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X ",= X+ O Y ~ +U iy Then he shows that all possible modular categories are the slices of additive categories with kernels.with, furthermore, m = m + U and n = n + U, up to isomorphism.PrQgf. Apply the axiom t to the left hand square, then the right hand square is a pullback and K U j J T isomorphic to J + U :Conversely, let us suppose that the left hand square is a pullback :

We explore some properties of Schreier split epimorphisms between monoids, which correspond to monoid actions. In particular, we prove that the split short five lemma holds for monoids, when it is restricted to Schreier split epimorphisms, and that any Schreier reflexive relation is transitive, partially recovering in monoids a classical property of Mal'tsev varieties.

The fibration p of pointed objects of a category E is shown to have some classifying properties: it is additive if and only if E is naturally Mal'cev, it is unital if and only if E is Mal'cev. The category E is protomodular if and only if the change of base functors relative to p are conservative. Mathematics Subject Classifications (1991). Primary 18D05; secondary 08B05, 20L17, 18G30. Key words; fibration, additive category, Mal'cev category, internal groupoid.It was shown in [1] that the category Grd E of internal groupoids in E is monadic above the category Pt E of split epimorphisms and commutative squares. A paper by Carboni on modular categories [4] led me to focus my attention rather on the fibration of pointed objects, that is the forgetful functor p: Pt E --4 E associating to each split epimorphism its codomain. Indeed modular categories were shown [2] to be exactly those categories whose fibration of pointed objects is such that every change of base functor is an equivalence and whose terminal object satisfies a certain modularity condition.Further works about naturally Mal'cev categories [8] by Johnstone and Mal'cev categories [5,6] by Carboni, Pedicchio and others allow to show, here, that this fibration of pointed objects has actually a larger class of classifying properties. Naturally Mal'cev categories were introduced to characterize the categories E for which the forgetful functor L: Grd E -4 Refl E from internal groupoids to reflexive graphs is an equivalence (the Lawvere condition). They appear to be characterized by the fact that their fibrations p of pointed objects are additive (every fiber is additive). In the Mal'cev categories any reflexive relation is an equivalence relation. Mal'cev categories are characterized by the fact that their fibrations p of pointed objects are unital (every fiber is unital), where a unital category satisfies the following property (which is fulfilled by the category of unitary magmas as well): it is pointed and any product X x Y is "generated" by X × 1 and 1 × Y. Actually this works leads up to five characterizations of Mal'cev categories. The main ones are the following: the fibration p is unital, every change of base functor, in respect to p, along a split epimorphism is saturated on subobjects, the forgetful functor L: Grd E --4 Refl E is saturated on subobjects. All these characterizations being thought and described as specific

The classical 3 = 3 lemma and snake lemma, valid in any abelian category, still Ž . hold in any quasi-pointed the map 0 ª 1 is a mono , regular, and protomodular category. Some applications are given, in this abstract context, concerning the Ž denormalization of kernel maps and the normalization of internals groupoids i.e., . associated crossed modules . ᮊ