By Andre Joyal, Myles Tierney

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Direct calculation shows that h is the universal solution to the problem of adding a complement to a. The injectivity of h a is equivalent to the statement that x A a = yA a and x V a = y V a entails x = y, which is true. Now let I be the directed set of all finite subsets of We have a directed system Ap -• Ap A and w h en ( F^FeI' is injective, since AG = (A p ) G . —• Fel G F S^A. £=L , the transition mapping Clearly, A,-, = lim A r . S F A. JOYAL $ M. TIERNEY 26 The injectivity of Definition.

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.

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. JOYAL $ M. TIERNEY Theorem 1. e. to X is isomorphic to the 0(X)' . In particular, we obtain the result of Dowker and Papert [10], and Isbell [11] : the lattice of local operators on a locale 0(X) is itself a (O-dimensional in the obvious sense) locale - namely 0(X) f .