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By Andre Joyal, Myles Tierney
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Additional info for An Extension of the Galois Theory of Grothendieck
Let X be a space in S. Then the category of sheaves on X is equivalent to the category of etale spaces E -• X. Proof: The category of sheaves on X is the category sh(0(X)), which is equivalent to the category of discrete spaces Y of sh(0(X)) which by Theorem 1 §5 Chapter V is the dual of the full subcategory of locales 0(Y) in sh(0(X)) such that fi + 0(Y) and O(Y)00(Y) -> 0 (Y) have linear left adjoints. Under T these are taken to locale extensions 0(X) + 0(E) such that 0(X) -> 0(E) and 0(E) 0 adjoints.
I Ixi in infimum is the colimit over s£ I is also the colimit in Loc, since the of the infima A. 0 A. A > A. l l i This is, of course, also true for a direct system of arbitrary commutative monoids. In any case, Proposition 2 chapter I §2 yields Proposition 1. for (A. : A. + A. ^e a direct system of locales such that Then, for any i e I the For an arbitrary family of locales, the coproduct _LLA. iel 1 is canonical map is injective. A. •*• lim A. is injective. I lim A. 0A. V^1 "i 3. H The free locale The free locale L(X) on a set X e S can be constructed in two steps.
Conversely, if op L: A *Loc(S) satisfies the conditions, then the supremum in each L(A) was calculated in the proof of Proposition 1, and the distributive law is a direct consequence of Frobenius reciprocity. Suppose now that A = Z is an inf-semilattice. A Z-module is a sup-lattice M together with an operation ZxM + M, written a«x for a e Z and x e M, such t h a t a < b =^ a«x < b ' x a x a - (iel V * ix) = i V ' i el a* (b»x) = (a Ab) »x 1 «x = x Clearly, Z-module structures on M are in 1-1 correspondence with ordinary P(Z)-module structures, where P(Z) is the free locale on the inf-semilattice Z (Chapter III §3).