In a categorical setting we
generalize the concept of radical as defined for groups and for rings. We define
semiabelian and co-semi abelian categories. Such categories lack the convenient
additive structure of the sets of morphisms between two objects, which may be
derived from the duality of the axioms for abelian categories, but, for example, the
concept of semi-abelian categories permits one to consider the categories of abelian
groups, all groups, commutative rings with identity all rings, rings with minimum
condition, Lie algebras and compact Hausdorff spaces with base points and
continuous maps under the same categorical formulation. Generalizations of the
classical radical properties are proved; for example, the fact that any object in a
semi-abelian category is the extension of a radical object by a semi-simple object and
the dual statement.
