I have three questions: Question 1: Is there an intrinsic characterization of the stably flat maps of locales, i.e., those continuous maps f: X->Y such that id_Z x f: Z x X -> Z x Y is flat for all Z? (Recall that f: X->Y is flat iff the right adjoint f_*: OX -> OY of f^*: OY -> OX preserves finite joins. Preservation of merely the empty join amounts to density, and hence flatness is a rather strong form of density. For example, the embedding of a completely regular locale into its Stone-Cech compactification is strongly dense in this sense.) I don't think all flat maps are stably flat, but I may be mistaken. Specifically, for a locale X let JX denote the spectral locale defined by OJX = ideals of OX. Then there is a sublocale embedding eta: X->JX defined by eta^*(I) = join I, which is known to be flat. Question 2: Is eta: X->JX stably flat for every X? If so, then compact Hausdorff locales would be exponentiating. Here a locale Y is called exponentiating iff the exponential Y^X exists for every X. Recall also that X is exponentiable iff the exponential Y^X exists for every Y, and that this is the case iff the frame OX is a continuous lattice. Now, JX is exponentiable because OJX is an algebraic lattice. We claim that if eta_X were stably flat, then for every compact Hausdorff locale Y, the exponential Y^JX would have the universal property of an exponential Y^X with respect to a suitably defined evaluation map and a suitable construction of exponential transposition. Define e: Y^JX x X -> Y as the restriction of the original evaluation map ev: Y^JX x JX -> Y, that is id x eta ev Y^JX x X ----------> Y^JX -> JX -------> Y. ------------------------------> e Now, to show that the pair (Y^JX,e) has the universal property of an exponential Y^X, given f: Z x X -> Y, we have to construct a transpose f': Z -> Y^JX such that e(f' x id_X) = f. Consider the diagram id_Z x eta_X Z x X --------------> Z x JX \ . \ . \ . \ . \ . f \ . f'' \ . \ . \ . v v Y. If eta_X were stably flat, then, by "Joyal's Lemma", which says that compact Hausdorff locales are orthogonal to flat embeddings, there would be a unique f'' making the diagram commute. Then its Y^JX-transpose f': Z -> Y^JX with respect to the original evaluation map would give the unique required Y^X-transpose of our given f: Z x X -> Y with respect to our constructed evaluation map, as an easy calculation shows, using the universal property of Y^JX with respect to the original evaluation map. This shows that if Question 2 had a positive answer then compact Hausdorff locales would be exponentiating. Question 3. But surely compact Hausdorff locales cannot possibly be exponentiating, can they? (These questions make sense for topological spaces too. ) MHE