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84 Chapter 7. Special Schreier and special homogeneous surjections horizontal quotients:
R[f]×X R[f]oo p1 ////R[f] f¯ ////Q
OO (d0p0,p) OO (p,d1p1) d0 s0 d1
OO ψ σ
p0
R[f] //XfY
oo d1 // d0
¯¯
(ψ,σ) such that ψf = fd0 = fd1 and σf = fs0. Moreover, since any of the
left hand side commutative squares is a pullback, by the Barr-Kock Theorem (Theorem 9.2.2) the right hand side one is a pullback as well.
Proposition 7.3.2. Let (f,k) be a special Schreier (resp. homogeneous) exten- sion with abelian kernel A. The split epimorphism (ψ,σ) is a Schreier (resp. homogeneous) one and its kernel is isomorphic to the abelian group A. Accord- ingly R[ψ] is a Schreier (resp. homogeneous) equivalence relation.
Proof. The right hand side square being a pullback, the kernel of ψ is iso- morphic to the kernel of d0, which is itself isomorphic to the kernel A of f. Moreover, since (d0,s0) is a Schreier (resp. homogeneous) split epimorphism, so is the split epimorphism (ψ, σ), by Corollary 2.3.6.
Definition 7.3.3. The split epimorphism (ψ, σ) in the proposition above is called the direction of the special Schreier (resp. homogeneous) extension (f,k).
The name direction was first used, in a more general context, in [9], being in- spired by the fact that the direction of an affine space is the associated vector space (both affine spaces and extensions are particular cases of the general con- text studied in [9]).
We have the immediate corollary
Corollary 7.3.4. Given any special Schreier extension with abelian kernel A, it is homogeneous if and only if so is its direction.
The direction of a Schreier extension with abelian kernel has an alternative construction, giving more directly the structure of Y -module on A. It is based on the following observation:
Proposition 7.3.5. Given a special Schreier extension with abelian kernel A, the classifying map φ: X → End(A) of the Schreier split epimorphism (d0,s0) coequalizes the equivalence relation R[f].
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The universal property of the quotients produces a unique split epimorphism