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50 Chapter 5. Schreier and homogeneous representability
Proof. The kernel of θX is given, up to isomorphism, by the map ιX : X Hol(X) defined by ι(x) = (x,1X). We can define a map q: Hol(X) → X by q(x, γ) = x and then we have:
and
(x, γ) = (x, 1X ) · (1, γ) = ιX q(x, γ) · ρX θX (x, γ) ιX q((x, 1X ) · (1, γ)) = ιX q(x, γ) = (x, 1X ).
Theorem 5.1.2. For any Schreier split epimorphism with kernel X: Aoo s //B
f
there exists a unique morphism φ: B → End(X) such that the following dia- gram is commutative and its right hand side part is a pullback of split epimor- phisms:
X oo qf // A oo s // B kf

X
ιX
// Hol(X)
// End(X).
φ¯ φ ooq ooρX
The explicit definition of φ is φ(b)(x) = qf (s(b) · x). The pair (k, s) being jointly strongly epimorphic, the homomorphism φ¯ is completely determined by the commutativity of this diagram, and we have: φ¯(a) = (qf (a), φf (a)).
Proof. We start by observing that, for any b ∈ B, φ(b) ∈ End(X). Indeed, for any b ∈ B and any x1,x2 ∈ X:
φ(b)(x1 · x2) = qf (s(b) · x1 · x2), φ(b)(x1) · φ(b)(x2) = qf (s(b) · x1) · qf (s(b) · x2)
and the two expressions are equal, because:
qf(s(b)·x1 ·x2) = qf(s(b)·x1)·qf(sf(s(b)·x1)·x2) = qf(s(b)·x1)·qf(s(b)·x2).
Moreover, φ is a monoid homomorphism. Indeed, for any b1,b2 ∈ B and any x ∈ X:
φ(b1b2)(x) = qf (s(b1b2) · x) = qf (s(b1) · s(b2) · x), which, thanks to Proposition 2.1.5, is equal to
qf (s(b1)) · qf (sfs(b1) · qf (s(b2) · x)) = 1 · qf (s(b1) · qf (s(b2) · x)) =
θX