**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 j_{X} 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

j_{X}= t_{X}: 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, j_{X} 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 Mal’cev 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, Mal’cev 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).