multiverse concepts?

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Mar 3, 2021 at 9:22	comment	added	Dustin Clausen		Apparently, these very special complete boolean algebras are not so interesting from a forcing perspective.
Mar 3, 2021 at 9:22	comment	added	Dustin Clausen		Related to this, it might be relevant to note that to specify a condensed set one need only specify its values on a very special class of complete boolean algebras, namely ones that are the boolean algebra of subsets of some set S (corresponding to the extr. disc. space given by the Stone-Cech compatification of S). The values on more exotic complete boolean algebras are determined from these, because they are retracts... but only in the category of boolean algebras, not in the category of complete boolean algebras and sup-preserving maps (cf the distinction Simon draws).
Mar 1, 2021 at 14:33	history	edited	Simon Henry	CC BY-SA 4.0	added 92 characters in body
Mar 1, 2021 at 13:55	comment	added	Tim Campion		I wonder if there's some kind of gros topos / petit topos distinction to be made. Maybe the site of Stonean spaces and all maps could be our big topos and the site of Stonean spaces and open maps could be our little topos...
Feb 28, 2021 at 23:19	comment	added	David Roberts♦		The (co)lax (co)limit is, I think, the same as the total topos I mentioned in a comment on the question. But you make a very good point about what sort of morphisms are allowed between Boolean algebras.
Feb 28, 2021 at 19:46	comment	added	Simon Henry		Absolutely. In the way I phrased the above, Stonean locales enter the picture beacause the category of Stonean locale is equivalent to the opposite of the category of complete boolean algebra with arbitrary boolean algebra map between them. But they are indeed also exactly the Stone-Czech compactification of Boolean locales. That's probably relevant as well, but so fat that fact doesn't play a role in the way I phrased things above (at least not in a clear way).
Feb 28, 2021 at 19:42	comment	added	მამუკა ჯიბლაძე		Not sure I understand your viewpoint but I believe extremally disconnected spaces mentioned in the OP must enter your picture one way or another. In other words, along with the Boolean locales you have the Stonean locales, whose frames of opens are the frames of all ideals of some complete Boolean algebra. I believe they are some sort of compactifications of Boolean locales and must play some rôle, no?
Feb 28, 2021 at 19:17	history	edited	Simon Henry	CC BY-SA 4.0	added 612 characters in body
Feb 28, 2021 at 14:29	history	answered	Simon Henry	CC BY-SA 4.0