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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