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
Martin Escardo writes:
Question 3. But surely compact Hausdorff locales cannot possibly be exponentiating, can they?
(These questions make sense for topological spaces too. )
Clarification: Alex Simpson points out, in a reply not sent to the list, that in the ambient category of topological spaces, compact Hausdorff spaces are not exponentiating. For example the exponential 2^(N^N) doesn't exist, where 2 is the two-point discrete space and N is the discrete space of natural numbers, and, of course, the exponential N^N does exist. In fact, Alex emphasized this years ago, when I subjected him to the ideas presented in the previous message. The point is that the locale product of two (sober) topological spaces doesn't coincide with the topological product, and hence exponentials are potentially different, as they are defined with respect to products. You may say: well, in any case, it is a fact that a sober space is exponentiable in Top iff it is exponentiable in Loc, so Alex's counter-example should work in Loc too. But then I reply: this coincidence has to do with the fact that the locale product coincides with the topological product if one of the factors is locally compact. In our case, because the exponent is NOT necessarily locally compact, this coincidence fails. We are considering exponentiating rather than exponentiable spaces. So, in principle, it may be that compact Hausdorff locales are exponentiating in Loc - although I very much doubt that this would be the case, as should be clear from the previous message. The point is that I just don't know, and I am looking forward to be enlightened after my failed attempts to decide the question either way. In light of Alex's observation, my conclusion to the previous message should have been:
Question 3. But surely compact Hausdorff locales cannot possibly be exponentiating, can they?
It would be rather amazing if they were, because compact Hausdorff spaces are not exponentiating in the category of topological spaces. But the known arguments in the case of topological spaces don't seem to apply here.
One further comment: it is plausible that eta: X->JX is stably flat for X exponentiable. If Alex's topological counter-example applies to locales, then eta_X is not stably flat for X=N^N. MHE
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Martin Escardo