Conjecture about algebras of the double power locale monad

Conjecture about algebras of the double power locale monad

[Townsend, Christopher]

Hello all,

I have the following conjecture that is beginning to annoy me as I can't prove or disprove it. It basically says that every algebra of the double power locale monad is an exponential.

Conjecture: Let P be the double power locale monad on the category of locales and let (X,a:PX->X) be a P-algebra. In the presheaf category [Loc^op,Set] form the equalizer, E, of S^a:S^X->S^PX (where S is the Sierpinski locale) and []:S^X->S^PX, the univeral map obtained via (double) exponential transpose of the identity on PX (recall PX iso. S^(S^X)). Then the exponential S^E exists in [Loc^op,Set], is representable, and is represented by X.

[Note: S^X is the presheaf that sends a locale Y to Loc(YxX,S)]

If someone can find a counterexample, please let me know so that I can stop  wasting time on it ...

If someone wants some details on cases I think I've covered so far, let me know.

Thanks, Christopher


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