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By Andre Joyal, Myles Tierney

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F = n f. ) «- iel commutes, since the Z are the canonical injections in a coproduct i of sup-lattices. Hence, f~(Z (1)) = Z (1), from which the final p y i i statement follows: any diagram of the form y 0(W) 0(U) where 3. u e 0(W), v e 0(Z) and q > 0(Z) > 0(V) q (u) = v, is a pushout of locales. Conditions for openness Proposition 1. For any space X, the map p": P(l) + 0(X) has a left adjoint. p: X -*• 1 is open iff Proof: Suppose p" has a left adjoint 3: 0(X) -*• P(l). We have to prove that for any y e P(l) and x e 0(X), Y A 3(x) -3(p"(y) A X).

Iel is an epimorphism. This is the dual of Proposition 1 Chapter III §2. Note the striking difference here with the theory of classical spaces, even when AC is satisfied in S , where an instructive counterexample is given by the poset of finite partial injections from a large set to a smaller infinite one. GALOIS THEORY 5. 31 The splitting space If X is a space, write h: X' + X for the dual of the solution 0(X) + 0(X) f to the problem of adding complements to every element of 0(X). We say X' is the splitting space of X.

Let L e s£(S), and express L as the coequalizer of a pair PX r . XxY. Applying p*, we obtain relations p*R, p*S C p*Xxp*Y, hence two morphisms p*X- of s£(E). n Q Their coequalizer is p (L). Proposition 2. We have: p*Y A. JOYAL § M. TIERNEY 44 1) Hom(L,p*(M)) - p*Hom(p (L),M) 2) p (PX) * ftP A 3) p # (L®M) = p # (L)0p # (M) 4) L®p*(M) - p*(p # (L)®M). Proof: The first identity expresses the (strong) adjointness p lp*2) and 3) are immediate consequences of 1 ) , and 4) follows from 1) using the formula L®M = Hom(L,M°)°.

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