Re: Opposite of objects in a bicategory?

[Michael Shulman <shulman@uchicago.edu>]

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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