By Andre Joyal, Myles Tierney
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Additional resources for An Extension of the Galois Theory of Grothendieck
A locale L e Loc(S^. ) is a functor L: A0-^—> Loc(S) satisfying conditions 1) and 2) of Proposition 1, but, where, in addition, the left adjoints Z of condition 1) satisfy Frobenius reciprocity: Za(L(a)(x)Ay) = xAZ a (y) . Proof: As in the proof of Proposition 1, the necessity of the conditions follows from the preliminary remarks of §1. 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.
By the same result, if A is an open surjection, p1 = p 2 , p is monic, and the lemma follows. Lemma 2. u = \/u.. iel x Proof: Suppose Let p: X -*• 1 J = Uflu. = 1}. Then We have to show: enough to show is open, and x u e 0 (X) u = satisfies \/ u1-- ieJ Vi e I, u. £ V / u . But u. <_ p~31u - , so it is 1 1 jeJ 3 u i A p"3 U i <_ X / u . J j eJ 3 u . < p*(u. ) 1 x jeJ 3 By the usual argument, it is enough to show that if the left hand side of the bottom line equals 1, then the right hand side equals 1, but this is obvious.
The resulting formulas provide an important technique for investigating the stability of various kinds of geometric morphisms under change of base. 1. Change of base for sup-lattices and locales We begin with a few preliminary remarks. Namely, let p: E -*- S topos defined over S. If M e s£(E) and I e E, then M e s£(E). Furthermore, if a: I -• J, then a* = Ma : M J -> MI has a left adjoint £ : M1 -+ M J defined by a J £a (f)(j) = V f(i). a(i)=j Moreover, for any pullback square 42 be a GALOIS THEORY 43 -* I J» in > J E we have Za',3'* = 3*Z a' , as is immediate from the above formula.