A note on split extension classifiers of perfect objects

TL;DR

This paper investigates the properties of split extension classifiers of perfect objects within pointed protomodular categories, revealing that under certain conditions, these classifiers have trivial centers, which advances understanding of their algebraic structure.

Contribution

It establishes that for perfect objects with existing split extension classifiers, the centralizer of the conjugation morphism is trivial, highlighting a new structural property.

Findings

Centralizer of conjugation morphism is trivial for perfect objects.

Split extension classifiers of perfect objects have trivial centers.

Provides conditions under which these properties hold.

Abstract

We show that for a pointed protomodular category $\mathbb{C}$ satisfying a certain condition on those Huq commutators which exist, if $X$ is a perfect object in $\mathbb{C}$ such that the split extension classifier $[X]$ exists, then the centralizer of the \emph{conjugation} morphism $c_X : X\to [X]$ is trivial and hence $[X]$ has trivial center.