Topos theory for spaces of connected components

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

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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