Re: Topos theory for spaces of connected components

[Marta Bunge <martabunge@hotmail.com>]

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.


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