Page 90 - Textos de Matemática Vol. 45

82 Chapter 7. Special Schreier and special homogeneous surjections Proposition 7.2.1. Consider a commutative diagram of special Schreier (resp.
homogeneous) extensions:
k′
The map u is an isomorphism if and only if K(u) is an isomorphism.
Proof. The fact that K(u) is an isomorphism as soon as so is u holds in any pointed category with pullbacks. For the converse, complete the diagram with the kernel equivalence relations:
K[f] (0,k) //R[f]oo p0 ////X f ////Y p1
K[f]//k //Xf////Y
K(u)
u

K[f′] // //X′ ////Y

f′
u
′ // ′oop′0// ′ ////
K(u) R(u)
K[f] ′ R[f] //X ′ Y (0,k ) p′ f
1
The two horizontal left hand side maps are the kernels of p0 and p′0, respectively. According to Theorem 2.3.7, since K(u) is an isomorphism and the split epi- morphisms (p0, s0) and (p′0, s′0) are Schreier ones, the associated middle square is a pullback. Thanks to the Barr-Kock Theorem (see Proposition 9.2.2 in the Appendix), the right hand side square is a pullback and consequently u is an isomorphism.
More precisely, we have
Proposition 7.2.2. Consider a commutative diagram of special Schreier exten- sions:
K[f]//k //Xf////Y
K(u)
u

K[f′] // //X′ ////Y

f′
k′
The map u is a monomorphism (resp. a surjective homomorphism) if and only
if so is K(u).
Proof. If u is a monomorphism, then k′K(u) = uk also is, and this implies that K(u) is a monomorphism. The fact that K(u) is a surjective homomorphism as soon as so is u holds in any pointed variety of universal algebra. As for the