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62 Chapter 9. Products. Completeness of the category Loc Proof. By Proposition 9.5,
(f1 ⊕f2)(E) = {(a,b)|b≤f2pL2((f1pL1)∗(a)→E)}
= {(a, b) | (f1pL1 )∗(a) ∩ (f2pL2 )∗(b) ⊆ E}
= {(a,b)|(f1∗(a)⊕1)∩(1⊕f2∗(b))⊆E} = {(a, b) | (f1∗(a) ⊕ f2∗(b)) ⊆ E} .
The latter set, being saturated, coincides with {a ⊕ b | f1∗(a) ⊕ f2∗(b) ⊆ E}, which is easily seen to be equal to {f1(a)⊕f2(b) | a⊕b ⊆ E} by the adjunction f∗ ⊣ f.
9.6. Completeness. The products are the limits in the category Loc that are not quite easy to construct. The rest of the completeness is easy.
Theorem. The category Loc is complete.
Proof. It is a standard fact in category theory that a category has all limits (≡ is complete) iff it has products and equalizers. Thus, it suffices to construct the latter.
Let f, g : L → M be localic maps. Recall the sublocale Asloc from 7.4 and set
Eq(f, g) = {x | f (x) = g(x)}sloc (this is correct: since f, g are right adjoints we have
f(xi) = g(xi) ⇒ f(xi) = f(xi) = g(xi) = g(xi) iiii
and {x | f(x) = g(x)} is closed under meets). Now if we have fh = gh for a localic map h : K → L, we have h[K] ⊆ {x | f(x) = g(x)} and since (by 7.1.1) h[K] is a sublocale and {x | f(x) = g(x)}sloc is the largest sublocale contained in {x | f(x) = g(x)}, h[K] ⊆ Eq(f,g), and the embedding of Eq(f,g) into L is the equalizer morphism.
9.7. Cocompleteness. The category Frm is a variety of algebras and hence it is, trivially, complete. Thus we obtain an immediate
Proposition. The category Loc is cocomplete. Note. The injections ιi of Li into the (coproduct in Loc) j∈J Lj have a very
simple formula. Since we have
xi = pi((xj)j) ≤ x iff (xj)j ≤ ιi(x)
we obtain
x for j = i, (ιi (x))j = 1 otherwise.