Page 67 - Textos de Matemática Vol. 41

8.3. Factorization Theorem 53 For any congruence E, each E-class contains precisely one E-saturated
element. Thus the L/E above is equivalent to the L/E of 5.3.2.
8.3. Factorization Theorem. Let R be a binary relation on a locale L. Let a
localic map f :M →L be such that
aRb ⇒ f∗(a) = f∗(b).
Then f[M] ⊆ L/R and hence there is a localic map f : M → L/R such that jf = f (where j : L/R → L is the embedding).
M f // L/R
CCCCCCCC
f CCCC
!!
L
Proof. It suffices to show that each f(x) is saturated (then, we finish using 4.6).
For aRb we have
a∧c≤f(x) iff f∗(a)∧f∗(c)=f∗(a∧c)≤x
iff f∗(b)∧f∗(c)=f∗(b∧c)≤x iff b∧c≤f(x). 8.3.1. Note that for the left adjoint f∗ of f we have f∗(a) = f∗(a) for all
a ∈ L/R. Indeed, f∗(a) = f∗(ν(a)) = f∗j∗(a) = f∗(a).
In the language of frame homomorphisms we can summarize the facts
above as follows:
Theorem. Let R be a binary relation on a locale L. Then μR : L → L/R is a
frame homomorphism such that
aRb ⇒ μR(a) = μR(b),
and for every frame homomorphism h : L → M such that
aRb ⇒ h(a) = h(b)
there is a frame homomorphism h : L/R → M such that hμR = h; moreover,
h = h|(L/R), that is, for every a ∈ L/R, h(a) = h(a).
8.4. A more transparent formula for saturatedness. The relations occurring in constructions of sublocales, although typically by far not congruences, are sometimes of a special nature which yields a more transparent formula for sat- uratedness.
j