Fundamental groups in E-semi-abelian categories

The main aim of this thesis is to begin the study of the notion of fundamental group, coming from the categorical Galois theory, in the framework of E-semi-abelian categories. This new context is sufficiently wide to include several categories of « algebraic » nature such as the semi-abelian categories, the integral almost abelian categories and the categories of topological semi-abelian algebras. At the abstract level, the fundamental groups are described by generalisations of the Brown-Ellis-Hopf formulae for the integral homology of groups, where the homological closure operators naturally occur. As an application, we give various descriptions of the fundamental groups corresponding to many adjunctions in the categories of groups, rings, topological groups and compact groups. This thesis yet provides another approach to the homology of non-abelian structures.(SC - Sciences) -- UCL, 201

A Seifert-van Kampen theorem in non-abelian algebra

We prove a variation on the Seifert-van Kampen theorem in a setting of non-abelian categorical algebra, providing sufficient conditions on a functor F, from an algebraically coherent semi-abelian category with enough projectives to an almost abelian (= Raikov semiabelian) category, for the preservation of pushouts of split monomorphisms by the left derived functor of F

A description of the fundamental group in terms of commutators and closure operators

A connection between the Galois-theoretic approach to semi-abelian homology and the homological closure operators is established. In particular, a generalised Hopf formula for homology is obtained, allowing the choice of a new kind of functors as coefficients. This makes it possible to calculate the fundamental groups corresponding to many interesting reflections arising, for instance, in the categories of groups, rings, compact groups and simplicial loops