March 7, 2017

An Extension of the Galois Theory of Grothendieck by Andre Joyal, Myles Tierney

By Andre Joyal, Myles Tierney

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TIERNEY 30 Clearly, Sp(X,$) = Loc(L (1),0(X)) = 0(X). This can be formulated as follows: let s e 0($) be the free generator; for any u e 0(X) there is exactly one continuous map f: X •* $ such that f~(s) = u. $ is called the Sierpinski space. The free locale L(I) on I is the coproduct of I copies of L(l). Thus 0($ ) = L(I). We conclude that any space X can be expressed as the equalizer of a pair $ + $ of continuous maps. 4. Pullbacks and proj ective limits The locale of open parts of the product X*Y 0(X)00(Y).

R provides a natural (in N) retraction rN = r®N: B8N -• A0N - N for A A A each nN, so, for each N, N ^ \ nN > B8N > B8B0N * rw ~ T A n(B®N) A A A < 18rN is a split equalizer of A-modules. Thus, f is a descent morphism, which, in any case, is clear from Theorem 1. In addition, however, let (M,6) e Des(f) and consider the diagram M — > B8M I A r ) rMJfnM * e $(M,6) >U l®nM 0 ; B0B8M A A r(B®M) I t n (B8M) A v I A > B®M > TIM A By naturality, both right hand squares (with the r's) commute, providing a unique r: M -> $(M,8) such that er = rM°e.

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. Define (j>: 0(X) + P(A) by (u) = {a e A| a <_ u} . ) = U iel x iel x a <_ v/u. —^> J i e l a <_ 1u. So iel 1 Furthermore, has a left adjoint o(I} <|> (u A v ) = (u) H 0 0 » an<* because, since a - 1, is a morphism of locales. a: P(A) + 0(X) defined by = \ A a l a e I}. (1) - A. 1} = are both isomorphic to On the other hand, and A by adjointness I CI4>c(I), and if a' e <|>a(I), > ^ a e l a' £ a = ^ a' = a, since a' and a 1.

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