Page 59 - Textos de Matemática Vol. 41
P. 59

7.3. Open maps 45 (iv)⇒(v) is obvious.
(v)⇒(iv): If c ≤ f(a)∨b then f∗(c) ≤ f∗(f(a)∨b) = f∗f(a)∨f∗(b) ≤ a∨f∗(b). On the other hand, if f∗(c) ≤ a∨f∗(b) then f∗(c)∨a∨f∗(b) = a∨f∗(b), that is, f∗(c∨b)∨a = a∨f∗(b). Thus, by hypothesis, f(a)∨b = f(a)∨b∨c, which implies f(a) ∨ b ≥ c. 
7.3. Open maps. A localic map f : L → M is open if the image of each open sublocale of L is open.
Proposition. For a localic map f : L → M the following are equivalent: (i) f is open.
(ii) f∗ : M → L is a complete Heyting homomorphism.
(iii) f∗ admits a left adjoint f! that satisfies the (Frobenius) identity
f!(a∧f∗(b))=f!(a)∧b for all a∈L and b∈M. (iv) f∗ admits a left adjoint f! that satisfies the identity
(7.3.1)
f(a→f∗(b))=f!(a)→b for all a∈L and b∈M.
Proof. (i)⇔(ii): f is open iff for every a ∈ L there is a b ∈ M (uniquely defined
by 6.2.3) such that
f[o(a)] = o(b). Denote this b by φ(a). Thus, by 6.2.1(4),
x∧φ(a)=y∧φ(a) iff f∗(x)∧a=f∗(y)∧a. (∗) This formula is obviously equivalent to
x∧φ(a)≤y∧φ(a) iff f∗(x)∧a≤f∗(y)∧a (∗∗) and this, in turn, to
x∧φ(a)≤y iff f∗(x)∧a≤f∗(y). (∗∗∗)
Setting in particular x = 1 we obtain φ(a) ≤ y iff a ≤ f∗(y); thus, f∗ is a right adjoint and hence it also preserves all meets and is complete.
Returning to (∗∗∗) and using the Heyting adjunction we obtain
a≤f∗(x→y) iff φ(a)≤x→yiffx∧φ(a)≤y
iff f∗(x) ∧ a ≤ f∗(y) iff a ≤ f∗(x)→f∗(y)
(7.3.2)















































































   57   58   59   60   61