Page 74 - Textos de Matemática Vol. 41

60 Chapter 9. Products. Completeness of the category Loc
Hence φ preserves meets, and we have a localic map g : M → ′i∈J Li adjoint to φ.
Now recall the relation R from Observation 9.2. Since fi∗((a ∗α u)i) = fα∗(a) ∧ fi∗(ui)
we have
i∈J i̸=α
φ↓(xk∗αu) = (fα∗(xk)∧fi∗(ui)) k k i̸=α
Therefore by 8.3 and 8.3.1 we have a
f : M → Lj
= = = =
((fα∗(xk))∧fi∗(ui) k i̸=α
fα∗(xk)∧fi∗(ui) k i̸=α
∗ k
f(( x)∗αu)i)
ik
φ↓(( xk)∗αu). k
localic map
j∈J
such that f∗ is given by the same formula (∗) as φ. Thus in particular (pif)∗(x) =
(f∗ιi)(x) = f∗(x) and consequently pif = fi. Finally, such an f is unique since
a homomorphism h such that hιi = fi is obviously unique (since the subset ∗
{ιi(x) | i ∈ J, x ∈ Li} obviously generates j∈J Lj by meets and joins). 9.3.4. We obviously have, for every saturated U,
U={⊕iai |(ai)i ∈U}={⊕iai |(ai)i ∈U}={⊕iai | ⊕iai ⊆U}.
Thus, the set {⊕iai | (ai)i ∈ ′i∈J Li} generates the frame i∈J Li by joins.
9.4. A useful simple distributive rule. In the case of products of two locales we have a useful formula following immediately from the observation in Remark 2 of 9.3.2 and from the fact that the mappings ιi are homomorphisms.
Proposition. i∈J (ai ⊕ b) = (i∈J ai) ⊕ b and i∈J (a ⊕ bi) = a ⊕ (i∈J bi). Consequently ai⊕bi= (ai⊕bj);
i∈J i∈J i,j∈J