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1.5. Mal’tsev categories 9
between such split epimorphisms. We denote by ¶E : P tE → E the functor associating with any split epimorphism its codomain. As soon as the category E has pullbacks, this functor is a fibration, which is called the fibration of points (see the Appendix). Given any object X in E, the fiber PtXE is the category of split epimorphisms with codomain X, while, given any map h: X → Y , the change-of-base functor h∗ : P tY E → P tX E is given by the pullback along the map h.
Proposition 1.4.1. If E is a unital category, then so is PtE.
Proof. Given two split epimorphisms (A,B,f,s) and (A′,B′,f′,s′) in E, their product in PtE is the split epimorphism (A×A′,B×B′,f×f′,s×s′). Consider then the following diagram:
Aoo πA //A×A′ oo πA′ //A′
OO ⟨1A,0⟩ OO ⟨0,1A′⟩ OO
f s f×f′ s×s′ f′ s′ oo πB ′ πB′ // ′
B
//B×B oo B ⟨1B ,0⟩ ⟨0,1B′ ⟩
Since the two pairs (⟨1A,0⟩,⟨0,1A′⟩) and (⟨1B,0⟩,⟨0,1B′⟩) are jointly strongly epimorphic in E, the pair of morphisms in PtE they define is jointly strongly epimorphic, too.
1.5 Mal’tsev categories
We briefly recall here some concepts that will be developed later in more details. A category C is said to be a Mal’tsev category [16, 17] when any internal reflexive relation R on an object X in C is an equivalence relation. The category
Gp of groups is a Mal’tsev one. The preorder ON on the natural numbers:
where
p0 // ON oo s0 // N
p1
ON ={(x,y)∈N×N|x≤y}
shows that the category Mon is no longer a Mal’tsev one.
From [8], it appears that being a Mal’tsev category is equivalent to the
property that any fiber PtXC of the fibration of points ¶C is unital. Conse- quently it is not true that any fiber PtX(Mon) is unital.
However we shall show here that these fibers are unital relatively to a certain full subcategory of specific split epimorphims and we shall make explicit what is remaining of the classical Mal’tsev results in this new structural context.