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Brendan Murphy (Feb 22 2024 at 15:39):A double bicategory is defined as a pair of bicategory structures on the same set of objects, plus a bunch more data and properties. It would be intuitive to me that we could instead ask for a pair of bicategories with a specified equivalence between them, plus a bunch more data and properties. Has anyone worked this out? I guess what I'm hoping for with this is that we could recognize double bicategories as something more natural looking if we think of them instead as a sort of pairing between two bicategories. The defintion is pretty hefty and it would be nice to have a more conceptual framing for it
Mike Shulman (Feb 22 2024 at 15:49):Do you similarly feel that a double category would be more intuitive as a pairing between two categories rather than as two category structures on the same set of objects?
Brendan Murphy (Feb 22 2024 at 15:51):Hmm, I guess not. Maybe that's a sign I'm barking up the wrong tree here
Brendan Murphy (Feb 22 2024 at 15:52):I think I may try and write out a defintion of a double category in this decoupled way just to see what it looks like though
Brendan Murphy (Feb 22 2024 at 16:11):Yeah thinking more about it the boundary of a square really wants to have all its vertices in one set
Brendan Murphy (Feb 22 2024 at 16:13):Is there a nice high level definition of a double bicategory? I seem to recall there's a connection between Segal spaces in Grpd and pseudo double categories, maybe that's a place to start
Dylan Braithwaite (Feb 22 2024 at 16:15):For what it's worth, Pare and Grandis' paper on intercategories explains that double bicategories correspond with a certain class of intercategories (so a pair of double categories with some extra properties and structure). I don't know if it's a more intuitive picture, but it at least avoids having to talk about the two categories having the same class of objects
Mike Shulman (Feb 22 2024 at 17:56):I don't know of a high-level definition that gives exactly double bicategories.