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28 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
Proof. We have that
(a, a′) = (q(a) · sf(a), a′) = (q(a) · sf(a′), a′) = (q(a), 1) · (sf(a′), a′).
Hence, if m gives a monoid structure in PtB(Mon), it satisfies the following equalities :
m(a, a′) = m(q(a), 1) · m(sf(a′), a′) = q(a) · a′,
having m(sf(a′),a′) = a′ since m, being a binary operation on the object
(A,B,f,s) in the fiber SPtB(Mon), has to preserve units. Corollary 2.4.9. A Schreier split epimorphism (A,B,f,s) is endowed with a
group structure in PtB(Mon) if and only if its kernel K[f] is an abelian group.
Proof. The kernel functor, being left exact, preserves the group structure. Con- versely suppose K[f] is an abelian group. It is a commutative monoid, and con- sequently the Schreier split epimorphism (A, B, f, s) is endowed with a monoid structure. Moreover the kernel functor SPtB(Mon) → Mon is left exact and conservative; accordingly it reflects pullbacks, and consequently its extension to internal monoids reflects groups. See the Appendix for more details.