N-fold groupoids and n-groupoids in semi-abelian categories

The notion of semi-abelian category, introduced by G. Janelidze, L. Márki and W. Tholen in 2002, generalizes that of abelian category, and captures the homological properties that the categories of groups, associative algebras, Lie algebras and cocommutative Hopf algebras over a field have in common. Internal structures behave surprisingly well in semi-abelian categories. For example, any internal reflexive graph admits at most one internal category structure, and the categories of internal categories and internal groupoids are isomorphic. Moreover, the category of internal groupoids is equivalent to the category of internal crossed modules. The notion of internal crossed module in any semi-abelian category was introduced by G. Janelidze in 2003, and recovers in particular the classical notion of crossed module of groups. The fact that the category of internal groupoids in a semi-abelian category is itself semi-abelian implies that also the category of internal n-fold groupoids is well-behaved.
 
In this talk, I will prove that the full subcategory of internal n-groupoids in a semi-abelian category is a Birkhoff subcategory of the category of internal n-fold groupoids, and provide a simple description of the corresponding reflection for n=2. In the abelian context, the internal n-groupoids yield a torsion-free subcategory of the category of internal n-fold groupoids, and it is possible to characterize (higher) central extensions and compute generalized Hopf formulae for homology. 
 
Part of this talk is based on joint work with Marino Gran