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100 Chapter 8. Conclusion
Proposition 8.2.1. Let C be a finitely complete pointed category and C′ be a full subcategory of C stable under finite limits and containing the terminal object. The category C is C′-unital if and only if it is ΠC′-protomodular.
Proof. Suppose C is C′-unital. Then any split epimorphism in ΠC′ is a strongly split epimorphism. Accordingly any map between such split epimorphisms is of the following form:
X×OOY f×g//X′×OOY′
πX ⟨1X ,0⟩ πX′ ⟨1X′ ,0⟩
//
X X,
f
(8.2.2)
which implies that ΠC′PtC is stable under equalizers; being already stable under finite products, it is stable under finite limits.
Conversely, suppose C is ΠC′-protomodular. The fact that any split epi- morphism in ΠC′ is a strongly split epimorphism means exactly that C is C′ -unital.
Here the fiber above 1 of this subfibration is C′, which is different from C if C is not unital.
8.3 Left exact conservative forgetful functors
Recall that a functor U is said to be conservative when it reflects the isomor- phims. Let U : C → D be a left exact functor between pointed finitely complete categories. The forgetful functor
U : SRng → CMon
is precisely a left exact conservative functor between pointed categories. More- over we noticed that the class of Schreier split epimorphisms in SRng are the inverse image of the class of Schreier split epimorphisms in CMon. We have the following very general result:
Proposition 8.3.1. Let U : C → D be a left exact functor between pointed finitely complete categories. Suppose, in addition, that the functor U is conservative. Suppose that the category D is S-protomodular. Then, setting Σ = U−1(S), the category C is Σ-protomodular.
Proof. Since U is left exact and conservative, it is straightforward that U re- flects the strongly split epimorphisms. Accordingly, any split epimorphism in Σ is a strongly split epimorphism when D is S-protomodular. The class Σ con- tains the isomorphisms since it is the case for the class S. Finally, Σ is stable under finite limits since U is left exact and S is stable under finite limits.
