We have two versions of Basecamp, each with their own API. For Basecamp Classic, your account will look like https://subdomain.basecamphq.com, while for the new Basecamp, your url will be https://basecamp.com/1234567
The Basecamp Classic API returns XML, and the docs are here. The new Basecamp API returns JSON, and the docs are here.
To get all of the categories from the Classic API, you'll want to use
https://#{subdomain}.basecamphq.com/projects/#{project_id}/categories.xml