RE: Triquotient assignments for geometric morphisms

RE: Triquotient assignments for geometric morphisms

[Townsend, Christopher]

Barney

As far as I am aware, no generalisation of localic triquotient assignments to geometric morphisms has been developed. That's not for want of trying on my part!

A naïve approach would be to define a weak triquotient assignment on a geometric morphism f:F->E to be a filtered colimit preserving functor that is required to interact with the inverse image of f in a manner that mimics the localic case. For this approach to work in a way that is similar to what happens for locales we would need to have a similar way of characterising such filtered colimit preserving functors which, as far as I am aware, is not available (essentially due to the technical difficulty that sheafification is 'two step' for toposes, but only 'one step' for locales). The technical problems here are, in my mind, the same as the more well known problems associated with constructing an upper power topos. 

My current view on how to solve this problem is to use localic representations of geometric morphisms. This requires us to re-state the theory of geometric morphisms as adjunctions between categories of locales and to develop 'topos theory' relative to these adjunctions. For example 'Grothendick topos' becomes 'category of localic diagrams of a localic groupoid' in this paradigm. The lower power topos construction should guide us to see how its action effects (the category of actions of) localic groupoids; then by upper/lower symmetry we know what the 'upper' case should be (and hence to weak triquotient assignments on geometric morphisms). Unfortunately getting the symmetry to work even at the much simpler level of bounded geometric morphisms is proving a headache. 

If you would like any further detail, please feel free to get in touch. 

Regards, Christopher 

-----Original Message-----
From: categories@mta.ca [mailto:categories@mta.ca] On Behalf Of Barney Hilken
Sent: 22 June 2009 16:49
To: categories
Subject: categories: Triquotient assignments for geometric morphisms

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.



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

This email is intended only for the use of the individual(s) to whom
it is addressed and may be privileged and confidential.

Unauthorised use or disclosure is prohibited. If you receive this
e-mail in error, please advise immediately and delete the original
message without copying, using, or telling anyone about its contents.

This message may have been altered without your or our knowledge and
the sender does not accept any liability for any errors or omissions
in the message.

This message does not create or change any contract.  Royal Bank of
Canada and its subsidiaries accept no responsibility for damage caused
by any viruses contained in this email or its attachments.  Emails may
be monitored.

RBC Capital Markets is a business name used by branches and
subsidiaries of Royal Bank of Canada, including Royal Bank of Canada,
London branch and Royal Bank of Canada Europe Limited. In accordance
with English law requirements, details regarding Royal Bank of Canada
Europe Limited are set out below:

ROYAL BANK OF CANADA EUROPE LIMITED
Registered in England and Wales 995939
Registered Address: 71 Queen Victoria Street, London, EC4V 4DE.
Authorised and regulated by the Financial Service Authority.
Member of the London Stock Exchange



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

Re: Triquotient assignments for geometric morphisms

[Barney Hilken]

Hi Christopher & Steve, thanks for your replies.

The version I need is the generalisation of open & proper morphisms
(i.e. some kind of map  f_*\Omega_F  -> \Omega_E) rather than the
generalisation of locally connected & tidy morphisms (some kind of
functor F -> E). As Steve says, I think the results I want follow from
the stability of the hyperconnected-localic factorisation, but I was
hoping someone had written out the details in the style of sections
C3.1 & C3.2 of the Elephant. Looks like I'll have to do it myself.

Barney.



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

Re: Triquotient assignments for geometric morphisms

[Steve Vickers]

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.
>


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

Triquotient assignments for geometric morphisms

[Barney Hilken]

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.



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