What is the correct reply to Where do you study?
- At the library.
- The library.
You could be at the library, in the library or just the library. English is pretty flexible with these things.
The full sentence is "I study at the library". "At the library" is ellipsis already. So there's no reason to not elide even further and say "The library". And indeed people often do just that. This is not limited to English, by the way.