Diana Rodelo

A new characterisation of Goursat categories

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 existence of the quaternary operations characterising 3-permutable varieties.

Higher central extensions and cohomology

Abstract We establish a Galois-theoretic interpretation of cohomology in semi-abelian categories: cohomology with trivial coefficients classifies central extensions, also in arbitrarily high degrees. This allows us to obtain a duality, in a certain sense, between “internal” homology and “external” cohomology in semi-abelian categories. These results depend on a geometric viewpoint of the concept of a higher central extension, as well as the algebraic one in terms of commutators.

Directions for the long exact cohomology sequence in Moore categories

A new method for realizing the first and second order cohomology groups of an internal abelian group in a Barr-exact category was introduced by Bourn (Cahiers Topologie Géom Différentielle Catég XL:297–316, 1999; J Pure Appl Algebra 168:133–146, 2002). 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 categories we show that they represent a good context for non-abelian cohomology, in particular for the Baer Extension Theory.

The Cuboid Lemma and Mal'tsev categories

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 characterisation of regular categories that are Mal’tsev categories (= 2-permutable) is similar to the one previously obtained for Goursat categories (= 3-permutable). We also analyse the “relative” version of the Cuboid Lemma and extend our results to that context.

Approximate Hagemann–Mitschke Co-operations

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 co-operation [2]. In particular, we extend characterisation theorems for n-permutable varieties due to J. Hagemann and A. Mitschke [8, 9] to regular categories with binary coproducts.

Some remarks on connectors and groupoids in Goursat categories

We prove that connectors are stable under quotients in any (regular) Goursat category. As a consequence, the category

The third cohomology group classifies double central extensions

\mathsf{Conn}(\mathbb{C})

Cohomology without projectives

of connectors in

Comprehensive factorization and I-central extensions

\mathbb{C}

Higher central extensions via commutators

is a Goursat category whenever