If you refer to a building or place ("Westminster Abbey", "Dodger Stadium", "Cedars Hospital", "Library of Alexandria") whose name is a proper noun which combines a unique identifier ("Westminster") with a category noun ("abbey") by its full name, you would capitalize all portions of the noun. If you omit the generic / categorical portion of the name, you continue to capitalize the remaining portion of the name ("I'm going to Cedars", "They held the coronation at Westminster").
But when omitting the unique identifier, and using the categorical noun as a metonym, it's less clear what should be done. If you're in a place with exactly one cathedral, should you say "I'm going to the cathedral" or "I'm going to the Cathedral" when "Cathedral" is part of the building's name?
Trying out various examples I can't seem to find a consistent pattern for what feels right in my head, and it's not clear.