Central reflections and nilpotency in exact Mal’tsev categories

Abstract

Listen

Translate article

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.