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, MatÃas. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]