Page 105 - Textos de Matemática Vol. 45
P. 105
Chapter 8
Conclusion
We shall sum up here the rich structural equipment we made explicit on the categories Mon of monoids and SRng of semirings. We recalled that a category C is a Mal’tsev category when any fiber PtXC of the fibration of points ¶C is unital; the category C is said to be protomodular (see [4] for more details) when any change-of-base functor of this same fibration is conservative; this im- plies that the category C is a Mal’tsev one. The category Gp of groups is the paradigmatic example of a pointed protomodular category, but the category Rng of rings without unit is pointed protomodular as well.
We showed that the categories Mon of monoids and SRng of semirings are unital categories C equipped with a class S of split epimorphisms determining 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:
SPtC // j
//PtC ¶C
satisfying the following properties:
1) any object in SPtC is a strongly split epimorphism;
2) the fibers coincide on the terminal object: SPt1C = C = Pt1C;
3) any fiber SPtXC is stable under finite limits in the fiber PtXC;
4) the change-of-base functors of the fibration ¶SC = ¶C ◦ j are conservative; 5) any fiber P tX C is SP tX C-unital, which implies by 2) that any fiber SP tX C is unital;
6) the change of base functors αX∗ : PtXC → C with respect to ¶C along the initial maps αX : 1 → X reflect the commutativity of maps having their do-
¶S C
## }} C
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