By Andre Joyal, Myles Tierney
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Extra resources for An Extension of the Galois Theory of Grothendieck
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). z op Proposition 3. Let M e s£(S ). The sup-lattice M(l) e s£(S) i s A. JOYAL § M. TIERNEY 48 equipped with a canonical Z-module structure, and putting defines an equivalence of categories T: sl(SL Moreover, for any pair M,N e si(S zop zop TM = M(l) ) -> Mod(Z) ) , we have a natural isomorphism T(Hom(M,N)) « Homz(TM,TN).
F: X -*- Y is open iff the image of any Proof: An open subspace U c — > X is described by the local operator u + ( ) on 0(X). The image f(U) c —> Y is described by the local operator f * (u -* f" ( )) (Chapter IV §1). p(u) e 0(Y). Conversely, if f(U) is open, we can write f*(u - f"(y)) - f(u) - y, where f(u) e 0(Y). Then u < f"(y) 1 = (u 1 = f*(u - f ( y ) 1 = ( f (u) •> y) + f""(7)) f(u) 1 y showing that f Proposition 1. has a left adjoint satisfying condition 3) of §1 o ^ n e interior S of a subspace S^—> X is the largest open subspace contained in S.
E. P(P) represents an open space. )> then 3 1 P • But as we saw at the end of Chapter III §4, the locale quotient of P(P) is the same as the sup-lattice quotient of P(P) by the sup-lattice congruence generated by the set of pairs (R, + (x)) for x e P and R e Cov(x). Thus, 3 p passes to a sup-lattice morphism 3 iff -3pR = 3 P + (x) for each x e P and R e Cov(x). However, as a morphism of sup-lattices, 3 p is determined by the condition 3 p ^ (x) = 1 for each x e P. So 3 p passes to Q iff 3 p R = 1 for each R e Cov(x).
An Extension of the Galois Theory of Grothendieck by Andre Joyal, Myles Tierney