Galois theory and homology in quasi-abelian functor categories

Abstract

Given a finite category $T$, we consider the functor category $A^T$, where $A$ can 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 Fréchet 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 $A^T$ for a suitable category $T$. For a replete full subcategory $S$ of $T$, we define $F$ to be the full subcategory of $A^T$ whose objects are given by the functors $F:T\to A$ with $F\left(T\right)=0$ for all $T\notin S$. We prove that $F$ is a torsion-free Birkhoff subcategory of $A^T$. This allows us to study (higher) central extensions from categorical Galois theory in $A^T$ with respect to $F$ and generalized Hopf formulae for homology.