I have the following question : can the adjunction between topological spaces and locales be lifted to an adjunction between Generalised Spaces and Grothendieck toposes. I.e. we are looking for
locales:spaces = Groth. toposes:? (*)
Clearly, for any Grothendieck topos E the category of points of E is an accessible category. But if A is an accesible category then the accessible functors to Set are the preshaeves over A_pres the category of presentable objects of A (surely one has to fix a cardina, say alpha_0). So things don't work so easily.
I mean there is already a problem that (*) does not specify the ? (generalised spaces) uniquely.
So my question is whether there is a notion of accessible category + some further conditions which is an appropriate fill-in for ? in (*).
Maybe the answer is well-known but I simply couldn't find it easily in the literature.
Thomas Streicher