Page 58 - Textos de Matemática Vol. 41
P. 58
44 Chapter 7. Images and preimages the image of S
under (the localic map) f.
7.1.2. Proposition. Let f : L → M be a localic map and let S ∈ Sl(L), T ∈
Sl(M) such that f[S] ⊆ T. Then the restriction f|S : S → T is a localic map. Proof.Defineh:T→Sbyh=jS∗f∗jT.Thenh⊣f|S sincef∗⊣fandjS∗ ⊣jS:
h ( t ) ≤ s i ff j S∗ f ∗ ( t ) ≤ s i ff f ∗ ( t ) ≤ j S ( s ) i ff t ≤ f j S ( s ) = f ( s ) = f | S ( s ) . Moreover, h preserves finite meets since jS∗ and f∗ preserve them.
Note that the particular case S = L gives immediately the factorization L f //M
OO
jf [L]
L f // f[L] |L
A localic map f : L → M is closed if the image of each closed sublocale of L is closed.
Proposition. For a localic map f : L → M the following are equivalent: (i) f is closed.
(ii) For each a ∈ L, f[↑a] =↑f(a).
(iii) For each a∈L and b∈M, f(a∨f∗(b))=f(a)∨b.
(iv) Foreacha∈Landb,c∈M,c≤f(a)∨bifff∗(c)≤a∨f∗(b).
(v) For each a ∈ L and b,c ∈ M, f(a)∨b = f(a)∨c iff a∨f∗(b) = a∨f∗(c). Proof. (i)⇔(ii): f[↑a] =↑b for some b and since f(a) is obviously smallest in
f[↑a], b = f(a).
(ii)⇔(iii): We always have f(a∨f∗(b)) ≥ f(a)∨ff∗(b) ≥ f(a)∨b. If f is closed, f(a) ∨ b = f(x) for some x ≥ a. As f(x) ≥ b, we have x ≥ f∗(b) ∨ a, and f(a ∨ f∗(b)) ≤ f(x) = f(a) ∨ b.
Conversely, if x ≥ f(a) then x = f(a) ∨ x = f(a ∨ f∗(x)).
(iii)⇔(iv): f(a)∨b = f(a∨f∗(b)) ⇔ c ≤ f(a)∨biffc ≤ f(a∨f∗(b)) ⇔ c ≤ f(a) ∨ b iff f∗(c) ≤ a ∨ f∗(b).
of 4.6.
7.2. Closed maps.