Dominique Bourn, Split extension classifier, actions representation and associated classification properties
We shall show that, in any pointed protomodular category C,
when an object X has a split extension classifier,
as it is the case with the object AutX in the category Gp of groups, this classifier is actually underlying an internal
groupoid structure D*(X) which measures the obstruction to the abelianity of X, in the sense that the kernel
relation of its normalization jX is exactly the centre of X.
Furthermore, the existence of such a classifier appears to have a more global classifying power: for instance,
the pointed protomodular category C is additive if and only if, for all X, we have
jX= tX: X---->1 (i.e. the classifier is trivial), and antiadditive (i.e. with no non trivial abelian object) if and only if, for
all X, jX is a monomorphism.
Marino Gran, Internal groupoids and crossed modules
Internal categorical structures are very useful for a conceptual understanding
of the theory of commutators
in universal algebra. In these lectures, we shall first focus on the property of centrality in a Malícev category,
and explain the link with the construction and the properties of the algebraic commutator.
By studying the internal graphs, categories and groupoids in a general modular variety, it will then be
shown how one can characterize distributive, Malícev and arithmetical varieties. Finally, by restricting our attention
to the semi-abelian context, we shall consider two torsion theories in the category of internal groupoids,
and give explicit examples in the category of crossed modules.
Enrico Vitale, Schreier theory and obstruction theory via categorical groups
In my lecture, I will recall some classical facts from Schreier theory
and obstruction theory for groups.
Then I will show how these facts can be expressed using the language of categorical groups (or crossed modules).
Finally, I will discuss possible generalizations of Schreier and obstruction theory suggested by the formalism of
categorical groups (possibly in the context of categories with split extension classifier).