P.T. Johnstone (1983). Open locales and exponentiation. In: J.W. Gray (ed.) Mathematical Applications of Category Theory Contemporary Mathematics vol. 30, American Mathematical Society.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....D; p 2 Omega and a Omega Gamma2 p) mod E) Then E E 0 is strongly dense and hence a homeomorphism, so Omega E can be presented by those used for Omega E 0 , which are all of the required form. We now address the other point, about non emptiness of opens. Definition 1. 6 (Johnstone [6]) Let D be a locale. 1. An open a 2 Omega D is positive iff whenever a W S(S Omega D) then S is inhabited. 2. D is open iff every open a 2 Omega D is a join of positive opens. Classically, S Omega D is either empty or inhabited, and it follows that a is positive iff a 6= false. If ....

....a = W fag is a join of positives. Hence classically every locale is open. Proposition 1.7 A locale D is open iff Omega Gamma : Omega Omega D has a left adjoint 9 . Moreover, if these hold we then have for each a 2 Omega D; 9 a is the truth value of a is positive . Proof Johnstone [6]. Following Joyal and Tierney [10] Proposition 1.7 is normally taken as the definition of openness. Then Definition 1.6 (2) appears in Johnstone [6] as a Proposition. Johnstone also proves the useful result that if D is open, then for any S Omega D we have W S = W fa 2 S : a ....

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Peter T. Johnstone. Open locales and exponentiation. Contemporary Mathematics, 30:84--116, 1984.

....universal quantification bounded over finite sets is geometric. Condition (ii) rewrites the geometric axioms (i) and (iii) of 7.6 as a geometric formula. The only difficult part is (ii) in 7.6. Suppose S g(M) and n N. We have (i,x)S. n i i n) and from the finiteness of S it follows [7] that either (i,x)S. n i or (i,x)S. i n. In the first case, suppose we have (k, x) S with n k; let k be the least such: so (i,y)S. i n k i) Then S (n,x) g(M) The second case is when (i,x)S. i n. Suppose S = n i , x i ) 1 i m 1 , and choose x m such that M(x m ) 2 ....

P.T. Johnstone, Open locales and exponentiation, Contemporary Mathematics 30 (1984) 84-- 116.

....We shall henceforth often omit the subscript T when referring to these notions. Note that in the boolean situation, an element is positive iff it is not the bottom element; the present positive way of expressing this property is related to a notion of positive elements in locales considered in [8] and [18] In fact, for any poset with a top element, an element a is T compact iff every cover of a is inhabited. For general reasons, we have that b 0 b a a 0 implies b 0 a 0 , and that satisfies an interpolation property; we shall not recall the latter, since we in our special ....

....algebras for this lift monad T on LOC, we have the following result. Recall (Theorem 11 and Lemma 12) that a T continuous poset is a (weakly cocomplete) poset such that every element in it is a sup of positive elements. For frames locales, this in turn can be expressed: C is an open locale, cf. [8]. Theorem 13 Let C be a locale. Then t.f.a.e: 1) C admits a structure (necessarily unique) for T 2) the positive elements in C are stable under finite intersections; and C is T continuous. 3) for any locale D, the poset LOC(D;C) has a maximal element, preserved by composition with any locale ....

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P.T.Johnstone, Open locales and exponentiation, in Mathematical Applications of Category Theory, Contemporary Math. 30 (1984), 84-116

....a predicate defined on the base of a formal topology and satisfying Pos(a) a Delta U (9b ffl U)Pos(b) Pos(a) a Delta U a Delta U The original definition in [16] of formal topology included a positivity predicate. This notion corresponds to the notion of open locale in the theory of locales [8]. A formal space does not necessarily have a positivity predicate. However, the following notion of positivity can always be defined: POS(a) 8U) a Delta U ) U inhabited) Since the definition of POS involves quantification over subsets, POS(a) is a type but not a set. We first prove that if ....

P. T. Johnstone. Open locales and exponentiation. In Mathematical Applications of Category Theory, volume 30 of A.M.S. Contemporary Math. series, 1984.

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P.T. Johnstone (1983). Open locales and exponentiation. In: J.W. Gray (ed.) Mathematical Applications of Category Theory Contemporary Mathematics vol. 30, American Mathematical Society.

No context found.

Peter Johnstone. Open locales and exponentiation. Contemporary Mathematics, 30:84-116, 1984.

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