Page 30 - Textos de Matemática Vol. 45

22 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
2.3.2 The Schreier split short five lemma
We can now assert the Schreier split short five lemma and its variations.
Corollary 2.3.8. Consider a commutative diagram of Schreier split epimor- phisms:
K[f]oo q //Aoo f //B
s
′ f′ //
// A oo B
k′
the map u is an isomorphism if and only if K(u) is an isomorphism.
Proof. The fact that K(u) is an isomorphism as soon as so is u holds in any pointed category with kernels. The converse is an immediate consequence of the previous theorem since, the right hand square being a pullback, u appears to be the pullback of the isomorphism 1B.
Corollary 2.3.9. Consider a commutative diagram of split exact sequences of the form
Xoo q //Aoo s ////B kf
φ
Proof. If φ is an isomorphism, then the lower split epimorphism is obviously a Schreier one: the unique map q′ satisfying the Schreier condition is qφ−1. The converse is an immediate consequence of the previous corollary.
More generally we have:
Proposition 2.3.10. Consider the following commutative diagram, where the two rows are Schreier split sequences:
K[f]oo q //Aoo s ////B kf
K(u) u v
k ′ooq′
u
K(u) K[f ]
s′
oo q′ k′
′oo s′
// A // // B,
X
where the upper split sequence is a Schreier one. The morphism φ is an iso-
f′
morphism if and only if the lower split sequence is a Schreier one.
′ oo q′
K[f ]
k′
′ oo s′ ′
// A
f′
// // B .