Page 37 - Textos de Matemática Vol. 41

4.6. A factorization system in Loc 23
can be completed into a commutative diagram
A ef //C mf //B uwv
′ // ′ // ′ A eg C mg B
(Note that choosing in (3) u = idA and v = idB we immediately see that the decomposition f = mf ef is unique up to isomorphism.)
We will show that the couple of classes
(onto localic maps, one-one localic maps)
constitutes a factorization system in Loc.
First, realize that it suffices to prove (2): (1) is trivial, and (3) will imme-
diately follow from the already established facts that
• the onto localic maps are epimorphisms and
• the one-one localic maps are strong monomorphisms.
Let h : L → M be a localic map. Consider the natural set-map decompo- sition
L e=(a→h(a)) // h[L] m=⊆ // M.
(at the start we do not know that e,m are localic maps, not even that h[L] is a locale). Consider the associated frame homomorphism h∗ : M → L and decompose it into
M ε=(b→h∗(b)) // h∗[M] μ=⊆ // L.
Here there is no problem to see that h∗[M] is a frame and that ε,μ are frame
homomorphisms. Set
α=eμ:h∗[M]→h[L], β=εm:h[L]→h∗[M].
Both α and β are monotone. Since we have
αβ(h(a)) = eμεmh(a) = meμεmh(a) = hh∗h(a) = h(a)