This may be worth cross-posting to /r/whatisthisthing.

The style of cap looks to be a Kepi. Your particular one was made by M.C. Lilley & Company, of Columbus, Ohio, which manufactured all manner of hats, swords, and sundries. My first instinct when I see the letters "P.C." on a hat like this is "Police Constable" - a fairly British term in my mind, but certainly usable in parts of the US. I could imagine it being a police officer's hat, to be sure - with as many departments as there are across the nation, it may take a very dedicated specialist to identify it specifically to one, if that is in fact what it is.

Could you take some clearer photos of the button and stamp inside the hat?