A Seifert-van Kampen theorem in non-abelian algebra
The authors prove a Seifert-van Kampen theorem for nonabelian categories. The Seifert-van Kampen theorem is well-known in algebraic topology, and it describes the fundamental group of a topological space in terms of the fundamental groups of open, path connected subspaces which cover the space. Loosely, the theorem says that the fundamental group functor sends pullbacks and pushouts of pointed topological spaces to pushouts in the category of groups. In this article, the authors' objective is to generalise the result to the more general setting of a regular epi-reflection. Given a functor \(F\) from an algebraically coherent, semi-abelian category \(\mathcal C\) with enough regular projectives, to an almost abelian category \(\mathcal X\), the authors give sufficient conditions for the fundamental group functor \(\pi_1^F\) relative to the functor \(F\) to preserve pushouts and split monomorphisms. When the category \(\mathcal X\) is abelian, the fundamental group functor \(\pi_1^F\) coincides with the classical left derived functor.
algebraically coherent semi-abelian category
scientific article; zbMATH DE number 6908503