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Jacques Carette (Jun 06 2021 at 15:33):I doubt that the above is existing terminology, but I think it is apt: has anyone studied double categories where both vertical and horizontal categories are the same? The reason one might want to do that is to put the focus on commutative squares as being the basic object of study by making them front-and-center. It is of course possible to study them in other ways, but that means picking them out (perhaps using some kind of predicate), which is awfully indirect.
Certainly one lesson that I've learned is that every single layer of 'encoding' that's done to study things tends to obscure the results somewhat.
Amar Hadzihasanovic (Jun 06 2021 at 18:46):I think you may want to check the case of cubical -categories. The go-to reference is this.
James Wood (Jun 06 2021 at 18:53):Are these the same as n-fold categories?
Jacques Carette (Jun 06 2021 at 19:02):Wow, that reference is written in a maximally-distasteful style for someone who prefers type theory over set theory. The amount of decoding necessary to go from that presentation to something I'd be able to use would be quite substantial.
Mike Shulman (Jun 06 2021 at 19:08):I think a keyword to look for is "edge-symmetric double categories".
Jacques Carette (Jun 06 2021 at 19:09):Indeed, that does seem like an interesting thread to pull - thank you.
Chad Nester (Jun 07 2021 at 09:14):In this the authors use an edge-symmetric double category (with one object) to explain stateful morphism sequences.