RE: Triquotient assignments for geometric morphisms

["Townsend, Christopher" <Christopher.Townsend@rbccm.com>]

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.



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