Page 35 - Textos de Matemática Vol. 41

4.4. Epimorphisms in Loc 21 every strong monomorphism (epimorphism) is extremal.
(Suppose m = fe, that is m · id = f · e; if m is strong, there is a w such that we = id which together with e being an epimorphism makes e an isomorphism.)
4.4. Epimorphisms in Loc (i.e. monomorphisms in Frm). The chain 3 = {0 < a < 1}
is called the Sierpin´ski locale.
(Note that it is the open set lattice Ω(S) of the well- known Sierpin´ski space whose role it takes in our con- text).
We have the trivial
Observation. For each frame L and each element x ∈ L, the mapping φx =(0→0,a→x,1→1):3→L
is a frame homomorphism.
Corollary. 3 is a generator in the category Frm (that is, whenever h1, h2 : L → M are distinct there is a φ : 3 → L such that h1φ ̸= h2φ). Consequently, epimorphisms in Loc are precisely the onto localic maps.
(That is, monomorphisms in Frm are precisely the one-one frame homomor- phisms: Obviously the one-one homomorphisms are monomorphisms. Now if h is not one-one, take x ̸= y with h(x) = h(y) and consider the equality hφx = hφy.)
4.5. Extremal resp. strong monomorphisms in Loc. The structure of monomor- phisms in Loc (epimorphisms in Frm) is by far not transparent (for instance there exist monomorphisms m : L → M with fixed M and arbitrarily large L). Luck- ily enough, for our purposes we will need, rather, the extremal ones, and those are easy.
Lemma. For a frame homomorphism h the following statements are equivalent: (i) h is onto,
(ii) h is a strong epimorphism, (iii) h is an extremal epimorphism.