Page 9 - Textos de Matemática Vol. 45

Introduction
Many intrinsic properties of the category Gp of groups have already been pointed out (for instance, see [4]), emphasizing and explaining the similari- ties with other algebraic structures, like rings and Lie algebras. The category Mon of monoids, however, did not seem to have been already investigated for itself, namely from the point of view of its internal structural properties. One of the aims of this work is to fill this gap.
It is true that, besides the property of being unital, a property which, among other things, controls the algebraic notion of commutative pair of sub- objects and, more generally, of commutative pair of morphisms [8], this cate- gory was not yet showed to satisfy strong categorical schemes. However, recent works about some aspects of the semidirect product in this setting [26] brought us a new pertinent tool with the notion of Schreier split epimorphism, which will make explicit a strong and meaningful parallelism with the category Gp of groups.
On one hand, from this point of view, in the same way as the group AutG of automorphisms of a group G classifies the class of split exact sequences with kernel G (see [5], [6] and also [3]), the monoid End(M) of endomorphisms of a monoid M will appear to classify a certain class of so-called Schreier split exact sequences with kernel M, while the group Aut(M) will appear to classify the subclass of so-called homogeneous split exact sequences with kernel M. In other words, the notions of Schreier split epimorphism and homogeneous split epimorphism allow us to read, in the category Mon, the tracks of the property of action representability satisfied by the category Gp.
On the other hand, the category Gp of groups is a Mal’tsev category in the sense of [16, 17], namely a category in which any reflexive relation is an equivalence relation; this, among other things, forbids the existence of internal preorders. On the contrary, there are preorders in the category Mon which reveal that its weaker categorical structuration gives it an interesting flexibility in comparison with the rigidity of the paradigmatic category Gp.
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