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7.4. The sublocale Asloc 47
7.4. The sublocale Asloc. Let A ⊆ L be asubset closed under meets. Then {1}⊆AandifSi ⊆Aforeveryi∈J then i∈J Si ⊆A.Thus,thereexiststhe largest sublocale contained in A. It will be denoted by
Asloc.
7.5. Preimages. Let f : L → M be a localic map and let T be a sublocale of M. Since f preserves meets, the set-theoretic preimage f−1[T] is closed under meets. However, it is not necessarily a sublocale of L. Let f−1[T] denote the largest sublocale contained in f−1[T], that is
f−1[T] = (f−1[T])sloc. the preimage of T
It will be referred to as
under (the localic map) f. It is right adjoint to the image as it should be:
7.5.1. Proposition. For every localic map f : L → M, the function f−1[−] is a right Galois adjoint to f[−] : Sl(L) → Sl(M).
Proof. We have f[S] ⊆ T iff S ⊆ f−1[T]. Since f−1[T] = (f−1[T])sloc is the largest sublocale contained in f−1[T], then f[S] ⊆ T iff S ⊆ f−1[T].
7.5.2. Proposition. The preimage of a closed (resp. open) sublocale under a localic map is closed (resp. open). More precisely, for any localic map f : L → M and any a ∈ M,
f−1[c(a)] = f−1[c(a)] = c(f∗(a)) and f−1[o(a)] = o(f∗(a)). Proof. The adjunction formula
can be rewritten as
f∗(a) ≤ x iff a ≤ f(x)
x ∈ c(f∗(a)) iff x ∈ f−1[↑a].
Now since f(f∗(a)→y) = a→f(y) by 2.2.1 we have o(f∗(a)) ⊆ f−1[o(a)]. Let S be a sublocale, S ⊆ f−1[o(a)]. We will prove that S ⊆ o(f∗(a)). If s ∈ S than any x→s is in S, and f(x→s) is in o(a). Hence by 2.2.1 and 1.3.1(H8),
f(x→s) = a→f(x→s) = f(f∗(a)→(x→s)) = f((f∗(a) ∧ x)→s) and in particular for x = f∗(a)→s we obtain, by 1.3.1(H8) and (H3),
f((f∗(a)→s)→s) = f((f∗(a) ∧ s)→s) = f(1) = 1