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Chapter 4
Mal’tsev aspects of Mon related to Schreier split epimorphisms
4.1 Mal’tsev categories
We recalled in Section 1.5 that a category C is a Mal’tsev category [16, 17] when any reflexive relation is an equivalence relation; this fact appears to be equivalent to the property that any fiber PtXC of the fibration ¶C is unital, see [8]. The category Gp of groups is a Mal’tsev one. The preorder ON on the natural numbers (Example 2.2.7) shows that the category Mon is not a Mal’tsev one. However, we pointed out a meaningful structural observation: the category Mon of monoids is equipped with a fully faithful subfibration ¶S of the fibration of points ¶:
SPtMon //
¶S && yy
Mon
such that any fiber PtBMon is SPtBMon-unital and, consequently, any fiber SPtBMon is unital. In this chapter, we shall be interested in what is remaining of the Mal’tsev results in this new structural context.
// PtMon
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¶
