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56 Chapter 5. Schreier and homogeneous representability sending a pair (x, b) to the element x · s(b). Indeed:
φψ(a) = φ(q(a), f (a)) = q(a) · sf (a) = a ψφ(x, b) = ψ(x · s(b)) = (q(x · s(b)), b) = (x, b),
and
where the last equality comes from Proposition 2.1.4. Finally, φ (and hence ψ)
is a homomorphism; indeed, using the commutativity of A, we have: φ((x1,b1)·(x2,b2))=φ(x1 ·x2,b1 ·b2)=(x1 ·x2)·s(b1 ·b2)=
=x1 ·x2 ·s(b1)·s(b2)=x1 ·s(b1)·x2 ·s(b2)=φ(x1,b1)·φ(x2,b2),
and obviously φ(1, 1) = 1.
Corollary 5.5.2. Given a Schreier split epimorphism (A, B, f, s), if A is a com- mutative monoid, then the Schreier retraction q is a homomorphism of monoids (and not only a set-theoretical map).
Proof. Consider the following commutative diagram:
K[f] oo q // A oo s k OO f
ψφ
ooπK[f] oo⟨0,1⟩
// B // B,
Proposition 5.5.1 implies that, in the category CMon of commutative monoids, the unique Schreier split epimorphisms are the direct product projec- tions (which are also homogeneous). Then we have the following
Proposition 5.5.3. In the category CMon, the trivial monoid 0 acts as a Schreier (and homogeneous) split extension classifier for every commutative monoid X.
Proof. Consider the following diagram, where the upper row is a Schreier split sequence in CMon:
Xoo q //Aoo s //B kf
q
1X
// K[f] × B
Then q = πK[f]ψ is a homomorphism.
K[f]
where φ and ψ are the morphisms defined in the proof of Proposition 5.5.1.
⟨1,0⟩
πB
oo 1X oo
// X // 0;
X
since A is isomorphic to X × B, the square on the right is a pullback.