Re: Topos theory for spaces of connected components

Re: Topos theory for spaces of connected components

[Matias M]

Dear colleagues,

I have some information that may be relevant to the thread started by Steve
Vickers. (Details may be found in my article in the recent  Freyd-Lawvere
issue of the Tbilisi journal.)

Steve Vickers <s.j.vickers@cs.bham.ac.uk> escribió:

> 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.)

Let  p:E ---> S be a hyperconnected and local geometric morphism.
(The intuition is that E is a topos of spaces and that the inverse image
p^* : S ---> E is the full subcategory of discrete spaces.)
A construction suggested by Lawvere produces a finite-product preserving
and idempotent monad pizero : E ---> E which, I think, is relevant to
Steve's question. Indeed, the paper mentioned above gives evidence to
support the intuition that:
1) pizero assigns, to each space, its associated space of connected
components, and
2)  the full subcategory of E given by the pizero-algebras is  the
subcategory of totally separated spaces.

Let me repeat some of that evidence here.

If p : E ---> S is, moreover, locally connected then
pizero = p^* p_! : E ---> E; that is,
pizero X   is the discrete space of connected components of X.
In other words, if p is lc then the pizero construction produces
essentially the left adjoint to p^*.


A motivating example that is not locally connected is Johnstone's
topological topos
p: J ---> Sets.
For each X in J, pizero X is the totally separated space of
`quasi-components' of X. The pizero-algebras are exactly the totally
separated sequential spaces.

(The construction works in categories that need not be toposes so, for
instance, it gives the `correct' result in the case of compactly generated
Hausdorff spaces.)

Of course, the inclusion of pizero-algebras into E has a finite-product
preserving left adjoint. George's mail suggests the question if this
reflection is semi-left-exact. It also raises the question if the explicit
construction that George gives of the left adjoint to

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).

I must admit that I don't know how the above connects with the work of
Bunge-Carboni-Funk, but Marta mentions

the double exponentiation O^O^X (even if X not necessarily exponentiable)
> where O is the Sierpinski locael.


and that already suggests a connection.

Best regards, Matías.


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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.

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Re: Topos theory for spaces of connected components

[Marta Bunge]

Dear Steve,

Curiously enough, what you describe about how to get a point of MX (a cosheaf, or distribution) is precisely what did in 1990.  I communicated it privately to Lawvere at Como 1990, as he was the one who had left the question as an open one. My method to do that was that of "forcing topologies" (Tierney)and not unlike what is done in the Joyal-Tierney paper except for forgetting the lex part. Unfortunately, when I sent a paper "Cosheaves and distributions on toposes" containing this result to Peter Freyd for the JPAA, whoever refereed it rejected it without giving a reason, but I know that it was my fault as the paper was not too clearly written. I subsequently sent it to Algebra Universalis where it did appear (same title and alas, still not too clearly written) but very much later (1995).

That sabbatical year 1995-96 I was spent at the Universita di Genova where (not coincidentally) Aurelio (Carboni) had just taken a job away from Milan. Aurelio immediately understood my construction, but thought that it would  be "wiser" to set it in algebraic terms. This resulted in our joint paper "The symmetric topos", which this time it did appear in JPAA (1995). We did  more than that in that paper, namely to extend it to a KZ-monad and characterize its algebras. I therefore abandoned the fibrational point of view which, as you say, found its way again in my work with Jonathon (Funk) as the  complete spreads with a locally connected domain. However, just this morning (and before reading what you just wrote) I was thinking of using the fibrational approach again for constructing the topos NX corresponding to the upper power locale, just as the topos MX corresponds to the lower power locale. 

It seems to me now that modulo some differences this is what you are trying  to do yourself. Of course it would be "wiser" as Aurelio would have said, to do it "algebraically" and this is what I did for the coherent monad in my "Pitts paper" (2015). So I am pursuing that line as well. I will read the  rest of your message later and maybe respond to your question about it privately. It may take a few days as other things are interfering with my work  at present. 

Many thanks for your remarks. 

All the best,
Marta     

----- Original Message -----
From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk>
To: bunge@math.mcgill.ca
Cc: categories@mta.ca
Sent: Tuesday, February 6, 2018 4:19:50 AM
Subject: Re: categories: Topos theory for spaces of connected components

Dear Marta,

Here's my thinking on connected components.

For M, the paradigm example for how to get a point of MX (a cosheaf, or distribution) is to take locally connected space Y with map p: Y -> X, and then to each sheaf U over X assign the set of connected components of p*U. This gives a covariant functor from SX to Set, and it preserves colimits. If X  is an ungeneralized space, then it suffices to do that for opens U, and the extension to sheaves follows. Your theory of complete spreads shows that that paradigm example is in fact general.

The extreme case of p is when X is itself locally connected and we can take  p to be the identity. The corresponding cosheaf is terminal in a strong sense: as global point of MX it provides a right adjoint to the map MX -> 1. The unit of the adjunction provides a unique morphism from the generic cosheaf to the terminal one.

If X is exponentiable, then (always? In favourable cases?) the cosheaf as described above can be got by taking points for a map R^X -> R, where R is (following your notation) the object classifier. This points out Lawvere's analogy with integration, where R would be the real line. Then just as Riesz  picks out the linear functionals as the distributions, we are interested in the colimit-preserving ones.

In the above account, the role of local connectedness is to ensure that the  connected components genuinely do form a set, a discrete space. What happens if we look for a Stone space instead? Here is my conjecture.

1. For ungeneralized X we should be looking for a Stone space of connected components of p*U for each _closed_ U. Y will need a suitable condition (strongly compact?) as analogue of local compactness. (By Stone duality that could also be expressed by assigning (covariantly) a Boolean algebra to each  open.)

2. Noting that a closed embedding is fibrewise Stone, that assignment will extend to U an arbitrary fibrewise Stone (entire) bundle over X - that is to say, by Stone duality and contravariantly, a sheaf of Boolean algebras.

3. For generalized X that will provide our Stone notion of cosheaf. The assignment from entire bundles to Stone spaces should preserve finite colimits  and cofiltered limits. There's an obvious technical hurdle of how to express that directly in terms of sheaves instead of entire bundles.

4. If X is exponentiable then this time, by Stone duality, we are looking for maps [BA]^X -> [BA] where [BA] is the classifier for Boolean algebras. They must preserve filtered colimits (automatic for maps) and finite limits.  NX would exist for arbitrary X, and classify those maps. 

Obviously there's lots to go wrong there, but do you think your coherent monad fits any of those points for coherent X?

By the way, although I haven't mention the effective lax descent and relatively tidy maps, I am interested in them. They are connected with stable compactness and Priestley duality.

All the best,


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Re: Topos theory for spaces of connected components

[Steve Vickers]

Correction - In Conjecture 1 I mistakenly wrote "local compactness" for
"local connectedness".
------

Dear Marta,

Here's my thinking on connected components.

For M, the paradigm example for how to get a point of MX (a cosheaf, or
distribution) is to take locally connected space Y with map p: Y -> X,
and then to each sheaf U over X assign the set of connected components
of p*U. This gives a covariant functor from SX to Set, and it preserves
colimits. If X is an ungeneralized space, then it suffices to do that
for opens U, and the extension to sheaves follows. Your theory of
complete spreads shows that that paradigm example is in fact general.

The extreme case of p is when X is itself locally connected and we can
take p to be the identity. The corresponding cosheaf is terminal in a
strong sense: as global point of MX it provides a right adjoint to the
map MX -> 1. The unit of the adjunction provides a unique morphism from
the generic cosheaf to the terminal one.

If X is exponentiable, then (always? In favourable cases?) the cosheaf
as described above can be got by taking points for a map R^X -> R, where
R is (following your notation) the object classifier. This points out
Lawvere's analogy with integration, where R would be the real line. Then
just as Riesz picks out the linear functionals as the distributions, we
are interested in the colimit-preserving ones.

In the above account, the role of local connectedness is to ensure that
the connected components genuinely do form a set, a discrete space. What
happens if we look for a Stone space instead? Here is my conjecture.

1. For ungeneralized X we should be looking for a Stone space of
connected components of p*U for each _closed_ U. Y will need a suitable
condition (strongly compact?) as analogue of local connectedness. (By
Stone duality that could also be expressed by assigning (covariantly) a
Boolean algebra to each open.)

2. Noting that a closed embedding is fibrewise Stone, that assignment
will extend to U an arbitrary fibrewise Stone (entire) bundle over X -
that is to say, by Stone duality and contravariantly, a sheaf of Boolean
algebras.

3. For generalized X that will provide our Stone notion of cosheaf. The
assignment from entire bundles to Stone spaces should preserve finite
colimits and cofiltered limits. There's an obvious technical hurdle of
how to express that directly in terms of sheaves instead of entire bundles.

4. If X is exponentiable then this time, by Stone duality, we are
looking for maps [BA]^X -> [BA] where [BA] is the classifier for Boolean
algebras. They must preserve filtered colimits (automatic for maps) and
finite limits. NX would exist for arbitrary X, and classify those maps.
Obviously there's lots to go wrong there, but do you think your coherent
monad fits any of those points for coherent X?

By the way, although I haven't mention the effective lax descent and
relatively tidy maps, I am interested in them. They are connected with
stable compactness and Priestley duality.

All the best,

Steve.


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

Re: Topos theory for spaces of connected components

[Steve Vickers]

Dear Marta,

Here's my thinking on connected components.

For M, the paradigm example for how to get a point of MX (a cosheaf, or distribution) is to take locally connected space Y with map p: Y -> X, and then to each sheaf U over X assign the set of connected components of p*U. This gives a covariant functor from SX to Set, and it preserves colimits. If X is an ungeneralized space, then it suffices to do that for opens U, and the extension to sheaves follows. Your theory of complete spreads shows that that paradigm example is in fact general.

The extreme case of p is when X is itself locally connected and we can take p  to be the identity. The corresponding cosheaf is terminal in a strong sense: as global point of MX it provides a right adjoint to the map MX -> 1. The unit of the adjunction provides a unique morphism from the generic cosheaf to  the terminal one.

If X is exponentiable, then (always? In favourable cases?) the cosheaf as described above can be got by taking points for a map R^X -> R, where R is (following your notation) the object classifier. This points out Lawvere's analogy with integration, where R would be the real line. Then just as Riesz picks out the linear functionals as the distributions, we are interested in the  colimit-preserving ones.

In the above account, the role of local connectedness is to ensure that the connected components genuinely do form a set, a discrete space. What happens if we look for a Stone space instead? Here is my conjecture.

1. For ungeneralized X we should be looking for a Stone space of connected components of p*U for each _closed_ U. Y will need a suitable condition (strongly compact?) as analogue of local compactness. (By Stone duality that could also be expressed by assigning (covariantly) a Boolean algebra to each open.)

2. Noting that a closed embedding is fibrewise Stone, that assignment will extend to U an arbitrary fibrewise Stone (entire) bundle over X - that is to say, by Stone duality and contravariantly, a sheaf of Boolean algebras.

3. For generalized X that will provide our Stone notion of cosheaf. The assignment from entire bundles to Stone spaces should preserve finite colimits and cofiltered limits. There's an obvious technical hurdle of how to express that directly in terms of sheaves instead of entire bundles.

4. If X is exponentiable then this time, by Stone duality, we are looking for maps [BA]^X -> [BA] where [BA] is the classifier for Boolean algebras. They must preserve filtered colimits (automatic for maps) and finite limits. NX  would exist for arbitrary X, and classify those maps. 

Obviously there's lots to go wrong there, but do you think your coherent monad fits any of those points for coherent X?

By the way, although I haven't mention the effective lax descent and relatively tidy maps, I am interested in them. They are connected with stable compactness and Priestley duality.

All the best,

Steve.

> On 5 Feb 2018, at 18:07, bunge@math.mcgill.ca wrote:
> 
> 
> Dear Steve,
> 
> This is response to your message reproduced below. 
> 
> I am aware of Johnstone’s results on the lower bagdomain. However,  both the symmetric monad M and the lower bagdomain monad B_L on BTop_S are “on the same side” as the lower power locale monad P_L on Loc_S, and the latter is the localic reflection for both. The upper power locale monad P_U on Loc_S is “on the other side”, in a sense that is  explained in my ‘Pitts monads paper”.In it I deduce effective lax descent theorems in a general setting of what I call "Pitts KZ-monads" and "Pitts co-KZ-monads" on a “2-category of spaces”. 
> 
> In the case of M on BTop_S, it is the S-essential surjective geometric morphisms that are shown to be of lax effective descent (a result originally due to Andy Pitts). In the case of P_L on Loc-S it is the open surjections of locales that are shown to be of lax effective descent (a result originally due to Joyal-Tierney). Both M and P_L are instances of "Pitts KZ-monads". Now,   P_U on Loc_S is instead an instance of a" Pitts co-KZ-monad" and the result recovered from my general setting is that proper surjections of locales are of effective lax descent (a result originally due to Jaapie Vermeulen). What I seek is a co-KZ-monad N (or perhaps B_U) on BTop_S for which my general  theorem would give me that relatively tidy surjections of toposes are of effective lax descent (a result due to I. Moerdijk and J.C.C.Vermeulen). 
> 
> In my Pitts paper there is another consequence of the general theorem proved therein and it is that coherent surjections between coherent toposes are of effective lax descent (a result proven by different methods and by several  people,  such asM. Zawadowsky 1995, D.Ballard and W.Boshuck 1998, and I.Moerdijk and J.C.C.Vermeulen 1994,thus establishing a conjecture of Pitts 1985 (in the Cambridge Conference whose slides you have requested to Andy). It is of interest for what we are discussing to point out that the “coherent monad C” that I use therein to deduce the latter from my general theorem is a Pitts co-KZ-monad, hence on the “same side” as P_U for Loc_S.  For a coherent topos E, the coherent monad C(E) applied to it  classifies pretopos morphisms E_{coh} —> S. where E_{coh} is the full subcategory of E of coherent objects with the topology of finite coverings. This theorem is perhaps all I can get in my setting when searching for the still elusive N or B_U but I have not given up yet. 
> 
> Also in my 2015 Pitts paper there are characterizations of the algebras for a Pitts KZ-monad M (dually for a Pitts co-KZ-monad N) as the "stably M-complete objects" ("stably N-complete objects"), where the former is stated in terms of pointwise left Kan extensions along M-maps, and the latter in terms of pointwise right Kan extensions along N-maps. These notions owe much to the  work of M, Escardo, in particular to his 1998 "Properly injective spaces and function spaces”. 
> 
> I will say more when i know more myself. Thanks very much for your pointers. I will most certainly look into them even if I do not at the moment think  they are what I need. 
> 
> 
> Best regards,
> Marta
> 

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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/ ]

Re: Topos theory for spaces of connected components

[Marta Bunge]

Dear Steve,

This is response to your message reproduced below. 

I am aware of Johnstone’s results on the lower bagdomain. However, both the symmetric monad M and the lower bagdomain monad B_L on BTop_S are “on the same side” as the lower power locale monad P_L on Loc_S, and the latter is the localic reflection for both. The upper power locale monad P_U on Loc_S is “on the other side”, in a sense that is explained in my ‘Pitts monads paper”.In it I deduce  effective lax descent theorems in a general setting of what I call "Pitts KZ-monads" and "Pitts co-KZ-monads" on a “2-category of spaces”. 

In the case of M on BTop_S, it is the S-essential surjective geometric morphisms that are shown to be of lax effective descent (a result originally due to Andy Pitts). In the case of P_L on Loc-S it is the open surjections of  locales that are shown to be of lax effective descent (a result originally  due to Joyal-Tierney). Both M and P_L are instances of "Pitts KZ-monads". Now,  P_U on Loc_S is instead an instance of a" Pitts co-KZ-monad" and the result recovered from my general setting is that proper surjections of locales are of effective lax descent (a result originally due to Jaapie Vermeulen). What I seek is a co-KZ-monad N (or perhaps B_U) on BTop_S for which my  general theorem would give me that relatively tidy surjections of toposes are of effective lax descent (a result due to I. Moerdijk and J.C.C.Vermeulen). 

In my Pitts paper there is another consequence of the general theorem proved therein and it is that coherent surjections between coherent toposes are of effective lax descent (a result proven by different methods and by several people,  such asM. Zawadowsky 1995, D.Ballard and W.Boshuck 1998, and I.Moerdijk and J.C.C.Vermeulen 1994,thus establishing a conjecture of Pitts 1985 (in the Cambridge Conference whose slides you have requested to Andy). It is of interest for what we are discussing to point out that the “coherent monad C” that I use therein to deduce the latter from my general theorem is a Pitts co-KZ-monad, hence on the “same side” as P_U for Loc_S.  For a coherent topos E, the coherent monad C(E) applied to it classifies pretopos morphisms E_{coh} —> S. where E_{coh} is the full subcategory of E of coherent objects with the topology of finite coverings. This theorem is perhaps all I can get in my setting when searching for the still elusive N or B_U but I have not given up yet. 

Also in my 2015 Pitts paper there are characterizations of the algebras for  a Pitts KZ-monad M (dually for a Pitts co-KZ-monad N) as the "stably M-complete objects" ("stably N-complete objects"), where the former is stated in  terms of pointwise left Kan extensions along M-maps, and the latter in terms of pointwise right Kan extensions along N-maps. These notions owe much to the work of M, Escardo, in particular to his 1998 "Properly injective spaces and function spaces”. 

I will say more when i know more myself. Thanks very much for your pointers. I will most certainly look into them even if I do not at the moment think  they are what I need. 


Best regards,
Marta

________________________________________________

From: Steve Vickers <s.j.vickers@cs.bham.ac.uk>
Sent: February 5, 2018 9:03 AM
To: martabunge@hotmail.com
Cc: categories@mta.ca
Subject: Re: categories: Topos theory for spaces of connected components
 
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.


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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
>
>



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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
>
>

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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


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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.
>

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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/ ]

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/ ]