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14 Chapter 3. The spectrum adjunction. Spatiality σL is a localic map.
We immediately see that
and hence
σ L∗ σ L = i d ( 3 . 2 . 1 ) each σL is one-one.
Note. Recall Lemma 2.3. We can use the localic map σL and Proposition 3.1 for a very easy proof that each Pt(L) is sober: if Σa is meet-irreducible then b = σL(Σa) is meet-irreducible, hence b ∈ Pt(L) and Σa = Σb = Pt(L) {b}.
Proposition. σ = (σL)L is a natural transformation LcPt → Id.
Proof. This is easily seen in the adjoint form: for any localic map f : L → M
LcPt(f)∗(σM∗ (b)) = Pt(f)−1[Σb] = Σf∗(b) = σL∗ (f∗(b)),
using (2.3.1).
3.3. The natural transformation λ : Id → PtLc. For a space X define λX : X → Pt(Lc(X))
by setting We have
λ X ( x ) = x .
λ−1 [ΣU ] = {x | U X {x}} = {x | x ∈ U } = U
X
and λX is continuous.
Proposition. λ = (λX )X is a natural transformation Id → PtLc. Proof. By (2.4.1), if f : X → Y is a continuous map,
PtLc(f)λX(x) = Lc(f)(x) = f(x) = λY (f(x)). 3.4. The spectrum adjunction.
Proposition. The transformations σ and λ constitute an adjunction Lc ⊣ Pt of Lc and Pt.
Proof. For any a ∈ Pt(L), σL(Σa) = a. Therefore, by Proposition 3.1, Pt(σL)(λPt(L)(a)) = σL(Pt(L) {a}) = σL(Σa) = a.