Page 73 - Textos de Matemática Vol. 41

9.3. The product i∈J Li 59 Then
for every saturated U, n ⊆ U.
It is easy to check that for every a ∈ ′i∈J Li, ↓ a ∪ n is saturated. Thus,
↓a ∪ n = ν(↓a). This element will be denoted by ⊕iai,
in case of small systems
a1 ⊕a2, a⊕b, a1 ⊕a2 ⊕a3, etc.
9.3.2. The product projections. Define maps
pi :Lj →Li, ιi :Li →Lj
j∈J j∈J
by setting
pi(U)={x|x∗i1∈U}, ιi(x)=↓(x∗i1)∪n.
We have piιi(x) = x and, since U is saturated, ιipi(U) ⊆ U. Thus, ιi is the left
adjoint of pi and obviously ιi(x ∧ y) = ιi(x) ∩ ι(y); hence pi is a localic map. Remarks. (1) Note that for the inclusion ιipi(U) ⊆ U it is essential that U is
saturated.
(2) Obviously ⊕iai = i ιi(ai) (this meet is, essentially, finite: all but finitely
many ai are 1); in particular, a ⊕ b = ι1(a) ∧ ι2(b).
9.3.3. Theorem. pi : j∈J Lj → Lii∈J is the product of the system Li, i ∈ J, in the category Loc.
Proof.Letfi :M→Li belocalicmapsandletfi∗ :Li →Mbetheirleft
adjoints. Define
by setting
Obviously φ preserves joins; moreover,
φ :
′ j∈J
Lj → M
φ(U) = { fi∗(xi) | x ∈ U}. (∗) i∈J
φ(U)∩φ(V) = {fi∗(xi)∧fi∗(yi)|x∈U,y∈V} i∈J i∈J
≤ {fi∗(zi)|z=x∧y∈U∩V} i∈J
≤ φ(U ∩V)≤φ(U)∧φ(V).