Re: Reflection to 0-dimensional locales

[Matias M <matias.menni@gmail.com>]

Dear colleagues,

this is to clarify my references to one of George Janelidze's messages in
this thread and to add a couple of comments about preservation of finite
products by pizero.


2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:

> 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 did not know.

2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:

> 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)".


Apologies, I inserted my question in the wrong place.
What I meant to ask is the following.
Let L : Locales ---> Locales be the monad determined by the inclusion
0-Dimensional locales--->Locales.
Is it the case that the unit
X ---> L X
is the first component of a factorization
X ---> L X ---> 2^(2^X)
of the canonical map to a double dualization, where 2 is a suitable
exponentiating object in Locales?


  2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:

> Note that Matias speaks of
> preservation of finite products while the reflection
> Locales--->0-Dimensional locales
> does not preserve them.
>

I did not mean to suggest that a positive answer to my question would
entail an answer to George's question. I simply wanted to understand the
construction of
L : Locales ---> Locales
in terms that I find more familiar.
(The finite-product preservation result I mentioned is not applicable to
Locales.)

  2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:

> Note also the big (and well known) difference between semi-left-exactness
> and preservation of finite products: for every 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)...
>

Concerning local connectedness, finite-product preservation of pizero and
even some homotopy theory, let me add a couple of related remarks.

The reason I looked for a finite-product preserving pizero in the Tbilisi
paper comes from
  "Axiom 1" in Lawvere's "Categories of spaces ..." paper, TAC Reprint 9,
which postulates an essential geometric morphism
Gamma: E ---> S
such that the leftmost adjoint
Gamma_! : E ---> S
preserves finite products.
Then Lawvere says:
"The axiom is necessary for the naive construction of the homotopic passage
from
quantity to quality; namely, it insures that (not only Gamma_* but also)
Gamma_! is a closed functor thus inducing a second way of associating an
S-enriched category to each E-enriched category
[...]
For example, E itself as an E-enriched category gives rise to a homotopy
category in which
[E](X, Y )=Gamma_!(Y^X)."

Axiomatic Cohesion (TAC 2007) continued the work in "Categories of spaces
..." introducing, among other things, the axiom:

(Nullstellensatz)  The canonical Gamma_* ---> Gamma_! is epic

which captures the intuition that every piece has a point.

The above is one source of inspiration for Johnstone's TAC 2011 paper where
he studies locally connected geometric morphisms
p: E ---> S
such that the leftmost adjoint p_! : E ---> S preserves finite products.
Among other things he shows that:
if p: E ---> S is lc, hyperconnected and local then p_! preserves finite
products.

It also follows from Johnstone's results that:
if p: E ---> S is lc and local then, p is hyperconnected iff
the Nullstellensatz holds.

A rough conclusion that one may arrive at is that:
if pizero does not preserve finite products then there is some connected
space without points.

Apologies again for the lack of clarity in my previous message.
I look forward to see how the different aspects of the 'pizero idea' evolve
and I am glad that Steve Vickers' message brought them up.

All the best,

Matías


2018-02-11 18:38 GMT-03:00 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 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/Commentat
>> MathUnivCarolRetro_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
>
> Disclaimer - University of Cape Town This email is subject to UCT policies
> and email disclaimer published on our website at
> http://www.uct.ac.za/main/email-disclaimer or obtainable from +27 21 650
> 9111. If this email is not related to the business of UCT, it is sent by
> the sender in an individual capacity. Please report security incidents or
> abuse via https://csirt.uct.ac.za/page/report-an-incident.php.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]