I was motivated for quite a while by the search for non-abelian analogues of the locale case. Check out the (physics) papers by the Isham group on this idea – eg. by Raptis.

]]>I’ve been studying localic topoi recently and find the process of recovering a topological space from a locale to be quite powerful. A point p: 1 -> X of a locale X can be expressed as a frame morphism to the initial frame {0,1}. This standard definition is reminiscent of Connes’ use of the two point space in the NCG Standard Model.

For M-theory, one may have to extend the initial frame to {0,1,2}, so that any point in the M-theory locale is defined to be a morphism of frames p^{-1}: O(X) -> {0,1,2}.

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