Date: Mon, 3 Jul 1995 14:40:00 -0300 (ADT)
Subject: generalised spaces 

Date: Mon, 03 Jul 1995 18:01:40 MESZ
From: Thomas Streicher <streicher@mathematik.th-darmstadt.de>

I have the following question : can the adjunction between topological spaces
and locales be lifted to an adjunction between GENERALISED SPACES and Grothen-
dieck 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