By Andre Joyal, Myles Tierney

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**Extra info for An Extension of the Galois Theory of Grothendieck**

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X, iel x and N is a sheaf. Conversely, let the action that M: A 0 ^ + s£(S) AxM(l) -> M(l) be a sheaf. e. -x. e. i i paa W i The partial order on where iel M < ~ P\a i (y). is a subsheaf of MxM, so we can conclude that But then v l 1, . E a P a (x) „ £ y or a-x£y, and we are done. Proposition 2. The category Loc(sh(A)) is naturally equivalent to the category of locale extensions A -*• B of A in 5. Proof: The fact that the functor T of Proposition 3 §2 preserves Horn, means that in Proposition 1, it also preserves ©-product.

Thus h (S) is closed. Any element of 0(S) becomes complemented in 0(h~ (S)), since 0(X)--* 0(S) is surjective, and every element in 0 (X) becomes complemented in 0(X)'. This shows h (S) - S' - just check the universal property. Also, in the pullback square h _ 1 (S) >S X' > X , the top line is an epimorphism, since h (S) = S'. But then h(h (S)) = S. Finally, a closed subspace of X 1 is described by a condition v = 0 for some v e 0(X)'. )'. x x iel Going backwards, this means that the closed subspace h" S, where 0(S) is the quotient of generated by the set of pairs Thus, we have (u^u^A CV is equal to 0(X) by the congruence relation v i ^ iel" 32 A.

V and a 0(V A CU) = {x e 0 ( X ) | u A v < x < v}. Finally, let us remark that a continuous mapping f: X -*• Y can be factored uniquely as an epimorphism followed by a subspace inclusion: X f > Y f(X) The lattice 0(f(X)) is the set of all closed elements for the local operator f *f" : 0(Y) -»• 0 (Y) . We say f(X) is the image of f. It is the smallest subspace of Y through which f factors. £—• X is a subspace of X, its image f(X£) is the subspace of Y determined by the local operator f *£f": 0 (Y) -*• 0(Y).