Torsion theories and radicals in normal categories
Abstract: We introduce a relativized notion of
(semi)normalcy for categories that come equipped with a proper stable
factorization system, and we use radicals and normal closure operators
in order to study torsion theories in such categories. Our results
generalize and complement recent studies in the realm of semi-abelian
and, in part, homological categories. In particular, we characterize
both, torsion-free and torsion classes, in terms of their closure under
extensions. We pay particular attention to the homological and, for our
purposes more importantly, normal categories of topological algebra,
such as the category of topological groups. But our applications go far
beyond the realm of these types of categories, as they include, for
example, the normal, but non-homological category of pointed
topological spaces, which is in fact a rich supplier for radicals of
topological groups.
