By Andre Joyal, Myles Tierney

ISBN-10: 0821823124

ISBN-13: 9780821823125

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Hence, f~(Z (1)) = Z (1), from which the final p y i i statement follows: any diagram of the form y 0(W) 0(U) where 3. u e 0(W), v e 0(Z) and q > 0(Z) > 0(V) q (u) = v, is a pushout of locales. Conditions for openness Proposition 1. For any space X, the map p": P(l) + 0(X) has a left adjoint. p: X -*• 1 is open iff Proof: Suppose p" has a left adjoint 3: 0(X) -*• P(l). We have to prove that for any y e P(l) and x e 0(X), Y A 3(x) -3(p"(y) A X). The usual argument (Chapter III §4 Proposition 3) shows that it is enough to prove that the right hand side is 1 when the left hand side is 1.

Proposition 1. A commutative monoid 2 Va c A, a _< 1 and a = a. A is a locale iff A satisfies Proof: The conditions are clearly necessary, so suppose they are satisfied in A. Then a»b _< a» 1 = a a«b <_ 1-b = b, since multiplication is order preserving. and c £ b, then c = c«c _< c«b £ a»b, so 2• On the other hand, if a»b = aAb . c£ a Limits and colimits Limits in Loc are easy to calculate: just take the limit of the underlying sets. The coproduct of two locales A and B is their coproduct as commutative monoids: A®B.

N Q Their coequalizer is p (L). Proposition 2. We have: p*Y A. JOYAL § M. TIERNEY 44 1) Hom(L,p*(M)) - p*Hom(p (L),M) 2) p (PX) * ftP A 3) p # (L®M) = p # (L)0p # (M) 4) L®p*(M) - p*(p # (L)®M). Proof: The first identity expresses the (strong) adjointness p lp*2) and 3) are immediate consequences of 1 ) , and 4) follows from 1) using the formula L®M = Hom(L,M°)°. The third identity of Proposition 2 implies that p u u preserves commutative monoids and locales, as does p*. In fact, p is the left adjoint to p*: Loc(E) + Loc(S).

### An Extension of the Galois Theory of Grothendieck by Andre Joyal, Myles Tierney

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