Re: Opposite of objects in a bicategory?

[Michael Shulman <shulman@uchicago.edu>]

Hi David,

The characterization Steve describes is nice and applies more generally
to enriched categories.  You can eliminate the dependence on Prof by
first constructing Prof from Cat: profunctors can be identified with
two-sided discrete fibrations, or two-sided discrete cofibrations, in
the bicategory Cat.  Mark Weber's paper "Yoneda structures in 2-toposes"
studies duality involutions characterized using discrete fibrations.
However, this property in Prof only characterizes the opposite up to
Morita equivalence (i.e. up to Cauchy completion, aka
idempotent-splitting), since Morita-equivalent categories are equivalent
objects of the bicategory Prof.

There is, in fact, also a way to describe opposites purely in terms of
Cat.  First we note that Cat is an "exact 2-category" -- this means that
among other things, every "2-congruence" has a quotient, where a
2-congruence is an internal category for which (source,target) is a
two-sided discrete fibration.  Now for any category A, let core(A) be
its core, i.e. its maximal subgroupoid.  The core can be characterized
in 2-categorical terms as a right adjoint to the inclusion of Gpd into
Cat_g, where Gpd is the 2-category of groupoids (= the category of
groupoidal objects in Cat) and Cat_g is the 2-category of categories,
functors, and natural isomorphisms.  However, of more importance for
this discussion is that core(A) is groupoidal and the map p:core(A) -->
A is "strong epic" in the sense that it is left orthogonal (in the
2-categorical sense) to all (representably) fully-faithful morphisms.
(In Cat, this condition is equivalent to p being essentially surjective
on objects.)  Since Cat is exact, this implies that A is equivalent to
the quotient of the "kernel" of p, which is the 2-congruence given by
the comma object (p/p) with its two projections to core(A).  However,
since core(A) is groupoidal, we can switch these two projections and
still have a 2-congruence.  The quotient of this "opposite"
2-congruence, which exists since Cat is exact, is (up to equivalence)
the opposite category of A.  This construction generalizes to any exact
2-category which "has cores," in the sense that every object admits a
strong epic from a groupoidal one.

The notion of exact 2-category is due to Ross Street in "A
characterization of bicategories of stacks."  I have not seen the above
construction of opposites written down anywhere, so I wrote it down
myself here:
http://ncatlab.org/michaelshulman/show/duality+involution#GpdFixUniq
but if anyone has seen it before, I would love to hear some references.

Best,
Mike

David Leduc wrote:
> Dear all,
>
> The same way monads on categories can be generalized to monads on
> objects of a bicategory, is there a way to generalize opposites of
> categories to opposites of objects in a bicategory?
>
> David


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