This was going to be a comment but got too long. First of all your framework is I think slightly off. You want to associate to a topological group G its classifying topos of continuous G-sets and its canonical point (the underlying set functor). The underlying set functor is pro-representable. Thus I think you really want to look at the category consisting of a pro-representable functor functors and you want to assign the automorphism group of this functor which is a strict pro-discrete group, whose projective limit should be seen in the category of locales. There is a paper of Moerdjik, Prodiscrete groups and Galois toposes, giving the corresponding Galois theory.
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This was going to be a comment but got too long. First of all your framework is I think slightly off. You want to associate to a topological group G its classifying topos of continuous G-sets and its canonical point (the underlying set functor). The underlying set functor is pro-representable. Thus you really want to look at the category consisting of a pro-representable functor and you want to assign the automorphism group of this functor which is a strict pro-discrete group, whose projective limit should be seen in the category of locales. There is a paper of Moerdjik, Prodiscrete groups and Galois toposes, giving the corresponding Galois theory. |
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