Hi David, The characterization Steve gave works for categories enriched over any category V with sufficient structure to define and compose profunctors, and characterizes the usual opposite V-category. It suffices for V to be cocomplete and closed symmetric monoidal (symmetry, or at least braiding, is necessary for opposites to exist). It also works for internal categories (in a suitably nice category, like a topos), fibered categories, stacks, and almost any other sort of category you can think of. It should also work to characterize op for 2-categories in the 3-category of 2-categories and 2-profunctors (of whatever strictness you like), but I don't know how to make it give co or coop (does anyone?). The construction I gave does not work in most enriched situations; V-Cat is rarely exact. It does work for internal categories in a suitably nice category, and it also works for fibered categories and for stacks (in categories) over any site, or indeed over any (2,1)-site (i.e. a locally groupoidal category with a suitable notion of Grothendieck topology). The 2-category of stacks on an arbitrary 2-site is still exact (this is part of Street's Giraud-type characterization of such 2-categories), but in general it doesn't have cores, so the construction fails there. I don't know for sure what an "exact 3-category" is (I haven't thought about it a whole lot; has anyone?), but it seems possible that there is a definition which would allow this sort of construction to go through in any such 3-category with cores. Depending on the notion of exactness, it might be possible to recover all three of op, co, and coop. Mike David Leduc wrote:
Thank you Steve and Mike for your interesting replies. I will definitely study them further.
When applying your definitions to other bicategories, do your definitions reduce to well-known notions?
Now if we move a bit higher and consider dual bicategories, we get 3 possibilities: op, co and coop. Do your definitions generalize to include those 3 ways to dualize?
David
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