Opposite of objects in a bicategory?

Opposite of objects in a bicategory?

[David Leduc]

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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Re: Opposite of objects in a bicategory?

[Steve Lack]

Dear David,

There is a way, but more structure, and possibly a change of perspective
is required.

I don't know how to describe opposites in terms of the category Cat. But
if you work instead with the monoidal bicategory Prof/Mod/Dist (all three
names are used) you can.

I'll write A* for the opposite of a category A.

I'll write Prof for the bicategory whose objects are (small) categories,
and whose morphisms from A to B are functors A-->[B*,Set]. These are
called profunctors (or modules or distributors) from A to B. They are
composed using colimits; one easy way to see the composition is that
up to equivalence, we can regard such profunctors as being cocontinuous
functors [A*,Set]-->[B*,Set] and from this latter point of view we simply
use ordinary composition of (cocontinuous) functors.

Prof is monoidal, via the cartesian product of categories AxB. Note that
this is not a cartesian monoidal structure on Prof, although it does have
some features of cartesianness; it is an example of what is called a
cartesian bicategory (studied by Carboni, Walters, Wood, and others).

Anyway, in Prof, the opposite of a category B is dual to B, in the sense
of monoidal (bi)categories, since functors AxB-->[C*,Set] correspond to
functors A-->[BxC*,Set], and so to functors A-->[(CxB*)*,Set]. Thus in Prof,
morphisms AxB-->C correspond to morphisms A-->CxB*.

Steve Lack.


On 7/03/10 11:28 PM, "David Leduc" <david.leduc6@googlemail.com> 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
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Opposite of objects in a bicategory?

[Michael Shulman]

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


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Opposite of objects in a bicategory?

[Michael Shulman]

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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