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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)
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)
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)
In 1970, M. Gerstenhaber introduced a list of axioms defining Moore categories in order to develop the Baer Extension Theory. In this paper, we study some implications between the axioms and compare them with more recent notions, showing that, apart from size restrictions, a Moore category is a pointed, strongly protomodular and Barr-exact category with… (More)
The characterisation of double central extensions in terms of commuta-tors due to Janelidze (in the case of groups), Gran and Rossi (in the case of Mal'tsev varieties) and Rodelo and Van der Linden (in the case of semi-abelian categories) is shown to be still valid in the context of exact Mal'tsev categories. In his article , George Janelidze gave a… (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)
We show that the adjunction between monoids and groups obtained via the Grothendieck group construction is admissible, relatively to surjective homomorphisms, in the sense of categorical Galois theory. The central extensions with respect to this Galois structure turn out to be the so-called special homogeneous surjections.