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Nathanael Arkor (Feb 22 2024 at 22:32):Every pseudo double category is equivalent to a strict double category. Is every pseudo double category with companions and conjoints equivalent to a strict double category with a strictly functorial choice of companions and conjoints?
Mike Shulman (Feb 22 2024 at 22:42):Yes, I think so, for a suitably weak meaning of "equivalence": https://mathoverflow.net/a/422798/49.
Nathanael Arkor (Feb 22 2024 at 22:44):Ah, perfect, thank you!
Mike Shulman (Feb 22 2024 at 22:48):Note that the equivalence I gave there requires modifying the loose bicategory and the tight 2-category both up to biequivalence, even if the tight composition started out strict. So it's not an equivalence in the usual 2-category of pseudo double categories, since that would require an ordinary 1-equivalence of tight 1-categories.
Nathanael Arkor (Feb 22 2024 at 22:53):Yes, that does raise an interesting question regarding the appropriate notion of equivalence of double categories...
Nathanael Arkor (Feb 22 2024 at 23:06):I wonder whether it would be possible to instead fatten up the loose-cells, e.g. defining a loose-cell from to to be a triple in the original double category.
Mike Shulman (Feb 22 2024 at 23:23):I don't know. In general, I don't think it is possible to strictify a pseudofunctor by modifying only the codomain, but I don't have a concrete counterexample to that, and this is not a generic pseudofunctor either. So maybe...