  We introduce the notions of proto-complete, complete, complete* and
strong-complete objects in pointed categories. We show under mild conditions on
a pointed exact protomodular category that every proto-complete (respectively
complete) object is the product of an abelian proto-complete (respectively
complete) object and a strong-complete object. This together with the
observation that the trivial group is the only abelian complete group recovers
a theorem of Baer classifying complete groups. In addition we generalize
several theorems about groups (subgroups) with trivial center (respectively,
centralizer), and provide a categorical explanation behind why the derivation
algebra of a perfect Lie algebra with trivial center and the automorphism group
of a non-abelian (characteristically) simple group are strong-complete.
