Re: Topos theory for spaces of connected components

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

Dear Marta,

Johnstone showed that B_L(X) is a partial product of X against the
"generic local homeomorphism", a geometric morphism p from the
classifier of pointed objects to the object classifier. A point of
B_L(X) is a family of points of X, indexed by elements of a set.

He also proposed other partial products, for example those against the
generic entire map, which goes to the classifier for Boolean algebras
from the classifier of Boolean algebras equipped with prime filter.
Wouldn't that be your B_U? A point would be a family of points of X,
indexed by points of a Stone space.

Steve.


On 04/02/2018 16:48, martabunge@hotmail.com wrote:
> 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
>
>

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