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32 Chapter 3. Schreier internal relations, categories and groupoids Example 3.1.9. Consider the homogeneous split epimorphism
Z∗oo i //N∗ abs
described in Example 2.2.8. The kernel equivalence relation R[abs], defined by:
xR[abs]y ⇔ |x| = |y|
is homogeneous, as well. This can be seen directly, but it also comes from Propo- sition 3.1.12 below, since the kernel of abs is K[abs] = {±1}, which is a group.
Example 3.1.10. Let G be a group and H be a normal subgroup; denote by
RH the equivalence relation on G associated with H, given by the following:
ifg,g ∈G,thengR g ifandonlyifgg−1 ∈H.GivenanymonoidM 12 1H2 12
such that H ⊆ M ⊆ G, then the restriction RM of RH on M is homogeneous. Indeed, the Schreier retraction q′ of the relation RM is nothing but the restric- tion of the Schreier retraction q of RH (since the kernels of the corresponding split epimorphisms are both isomorphic to H); this proves that RM is right homogeneous. The proof that it is also left homogeneous is analogous.
Example 3.1.11. Some particular cases of the general situation described in Example 3.1.10 are obtained when G = C∗ is the multiplicative group of non- zero complex numbers, M ={a+ib | a,b∈Z, (a,b)̸=(0,0)} and H is either {±1} or {±1, ±i}.
Proposition 3.1.12. Given a split epimorphism (X,Y,f,s), its kernel equiva- lence relation R[f] is a Schreier one if and only if the split epimorphism is itself a Schreier one and the kernel K[f] is a group. Its kernel equivalence rela- tion is homogeneous if and only if the split epimorphism is itself homogeneous and the kernel K[f] is a group.
Proof. Let (X,Y,f,s) be a split epimorphism. Suppose R[f] is a Schreier equiv- alence relation, then K[p0] ≃ K[f] is a group. On the other hand, consider the following diagram of split epimorphisms, which is always a pullback:
X s1 // R[f] (3.1.1)
f s p0 s0 //
Y s X,
where the morphism s1 is given by s1(x) = (sf(x),x) (see the Appendix). Then the left hand side split epimorphism is a Schreier one, since so is the right hand side one. In the same way, when R[f] is a homogeneous equivalence relation, the split epimorphism in question is homogeneous.
OO
OO