Gran, MarinoRodelo, DianaNguefeu, Idriss Tchoffo2020-07-242020-07-242019-030002-52401420-8911http://hdl.handle.net/10400.1/14408We prove that Mal'tsev and Goursat categories may be characterized through variations of the Shifting Lemma, that is classically expressed in terms of three congruences R, S and T, and characterizes congruence modular varieties. We first show that a regular category C is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in C. Moreover, we prove that a regular category C is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation S and reflexive and positive relations R and T in C. In particular this provides a new characterization of 2-permutable and 3-permutable varieties and quasi-varieties of universal algebras.engMal'tsev categoriesCategoriesShifting LemmaCongruence modular varieties3-permutable varietiesjournal article10.1007/s00012-018-0575-z