Re: Triquotient assignments for geometric morphisms

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear Barney,

The weak triquotient assignments go along with the double powerlocale
monad PP, since a weak triquotient assignment for f: X -> Y is a map g:
Y -> PP(X) satisfying certain conditions that relate to the strength of
PP. I believe Townsend has published some work on this.

This is similar to how open maps go along with the lower powerlocale P_L
(see my "Locales are not pointless"), though with P_L it is made tighter
using an adjunction that is not available in the PP case, so the open
analogue of triquotient assignment, the map from Y to P_L(X), is
characterized uniquely.

For open maps we know of a trivial generalization to toposes: a
geometric morphism is open if its localic part is an open locale map.
You could probably play the same trick with triquotient assignments, and
 then I think your stability property follows from stability of the
hyperconnected-localic factorization.

However, there is also a more interesting generalization in the case of
open maps, got by generalizing P_L to the symmetric topos construction
M. (This is described in the Elephant, but also, in much more detail, in
the Bunge-Funk book "Singular coverings of toposes". See also my paper
"Cosheaves and connectedness in formal topology".) In fact, Bunge and
Funk have proved that for a locale X, P_L(X) is the localic reflection
of M(X). The relationship between P_L and open maps transfers to one
between M and locally connected geometric morphisms.

Since PP is the composite of (commuting) monads P_U and P_L, where P_U
is the upper powerlocale, one natural approach to a topos generalization
would be try also to generalize P_U.

This generalization seems to be missing in our current state of
knowledge, though I've had some thoughts about it and firmly believe
that it exists.

Regards,

Steve.

Barney Hilken wrote:
> Has anyone generalised the theory of (weak) triquotient assignments
> from locale maps to geometric morphisms? In particular, does the
> pullback (assuming boundedness) of a geometric morphism with a
> triquotient assignment have a unique triquotient assignment satisfying
> the Beck-Chevalley condition?
>
> Also, if f:X->Y is a continuous function between topological spaces,
> are there any reasonable conditions (other than openness) under which
> the interior of the direct image along f is a weak triquotient
> assignment for the inverse image map?
>
> Thanks,
>
> Barney.
>


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