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5.3. Alternative representations of sublocales 27
5.3. Alternative representations of sublocales. The reader will find in the literature typically some of the following representations of subobjects in the point-free context. They are, in a way, more algebraic than the one above. Sometimes we will use some of them too for technical purposes.
5.3.1. Sublocale homomorphisms. Recall 4.1. Up to isomorphism, the sublocales can be represented by sublocale homomorphisms, that is, onto frame homomor- phisms. The translation is as follows:
jS →jS∗ :L→S forjS :S⊆L, and
h → h∗[M] for an onto h : L → M and h∗ the corresponding right adjoint.
Note that the inclusion of sublocales S1 ⊆ S2 is represented by the existence of a frame homomorphism h such that hh2 = h1 for the corresponding
h1 =jS∗
:L→S1, h2 =jS∗ 12
:L→S2 :
L h1
// S1 OO
???? ????
h h2 ??? S2
5.3.2. Frame congruences. A frame congruence is, of course, an equivalence re- lation on a frame respecting all joins and finite meets. The translation between sublocale homomorphisms and frame congruences is
h→Eh ={(x,y)|h(x)=h(y)}, E→hE =(x→E[x]):L→L/E
(where L/E denotes the quotient frame defined by the congruence E, just as quotients are always defined for algebraic systems, and E[x] denotes the E-class {y | (x, y) ∈ E} of x ∈ L).
Here the inclusion of sublocales is represented by the inverted inclusion of equivalence relations:
S1⊆S2iffEh2 ⊆Eh1.
Note. Trivially, any intersection of frame congruences is a frame congruence. Thus, frame congruences constitute a complete lattice, and hence so do also the sublocales of a locale. We will present an explicit description of its structure shortly.