Page 63 - Textos de Matemática Vol. 41

7.6. f−1[−] as a coframe homomorphism 49 7.6.2. Proposition. Let f : L → M be a localic map. Then the preimage map
f−1[−] : Sl(M) → Sl(L) preserves all meets and all finite joins.
Proof. The preservation of meets is given by the adjunction, and f−1[O] = O since O = c[{1}]. Thus, we only need to show that f−1 preserves binary joins. Let S,T be in Sl(M). By 6.5 we have
S = (c(xi) ∨ o(yi)) and T = (c(uj) ∨ o(vj)). i∈I j∈J
Thus, by 7.6.1, 7.5.2, 6.2.4, and 7.5.2 and 7.6.1 again
f−1[S ∨ T ]
= f−1 (c(xi ∧ uj) ∨ o(yi ∨ vj))
i,j = f−1[c(xi ∧ uj)] ∨ f−1[o(yi ∨ vj)]
i,j
= c(f∗(xi ∧ uj)) ∨ o(f∗(yi ∨ vj)) i,j
= c(f∗(xi) ∧ f∗(uj)) ∨ o(f∗(yi) ∨ f∗(vj)) i,j
= c(f∗(xi)) ∨ c(f∗(uj)) ∨ o(f∗(yi)) ∨ o(f∗(vj)) i,j
= f−1[c(xi)] ∨ f−1[c(uj)] ∨ f−1[o(yi)] ∨ f−1[o(vj)] i,j
= f−1[c(xi)∨o(yi)]∨f−1[c(uj)∨o(vj)] ij
= f−1[S]∨f−1[T].
7.6.3. Corollary. The preimage function preserves complements.
7.6.4. Corollary. In the notation of 5.6, the image function viewed as a mapping
is a localic map.

f[−] : C(L) → C(M)