Re: Topos theory for spaces of connected components

["George Janelidze" <George.Janelidze@uct.ac.za>]

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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