We prove that all semi-abelian categories with the the Smith is Huq property satisfy the Commutator Condition (CC): higher central extensions may be charac-terised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commuta-tors suffice. As a consequence, in the presence of enough projectives we obtain explicit Hopf formulae for homology… (More)
We characterise the double central extensions in a semi-abelian category in terms of commutator conditions. We prove that the third cohomology group H 3 (Z, A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A.
We present a new characterisation of Goursat categories in terms of special kind of pushouts, that we call Goursat pushouts. This allows one to prove that, for a regular category, the Goursat property is actually equivalent to the validity of the denormalised 3-by-3 Lemma. Goursat pushouts are also useful to clarify, from a categorical perspective, the… (More)
We prove that a regular category C is a Mal'tsev category if and only if a strong form of the denormalised 3 × 3 Lemma holds true in C. In this version of the 3 × 3 Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new… (More)
We show that varietal techniques based on the existence of operations of a certain arity can be extended to n-permutable categories with binary coproducts. This is achieved via what we call approximate Hagemann– Mitschke co-operations, a generalisation of the notion of approximate Mal'tsev cooperation . In particular, we extend characterisation theorems… (More)
We investigate 3-permutability, in the sense of universal algebra, in an abstract categorical setting which unifies the pointed and the non-pointed contexts in categorical algebra. This leads to a unified treatment of regular subtractive categories and of regular Goursat categories, as well as of E-subtractive varieties (where E is the set of constants in a… (More)
A new method for realizing the first and second order cohomology groups of an internal abelian group in a Barr-exact category was introduced in  and . The main role, in each level, is played by a direction functor. This approach can be generalized to any level n and produces a long exact cohomology sequence. By applying this method to Moore… (More)
In this talk we give an overview of the work developed in the past three years concerning the topic of the title, one of the tasks of the project " Categorical Methods in Non Abelian Algebra ". We mainly explored properties of n-permutable categories. For n = 2, we obtained a new homological lemma, called the Cuboid Lemma, which characterises Mal'tsev (=… (More)