to love him.

This is a lot of information for a hat, but Holmes can tell a lot about a piece of clothing from how it is worn.  He can tell things about the owner, and where the piece has been.  This is the work of a detective.