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

By Andre Joyal, Myles Tierney

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Thus, we get a commutative diagram 0(Y) 0 0(X) 0(Z) -om i€L 0(Y) Moreover, l®3r 1 <_ f 3r an is a left adjoint to d 3^^ £ 1 3-e' (f ~(y) A u) = y A ] r , ( u ) 103c 1 is 0(Y) 18f since the adjunctions are preserved under 8-product. f 1 •0(1) for y e 0(Y) and linear, which is true because 3 ^ u e 0(YxX) Z Proof: Let Then p We must show that 0(Z) > — E - * 0(X) is an equalizer. We have So let u e 0(X) ^ O(XxX) I* Z be such that P'GnCu)) = 3 n P2 p. (u) = p2 (u) Cpn'Cu}) 3 P ((Pi(u)) 2 since means that is 0(Z)-linear.

A. JOYAL \$ M. TIERNEY 24 are characteristic maps Xc ,Xcb : PlxPl -• PI b l 2 of subsets S ^ L ^ C P l x P l , and it is enough to check that x -*• y <_ I (x) -• I (y) . , C S . , for But for this it is enough to know that x <_ y =£> &(x) £ A(y), which is true. As we did for sup-lattices given a subset explicitly describe the quotient locale relation on a e A A and generated by R. we may suppose R of We say that (z,,z 2 ) e R, we have AA Q R C AxA, we want to A by the congruence R is inf-stable if given (a A z, , a A z«) e R.

If A is a locale of S, then sh(p # A) pt v p E > sh(A) > y S is a pullback square. Proof: Diaconescu's Theorem says that morphisms of topoi F \ are classified by left exact F f > sh(A) / S valued functors on A which take covers GALOIS THEORY to surjective families. 45 These are exactly locale morphisms A -*• q^fi, so the Proposition follows immediately from the universal property of pf The morphism is induced by the adjunction As an application of p A. A -* p*p A. p , and because we need the result in Chapter VII, we give here a new characterization of atomic topoi [5] over S.