Page 138 - Textos de Matemática Vol. 46

128 M. GRAN AND D. RODELO
whenever x, y, t, z are elements in X with (x, y) 2 R ^ T , (x, t) 2 S, (y, z) 2 S and (t,z) 2 R, it then follows that (t,z) 2 T:
xSt TRRT ySz
This notion has been extended to a categorical context in [6]. Indeed, the property expressed by the Shifting Lemma can be equivalently reformulated in any finitely complete category C by asking that a specific class of internal functors are discrete fibrations, as we are now going to recall. For any object X in C and any equivalence relations R, S, T on X with
R^STR
there is a canonical inclusion (i,j): T⇤S ! R⇤S of equivalence relations,
depicted as
T⇤S // j //R⇤S
⇡1 ⇡2 ✏✏ ✏✏//
⇡1 ⇡2 (3.1) //✏✏ ✏✏
T i R,
where T⇤S (respectively, R⇤S) is the largest double equivalence relation on T and S (respectively, on R and S) and ⇡1 and ⇡2 are the projections on T (respectively, on R).
Definition 3.1. [6] A finitely complete category C is called a Gumm category when any inclusion (i,j): T⇤S ! R⇤S as in (3.1) is a discrete fibration. This means that any of the commutative squares in (3.1) is a pullback.
Let us recall that a Mal’tsev category C is a finitely complete category such that every reflexive relation in C is an equivalence relation. A regular cate- gory C is a Mal’tsev category when the composition of (e↵ective) equivalence relations on any object in C is 2-permutable: RS = SR, where R and S are (ef- fective) equivalence relations on a same object (see [9, 8]). The strictly weaker 3-permutability property for (e↵ective) equivalence relations, RSR = SRS, de- fines the notion of regular Goursat categories [8]. Goursat categories, thus in particular Mal’tsev categories, have the property that every lattice of equiva- lence relations (on the same object) is modular (Proposition 3.2 in [8]). This fact implies that any regular Mal’tsev category and, more generally, any regular