Though groups like the Foundation and GOC are described on the 'Groups of Intrigue' page, I don't believe there's much content about how ones like the Three Moons Initiative or Arcadia may relate to the Library, so I'm creating this topic for if anyone has any ideas they'd like to share.
I've personally had the idea that Corbenic is banned from the Library for trying to edit books to be more 'moral' by their standards, and I believe it's been mentioned that the Circus has performed in the Library in the Dread and Circuses canon.
I also see the less… hostile varieties of the religious GoIs having churches somewhere in the Library.