Re: Topos theory for spaces of connected components

[Marta Bunge <bunge@math.mcgill.ca>]

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The first part (otherwise unpublished) reports my construction of the totally (paths) disconnected topos P_0(F) of path components of F by collapsing paths to a point.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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