  Given a finite category T, we consider the functor category [T,A], where A
can in particular be any quasi-abelian category. Examples of quasi-abelian
categories are given by any abelian category but also by non-exact additive
categories as the categories of torsion(-free) abelian groups, topological
abelian groups, locally compact abelian groups, Banach spaces and Fr\'echet
spaces. In this situation, the categories of various internal categorical
structures in A, such as the categories of internal n-fold groupoids, are
equivalent to functor categories [T,A] for a suitable category T. For a replete
full subcategory S of T, we define F to be the full subcategory of [T,A] whose
objects are given by the functors G with G(X)=0 for all objects X not in S. We
prove that F is a torsion-free Birkhoff subcategory of [T,A]. This allows us to
study (higher) central extensions from categorical Galois theory in [T,A] with
respect to F and generalized Hopf formulae for homology.
