Galois theory and homology in quasi-abelian functor categories

Given a finite category T, we consider the functor category dT, T , where d 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 Fr & eacute;chet spaces. In this situation, the categories of various internal categorical structures in d, such as the categories of internal n-fold groupoids, are equivalent to functor categories dT T for a suitable category T. For a replete full subcategory S of T, we define Fto be the full subcategory of dT T whose objects are given by the functors F : T -> d with F ( T ) = 0 for all T is an element of / S. We prove that Fis a torsion-free Birkhoff subcategory of dT. T . This allows us to study (higher) central extensions from categorical Galois theory in dT T with respect to Fand generalized Hopf formulae for homology. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.