Variations of the Shifting Lemma and Goursat categories

We 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. 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