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