Hi Andrej, There is an analogous problem when trying to "extend" Gelfand duality to locally compact Hausdorff spaces and (not necessarily unital) C*-algebras: everything works fine on the object level, but there are many (not necessarily unital) *-homomorphisms C_0(1) --> C_0(2), but only one continuous map 2 --> 1. The standard solution, IIRC, is to restrict the class of *-homomorphisms to those which "preserve the approximate unit". [As it turns out: every (not necessarily unital) C*-algebra has an "approximate unit" (even a canonical one); and, for a *-homomorphism between unital C*-algebras, preserving the approximate unit is equivalent to preserving the unit.] In any event, I have found it (paradoxically) illuminating to think of: locally compact Hausdorff spaces and proper maps as a subcategory, via the one-point compactification functor (here denoted ( )+1), of compact Hausdorff spaces; and, (not necessarily unital) C*-algebras as a subcategory, via the free functor (also denoted ( )+1), of unital C*-algebras. Since C(X+1)=C_0(X)+1 holds at the level of objects (where = means isomorphic), it remains to reverse-engineer the correct classes of arrows in order to piggyback the desired statement off the usual duality theorem. Of course, it's also possible to consider the b.o./f.f. factorisations of the two ( )+1 functors: that results in some class of partial maps on the topological side, as you suggest. I expect that something similar happens in the case of Stone duality and GBAs. Hope this helps! Cheers, Jeff. --- On Fri, 1/21/11, Andrej Bauer <andrej.bauer@andrej.com> wrote:
From: Andrej Bauer <andrej.bauer@andrej.com> Subject: categories: Stone duality for generalized Boolean algebras To: "categories list" <categories@mta.ca> Received: Friday, January 21, 2011, 8:19 AM The well known Stone duality says that there is an equivalence between Boolean algebras (BA) and the opposite of Stone spaces and continuous maps. Here a Stone space is a Hausdorff zero-dimensional compact space. Furthermore, Boolean algebras correspond to Boolean rings with unit.
How exactly does this extend to generalized Boolean algebras? A generalized Boolean algebra (GBA) is an algebra with 0, binary meet, binary join, and relative complement in which meets distribute over joins. Equivalently it is a Boolean ring (possibly without a unit). I have seen it stated that the dual to these are (the opposite of) locally compact zero-dimensional Hausdorff spaces and proper maps, e.g., it is stated in Benjamin Steinberg: "A groupoid approach to discrete inverse semigroup algebras", Advances in Mathematics 223 (2010) 689-727.
Another source to look at is Givant & Halmos "Introduction to Boolean algebras", but there this material is covered in exercises and the duality is stated separately for objects and for morphisms, and I can't find an exercise that treats the morphisms, so I wouldn't count that as a reliable reference. Stone's original work does not seem to speak about morphisms very clearly (to me).
Unless I am missing something very obvious, it cannot be the case that GBA's correspond to locally compact 0-dimensional Hausdorff spaces and proper maps, for the following reason.
The space which corresponds to the GBA 2 = {0,1} is the singleton. The space which corresponds to the four-element GBA 2 x 2 is the two-point discrete space 2. There are _four_ GBA homomorphisms from 2 to 2 x 2 (because a GBA homomorphism preserves 0 but it need not preserve 1), but there is only one continuous map from 2 to 1. Or to put it another way, there are _four_ ring homomorphisms from Z_2 to Z_2 x Z_2 (because they need not preserve 1), but there is only one continuous map from 2 to 1. So, either the spectrum of a GBA is not what I think it is (namely maximal ideals), or we should be taking a more liberal notion of maps on the topological side. For example, there are _four_ partial maps from 2 to 1.
If someone knows of a reliable reference, one would be much appreciated. I won't object to a direct proof of duality either.
With kind regards,
Andrej
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