On congruence modular varieties and Gumm categories

One of the main interest of the Shifting lemma is that, being freed of any inductive condition (as the existence of suprema for instance), it allows us to recover in any category V(E) of internal V-algebras in E all the projective properties of the variety V, when this variety is congruence modular. Being non-inductive, this shifting property has a meaning in any finitely complete category E; its categorical characterization was given in [11] where its first consequences were investigated, mainly concerning the centralization of equivalence relations, the internal groupoids and the internal categories. Later on, the categories satisfying the Shifting Lemma were called Gumm categories in [12] and [7]. Now, in the same way as the Mal’tsev categories [4], the Gumm categories were characterized in [7] by some properties of the pointed fibers PtY E (=split epimorphisms above the object Y ) of their fibrations of points ¶E, see Sections 1 and 2.1 below. The aim of this work is to start from this characterization to deepen the investigatation on the Gumm categories, to better understand the observations already made in [11] and to produce new examples. Indeed, the characterization theorem will provide us with a golden thread inside the Gumm categories: an intrinsic notion of abelian split epimorphism, see Section 4.2.