Page 107 - Textos de Matemática Vol. 45

8.2. Back to C′-unital categories 99
then also the upper one is in SPtBC, since ¶SC is a subfibration of ¶C. So the split epimorphism (πC , eC ) is a strongly split epimorphism. On the other hand, the right hand side square is still a pullback, so the map eAk is the kernel of πC. Accordingly, the pair (eAk,eC) is jointly strongly epimorphic. So this is equally the case for the pair (eA,eC).
2) Since any change-of-base functor with respect to ¶SC is left exact, it is enough to prove that it is conservative on monomorphisms (see the Appendix). Let us then consider the following diagram, where all the quadrangles are pull- backs and all the split epimorphisms are in SPtC:
′ kf′ //′ x // K[f] X X
%% K(m′)
%% ¯′ kf¯′
OO OO ′
mm //¯′ x¯ //¯
K[f] X X
DD
f′ s′ f s
EE
// ′ s¯ // 1YyY
f¯′ f¯
′ s¯

Suppose moreover that m′, and consequently K(m′), are isomorphisms. Cer-
tainly x¯k ¯′ ¯ ¯
f is the kernel of f , and, since the split epimorphim (f , s¯) is a strongly split epimorphism, the pair (x¯kf¯′ , s¯) is jointly strongly epimorphic. Accordingly, since K(m′) = K(m) is an isomorphism, so is m.
8.2 Back to C′-unital categories
The C′-unital categories can appear as a particular case of the previous situa- tion. Let C be a finitely complete pointed category. Let C′ be a full subcategory of C stable under finite limits and containing the terminal object. Let us de- note by ΠC′ the class of split epimorphims which are, up to isomorphisms, the canonically split direct product projections:
πX // ⟨1X ,0⟩
where the factor Y is in C′. It is clear that this class is stable under pullbacks.
The full subcategory j : ΠC′P tC P tC whose objects are those which are in
ΠC′ is such that the functor ¶ΠC′ = ¶C ◦ j: ΠC′PtC → C is a subfibration of C
¶C. On the other hand, the category ΠC′PtC is clearly stable under products, since so is C′. The fact that the terminal object 1 belongs to C′ implies that the isomorphisms are in ΠC′.
X×Y oo
X,