Page 59 - Textos de Matemática Vol. 45

5.1. The Schreier split extension classifier 51 = qf (s(b1) · qf (s(b2) · x)),
while
φ(b1)(φ(b2)(x)) = φ(b1)(qf (s(b2) · x)) = qf (s(b1) · qf (s(b2) · x)),
and the two expressions are equal.
Also we have to check that φ¯ is a monoid homomorphism, which is equiva-
lent to: qf (a·a′) = qf (a)·qf (sf(a)·qf (a′)), and this is true thanks to Proposition 2.1.5 (e). Moreover, the fact that the square θXφ¯ = φf is a pullback comes from Theorem 2.3.7.
It remains to prove the uniqueness of the pair (φ, φ¯). So, suppose we have another pair (ψ, ψ¯) satisfying the same conditions:
X oo X
qf // A oo s // B kf
ψ¯ ψ ooq ooρX
ιX
// Hol(X)
// End(X)
The commutativity of the square at the level of split epimorphisms implies that ψ¯(a) = (q¯(a),ψf(a)) for some set theoretical map q¯: A → X. The commutativ- ity at the level of the sections implies that q¯(s(b)) = 1. The fact that the square is a pullback of Schreier split epimorphisms is equivalent to the fact that the pair (f,ψ¯) is jointly monomorphic, which, itself, is equivalent to the fact that the pair (f, q¯) is jointly monomorphic. And finally, we have ψ¯k = ιX if and only if we have q¯k = 1X . The fact that ψ¯ is a monoid homomorphism is equivalent to q¯(a·a′) = q¯(a)·ψ(f(a))(a′). Whence q¯(s(b)·a′) = q¯(s(b))·ψ(b)(a′) = ψ(b)(a′) which gives the definition of ψ via q¯.
Let us check now that q¯(a)·sf(a) = a. For that we check it by composition with the jointly monomorphic pair (f,q¯). This is clear for f, and we have q¯(q¯(a) · sf(a)) = q¯(q¯(a)) · q¯(sf(a)) = q¯(q¯(a)) = q¯(a). The uniqueness of the factorization associated with the Schreier split epimorphism in question implies that q¯(a) = qf (a).
The previous proposition allows us to give the following definition, which is inspired by Definition 1.1 in [3]:
Definition 5.1.3. For any monoid X, the monoid End(X) will be called the Schreier split extension classifier of X. For any Schreier split epimorphism (A,B,f,s) with kernel X, the pair (φ,φ¯) will be called its classifying map or its index.
Proposition 3.1.12 has the following immediate corollary:
Theorem 5.1.4. R[θX] is a Schreier equivalence relation if and only if X is a group.
θX