A Categorical Approach to Commutator Theory

Semi-abelian exact completions

“…Since any internal reflexive relation in a protomodular category is an equivalence relation, protomodular categories also have all the nice properties of Maltsev categories [12], so that there is in particular a good theory of centrality of equivalence relations [7] [8] [22] [25].…”

On low-dimensional homology in categories

“…At the time when the Fröhlich school was investigating this subject, the categorical properties of groups that are essential in order to develop this theory had not all been discovered. Recent advances in categorical algebra [3,4,5,7,21,22,24,28] now make it possible to build a more general theory based on simpler arguments.…”

Arithmetical categories and commutator theory

“…In [15], the problem of finding a suitable "commutator theory" for equivalence relations in Maltsev categories, that maintains the main classical varietal properties (see [5,6] and [17]), is widely discussed and a definition in terms of universal property and adjoint functors is proposed. We recall the construction of such a commutator and refer to [15] for more details.…”

“…In particular, by using previous results of [15], we show that for Maltsev categories the notion of pregroupoid given by Kock [12], exactly corresponds to the case of a span with a trivial commutator for the induced kernel relations. This will allow to characterize Maltsev categories in terms of the behaviour of the corresponding categories of internal pregroupoids and of internal spans (see Theorem 1.11).…”