Complete objects in categories

TL;DR

This paper introduces new categorical notions of completeness for objects, generalizes classical group theorems, and explains properties of Lie algebras and simple groups through a categorical lens.

Contribution

It defines proto-complete, complete, complete*, and strong-complete objects, and relates these to classical group theory results within a categorical framework.

Findings

Every proto-complete object decomposes into an abelian proto-complete and a strong-complete object.

The trivial group is the only abelian complete group, recovering Baer's theorem.

Categorical explanations for properties of Lie algebras and simple groups are provided.

Abstract

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.