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Extra info for An Extension of the Galois Theory of Grothendieck
M are B0B-algebras, M0M has effective descent A B data 000 on it, and if 0 is an algebra morphism, then m is compatible and hence descends uniquely. Similar remarks apply to the unit. g. , etc.. Later we will have to consider descent of properties, which are not quite of this type, but we leave those until we need them. M CHAPTER III - LOCALES As the reader will recall, a locale A is a commutative monoid in si such that a«b = aAb . e. bAa < c b _< a + c Thus, a locale is a complete Heyting algebra.
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. Similarly if A e Loc(E) and I e E, then A e Loc(E). e. for f e A and g e A , £a(a*(g) A £) - g A Z a (f), which also follows by direct calculation from the formula for Z.
E. a cover subset of the down segment of subset of P generated by x. Let R: z e R R iff z <_ y for some the condition that these covers should satisfy is: R e Cov(x), then topology on RA + ( y ) 2 R ' , where is a given denote the downward closed if R1 e Cov(y) . y e R. y <_ x Then and This is not yet a P, since we have not required that singletons be covers, or that the local axiom is satisfied. It is a system of generators for a topology. Now consider the locale the subset of and P(P)xp(P) R e Cov(x).