Answer: I like just about all the questions, but especially ones that have the potential to let me bore the socks off of everyone in the netverse by drifting into related topics.

Which I won't do this time.

The guitar, like the piano, is fixed at 12 tones per octave, though music in theory involves many more, so each key can have any of 2 or 3 names. This ambiguity means that raising an A by 4 half steps could indeed take you to either C# or Db - what you call it is going to depend on the context. When using a capo you are transposing the entire piece to a new key, of course, so the only real context to be considered is: would you rather deal with a key signature of 4 sharps in C# minor, or - well, what would be the key signature for Db minor, anyway? The scale would be Db, Eb, Fb, Gb, Ab, Bbb, Cb. No key has a double flat in the key signature, so there's really no such thing as Db minor and that pretty much seals the deal: you'd call the new key C# minor.