Central reflections and nilpotency in exact Mal'tsev categories

TL;DR

This paper explores nilpotency in exact Mal'tsev categories using central extensions, establishing a connection with Goodwillie's calculus and characterizing n-nilpotent objects via folding and commutators.

Contribution

It introduces a nilpotency tower in Mal'tsev categories and characterizes the reflection into n-nilpotent objects as a universal functor of degree n.

Findings

Reflection into n-nilpotent objects is the universal endofunctor of degree n.

In semi-abelian categories, n-folded objects have vanishing Higgins commutator of length n+1.

The study links nilpotency, central extensions, and functor calculus in categorical algebra.

Abstract

We study nilpotency in the context of exact Mal'tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie's functor calculus. We show in particular that the reflection into the subcategory of $n$-nilpotent objects is the universal endofunctor of degree $n$ if and only if every $n$-nilpotent object is $n$-folded. In the special context of a semi-abelian category, an object is $n$-folded precisely when its Higgins commutator of length $n+1$ vanishes.