By Andre Joyal, Myles Tierney

ISBN-10: 0821823124

ISBN-13: 9780821823125

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**Extra resources for An Extension of the Galois Theory of Grothendieck**

**Example text**

TIERNEY 24 are characteristic maps Xc ,Xcb : PlxPl -• PI b l 2 of subsets S ^ L ^ C P l x P l , and it is enough to check that x -*• y <_ I (x) -• I (y) . , C S . , for But for this it is enough to know that x <_ y =£> &(x) £ A(y), which is true. As we did for sup-lattices given a subset explicitly describe the quotient locale relation on a e A A and generated by R. we may suppose R of We say that (z,,z 2 ) e R, we have AA Q R C AxA, we want to A by the congruence R is inf-stable if given (a A z, , a A z«) e R.

Dis for it. We have: 0(idis) = odJLi) = ow1 = (Pi)1 = p(i) and in < >dis Spi > S we have II ( ) dis —I | | Proposition 1. ( )ji<5 Proof; is full and faithful iff the unit of the adjunction ( )dis ( ),. is full and faithful. 1 I I is an isomorphism: So we want to show that any strong prime filter F on PI is principal. But I e F and I = (J {i}, so we can find i e l such that {i}e F. iel Suppose J e F. Then J H {i} e F and the above shows that j^3 e J H {i} such that {j} e F. Thus, {i}C_J, and F is the principal filter generated by i e l .

X x iel Going backwards, this means that the closed subspace h" S, where 0(S) is the quotient of generated by the set of pairs Thus, we have (u^u^A CV is equal to 0(X) by the congruence relation v i ^ iel" 32 A. JOYAL $ M. TIERNEY Theorem 1. e. to X is isomorphic to the 0(X)' . In particular, we obtain the result of Dowker and Papert [10], and Isbell [11] : the lattice of local operators on a locale 0(X) is itself a (O-dimensional in the obvious sense) locale - namely 0(X) f . CHAPTER V - OPEN MAPS OF SPACES In this chapter we introduce the central concept of an open mapping between spaces.

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

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