Relative group completions
Let \(\mathcal{C}\) be a category. A functor \(L: \mathcal{C} \rightarrow \mathcal{C}\) is called a pointed endofunctor if there exists a natural transformation \(l: Id_{\mathcal{C}} \rightsquigarrow L\). The first result of this paper shows that a pointed endofunctor on the category of groups, \(\mathcal{G}\), which is compatible with conjugation, induces a pointed endofunctor on the category of pairs of groups. \noindent The second result shows that if a pointed endofunctor on \(\mathcal{G}\) is a completion, then it is compatible with conjugation, so it admits a relative version. The authors remark that, since localizations are special cases of completions, then they admit a relative version. The authors explicit the universal property of relative group localizations and give some applications of their results, namely the study of the effect of plus-constructions on homotopy fibre sequences.
scientific article; zbMATH DE number 2157742
scientific article