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9.5. The diagonal morphism 61
in particular
(a1 ∨a2)⊕(b1 ∨b2)=(a1 ⊕b1)∨(a1 ⊕b2)∨(a2 ⊕b1)∨(a2 ⊕b2) (where the joins on the right hand sides are the suprema of saturated sets in
L1 ⊕ L2). 9.5. The diagonal morphism.
Proposition. Let fi : M → Li, i = 1,2, be localic maps. Then the associated f :M →L1 ⊕L2 is given by the formula
f(a) = {(x,y) | y ≤ f2(f1∗(x) → a)} = {(x,y) | x ≤ f1(f2∗(y) → a)}.
Proof. By the formula (∗) in the proof of 9.3.3,
f ∗ ( U ) = { f 1∗ ( x ) ∧ f 2∗ ( y ) | ( x , y ) ∈ U } .
Let f be as above. If f∗(U) ≤ a and (x,y) ∈ U then f1∗(x) ∧ f2∗(y) ≤ a, hence f1∗(x) ≤ f2∗(y) → a and hence x ≤ f1(f2∗(y) → a) and (x, y) ∈ f(a). On the other hand, let U ⊆ f(a) and (x, y) ∈ U. Then x ≤ f1(f2∗(y) → a), hence f1∗(x) ≤ f2∗(y)→a and f1∗(x) ∧ f2∗(y) ≤ a. Thus, f∗(U) ≤ a.
9.5.1. Corollary. In particular, the diagonal morphism ∆ : L → L ⊕ L is given by the formula
∆(a) = {(x, y) | x ∧ y ≤ a}.
9.5.2. Corollary. For any localic maps fi : Mi → Li (i = 1, 2), the unique f1 ⊕ f2 that makes the diagram
L oo pL1 L ⊕L pL2 //L 1122

f1 f1⊕f2

M1 oo pM1 M1 ⊕ M2
commutative is given by the formula
pM2
f2
// M2
(f1 ⊕f2)(E)={f1(a)⊕f2(b)|a⊕b⊆E}.