Page 36 - Textos de Matemática Vol. 41

22 Chapter 4. The basic structure of morphisms in Loc Proof. (i)⇒(ii): Let h : L → M be onto, let m : L′ → M′ be a monomorphism
and let mu = vh.
L h //M uv
′ // ′ LmM
For y ∈ M choose an x ∈ L such that h(x) = y and set w(y) = u(x). This uniquely defines a mapping w : M → L′: indeed, if also h(z) = y then m(u(z)) = vh(z) = vh(x) = m(u(x)), and m is one-one by Corollary 4.4. We have wh = u and since h, u are homomorphisms and h is onto, w is a homomorphism. Finally, mwh = mu = vh and hence mw = v.
(ii)⇒(iii) is the general fact we have already discussed.
(iii)⇒(i): If h : L → M is not onto consider K = h[L]. Then K is a frame and the embedding j : K ⊆ M is not an isomorphism while h = jh′ for the h′(x → h(x)) : L → K.
Corollary. In the category Loc, the extremal monomorphisms coincide with the strong monomorphisms, and those are precisely the one-one localic maps.
4.6. A factorization system in Loc. Recall that a factorization system in a category is a couple (E,M) of classes of morphisms such that
(1) each of E,M is closed under isomorphisms,
(2) every morphism f : A → B can be decomposed as
A ef //C mf //B with ef ∈E and mf ∈M, and
(3) each commutative diagram
A f //B uv
′ // ′ AgB