The building complex is always called “the Barbican”, but the surrounding map area and its tube station are named just “Barbican”. Also, the arts theatre place within the Barbican seems to be officially named “Barbican Centre” (but people always say “going to see X at the Barbican”).
That makes sense. I still find hard to imagine saying something like "I live in Barbican" if I didn't live in _The_ Barbican. But going to "Barbican" if traveling by tube would be obvious.
