Re: Reflection to 0-dimensional locales

["George Janelidze" <George.Janelidze@uct.ac.za>]

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/CommentatMathUnivCarolRetro_42-2001-2_13.pdf
>
> 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


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