Page 42 - Textos de Matemática Vol. 41

28 Chapter 5. Sublocales, that is, generalized subspaces
5.3.3. Nuclei. A nucleus in a locale L is a mapping ν : L → L such that (N1) a ≤ ν(a),
(N2) a≤b ⇒ ν(a)≤ν(b),
(N3) νν(a) = ν(a), and
(N4) ν(a∧b)=ν(a)∧ν(b).
The translation between nuclei and frame congruences resp. sublocale ho-
momorphisms is straightforward:
ν → Eν = {(x,y) | ν(x) = ν(y)},
E→νE = (x→E[x]):L→L,
ν→hν = therestriction ν:L→ν[L],
h → νh = (x → h∗h(x)) : L → L.
The nucleus representation is very important and hence it is worthwhile to briefly discuss the relation of sublocales and nuclei directly.
ForasublocaleS⊆L,withjS :S⊆L,set
ν S ( a ) = j S∗ ( a ) = { s ∈ S | a ≤ s } ,
and for a nucleus ν : L → L set
Sν = ν[L].
Proposition. The formulas S → νS and ν → Sν constitute a one-one correspon-
dence between sublocales of L and nuclei in L. Proof. We will use the fact that, for any nucleus ν,
∀a,b ∈ L, ν(a→ν(b)) = a→ν(b). (5.3.3)
(Indeed, by 1.3.1(H6),
ν(a→ν(b)) ∧ a ≤
= ν((a→ν(b)) ∧ a)
= ν(a ∧ ν(b))
≤ ν(ν(b)) = ν(b).
ν(a→ν(b)) ∧ ν(a)