Topos theory for spaces of connected components

Topos theory for spaces of connected components

[Steve Vickers]

Topos theory gives a solid account of local connectedness, where each open -  indeed, each sheaf - has a set (discrete space) of connected components. The definition of locally connected geometric morphism covers not only individual spaces but also bundles, considered fibrewise. It also covers generalized spaces as well as ungeneralized.

Is there an analogous theory for where the space of connected components is Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.)

The obvious example is any Stone space X, for instance, Cantor space, where X  is its own space of connected components. We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice  the Stone space aspects in the usual examples based on real analysis, since  they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components.

(By the way, if you wonder what brought me to this, it was from pondering the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for ungeneralized spaces its localic reflection is the lower powerlocale, which raises the question of whether there is a missing topos construction whose localic reflection is the upper powerlocale. On the other hand, the symmetric monad is related to local connectedness. Points of MX are cosheaves on X, and  X is locally connected if there is a terminal cosheaf in a strong sense, with that cosheaf providing the sets of connected components. Perhaps understanding the Stone space view of connected components would cast light on this missing construction.)

All the best,

Steve.


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

Re: Topos theory for spaces of connected components

[Marta Bunge]

Dear Steve,

I have nothing to say about your Stone spaces question in general, except for your remarks in the second part of your message about the symmetric monad M, where you suggest that the Stone locale view of connected components would perhaps cast light on the missing construction of a topos version N of  the upper power locale P_U, just as the symmetric monad M is a topos version of the lower power locale P_L.

In my paper “Pitts monads and a lax descent theorem” (2015), (Remark 7.6), I leave it as an open question (more or less) the construction of such an N. [ The name “Pitts monad” I gave on account on a condition which first appears in a theorem of A.M. Pitts whereby, in a lax pullback with bottom map an S-essential geometric morphisms, the top map is locally connected. The S-essential geometric morphisms are precisely the M-maps, and for the lower power locale monad P_L, the P_L-maps are the open maps. ]

However, toposes are more complicated than locales and a perfect analogue may not be what one should seek Indeed, one can view the symmetric monad M (classifier of distributions on toposes X,  or equivalently of complete spreads over X with a locally connected domain) as a topos version of the lower  power locale  P_L. There is however another such candidate, which is the bagdomain monad B_L (classifier of bags of points, or equivalently of branched coverings over X,  namely of those complete spreads that are purely locally equivalent to a locally constant cover). See M. Bunge and J. Funk, Singular Coverings of Toposes (2006), (Def. 9.32). In the same source SCT ( 8.3) there is a diagram which shows that there are two factorizations of the unit X—> M(X), namely one through the unit X—> B_L(X) and the other through the unit X—> T(X) where T (classifier of probability distributions, that is of distributions on X which preserve the terminal object, equivalently of complete spreads over X whose domains are locally connected and have totally connected components, the latter meaning that the connected components functor preserves pullbacks). In particular, M(X) is equivalent to B_L(T(X)).

It is therefore of interest (to me at least) to find, not just the N that I  mentioned above, but also a monad B_U, as both would presumably be topos versions of the upper power locale monad P_U. In addition, it is of interest  (to me at least) to find versions of a "single universe”, by which I mean an analogue to the double power locale monad P, which as you and C. Townsend have shown, is such that P(X) for X a locale, can be viewed either as a  composite in either direction of P_L and P_U applied to X,  or  as equivalent to the double exponentiation O^O^X (even if X not necessarily exponentiable) where O is the Sierpinski locael.

For O = the objects classifier in Top_S, the double exponential is in fact relevant already in my first (Algebra Universalis 1995) paper where I construct the symmetric topos by forcing methods, in that distributions on X can be seen as carved out of O^O^X (suitably interpreted via points). Similarly, an “upper” version N of  M can be constructed as the classifier N of local homomorphisms over toposes. The question then in my view is now how to deal with the “upper” version B_U of B_L. The analogues semiopen-open versus perfect-proper (or tidy-relatively tidy) are of course relevant to this and constitutes work in progress.

Best regards,
Marta




________________________________
From: Steve Vickers <s.j.vickers@cs.bham.ac.uk>
Sent: February 4, 2018 5:52 AM
To: categories@mta.ca
Subject: categories: Topos theory for spaces of connected components

Topos theory gives a solid account of local connectedness, where each open -  indeed, each sheaf - has a set (discrete space) of connected components.  The definition of locally connected geometric morphism covers not only individual spaces but also bundles, considered fibrewise. It also covers generalized spaces as well as ungeneralized.

Is there an analogous theory for where the space of connected components is  Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.)

The obvious example is any Stone space X, for instance, Cantor space, where  X  is its own space of connected components. We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to  notice  the Stone space aspects in the usual examples based on real analysis, since  they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components.

(By the way, if you wonder what brought me to this, it was from pondering the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for ungeneralized spaces its localic reflection is the lower powerlocale, which raises the question of whether there is a missing topos construction whose localic reflection is the upper powerlocale. On the other hand, the symmetric monad is related to local connectedness. Points of MX are cosheaves on X, and  X is locally connected if there is a terminal cosheaf in a strong  sense, with that cosheaf providing the sets of connected components. Perhaps understanding the Stone space view of connected components would cast light on this missing construction.)

All the best,

Steve.


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

Re: Topos theory for spaces of connected components

[George Janelidze]

Dear Steve,

Let me use this opportunity to ask a question 'at a lower level', referring
to papers listed at the end of this message. How seriously it is related to
your question? I don't know, but since I was going to ask it one day anyway,
let me ask it now:

As you know, taking connected components gives reflections:

(a) Locally connected spaces--->Sets,
(b) Compact Hausdorff spaces--->Stone spaces,

and although it is easy to put them together to involve all topological
spaces, there is no NICE such reflection. But what is "nice"? To me,
inspired by Galois theory, "nice" would mean "Grothendieck fibration", or,
equivalently in this case, it means "semi-left-exact" in the sense of [CHK].
The fact that (a) is semi-left-exact is used in Galois theory in my several
papers with and without co-authors, but I would rather call it a folklore
result (probably very old, and, for example, hidden in a sense in [BD]). The
fact that (b) is semi-left-exact and even has stable units in the sense of
[CHK], which is also easy, is explicitly stated and used in [CJKP], to
define the (compact) monotone-light factorization categorically; various
analogous results (but in different categories) are obtained by J. J. Xarez.
A more general story, but with weaker results (still sufficient for
something in Galois theory) are in [CJ]. Another kind of developments, very
interesting and involving toposes, are in several papers of M. Bunge, some
with J. Funk - I am not listing them since Marta can obviously do it better.

My question is a 'localic question' (this is what I mean by "lower level"),
but it might indeed be related to your 'topos-theoretic question':

As you know, a locale is called 0-dimensional if all its elements are joins
of complemented ones. By a morphism L--->L' of locales I shall mean a map
L'--->L that preserves all joins and finite meets (as usually). The
inclusion functor

0-Dimensional locales--->Locales

has a left adjoint F, for which

F(L)={x in L | x is a join of complementary elements}.

Question: Is F semi-left-exact?

I mentioned this question several times in past to several people... I am
very interested to know the answer, no matter whether it is YES or NO; if
NO, then I have weaker questions...

Best regards, George

References:
[BD] M. Barr and R. Diaconescu, On locally simply connected toposes and
their fundamental groups, Cahiers de Topologie et Geométrie Différentielle
Catégoriques 22-3, 1980, 301-314
[CHK] C. Cassidy, M. Hébert, and G. M. Kelly, Reflective subcategories,
localizations, and factorization systems, Journal of Australian Mathematical
Society (Series A), 1985, 287-329
[CJKP] A. Carboni, G. Janelidze, G. M. Kelly, and R. Paré, On localization
and stabilization of factorization systems, Applied Categorical Structures
5, 1997, 1-58
[CJ] A. Carboni and G. Janelidze, Boolean Galois theories, Georgian
Mathematical Journal 9, 4, 2002, 645-658 (Also available as Preprint
15/2002, Dept. Math. Instituto Superior Téchnico, Lisbon 2002)

--------------------------------------------------
From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk>
Sent: Sunday, February 4, 2018 12:52 PM
To: <categories@mta.ca>
Subject: categories: Topos theory for spaces of connected components

> Topos theory gives a solid account of local connectedness, where each
> open -  indeed, each sheaf - has a set (discrete space) of connected
> components. The definition of locally connected geometric morphism covers
> not only individual spaces but also bundles, considered fibrewise. It also
> covers generalized spaces as well as ungeneralized.
>
> Is there an analogous theory for where the space of connected components
> is Stone? ("Connected" is now defined by orthogonality with respect to
> Stone spaces instead of discrete spaces.)
>
> The obvious example is any Stone space X, for instance, Cantor space,
> where X  is its own space of connected components. We get Stone spaces of
> connected components more generally for any compact regular space - take
> the Stone space corresponding to the Boolean algebra of clopens. People
> tend not to notice  the Stone space aspects in the usual examples based on
> real analysis, since  they are also locally connected. Being a Stone space
> then just makes the set of connected components finite with decidable
> equality. For any compact regular space, we find that each closed subspace
> has a Stone space of connected components.
>
> (By the way, if you wonder what brought me to this, it was from pondering
> the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that
> for ungeneralized spaces its localic reflection is the lower powerlocale,
> which raises the question of whether there is a missing topos construction
> whose localic reflection is the upper powerlocale. On the other hand, the
> symmetric monad is related to local connectedness. Points of MX are
> cosheaves on X, and  X is locally connected if there is a terminal cosheaf
> in a strong sense, with that cosheaf providing the sets of connected
> components. Perhaps understanding the Stone space view of connected
> components would cast light on this missing construction.)
>
> All the best,
>
> Steve.
>

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

Re: Topos theory for spaces of connected components

[John Baez]

Steve Vickers wrote:

> We get Stone spaces of connected components more generally for any
compact
regular space - take the Stone space corresponding to the Boolean algebra
of clopens.
People tend not to notice  the Stone space aspects in the usual examples
based on
real analysis, since  they are also locally connected. Being a Stone space
then just
makes the set of connected components finite with decidable equality. For
any compact
regular space, we find that each closed subspace has a Stone space of
connected components.

Digressing a bit, this reminds me of some results David Roberts recently
pointed out.
However, they concern path-connected components rather than connected
components.
The set of path-connected components of a space X is a quotient set of X,
so we can give
it the quotient topology.   What can the resulting space be like?

Anything!     For every topological space X, there is a paracompact
Hausdorff space
whose space of path-connected components is homeomorphic to X.

D. Harris, Every space is a path-component space, Pacific J. Math. 91 no. 1
(1980) 95-104.
http://dx.doi.org/10.2140/pjm.1980.91.95

There is more here:

https://mathoverflow.net/questions/291443/paths-in-path-component-spaces

Best,
jb


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

Re: Topos theory for spaces of connected components

[Steve Vickers]

Dear Marta,

Johnstone showed that B_L(X) is a partial product of X against the
"generic local homeomorphism", a geometric morphism p from the
classifier of pointed objects to the object classifier. A point of
B_L(X) is a family of points of X, indexed by elements of a set.

He also proposed other partial products, for example those against the
generic entire map, which goes to the classifier for Boolean algebras
from the classifier of Boolean algebras equipped with prime filter.
Wouldn't that be your B_U? A point would be a family of points of X,
indexed by points of a Stone space.

Steve.


On 04/02/2018 16:48, martabunge@hotmail.com wrote:
> Dear Steve,
>
> I have nothing to say about your Stone spaces question in general,
> except for your remarks in the second part of your message about the
> symmetric monad M, where you suggest that the Stone locale view of
> connected components would perhaps cast light on the missing
> construction of a topos version N of the upper power locale P_U, just
> as the symmetric monad M is a topos version of the lower power locale
> P_L.
>
> In my paper ?Pitts monads and a lax descent theorem? (2015), (Remark
> 7.6), I leave it as an open question (more or less) the construction
> of such an N. [ The name ?Pitts monad? I gave on account on a
> condition which first appears in a theorem of A.M. Pitts whereby, in a
> lax pullback with bottom map an S-essential geometric morphisms, the
> top map is locally connected. The S-essential geometric morphisms are
> precisely the M-maps, and for the lower power locale monad P_L, the
> P_L-maps are the open maps. ]
>
> However, toposes are more complicated than locales and a perfect
> analogue may not be what one should seek Indeed, one can view the
> symmetric monad M (classifier of distributions on toposes X,  or
> equivalently of complete spreads over X with a locally connected
> domain) as a topos version of the lower power locale  P_L. There is
> however another such candidate, which is the bagdomain monad B_L
> (classifier of bags of points, or equivalently of branched coverings
> over X,  namely of those complete spreads that are purely locally
> equivalent to a locally constant cover). See M. Bunge and J. Funk,
> Singular Coverings of Toposes (2006), (Def. 9.32). In the same
> source SCT ( 8.3) there is a diagram which shows that there are two
> factorizations of the unit X?> M(X), namely one through the unit X?>
> B_L(X) and the other through the unit X?> T(X) where T (classifier of
> probability distributions, that is of distributions on X which
> preserve the terminal object, equivalently of complete spreads over X
> whose domains are locally connected and have totally connected
> components, the latter meaning that the connected components functor
> preserves pullbacks). In particular, M(X) is equivalent to B_L(T(X)).
>
> It is therefore of interest (to me at least) to find, not just the N
> that I mentioned above, but also a monad B_U, as both would presumably
> be topos versions of the upper power locale monad P_U. In addition, it
> is of interest (to me at least) to find versions of a "single
> universe?, by which I mean an analogue to the double power locale
> monad P, which as you and C. Townsend have shown, is such that P(X)
> for X a locale, can be viewed either as a composite in either
> direction of P_L and P_U applied to X,  or  as equivalent to the
> double exponentiation O^O^X (even if X not necessarily exponentiable)
> where O is the Sierpinski locael.
>
> For O = the objects classifier in Top_S, the double exponential is in
> fact relevant already in my first (Algebra Universalis 1995) paper
> where I construct the symmetric topos by forcing methods, in that
> distributions on X can be seen as carved out of O^O^X (suitably
> interpreted via points). Similarly, an ?upper? version N of  M can be
> constructed as the classifier N of local homomorphisms over toposes.
> The question then in my view is now how to deal with the ?upper?
> version B_U of B_L. The analogues semiopen-open versus perfect-proper
> (or tidy-relatively tidy) are of course relevant to this and
> constitutes work in progress.
>
> Best regards,
> Marta
>
>

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

Re: Topos theory for spaces of connected components

[Steve Vickers]

Dear John,

For point-set results like this it can be a bit delicate working out how
the point-free topos treatment goes.

Moerdijk has proved that for a connected, locally connected topos X, the
map ends: X^I -> XxX is an open surjection.

(Here I = [0,1] is the closed real interval, and if p: I -> X then
ends(p) = (p(0), p(1)).)

This is interpreted as the appropriate point-free way to say that X is
path-connected, so connected, locally connected => path connected -
which goes against the classical account. Part of the issue is that a
point-free surjection is not necessarily surjective on points.

Hence even for locally connected spaces, which are supposed to be the
well behaved ones, the path-connected components got from the topos
theory (which, by Moerdijk's result, agree with the connected
components) may be different from the ones got from point-set topology.

All the best,

Steve.

On 04/02/2018 20:57, John Baez wrote:
> Steve Vickers wrote:
>
>> We get Stone spaces of connected components more generally for any
> compact
> regular space - take the Stone space corresponding to the Boolean algebra
> of clopens.
> People tend not to notice  the Stone space aspects in the usual examples
> based on
> real analysis, since  they are also locally connected. Being a Stone space
> then just
> makes the set of connected components finite with decidable equality. For
> any compact
> regular space, we find that each closed subspace has a Stone space of
> connected components.
>
> Digressing a bit, this reminds me of some results David Roberts recently
> pointed out.
> However, they concern path-connected components rather than connected
> components.
> The set of path-connected components of a space X is a quotient set of X,
> so we can give
> it the quotient topology.   What can the resulting space be like?
>
> Anything!     For every topological space X, there is a paracompact
> Hausdorff space
> whose space of path-connected components is homeomorphic to X.
>
> D. Harris, Every space is a path-component space, Pacific J. Math. 91 no. 1
> (1980) 95-104.
> http://dx.doi.org/10.2140/pjm.1980.91.95
>
> There is more here:
>
> https://mathoverflow.net/questions/291443/paths-in-path-component-spaces
>
> Best,
> jb
>
>



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

Re: Topos theory for spaces of connected components

[Eduardo J. Dubuc]

For any topos in SGA4 SLN 169 IV Exercice 8.7 it is established that the
constant sheaf functor has a proadjoint. Thus the "connected components"
of any topos form a proset, which in the locally connected case is an
actual set. I do not remember a characterization of the category
Pro(Set), but I do remember that the category Pro(finiteSet) is the
category of stone spaces (this means that the inverse limit set with the
product topology wholy characterize the proset). Thus, the topos with an
Stone space of connected component are those in which the proset of
connected components is a proset of finite sets. This are exactly the
quasi-compact Topos (all covers of 1 admits a finite subcover). Obvious
question is if this can be extended to the general case, that is taking
the inverse limit of the proset with the product topology (that is,
totally disconnected topological spaces). We know this can not be the
case since the inverse limit may be empty, but may be the inverse limit
in the category of locales is worth to investigate.

Best   e.d.

On 04/02/18 07:52, Steve Vickers wrote:
> Topos theory gives a solid account of local connectedness, where each
> open -  indeed, each sheaf - has a set (discrete space) of connected
> components. The definition of locally connected geometric morphism
> covers not only individual spaces but also bundles, considered
> fibrewise. It also covers generalized spaces as well as
> ungeneralized.
>
> Is there an analogous theory for where the space of connected
> components is Stone? ("Connected" is now defined by orthogonality
> with respect to Stone spaces instead of discrete spaces.)
>
> The obvious example is any Stone space X, for instance, Cantor space,
> where X  is its own space of connected components. We get Stone
> spaces of connected components more generally for any compact regular
> space - take the Stone space corresponding to the Boolean algebra of
> clopens. People tend not to notice  the Stone space aspects in the
> usual examples based on real analysis, since  they are also locally
> connected. Being a Stone space then just makes the set of connected
> components finite with decidable equality. For any compact regular
> space, we find that each closed subspace has a Stone space of
> connected components.
>
> (By the way, if you wonder what brought me to this, it was from
> pondering the symmetric monad M on Grothendieck toposes. Bunge and
> Funk proved that for ungeneralized spaces its localic reflection is
> the lower powerlocale, which raises the question of whether there is
> a missing topos construction whose localic reflection is the upper
> powerlocale. On the other hand, the symmetric monad is related to
> local connectedness. Points of MX are cosheaves on X, and  X is
> locally connected if there is a terminal cosheaf in a strong sense,
> with that cosheaf providing the sets of connected components. Perhaps
> understanding the Stone space view of connected components would cast
> light on this missing construction.)
>
> All the best,
>
> Steve.
>



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

Reflection to 0-dimensional locales

[George Janelidze]

Dear Colleagues,

Let me repeat from my exchange of massages with Steve Vickers:

> As you know, a locale is called 0-dimensional if all its elements are
> joins
> of complemented ones. By a morphism L--->L' of locales I shall mean a map
> L'--->L that preserves all joins and finite meets (as usually). The
> inclusion functor
>
> 0-Dimensional locales--->Locales
>
> has a left adjoint F, for which
>
> F(L)={x in L | x is a join of complementary elements}.
>
> Question: Is F semi-left-exact?
>
> I mentioned this question several times in past to several people... I am
> very interested to know the answer, no matter whether it is YES or NO; if
> NO, then I have weaker questions...

Almost immediately after writing this I received the following message:

"...a sufficient condition for F failing to be semi-left exact is for the
coproduct of a connected frame Q and a Boolean frame X to have a
complemented element that is not in the image of the inclusion of X. I
believe such an example is described in chapter XIII, section 4, pages
260--266 of the book Frames and Locales by Picado and Pultr..."

The author is Graham Manuell, a PhD student at the University of Edinburgh
who did his MSc in Cate Town.

I looked at the book: it will take me a long time (which I don't have now)
to check the details, because understanding them will require carefully
reading every word of those pages... But if what the book says is correct (I
cannot imagine these good authors to be careless of course!), then what
Graham says is certainly correct, in spite of the fact that
semi-left-exactness is not mentioned in the book. The example, as the
authors say, was found by I. Kriz (I apologize for not using proper accents
on r, i, and z).

Moreover, most of the "weaker questions" I had in mind, are also answered...

But I still have a question: Kriz's example is presented as a
counter-example to a frame-theoretic counterpart of a purely topological
property, but now - thanks to Graham's simple remark - it is also a
counter-example to semi-left-exactness, whose topological counterpart also
fail (unless we restrict spaces to, say, locally connected, or compact). Is
there an easier counter-example?

George Janelidze


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

Re: Reflection to 0-dimensional locales

[Andrej Bauer]

On Tue, Feb 6, 2018 at 12:01 PM, George Janelidze
<George.Janelidze@uct.ac.za> wrote:
> Dear Colleagues,
>
> Let me repeat from my exchange of massages with Steve Vickers:
>
>> As you know, a locale is called 0-dimensional if all its elements are
>> joins
>> of complemented ones. By a morphism L--->L' of locales I shall mean a map
>> L'--->L that preserves all joins and finite meets (as usually). The
>> inclusion functor
>>
>> 0-Dimensional locales--->Locales
>>
>> has a left adjoint F, for which
>>
>> F(L)={x in L | x is a join of complementary elements}.
>>
>> Question: Is F semi-left-exact?

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.

With kind regards,

Andrej


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

Re: Topos theory for spaces of connected components

[Marta Bunge]

Dear Steve,

You wrote:


> Topos theory gives a solid account of local connectedness, where each
> open -  indeed, each sheaf - has a set (discrete space) of connected
> components.
[...]
>
> Is there an analogous theory for where the space of connected
> components is Stone? ("Connected" is now defined by orthogonality
> with respect to Stone spaces instead of discrete spaces.)

I have only partial answers to your question. 

Consider F a bdd S-topos, not necessarily locally connected. There are two instances of non-discrete localic generalizations of the discrete \Pi_0(F) of connected components that may be relevant. They were both reported in my  lecture "On two non-discrete localic generalizations of \pi_0” at the Colloque Internationale ‘Charles Ehresmann : 100 ans”, Amiens, 2005. An abstract is included in Cahiers de Top.et Geo.Diff.Cat 46-3 (2005). A fuller account of my lecture can be found in my Research Gate page. It consists of two unrelated parts. 

The first part (otherwise unpublished) reports my construction of the totally (paths) disconnected topos P_0(F) of path components of F by collapsing paths to a point. It was also the subject matter of a lecture that I gave at UNIGE Seminar in 2003, and of another that I gave at the Workshop on the Ramifications of category Theory, Firenze, 2003. 

     
The second part (in collaboration with J. Funk, published as “Quasicomponents in topos theory : the hyperpure-complete spread factorization”, Math.. Proc. Camb. Phil. Soc 142. 2007 ) contains a construction of  the zero-dimensional topos \P_0(F) of quasicomponents of F. 

Both reduce to the usual (discrete) in the case of a locally connected topos F. 

With best regards,
Marta



----- Original Message -----
From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk>
To: categories@mta.ca
Sent: Sunday, February 4, 2018 5:52:14 AM
Subject: categories: Topos theory for spaces of connected components

Topos theory gives a solid account of local connectedness, where each open -  indeed, each sheaf - has a set (discrete space) of connected components.  The definition of locally connected geometric morphism covers not only individual spaces but also bundles, considered fibrewise. It also covers generalized spaces as well as ungeneralized.

Is there an analogous theory for where the space of connected components is  Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.)

The obvious example is any Stone space X, for instance, Cantor space, where  X  is its own space of connected components. We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to  notice  the Stone space aspects in the usual examples based on real analysis, since  they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components.

(By the way, if you wonder what brought me to this, it was from pondering the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for ungeneralized spaces its localic reflection is the lower powerlocale, which raises the question of whether there is a missing topos construction whose localic reflection is the upper powerlocale. On the other hand, the symmetric monad is related to local connectedness. Points of MX are cosheaves on X, and  X is locally connected if there is a terminal cosheaf in a strong  sense, with that cosheaf providing the sets of connected components. Perhaps understanding the Stone space view of connected components would cast light on this missing construction.)

All the best,

Steve.

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

Re: Reflection to 0-dimensional locales

[George Janelidze]

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


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

Re: Reflection to 0-dimensional locales

[Matias M]

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/ ]