Dear Colleagues, Concerning Steve's messages started with "Topos theory for spaces of connected components" sent on February 4 and comments to them, I am not sure I understand what was the end of the story, but I would like to comment on a part of the story related to my question, in the 'chronological' order: 1. I think on February 6 I have written three messages, the first of which was not posted (which is reasonable since my second message contained its copy). In the third message, whose subject was "Reflection to 0-dimensional locales", I wrote that the answer to my question "Is the reflection Locales--->0-Dimensional locales semi-left-exact?" is NO. I also wrote that I know this from Graham Manuell, who explained to me that this follows from the existence of a counter-example, due to I. Kriz, presented in the book "Frames and Locales" by J. Picado and A. Pultr (Pages 260-266). And I asked if it is possible to construct a simpler counter-example. 2. My question above is mentioned (among many other things) in the message of Matias Menni posted on February 8, although it is not clear to me whether or not Matias already knew then that the answer to it is negative. I also don't understand what exactly does Matias mean by asking whether or not the inclusion functor 0-Dimensional locales--->Locales "is the result of a variant of Bill's construction (using an exponentiating object and a `good' factorization system)". Note that Matias speaks of preservation of finite products while the reflection Locales--->0-Dimensional locales does not preserve them. Note also the big (and well known) difference between semi-left-exactness and preservation of finite products: for every connected locally connected topos E with coproducts, the functor Pizero : E--->Sets is a semi-left-exact reflection - but if it were always finite product preserving, then, say, homotopy theory would not exist (all fundamental groups of 'good' spaces would be trivial)... 3. Andrej Bauer, in his message of February 9, also mentions my question and says:
Can Example 1 in
https://dml.cz/bitstream/handle/10338.dmlcz/119250/CommentatMathUnivCarolRet...
be put to some use to answer the question negatively? It shows that the zero-dimensional reflection in topological spaces does not preserve finite products. The example uses fairly nice subspaces of R and R^2.
I think the topological version does not help; note also that the non-semi-left-exactness there was known for a very long time. Summarizing, I thank again Graham for his help, and Matias and Andrej for their comments, but let me insist: the counter-example of Kriz is so complicated... can someone construct an easier one? George Janelidze [For admin and other information see: http://www.mta.ca/~cat-dist/ ]