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By Andre Joyal, Myles Tierney

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Y <_ x Then and This is not yet a P, since we have not required that singletons be covers, or that the local axiom is satisfied. It is a system of generators for a topology. Now consider the locale the subset of and P(P)xp(P) R e Cov(x). P(P) of downward closed subsets of consisting of pairs (R, +(x)) where P, and x e P Making this inf-stable, we obtain the set of pairs GALOIS THEORY 25 (R A S , +(x) A S) where S e P(P) and R e Cov(x) . By Proposition 4 the locale quotient Q of P(P) by the generated congruence relation is Q = {T e P(P) |Vx e P VR e Cov(x) VS e P(P) R A S C T iff + (x) A S C T} .

Then We have to show: enough to show is open, and x u e 0 (X) u = satisfies \/ u1-- ieJ Vi e I, u. £ V / u . But u. <_ p~31u - , so it is 1 1 jeJ 3 u i A p"3 U i <_ X / u . J j eJ 3 u . < p*(u. ) 1 x jeJ 3 By the usual argument, it is enough to show that if the left hand side of the bottom line equals 1, then the right hand side equals 1, but this is obvious. Proof of Theorem 1: An atom of X is an open subspace a c — > X such that a*a CZ A and 3 a = 1- Let A be the set of atoms. Each atom a defines a point of X, since a = 1 by Lemma 1.

Let M e s£(S ). The sup-lattice M(l) e s£(S) i s A. JOYAL § M. TIERNEY 48 equipped with a canonical Z-module structure, and putting defines an equivalence of categories T: sl(SL Moreover, for any pair M,N e si(S zop zop TM = M(l) ) -> Mod(Z) ) , we have a natural isomorphism T(Hom(M,N)) « Homz(TM,TN). e. M: Z o p -• si(S) is a functor satisfying conditions 1) and 2) of Proposition 1. If a <_ b, denote the morphism M(b) •* M(a) by p , and write E^ for its left adjoint. Then, the multiplication by a on TM = M(l) is given by the composite Ia p a : M(l) ->• M(l).