Re: Topos theory for spaces of connected components

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

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.
>



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