Page 106 - Textos de Matemática Vol. 45

98 Chapter 8. Conclusion mains in SPtXC.
8.1 Partial protomodularity
It is worth introducing the following definition:
Definition 8.1.1. A finitely complete pointed category C will be said to be S- protomodular when there is a class of split epimorphisms S which determines a subfibration ¶SC of the fibration of points ¶C, where SPtC is the full subcategory of PtC whose objects are those which are in S:
oo eC
OO πC OO
oo eC
OO πC OO
SPtC // j
//PtC ¶C
satisfying the following properties:
(1) any object in SPtC is a strongly split epimorphism;
(2) SPtC is stable under finite limits in PtC (in particular, it contains the terminal object 1 1 in P tC).
So, S is a class of strongly split epimorphisms. The fact that ¶SC is a subfibration of ¶C means that this class S is stable under pullbacks. Condition (2) implies that any fiber SPtXC is stable under finite limits in the fiber PtXC and that any change-of-base functor with respect to ¶SC is left exact. The fact that SPtC contains the terminal object is equivalent to the fact that the class S contains the isomorphisms and that any fiber SPtXC is pointed. From this definition, we get immediately:
Proposition 8.1.2. Let C be an S-protomodular category. Then:
1) any fiber PtXC is SPtXC-unital, which implies that any fiber SPtXC is unital by Condition (2) above;
2) any change-of-base functor with respect to the fibration ¶SC is conservative.
Proof. 1) Consider the following left hand side downward pullback of split epimorphisms, where the lower one is in the fiber SPtBC:
¶S C
## }} C
A×B C oos // //oos
A×B C
//C (8.1.1) πAeAgt eAt
//C
A//B. K[f]kA//B. ff