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80 Chapter 7. Special Schreier and special homogeneous surjections Example 7.1.2. According to Example 3.1.9, the morphism abs: Z∗ → N∗ is a
special Schreier surjection (actually a special Schreier split epimorphism).
Similarly to Schreier split epimorphisms, we get:
Proposition 7.1.3. Given any special Schreier (a fortiori special homogeneous) surjection f : X Y , the following sequence is exact:
K[f]//k //X f////Y, namely f is the cokernel of its kernel k.
Proof. Suppose we have a map h:X → Z such that hk = 0. Let us show that h coequalizes the kernel equivalence relation R[f]. For that, consider the following diagram:
R[f] oo s0 //oo X ;; p0
p1 f
(0,k)
;; K[f] // //X //// Y
kf
The map (0,k) is the kernel of p0. Accordingly, since R[f] is a Schreier equiva- lence relation, the pair (s0, (0, k)) is jointly strongly epimorphic. The equality hp0 = hp1 is checked by composition with this pair (since hk = 0). Since f is the quotient of R[f], we get a unique factorization h¯: Y → Z such that h = h¯f.
Proposition 7.1.4. The special Schreier (resp. special homogeneous) surjections are stable under products and pullbacks.
Proof. The first assertion comes from the stability of the Schreier (resp. homo- geneous) equivalence relations under products. On the other hand, in Mon as in any variety of algebras, the surjective homomorphisms are stable under pull- backs. Then consider the following diagram, where the right hand side square is a pullback:
p0 // f R[f]oo s0 //X ////Y
p1 R(x)
xy p′0//
R[f′]oo s′0 //X′ ////Y′.
p′ 1
f′
Then any of the left hand side ones are pullbacks (see the Appendix). Therefore, when R[f′] is a Schreier (resp. homogeneous) equivalence relation, so is R[f].