Page 31 - Textos de Matemática Vol. 45

2.4. The fibrations of Schreier and homogeneous points 23 (i) u is a regular epimorphism (i.e. a surjective homomorphism) if and only
if so are v and K(u);
(ii) u is a monomorphism if and only if so are v and K(u).
Proof. (i) If u is a regular epimorphism, then so is vf = f′u, and this implies that v is a regular epimorphism. The map q′ is surjective in Set, then so is q′u = K(u)q, and this implies that K(u) is surjective. Conversely, suppose that v and K(u) are regular epimorphisms. Consider a′ ∈ A′. There are α ∈ K(f) and b ∈ B such that u(α) = q′(a′) and v(b) = f′(a′). Accordingly u(α·s(b)) = u(α)·us(b) = q′(a′)·s′v(b) = q′(a′)·s′f′(a′) = a′. Hence the homomorphism u is surjective.
(ii) If u is a monomorphism, then so is us = s′v, which implies that v is a monomorphism. Similarly uk = k′K(u) is a monomorphism which implies that K(u) is a monomorphism. Conversely, suppose that K(u) and v are monomorphisms. Suppose now that u(a1) = u(a2). Then we have f′u(a1) = f′u(a2) and then vf(a1) = vf(a2), whence f(a1) = f(a2). From u(a1) = u(a2), we can conclude uq(a1) = q′u(a1) = q′u(a2) = uq(a2). Since the restriction K(u) of u to K[f] is a monomorphism, we get q(a1) = q(a2). With f(a1) = f(a2), we get a1 = a2.

2.4 ThefibrationsofSchreierandhomogeneouspoints
Let us consider the category Pt(Mon) and its full subcategories SPt(Mon) and HPt(Mon) whose objects are respectively the Schreier split epimorphisms and the homogeneous ones. According to Proposition 2.3.4, the restrictions ¶S and ¶H of the fibration ¶: Pt(Mon) → Mon to the full subcategories SPt(Mon) and HPt(Mon) are still fibrations, which we shall call the fibrations of Schreier points and of homogeneous points.
Proposition 2.4.1. The change-of-base functors of the fibrations ¶S and ¶H are conservative (i.e. they reflect isomorphisms).
Proof. The Schreier split short five lemma is equivalent to the fact that the change of base functor along the initial map αB∗ : SPtB(Mon) → Mon is con- servative. Now, given any homomorphism h : A → B we have hαA = αB and thus αA∗ h∗ = αB∗ , and since αB∗ is conservative, so is h∗.
Theorem 2.4.2. Given any monoid B, the fiber PtB(Mon) is SPtB(Mon)- unital and consequently HPtB(Mon)-unital.