Re: Topos theory for spaces of connected components

[Matias M <matias.menni@gmail.com>]

Dear colleagues,

I have some information that may be relevant to the thread started by Steve
Vickers. (Details may be found in my article in the recent  Freyd-Lawvere
issue of the Tbilisi journal.)

Steve Vickers <s.j.vickers@cs.bham.ac.uk> escribió:

> Topos theory gives a solid account of local connectedness, where each
> open -  indeed, each sheaf - has a set (discrete space) of connected
> components.
[...]
>
> 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.)

Let  p:E ---> S be a hyperconnected and local geometric morphism.
(The intuition is that E is a topos of spaces and that the inverse image
p^* : S ---> E is the full subcategory of discrete spaces.)
A construction suggested by Lawvere produces a finite-product preserving
and idempotent monad pizero : E ---> E which, I think, is relevant to
Steve's question. Indeed, the paper mentioned above gives evidence to
support the intuition that:
1) pizero assigns, to each space, its associated space of connected
components, and
2)  the full subcategory of E given by the pizero-algebras is  the
subcategory of totally separated spaces.

Let me repeat some of that evidence here.

If p : E ---> S is, moreover, locally connected then
pizero = p^* p_! : E ---> E; that is,
pizero X   is the discrete space of connected components of X.
In other words, if p is lc then the pizero construction produces
essentially the left adjoint to p^*.


A motivating example that is not locally connected is Johnstone's
topological topos
p: J ---> Sets.
For each X in J, pizero X is the totally separated space of
`quasi-components' of X. The pizero-algebras are exactly the totally
separated sequential spaces.

(The construction works in categories that need not be toposes so, for
instance, it gives the `correct' result in the case of compactly generated
Hausdorff spaces.)

Of course, the inclusion of pizero-algebras into E has a finite-product
preserving left adjoint. George's mail suggests the question if this
reflection is semi-left-exact. It also raises the question if the explicit
construction that George gives of the left adjoint to

The inclusion functor
> 0-Dimensional locales--->Locales


is the result of a variant of Bill's construction (using an exponentiating
object and a `good' factorization system).

I must admit that I don't know how the above connects with the work of
Bunge-Carboni-Funk, but Marta mentions

the double exponentiation O^O^X (even if X not necessarily exponentiable)
> where O is the Sierpinski locael.


and that already suggests a connection.

Best regards, Matías.


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